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Showing 428 questions for Mathematical Olympiad.

A company sells peanut butter in cylindrical jars. If the diameter of the jars is increased by 25\% ...

Difficulty: 3/10

There are 10 true-false questions in an examination. In how many distinct ways can these questions b...

Difficulty: 3/10

If n is a positive integer such that n^2 is divisible by 72, then the smallest possible value of n i...

Difficulty: 3/10

If p is a prime number greater than 3, what is the remainder when p^2 is divided by 12?...

Difficulty: 3/10

What is the units digit of 2^{2047} \times 3^{11}?...

Difficulty: 3/10

What is the least positive integer that is exactly divisible by all the integers from 1 to 10?...

Difficulty: 3/10

What is the remainder when 7^{2024} is divided by 13?...

Difficulty: 4/10

The equations 2x+7=3 and bx - 10 = -2 have the same solution. What is the value of b?...

Difficulty: 2/10

What is the sum of the first nine prime numbers?...

Difficulty: 2/10

What is the sum of all the positive divisors of 600?...

Difficulty: 3/10

If the diagonal of a square is 10, find its area....

Difficulty: 3/10

A store owner offers a successive discount of 20\% followed by another discount of 10\%. What is the...

Difficulty: 3/10

Find the remainder when 2^{100} is divided by 3....

Difficulty: 3/10

Sorted sequences x_i, y_i with zero sum and unit sum of squares. Prove Σ (x_i y_i - x_i y_{n+1-i}) ≥...

Difficulty: 10/10

Set S is overdetermined if |S| ≥ 2 and a polynomial of degree ≤ |S|-2 passes through it. Max overdet...

Difficulty: 8/10

Let n = 100. Given n distinct ordered pairs of non-negative integers (a_1, b_1), (a_2, b_2), \dots, ...

Difficulty: 7/10

Set A of integers a such that 1 ≤ a < p and both a and 4-a are quadratic non-residues. Find product ...

Difficulty: 9/10

A 2020 \times 2020 \times 2020 grid is made of unit cubes. A beam is a 1 \times 1 \times 2020 rectan...

Difficulty: 8/10

Let ABC be a triangle. Let P be a point on the segment BC. Prove that the circumcircle of \triangle ...

Difficulty: 7/10

Convex hexagon AB||DE, BC||EF, CD||FA and ABDE = BCEF = CD*FA. Prove circumcenters of ACE, BDF and o...

Difficulty: 10/10

Solve system of 2n equations: a_1 = 1/a_{2n} + 1/a_2, a_2 = a_1 + a_3, etc....

Difficulty: 8/10

Set S of positive integers where for each s in S and d|s, there's a unique t in S with gcd(s, t) = d...

Difficulty: 7/10

An n \times n board is initially empty. We are allowed to perform the following operations: Place St...

Difficulty: 9/10

Visitor in 3-regular planar graph park turns left/right alternatingly. Max times entering any vertex...

Difficulty: 8/10

Let a_n be a sequence of positive real numbers such that a_{n+1} \ge a_n^2 + \frac{1}{4} for all n \...

Difficulty: 7/10

2022 users, friendships. New friendship only if 2 common friends. Min friendships needed for everyon...

Difficulty: 10/10

Smallest k such that any 2022 numbers can be represented as sum of k essentially increasing function...

Difficulty: 8/10

Find all pairs of prime numbers (p, q) such that p - q and p q - q are both perfect squares....

Difficulty: 7/10

Find all functions f: \mathbb{R}_{>0} \to \mathbb{R}_{>0} such that for all x, y \in \mathbb{R}_{>0}...

Difficulty: 9/10

2b black rods, 2w white rods form 2n-gon. Convex 2b-gon B and 2w-gon W formed by translating rods. P...

Difficulty: 8/10

Let G be a simple graph with n vertices. Show that if G has no triangles (cycles of length 3), then ...

Difficulty: 8/10

ABC triangle, I incenter, I_a, I_b, I_c excenters. D on circumcircle. Circumcircles of DII_a and DI_...

Difficulty: 10/10

Row-valid and column-valid n x n tables of 1..n^2. For what n can any row-valid table be permuted in...

Difficulty: 8/10

Game with N integers. Alice replaces n with n+a. Bob replaces even n with n/2. Show the game always ...

Difficulty: 7/10

n x n board (n odd). Maximal domino configurations. Find all possible values of k(C), the number of ...

Difficulty: 9/10

Find all functions f: \mathbb{R}_{>0} \to \mathbb{R}_{>0} such that for all x, y \in \mathbb{R}_{>0}...

Difficulty: 8/10

Determine all positive integers n such that n \cdot 2^{n-1} + 1 is a perfect square....

Difficulty: 8/10

Find the largest c such that ΣΣ x_i x_j |A_i ∩ A_j|^2 / (|A_i| |A_j|) ≥ c (Σ x_i)^2 for l-large coll...

Difficulty: 10/10

D inside acute ABC, ∠DAC = ∠ACB, ∠BDC = 90° + ∠BAC. E on ray BD, AE = EC. M midpoint of BC. Show AB ...

Difficulty: 8/10

Necklace with mn beads (red/blue). No matter how it is cut into m blocks of n, each block has a dist...

Difficulty: 7/10

Triangulate a regular n-gon into n-2 triangles, each colored one of m colors, so each color has equa...

Difficulty: 9/10

100 finite sets S_i with non-empty intersection. For any T ⊆ {S_i}, |∩_{S∈T} S| is a multiple of |T|...

Difficulty: 8/10

Find all integers n ≥ 3 such that the gaps between consecutive divisors of n! are non-decreasing....

Difficulty: 7/10

Let m ≥ n be positive integers. m cupcakes around a circle, n people. Each person partitions the cir...

Difficulty: 10/10

Find all positive integers k such that for every n, the sum Σ_{i=0}^n (n choose i)^k is divisible by...

Difficulty: 8/10

Let H be the orthocenter of acute triangle ABC, F the foot of altitude from C, P the reflection of H...

Difficulty: 7/10

Alice and Bob play a game with cities and roads. Bob places cities with distance ≥ 1 and no three co...

Difficulty: 9/10

Let n > k ≥ 1 be integers. Let P(x) be a polynomial of degree n with no repeated roots and P(0) ≠ 0....

Difficulty: 8/10

Fix positive integers k and d. Prove that for all sufficiently large odd positive integers n, the di...

Difficulty: 7/10

Let a and b be positive integers such that φ(ab + 1) divides a^2 + b^2 + 1. Prove that a and b are F...

