Ria writes down the numbers $1,2,\dots ,101$ in red and blue pens. The largest blue number is equal to the number of numbers written in blue, and the smallest red number is equal to half the number of numbers written in red. How many numbers did Ria write with a red pen?

Let $R$ be the number of red numbers, and $B$ be the number of blue numbers.
Since the total numbers written is 101, we have:

Let the largest blue number be $L_b$ and the smallest red number be $S_r$.

• From the problem, the largest blue number equals the count of blue numbers:

• The smallest red number equals half the count of red numbers:

Since $S_r$ is a positive integer, $R$ must be an even number.

Now, consider the distribution of the numbers. All blue numbers are at most $L_b=B$. Thus, all numbers larger than $B$ must be written in red.
The numbers larger than $B$ are $B+1,B+2,\dots ,101$. There are exactly $101-B=R$ such numbers.
Therefore, the red numbers must be exactly the set of numbers from $B+1$ to $101$.
The smallest red number in this set is $S_r=B+1$.

Substitute $S_r=R/2$ and $B=101-R$:

Therefore, Ria wrote $68$ numbers with a red pen.