Let $ABC$ be an acute-angled triangle with integer side lengths and $AB<AC$. Points $D$ and $E$ lie on segments $BC$ and $AC$, respectively, such that $AD=AE=AB$. Line $DE$ intersects $AB$ at $X$. Circles $BXD$ and $CED$ intersect for the second time at $Y \neq D$. Suppose that $Y$ lies on line $AD$. Find the sum of all possible values of $BC$.
Ground Truth:336
#2 · 26de63
Unsolved
Define a function $f \colon \mathbb{Z}_{\geq 1} \to \mathbb{Z}_{\geq 1}$ by $f(n) = \sum_{i=1}^n \sum_{j=1}^n j^{1024} \lfloor \frac{1}{j} + \frac{n-i}{n} \rfloor$. Let $M=2\cdot3\cdot5\cdot7\cdot11\cdot13$ and let $N = f(M^{15}) - M^{15360} \cdot \frac{M^{1025}-1}{1025}$. Find the remainder when $N$ is divided by $10^6$.
Ground Truth:32951
#3 · 3a6e20
Unsolved
Find the number of positive integers $n \leq 600$ such that $\lfloor\log_2(2n)\rfloor$ is even.
Ground Truth:300
#4 · 4e5b6a
Unsolved
Let $\omega = e^{2\pi i/2026}$. Find the number of unordered pairs $\{A, B\}$ of subsets of $\{0,1,\ldots,2025\}$ such that $\sum_{a \in A} \omega^a = \sum_{b \in B} \omega^b$.
Ground Truth:94
#5 · 6c2da5
Unsolved
For a sequence of real numbers $a_0, a_1, a_2, \ldots$, define $H_n = \frac{1}{\binom{n}{0}a_0 + \binom{n}{1}a_1 + \cdots + \binom{n}{n}a_n}$ for each non-negative integer $n$. Suppose $H_0 = 1, H_1 = \frac{1}{3}, H_2 = \frac{1}{7}, H_3 = \frac{1}{15}$. Find $H_{10} \cdot 2^{55}$.