Difficulty: 10/10

Let ABC be a triangle. D, E, F on BC, CA, AB such that ∠AFE = ∠BDF = ∠CED. Let O_A, O_B, O_C be circ...

Difficulty: 8/10

A positive integer n is called solitary if, for any non-negative integers a and b such that a + b = ...

Difficulty: 7/10

Let ABC be an acute scalene triangle with no angle equal to 60°. Let ω be the circumcircle. Let l_B ...

Difficulty: 9/10

Annie is playing a game with a row of positive integers (powers of 2). She can replace two adjacent ...

Difficulty: 8/10

Fix an integer n ≥ 2. For which real numbers x is \lfloor nx \rfloor - \sum_{k=1}^{n} (⌊kx⌋ / k) max...

Difficulty: 7/10

Given BE and CF are the altitudes of the triangle ABC. P, Q are on BE and the extension of CF respec...

Difficulty: 5/10

Find value of sqrt(2 - sqrt(2 - sqrt(2 - sqrt(2 - \dots)))) + sqrt(2 + sqrt(2 + sqrt(2 + sqrt(2 + \d...

Difficulty: 3/10

In a circle, a pair of chords that subtend congruent (equal) central angles must be:...

Difficulty: 2/10

The value of cos(tan^-1(-1)) is ________....

Difficulty: 2/10

The expression sqrt(20 - sqrt(20 - sqrt(20 - sqrt(20 - \dots)))) is equal to ________....

Difficulty: 3/10

If f(x-3) = 9x^2 + 2 then f(5) is ________....

Difficulty: 3/10

The length of a rectangular screen is 15 cm. Its area is 180 sq. cm. Its width is ________....

Difficulty: 1/10

Area of a rhombus having diagonals equal to 6 cm and 8 cm is ________....

Difficulty: 2/10

Polygon with 9 sides is called...

Difficulty: 1/10

The inverse of the function f(x) = 2x-9 is ________....

Difficulty: 2/10

The solution of 2x-9 \ge 7 is ________....

Difficulty: 2/10

If A and B are 2 \times 2 square matrices, then the determinant \det(A+B) is generally:...

Difficulty: 2/10

Four non-parallel lines in a plan intersect at most at...

Difficulty: 2/10

Midpoint of (-1,1) and (1,-1) is ________....

Difficulty: 1/10

Find the number of factors of the product 5^{8} 7^{5} 2^{3} which are perfect squares....

Difficulty: 4/10

There are six movie parts numbered from 1 to 6. Find the number of ways in which they be arranged so...

Difficulty: 3/10

If nPr = 3024 and nCr = 126 then find n and r....

Difficulty: 3/10

In a colony, there are 55 members. Every member posts a greeting card to all the members. How many g...

Difficulty: 2/10

From a group of 7 men and 6 women, five persons are to be selected to form a committee so that three...

Difficulty: 3/10

If an arithmetic progression is 13, 11, 9\dots., then find 50th term of that arithmetic progression....

Difficulty: 2/10

If the measure of each interior angle of a regular polygon is 150°, then the number of its diagonals...

Difficulty: 3/10

ΔABC is inscribed in a circle with center O. If AB = 17 cm, BC = 10 cm and AC = 9 cm, then find the ...

Difficulty: 4/10

If the angles of a triangle, in degrees, are x, 3x + 20, and 6x, then the triangle must be:...

Difficulty: 2/10

The area of the triangle whose vertices are given by the coordinates (1,2), (-4, -3) and (4, 1) is _...

Difficulty: 2/10

A circle is circum-scribed of ΔABC. O is the center of the circle and CP is the tangent of the circl...

Difficulty: 4/10

In ΔXYZ, XY and XZ are 20 cm and 25 cm respectively, XP is angle bisector of ∠YXZ, YP = a cm and PZ ...

Difficulty: 4/10

How many prime numbers are less than 50?...

Difficulty: 2/10

The sum of first five prime numbers is ________....

Difficulty: 1/10

What is the least number which when divided by the numbers 3, 5, 6, 8, 10 and 12 leaves in each case...

Difficulty: 4/10

Three bells beep at an interval of 12, 20, and 35 minutes. If they beep together at 10 a.m., then th...

Difficulty: 3/10

The three numbers are in the ratio 1/2 : 2/3 : 3/4. The difference between greatest and smallest num...

Difficulty: 3/10

If the integer a is divisible by the integer b, which of the following notations correctly represent...

Difficulty: 1/10

Which of the following mathematical expressions is equal to 8?...

Difficulty: 1/10

If b \neq 0 is an integer in \mathbb{Z}, which of the following statements is mathematically correct...

Difficulty: 1/10

The number of ways of painting the faces of a cube with six different colors is ________....

Difficulty: 4/10

6 men and 4 women are to be seated in a row so that no two women sit together. The number of ways th...

Difficulty: 4/10

1 + 3 + 5 + 7 + ... + 17 is equal to ________....

Difficulty: 2/10

The number of rectangles that a chessboard has......

Difficulty: 3/10

Everybody in a room shakes hands with everybody else. The total number of shake hands is 66. The num...

Difficulty: 2/10

The product of r consecutive positive integers is divisible by...

Difficulty: 3/10

The number of 5 digit numbers all digits of which are odd is ________....

Difficulty: 2/10

In 3 fingers, the number of ways four rings can be worn is ... ways....

Difficulty: 3/10

The least common multiple of 345, 453, 234 is ________....

Difficulty: 3/10

Which of the following positive integers is a prime number?...

Difficulty: 1/10

Decimal representation of (A35B0F)_16 is...

Difficulty: 3/10

If x^2 - 2x + 1 = 0, then x^2 + 1/x^2 is ________....

Difficulty: 2/10

If p is a prime number and n is a positive integer, then the number of positive divisors of p^n is:...

Difficulty: 2/10

For any integers a, b, n in \mathbb{Z} such that n \mid a and n \mid b, which of the following must ...

Difficulty: 2/10

5^{5} / 5^{4} is equal to...

Difficulty: 1/10

6^2 + |6| + |-6| - 6^2 is equal to...

Difficulty: 1/10

Which of the fractions below is closest to 1?...

Difficulty: 2/10

If a b = a-b+ab, then 2 5 + 5 * 2 is...

Difficulty: 2/10

In ΔABC, AB = AC and D, E, F are on AB, BC, CA, such that DE = EF = FD. Prove that ∠DEB = 1/2 (∠ADF ...

Difficulty: 5/10

Let a, b and c be distinct nonzero real numbers such that a + 1/b = b + 1/c = c + 1/a. Prove that |a...

Difficulty: 5/10

The period of y = 2Sin(x - \pi/3) is ________....

Difficulty: 2/10

Find x if Log_x(9/25) = 2...

Difficulty: 2/10

Which of these lines is parallel to the line 2y = x+7...

Difficulty: 2/10

The inverse of the function f(x)=2x-3 is ________....

Difficulty: 2/10

Amir is a window cleaner. He uses the following formula to calculate the amount to charge his custom...

Difficulty: 1/10

17 students are present in a class. In how many ways, can they be made to stand in 2 circles of 8 an...

Difficulty: 4/10

There are 6 equally spaced points A, B, C, D, E and F marked on a circle with radius R. How many con...

Difficulty: 2/10

If nPr = 3024 and nCr = 126 then find n and r....

Difficulty: 3/10

Find the number of ways of arranging the letters of the words DANGER, so that no vowel occupies odd ...

Difficulty: 3/10

If arithmetic mean of 6, 7, 9, x is 10, then value of x is ________....

Difficulty: 1/10

How many numbers must be selected from the set {1, 2, 3, 4, 5, 6} to guarantee that at least one pai...

Difficulty: 3/10

Consider the recurrence relation a_n = a_{n-1} + n with initial condition a_1 = 1. The value of a_{1...

Difficulty: 3/10

How many odd 3-digit whole numbers are there? For example, 203 is acceptable but 023 is not....

Difficulty: 2/10

There are 20 points in a plane, how many triangles can be formed by these points if 5 are collinear?...

Difficulty: 3/10

In a colony, there are 55 members. Every member posts a greeting card to all the members. How many g...

Difficulty: 2/10

In an arithmetic progression the m times of m^th term is equal to n times the n^th term, its (m + n)...

Difficulty: 4/10

In a group of 50 students 24 like cold drinks and 36 like hot drinks and each student likes at least...

Difficulty: 2/10

A boy lives at X and wants to go to School at Z. From his home X he has to first reach Y and then Y ...

Difficulty: 2/10

A six-person committee composed of Alice, Ben, Connie, Dolph, Egbert, and Francisco is to select a c...

Difficulty: 3/10

The number obtained by interchanging the two digits of a two digit number is less than the original ...

Difficulty: 3/10

A certain number when divided by 222 leaves a remainder 35, another number when divided by 407 leave...

Difficulty: 3/10

When a particular positive number is divided by 5, the remainder is 2. If the same number is divided...

Difficulty: 3/10

If n is a positive integer, which one of the following numbers must have a remainder of 3 when divid...

Difficulty: 3/10

If each of the three nonzero numbers a, b and c is divisible by 3, then abc must be divisible by whi...

Difficulty: 2/10

A girl wrote all the numbers from 100 to 200. Then she started counting the number of one's that has...

Difficulty: 4/10

The remainders when 2^{50} is divided by 7....

Difficulty: 3/10

If p is a prime number and p \mid a^n (where a is an integer and n is a positive integer), then whic...

Difficulty: 2/10

gcd(2a + 1, 9a + 4) =...

Difficulty: 3/10

-123 = a(mod 10)...

Difficulty: 2/10

Positive divisor of both a and a + 1 is ________....

Difficulty: 2/10

If the integers d, a, b satisfy d \mid a and d \mid b, which of the following must be true?...

Difficulty: 2/10

Which of the following real numbers is NOT irrational (i.e., is rational)?...

Difficulty: 1/10

If x^2 - 4x + 4 = 0, then the value of x^2 is ________....

Difficulty: 2/10

The number of real solutions of the equation x^4 - 1 = 0 are ________....

Difficulty: 2/10

The function f: R \rightarrow R defined by f(x) = 2x^2 + x - 1 is ________....

Difficulty: 3/10

The quadratic equation corresponding to the roots 2 + sqrt(5) and 2 - sqrt(5) is ________....

Difficulty: 3/10

If x^2 + y^2 + z^2 = 125 and xy + yz + za = 250, then x + y + z is ________....

Difficulty: 3/10

The ratio of heights of a cone and a hemisphere with equal bases and equal volumes is ________....

Difficulty: 3/10

The distance between two parallel tangents of a circle of radius 6 cm is ________....

Difficulty: 2/10

Maximum number of intersection points of 5 non parallel distinct lines in a plane are ________....

Difficulty: 2/10

If the supplement of an angle is 3 times of its compliment, find the angle....

Difficulty: 2/10

How many minimum points are required to draw a unique line in a plane...

Difficulty: 1/10

In a right angle triangle length of the hypotenuse is 5cm and length of one side is 4cm. What is the...

Difficulty: 2/10

Surface Area of a sphere with radius 3cm is ________....

Difficulty: 2/10

Length of a diagonal of a rectangle with length 4 cm and width 3 cm is ________....

Difficulty: 2/10

Total number of circles that can pass through 2 pair of points in a plane are ________....

Difficulty: 2/10

Consider a triangle ABC with ∠B = 30° and ∠C = 60°. What is the type of ΔABC?...

Difficulty: 2/10

Find the slope of the line with equation 3x + 2y = 10...

Difficulty: 2/10

A chord of length 24cm is at a distance of 5cm from the center of a circle. The radius of the circle...

Difficulty: 2/10

If x + 1/x = 2, then the value of x^2 + 1/x^2 is:...

Difficulty: 2/10

The coordinates of the midpoint of points in plane with coordinates (-2,8) and (8,-2) is...

Difficulty: 2/10

Total number of lines passing through the point (0,0) are...

Difficulty: 1/10

Angles that sum up to 90° are known as...

Difficulty: 1/10

The distance between the points (-2, 2) and (1, 6) is...

Difficulty: 2/10

Tri\angle ABC is inscribed in a circle. P, Q and R are any points on arcs AB, BC and AC respectively...

Difficulty: 4/10

Prove that (sqrt(5)+sqrt(6)+sqrt(7))(sqrt(5)+sqrt(6)-sqrt(7))(sqrt(5)-sqrt(6)+sqrt(7))(-sqrt(5)+sqrt...

Difficulty: 4/10

If LCM of two number is 10 and GCD is 5 then the product of two numbers is ________....

Difficulty: 2/10

Which of the following real numbers is NOT irrational (i.e., is rational)?...

Difficulty: 1/10

The inverse of 3 modulo 7 is ________....

Difficulty: 3/10

The linear combination of gcd(252, 198) = 18 is ________....

Difficulty: 3/10

Simplify log_3(27) - log_3(9)....

Difficulty: 2/10

If the sides of two similar triangles are in the ratio 2:3, what is the ratio of their areas?...

Difficulty: 2/10

If p is inversely proportional to q, and p=8 when q=6, what is the value of q when p=12?...

Difficulty: 2/10

A recipe calls for a ratio of 2 cups of flour to 1 cup of sugar. If you want to use 4 cups of sugar,...

Difficulty: 1/10

Determine the sum of the geometric series: 4 + 2 + 1 + 1/2 + \dots....

Difficulty: 2/10

In a group of 8 people, how many ways can you choose a committee of 3 people if two specific individ...

Difficulty: 3/10

The LCM of two numbers is 7200, and their HCF is 180. If one number is 360, then the second number i...

Difficulty: 2/10

If A and B are square matrices of order three, then the determinant \det(AB) is equal to:...

Difficulty: 2/10

A company sells peanut butter in cylindrical jars. Marketing research suggests that using wider jars...

Difficulty: 6/10

You and five friends need to raise 1500 dollars in donations for a charity, dividing the fundraising...

Difficulty: 1/10

Compute the sum of all the roots of (2x+3)(x-4)+(2x+3)(x-6)=0...

Difficulty: 3/10

Cindy was asked by her teacher to subtract 3 from a certain number and then divide the result by 9. ...

Difficulty: 2/10

For how many positive integers m does there exist at least one positive integer n such that mn < m +...

Difficulty: 4/10

The sum of two numbers is S. Suppose 3 is added to each number and then each of the resulting number...

Difficulty: 3/10

Cities A and B are 45 miles apart. Alicia lives in A and Beth lives in B. Alicia bikes towards B at ...

Difficulty: 3/10

A ball is dropped from a height of 49 m. The quadratic equation d(t) = -t^2 + 49 provides informatio...

Difficulty: 2/10

Which of the following relations is NOT a mathematical function?...

Difficulty: 2/10

If n(A) = 20 and n(B) = 30 and n(A \cup B) = 40 then n(A \cap B) is ________....

Difficulty: 2/10

If 10 boys and 10 girls sit alternately in a row and then sit alternately along a circular table, wh...

Difficulty: 4/10

There are 27 points in a plane. 5, 10 and 15 points are collinear on distinct lines. By joining thes...

Difficulty: 4/10

What is the exact numerical value of the combination sum \binom{50}{4} + \sum_{r=1}^{6} \binom{56-r}...

Difficulty: 4/10

How many terms are in the expansion of (2x - 1/y)^10?...

Difficulty: 2/10

Let the players who play cricket be 12, the ones who play football 10, those who play only cricket a...

Difficulty: 3/10

What is the Cartesian product of set A and set B, if the set A = {1, 2} and set B = {a, b}?...

Difficulty: 2/10

Mayur travels from Mumbai to Jammu in 7 different ways. But he is allowed to return to Mumbai by any...

Difficulty: 2/10

There are 10 true-false questions in an examination. These questions can be answered in ________....

Difficulty: 4/10

The 6th and 17th terms of an A.P. are 19 and 41 respectively. Find the 40th term...

Difficulty: 3/10

From a group 7 men and 6 women, five persons are to be selected to form a committee so that at least...

Difficulty: 4/10

If n is a positive integer such that n^2 is divisible by 72, what is the smallest possible value of ...

Difficulty: 3/10

If p is a prime number greater than 3, what is the remainder when p^2 is divided by 12?...

Difficulty: 4/10

What is the value of 4^{234} mod 11...

Difficulty: 4/10

What is the units digit of 2^{2047} * 3^{11}?...

Difficulty: 3/10

What is the least number that is exactly divisible by all the numbers from 1 to 10?...

Difficulty: 4/10

What is the remainder when 7^{2024} is divided by 13?...

Difficulty: 4/10

How many distinct prime factors does 2^{16} 3^{4} 5^{2} have?...

Difficulty: 2/10

Sum of two different prime numbers is a ________....

Difficulty: 2/10

If a and b are coprime integers, which of the following pairs of integers must also be coprime?...

Difficulty: 3/10

If a straight line intersects two concentric circles with center O at the points A, B, C, and D in t...

Difficulty: 3/10

All the ________ in a plane are similar...

Difficulty: 1/10

Consider a triangle ABC with ∠A = 45° and ∠B = 5°. What is the type of ΔABC?...

Difficulty: 2/10

The tangents on the end points of a diameter of a circle are always ________....

Difficulty: 2/10

The angle between the tangents at the ends of two perpendicular radii is ________....

Difficulty: 2/10

How many different tangents can be drawn on a circle?...

Difficulty: 1/10

Let the central angle subtended by two points on a circle is 120°. Then the inscribed angle subtende...

Difficulty: 2/10

The points (1,-1), (4,2), (10,8) in the plane are ________....

Difficulty: 2/10

The equations 2x + 7 = 3 and bx - 10 = -2 have the same solution. What is the value of b?...

Difficulty: 4/10

What is the sum of the first nine prime numbers?...

Difficulty: 3/10

What is the sum of all the divisors of 600?...

Difficulty: 3/10

The center of the circle, ________....

Difficulty: 1/10

A quadrilateral with vertices at (0,-1), (3,1), (3,-1), (0,1) is a ________....

Difficulty: 2/10

Positive integers a, b, c satisfy \frac{ab}{a-b} = c. What is the largest possible value of a+b+c no...

Difficulty: 6/10

In a figure, 4 of the 6 disks arranged in a ring or hexagon are to be colored black and 2 are to be ...

Difficulty: 6/10

For a positive integer n, let \langle n \rangle denote the perfect square integer closest to n. If N...

Difficulty: 6/10

A total fixed amount of N thousand rupees is given to three persons A, B, C every year, each being g...

Difficulty: 5/10

Let ABCD be a parallelogram. Let E and F be the midpoints of AB and BC respectively. The lines EC an...

Difficulty: 5/10

If \sum_{k=1}^{40} \left( 1 + \frac{1}{k^2} + \frac{1}{(k+1)^2} \right)^{1/2} = a + \frac{b}{c} wher...

Difficulty: 5/10

How many two-digit numbers have exactly 4 positive factors? (Here 1 and the number n are also consid...

Difficulty: 5/10

The product 55 \times 60 \times 65 is written as the product of five distinct positive integers. Wha...

Difficulty: 4/10

Find the sum of all positive integers n for which |2^n + 5^n - 65| is a perfect square....

Difficulty: 4/10

Five students take a test on which any integer score from 0 to 100 inclusive is possible. What is th...

Difficulty: 4/10

Let ABC be a triangle with AB = 5, AC = 4, BC = 6. The internal angle bisector of C intersects the s...

Difficulty: 4/10

Let ABC be a triangle with AB = AC. Let D be a point on the segment BC such that BD = 48 and DC = 61...

Difficulty: 4/10

What is the least positive integer by which 2^5 \times 3^6 \times 4^3 \times 5^3 \times 6^7 should b...

Difficulty: 4/10

Find the number of integer solutions to ||x| - 2020| < 5....

Difficulty: 3/10

Let ABCD be a rectangle in which AB + BC + CD = 20 and AE = 9, where E is the midpoint of the side B...

Difficulty: 3/10

A bug travels in the coordinate plane moving only along the lines that are parallel to the x-axis or...

Difficulty: 3/10

A number N in base 10 is 503 in base b and 305 in base b + 2. What is the product of the digits of N...

Difficulty: 3/10

Let ABCD be a trapezium in which AB \parallel CD and AB = 3CD. Let E be the midpoint of the diagonal...

Difficulty: 3/10

A 12 \times 12 board is divided into 144 unit squares by drawing lines parallel to the sides. Two ro...

Difficulty: 4/10

In how many ways can four married couples sit on a merry-go-round with identical seats such that men...

Difficulty: 4/10

Suppose that P is the polynomial of least degree with integer coefficients such that P(\sqrt{7} + \s...

Difficulty: 4/10

Let P_0 = (3, 1) and define P_{n+1} = (x_{n+1}, y_{n+1}) for n \ge 0 by: x_{n+1} = \frac{3x_n - y_n}...

Difficulty: 4/10

For any real number t, let [t] (or \lfloor t \rfloor) denote the largest integer less than or equal ...

Difficulty: 4/10

Find the number of maps f : \{1, 2, 3\} \to \{1, 2, 3, 4, 5\} such that f(i) \le f(j) whenever i < j...

Difficulty: 4/10

Let x, y, z be positive real numbers such that x^2 + y^2 = 49, y^2 + yz + z^2 = 36, and x^2 + \sqrt{...

Difficulty: 4/10

In a parallelogram ABCD, the longer side is twice the shorter side. Let XYZW be the quadrilateral fo...

Difficulty: 4/10

Consider the set of all 6-digit numbers consisting of only 3 distinct non-zero digits a, b, c, such ...

Difficulty: 3/10

Ria writes down the numbers 1, 2, \dots, 101 in red and blue pens. The largest blue number is equal ...

Difficulty: 3/10

Three parallel lines L1, L2, L3 are drawn in the plane such that the perpendicular distance between ...

Difficulty: 3/10

Consider the set T of all triangles whose sides are distinct prime numbers which are also in arithme...

Difficulty: 3/10

Let N be the number of ways of distributing 52 identical balls into 4 distinguishable boxes such tha...

Difficulty: 6/10

In a triangle ABC, the median AD divides ∠BAC in the ratio 1 : 2. Extend AD to E such that EB is per...

Difficulty: 6/10

A binary sequence is a sequence in which each term is equal to 0 or 1. A binary sequence is called f...

Difficulty: 5/10

An ant is at a vertex of a cube. Every 10 minutes it moves to an adjacent vertex along an edge. If N...

Difficulty: 5/10

For an integer n  3 and a permutation  = (p1, p2, …, pn) of {1, 2, …, n}, we say p1 is a landmark ...

Difficulty: 5/10

Consider a string of n 1’s. We wish to place some + signs in between so that the sum is 1000. For in...

Difficulty: 5/10

Let m, n be natural numbers such that m + 3n – 5 = 5LCM(m, n) – 11GCD(m, n). Find the maximum possib...

Difficulty: 5/10

For a positive integer n > 1, let g(n) denote the largest positive proper divisor of n and f(n) = n ...

Difficulty: 5/10

Let a, b, c be real numbers satisfying: 3ab + 2 = 6b3bc + 2 = 5c3ca + 2 = 4aGiven that the product a...

Difficulty: 5/10

Let x, y be real numbers such that xy = 1. Let T and t be the largest and the smallest values of the...

Difficulty: 5/10

Let x, y, z be complex numbers such that x y z   9 yz zx xy x2 y2 z2    64 yz zx xy x3 y...

Difficulty: 5/10

Let ABC be a triangle let D be a point on the segment BC such that AD = BC. Suppose ∠CAD = x°, ∠ABC ...

Difficulty: 4/10

Given ABC with ∠B = 60° and ∠C = 30°, let P, Q, R be points on sides BA, AC, CB respectively such t...

Difficulty: 4/10

Let AB be a diameter of a circle \\omega and let C be a point on \\omega, different from A and B. Th...

Difficulty: 4/10

Consider the 10-digit number M = 9876543210. We obtain a new 10-digit number from M according to the...

Difficulty: 4/10

Two sides of an integer sided triangle have lengths 18 and x. If there are exactly 35 possible integ...

Difficulty: 4/10

Suppose the prime numbers p and q satisfy q^2 + 3p = 197p^2 + q. Write \frac{q}{p} as l + \frac{m}{n...

Difficulty: 4/10

Find the number of ordered pairs (a, b) such that a, b in {10, 11, …, 29, 30} and GCD(a, b) + LCM(a,...

Difficulty: 4/10

Let a, b be positive integers satisfying a3 – b3 – ab = 25. Find the largest possible value of a2 + ...

Difficulty: 4/10

Let m be the smallest positive integer such that m2 + (m + 1)2 + … + (m + 10)2 is the square of a po...

Difficulty: 3/10

Starting with a positive integer M written on the board, Alice plays the following game: in each mov...

Difficulty: 3/10

In a trapezoid ABCD, the internal bisector of angle A intersects the base BC (or its extension) at t...

Difficulty: 3/10

In a parallelogram ABCD, the point P on segment AB is taken such that \frac{AP}{AB} = \frac{61}{2022...

Difficulty: 3/10

A triangle ABC with AC = 20 is inscribed in a circle \\omega. A tangent t to \\omega is drawn throug...

Difficulty: 3/10

Let d(m) denote the number of positive integer divisors of a positive integer m. If r is the number ...

Difficulty: 5/10

A positive integer n > 1 is called beautiful if n can be written in one and only one way as n = a1 +...

Difficulty: 5/10

On each side of an equilateral triangle with side length n units, where n is an integer, 1 \le n \le...

Difficulty: 5/10

A quadruple (a, b, c, d) of distinct integers is said to be balanced if a + c = b + d. Let S be any ...

Difficulty: 5/10

In the land of Binary, the unit of currency is called Ben and currency notes are available in denomi...

Difficulty: 5/10

Find the least positive integer n such that there are at least 1000 unordered pairs of diagonals in ...

Difficulty: 5/10

A trapezium in the plane is a quadrilateral in which a pair of opposite sides are parallel. A trapez...

Difficulty: 5/10

In the coordinate plane, a point is called a lattice point if both of its coordinates are integers. ...

Difficulty: 5/10

In an equilateral triangle of side length 6, pegs are placed at the vertices and also evenly along e...

Difficulty: 5/10

For n \in \mathbb{N}, consider non-negative integer-valued functions f on \{1, 2, \dots, n\} satisfy...

Difficulty: 5/10

For any finite non empty set X of integers, let max(X) denote the largest element of X and |X| denot...

Difficulty: 5/10

For n in , let P(n) denote the product of the digits in n and S(n) denote the sum of the digits in n...

Difficulty: 5/10

Let P be a convex polygon with 50 vertices. A set F of diagonals of P is said to be minimally friend...

Difficulty: 4/10

Consider the set S = {(a, b, c, d, e): 0 < a < b < c < d < e < 100} where a, b, c, d, e are integers...

Difficulty: 4/10

The six sides of a convex hexagon A1A2A3A4A5A6 are colored red. Each of the diagonals of the hexagon...

Difficulty: 4/10

Let ABCD be a unit square. Suppose M and N are points on BC and CD respectively such that the perime...

Difficulty: 4/10

Let ABC be a triangle in the xy plane, where B is at the origin (0, 0). Let BC be produced to D such...

Difficulty: 4/10

The ex-radii of a triangle are 10 , 12 and 14. If the sides of the triangle are the roots of the cub...

Difficulty: 4/10

Let P(x) = x3 + ax2 + bx + c be a polynomial where a, b, c are integers and c is odd. Let \pi be the...

Difficulty: 4/10

A positive integer m has the property that m2 is expressible in the form 4n2 – 5n + 16 where n is an...

Difficulty: 4/10

The sequence an n  0 is defined by a0 = 1, a1 = – 4 and an + 2 = – 4an + 1 – 7an, for n  0 . Find ...

Difficulty: 4/10

Find the number of triples (a, b, c) of positive integers such that (a) ab is a prime; (b) bc is a p...

Difficulty: 4/10

Given a 2 × 2 tile and seven dominoes (2 × 1 tile), find the number of ways of tiling (that is, cove...

Difficulty: 4/10

Unconventional dice are to be designed such that the six faces are marked with numbers from 1 to 6 w...

Difficulty: 4/10

Let X be the set of all even positive integers n such that the measure of the angle of some regular ...

Difficulty: 3/10

In a triangle ABC, let E be the midpoint of AC and F be the midpoint of AB. The medians BE and CF in...

Difficulty: 3/10

Let x, y be positive integers such that x 4 = ( x − 1)(y 3 − 23) − 1 . Find the maximum possible val...

Difficulty: 3/10

Let \alpha and \beta be positive integers such that: \frac{16}{37} < \frac{\alpha}{\beta} < \frac{7}...

Difficulty: 3/10

Let n be a positive integer such that 1 \le n \le 1000. Let M_n be the number of perfect squares in ...

Difficulty: 3/10

Find the number of elements in the set \{(a, b) \in \mathbb{N} : 2 \le a, b \le 2023, \log_a(b) + 6\...

Difficulty: 3/10

Let ABC be a right-angled triangle with ∠B = 90° . Let the length of the altitude BD be equal to 12....

Difficulty: 5/10

Let n = 219312. Let M denote the number of positive divisors of n2 which are less than n but would n...

Difficulty: 5/10

Find the largest positive integer n < 30 such that ( 1 8 ) n + 3n 4 − 4 is not divisible by the squa...

Difficulty: 5/10

In a triangle ABC, a point P in the interior of ABC is such that ∠BPC − ∠BAC = ∠CPA − ∠CBA = ∠APB − ...

Difficulty: 5/10

The sum of [x] for all real numbers x satisfying the equation 16 + 15x + 15x2 = [x]3 is:...

Difficulty: 5/10

A finite set M of positive integers consists of distinct perfect squares and the number 92. The aver...

Difficulty: 5/10

Consider the set F of all polynomials whose coefficients are in the set of {0, 1}. Let q(x) = x3 + x...

Difficulty: 5/10

Consider the fourteen numbers, 14, 24 ,….,144. The smallest natural number n such that they leave di...

Difficulty: 5/10

In a triangle ABC, \angle BAC = 90^\circ. Let D be the point on segment BC such that AB + BD = AC + ...

Difficulty: 5/10

An integer n is such that \lfloor n/9 \rfloor is a three-digit number with equal digits, and \lfloor...

Difficulty: 5/10

On a natural number n you are allowed two operations: (1) multiply n by 2 or (2) subtract 3 from n. ...

Difficulty: 5/10

Consider five points in the plane, with no three of them collinear. Every pair of points among them ...

Difficulty: 5/10

Let p, q be two-digit numbers neither of which are divisible by 10. Let r be the four-digit number b...

Difficulty: 4/10

Consider an isosceles triangle ABC with sides BC = 30, CA = AB = 20. Let D be the foot of the perpen...

Difficulty: 4/10

Let X be the set consisting of twenty positive integers n, n + 2,…., n + 38. The smallest value of n...

Difficulty: 4/10

Initially, there are 3^{80} particles at the origin (0, 0). At each step, the particles are moved to...

Difficulty: 4/10

Three positive integers a, b, c with a > c satisfy the following equations: ac + b + c = bc + a + 66...

Difficulty: 4/10

Consider a square ABCD of side length 16. Let E, F be points on CD such that CE = EF = FD. Let the l...

Difficulty: 4/10

The positive real numbers a, b, c satisfy: \frac{a}{2b+1} + \frac{2b}{3c+1} + \frac{3c}{a+1} = 1\fra...

Difficulty: 4/10

Determine the number of positive integral value of p for which there exists a triangle with side a, ...

Difficulty: 4/10

Consider the grid of points X = ( m, n ) | 0 \le m, n \le 4 . We say a pair of points {(a, b), (c,...

Difficulty: 4/10

Let n, be the smallest integer such that the sum of digits of n is divisible by 5 as sell as the sum...

Difficulty: 4/10

Determine the sum of all possible surface areas of a cube two of whose vertices are (1, 2, 0) and (3...

Difficulty: 4/10

Find the number of triples of real numbers (a, b, c) such that a 20 + b 20 + c 20 = a 24 + b 24 + c ...

Difficulty: 3/10

Let a = \frac{x}{y} + \frac{y}{z} + \frac{z}{x}, let b = \frac{y}{x} + \frac{z}{y} + \frac{x}{z} and...

Difficulty: 3/10

Let ABCD be a quadrilateral with ∠ADC = 70°, ∠ACD = 70°, ∠ACB = 10° and ∠BAD = 110° . The measure of...

Difficulty: 3/10

The number obtained by taking the last two digits of 5 2024 in the same order is:...

Difficulty: 3/10

The number of four-digit odd numbers having digits 1, 2, 3, 4, each occurring exactly once, is:...

Difficulty: 3/10

The smallest positive integer that does not divide 1 \times 2 \times 3 \times 4 \times 5 \times 6 \t...

Difficulty: 3/10

Find the perimeter of a regular hexagon of side length 6....

Difficulty: 2/10

Find the value of \binom{10}{3} (10 choose 3)....

Difficulty: 3/10

Find the sum of the first 20 positive integers....

Difficulty: 2/10

What is the sum of all prime numbers between 10 and 20?...

Difficulty: 2/10

In a triangle ABC, the base BC = 10 and the altitude from vertex A to base BC is 8. Find the area of...

Difficulty: 2/10

If 3 x + 2 y = 12 and 2 x + 3 y = 13, find the value of x + y....

Difficulty: 2/10

Find the greatest common divisor of 123456 and 654321....

Difficulty: 3/10

In how many distinct ways can 5 people be seated around a circular table? (Two seatings are consider...

Difficulty: 3/10

The interior angles of a triangle are in the ratio 1:2:3. Find the measure of the largest angle in d...

Difficulty: 2/10

If the 5th term of an Arithmetic Progression is 15 and the 9th term is 27, find the common differenc...

Difficulty: 2/10

If x + \frac{1}{x} = 4 for a non-zero real number x, what is the value of x^2 + \frac{1}{x^2}?...

Difficulty: 2/10

In a right-angled triangle, the lengths of the two legs are 9 and 12. Find the length of the hypoten...

Difficulty: 2/10

Find the number of ways to arrange the letters of the word OLYMPIAD such that the vowels (O, Y, I, A...

Difficulty: 3/10

If \log_2(x) + \log_4(x) + \log_{16}(x) = 7 for a positive real number x, find the value of x....

Difficulty: 3/10

Find the number of positive divisors of the year 2025....

Difficulty: 2/10

If the roots of the quadratic equation x^2 - px + q = 0 are consecutive integers, find the value of ...

Difficulty: 5/10

Find the number of pairs of positive integers (x, y) that satisfy the equation: \frac{1}{x} + \frac{...

Difficulty: 5/10

In a circle, chords AB and CD intersect at an interior point P. If AP = 4, PB = 6, and CP = 3, find ...

Difficulty: 5/10

If the circumference of a circle is increased by 50\%, by what percentage does the area of the circl...

Difficulty: 4/10

Find the sum of all two-digit positive integers that are multiples of 7....

Difficulty: 4/10

A bag contains 10 red, 15 blue, and 20 green balls. What is the minimum number of balls one must dra...

Difficulty: 4/10

What is the remainder when 3^{2025} is divided by 10?...

Difficulty: 3/10

Let x, y be real numbers such that x^2 + y^2 - 4x - 6y + 13 = 0. Find the value of x^y....

Difficulty: 3/10

The age of a person (in years) in 2025 is a perfect square. His age (in years) was also a perfect sq...

Difficulty: 4/10

Four sides and a diagonal of a quadrilateral are of lengths 10, 20, 28, 50, 75, not necessarily in t...

Difficulty: 4/10

How many 3-digit numbers \overline{abc} in base 10 are there with a \neq 0 and c = a + b?...

Difficulty: 3/10

How many isosceles integer-sided triangles are there with perimeter 23?...

Difficulty: 3/10

The area of an integer-sided rectangle is 20. What is the minimum possible value of its perimeter?...

Difficulty: 2/10

Find the number of positive integers n less than or equal to 100, which are divisible by 3 but are n...

Difficulty: 2/10

If 60\% of a number x is 40, then what is x\% of 60?...

Difficulty: 2/10

Consider a 2025 \times 2025 grid of unit squares. Matilda wishes to place on the grid some rectangul...

Difficulty: 10/10

Alice and Bazza play a game (called Inekoalaty) whose rules depend on a positive real number \lambda...

Difficulty: 7/10

An infinite sequence a_1, a_2, \ldots of positive integers is defined as follows: each a_n has at le...

Difficulty: 5/10

A function f : \mathbb{Z}_{>0} \to \mathbb{Z}_{>0} is called bonza if f(a) \mid b^a - f(b)^{f(a)}for...

Difficulty: 9/10

Let \omega_1 and \omega_2 be circles with centres O_1 and O_2, respectively, such that the radius of...

Difficulty: 7/10

A line in the plane is called sunny if it is not parallel to any of the x-axis, the y-axis, or the l...

Difficulty: 5/10

Let \mathbb{Q} denote the set of rational numbers. A function f : \mathbb{Q} \to \mathbb{Q} is calle...

Difficulty: 10/10

Turbo the snail plays a game on a board with 2024 rows and 2023 columns. There are hidden monsters i...

Difficulty: 7/10

Let ABC be a triangle with AB < AC < BC. Let the incircle of ABC be tangent to BC, CA, and AB at D, ...

Difficulty: 5/10

Let a_1, a_2, a_3, \ldots be an infinite sequence of positive integers, and let N be a positive inte...

Difficulty: 9/10

Determine all pairs (a, b) of positive integers for which there exist positive integers g and N such...

Difficulty: 7/10

Determine all real numbers \alpha such that, for every positive integer n, the integer \lfloor \alph...

Difficulty: 5/10

Let ABC be an equilateral triangle. Let A_1, B_1, C_1 be interior points of ABC such that BA_1 = A_1...

Difficulty: 10/10

Let n be a positive integer. A Japanese triangle consists of 1 + 2 + \cdots + n circles arranged in ...

Difficulty: 7/10

Let x_1, x_2, \ldots, x_{2023} be pairwise different positive real numbers such that a_n = \sqrt{\le...

Difficulty: 5/10

For each integer k \geq 2, determine all infinite sequences of positive integers a_1, a_2, \ldots fo...

Difficulty: 9/10

Let ABC be an acute-angled triangle with AB < AC. Let \Omega be the circumcircle of ABC. Let S be th...

Difficulty: 7/10

Determine all composite integers n > 1 that satisfy the following property: if d_1 < d_2 < \cdots < ...

Difficulty: 5/10

Let n be a positive integer. A Nordic square is an n \times n board containing all the integers from...

Difficulty: 10/10

Find all triples (a, b, p) of positive integers with p prime and a^p = b! + p....

Difficulty: 7/10

Let ABCDE be a convex pentagon such that BC = DE. Assume that there is a point T inside ABCDE with T...

Difficulty: 5/10

Let k be a positive integer and let S be a finite set of odd prime numbers. Prove that there is at m...

Difficulty: 9/10

Find all functions f : \mathbb{R}^+ \to \mathbb{R}^+ such that for each x \in \mathbb{R}^+, there is...

Difficulty: 7/10

The Bank of Oslo issues two types of coin: aluminium (denoted A) and bronze (denoted B). Marianne ha...

Difficulty: 5/10

Let m \geq 2 be an integer, A a finite set of (not necessarily distinct) integers, and B_1, B_2, \ld...

Difficulty: 10/10

Two squirrels, Bushy and Jumpy, have collected 2021 walnuts for the winter. Jumpy numbers the walnut...

Difficulty: 7/10

Let \Gamma be a circle with centre I, and ABCD a convex quadrilateral such that each of the segments...

Difficulty: 5/10

Let D be an interior point of the acute triangle ABC with AB > AC such that \angle DAB = \angle CAD....

Difficulty: 9/10

Show that the inequality \sum_{i=1}^{n} \sum_{j=1}^{n} \sqrt{|x_i - x_j|} \leq \sum_{i=1}^{n} \sum_{...

Difficulty: 7/10

Let n \ge 100 be an integer. Let x_1, x_2, \dots, x_n be positive real numbers such that x_1 + x_2 +...

Difficulty: 7/10

Prove that there exists a positive constant c such that the following statement is true: Consider an...

Difficulty: 10/10

A deck of n > 1 cards is given. A positive integer is written on each card. The deck has the propert...

Difficulty: 7/10

There is an integer n > 1. There are n^2 stations on a slope of a mountain, all at different altitud...

Difficulty: 5/10

There are 4n pebbles of weights 1, 2, 3, \ldots, 4n. Each pebble is coloured in one of n colours and...

Difficulty: 9/10

Let a, b, c, d be positive real numbers such that a \geq b \geq c \geq d > 0 and a + b + c + d = 1. ...

Difficulty: 7/10

Consider a convex quadrilateral ABCD. The point P is in the interior of ABCD such that \angle PAD = ...

Difficulty: 7/10

Let I be the incentre of acute triangle ABC with AB \neq AC. The incircle \omega of ABC is tangent t...

Difficulty: 10/10

The Bank of Bath issues coins with an H on one side and a T on the other. Harry has n of these coins...

Difficulty: 7/10

Find all pairs (k, n) of positive integers such that k! = (2^n - 1)(2^n - 2)(2^n - 4) \cdots (2^n - ...

Difficulty: 5/10

A social network has 2019 users, some pairs of whom are friends. Whenever user A is friends with use...

Difficulty: 9/10

In triangle ABC, point A_1 lies on side BC and point B_1 lies on side AC. Let P and Q be points on s...

Difficulty: 7/10

Let \mathbb{Z} be the set of integers. Determine all functions f : \mathbb{Z} \to \mathbb{Z} such th...

Difficulty: 5/10

A convex quadrilateral ABCD satisfies AB \cdot CD = BC \cdot DA. Point X lies inside ABCD such that ...

Difficulty: 9/10

Let a_1, a_2, \ldots be an infinite sequence of positive integers. Suppose that there is an integer ...

Difficulty: 7/10

A site is any point (x, y) in the plane such that x and y are both positive integers less than or eq...

Difficulty: 6/10

An anti-Pascal triangle is an equilateral triangular array of numbers such that, except for the numb...

Difficulty: 10/10

Find all integers n \geq 3 for which there exist real numbers a_1, a_2, \ldots, a_{n+2} satisfying a...

Difficulty: 7/10

Let \Gamma be the circumcircle of acute triangle ABC. Points D and E are on segments AB and AC respe...

Difficulty: 5/10

An ordered pair (x, y) of integers is a primitive point if \gcd(x, y) = 1. Given a finite set S of p...

Difficulty: 10/10

A positive integer N \geq 2 is given. A collection of N(N+1) soccer players, no two of the same heig...

Difficulty: 8/10

Let R and S be different points on a circle \Omega such that RS is not a diameter. Let \ell be the t...

Difficulty: 7/10

A hunter and an invisible rabbit play a game in the Euclidean plane. The rabbit's initial point coin...

Difficulty: 9/10

Determine all functions f: \mathbb{R} \to \mathbb{R} such that, for any real numbers x and y, f(f(x)...

Difficulty: 7/10

For each integer a_0 > 1, define the sequence a_0, a_1, a_2, \dots by a_{n+1} = \begin{cases} \sqrt{...

Difficulty: 6/10

There are n \geq 2 line segments in the plane such that every two segments cross and no three segmen...

Difficulty: 10/10

The equation (x-1)(x-2) \cdots (x-2016) = (x-1)(x-2) \cdots (x-2016)is written on the board, with 20...

Difficulty: 7/10

A set of positive integers is called fragrant if it contains at least two elements and each of its e...

Difficulty: 5/10

Let P = A_1 A_2 \cdots A_k be a convex polygon in the plane. The vertices A_1, A_2, \ldots, A_k have...

Difficulty: 9/10

Find all integers n for which each cell of an n \times n table can be filled with one of the letters...

Difficulty: 7/10

In convex pentagon ABCDE with \angle B > 90^\circ, let F be a point on segment AC such that \angle F...

Difficulty: 5/10

The sequence a_1, a_2, \ldots of integers satisfies the following conditions: (i) 1 \leq a_j \leq 20...

Difficulty: 10/10

Let \mathbb{R} be the set of real numbers. Determine all functions f : \mathbb{R} \to \mathbb{R} sat...

Difficulty: 7/10

Triangle ABC has circumcircle \Omega and circumcenter O. A circle \Gamma with center A intersects se...

Difficulty: 5/10

Let ABC be an acute triangle with AB > AC. Let \Gamma be its circumcircle, H its orthocenter, and F ...

Difficulty: 9/10

Determine all triples (a, b, c) of positive integers such that each of the numbers ab - c, bc - a, c...

Difficulty: 7/10

We say that a finite set S in the plane is balanced if, for any two different points A, B in S, ther...

Difficulty: 7/10