In acute $\triangle ABC,$ let $I$ denote the incenter and suppose that line $AI$ intersects segment $BC$ at a point $D.$ Given that $AI=3, ID=2,$ and $BI^2+CI^2=64,$ compute $BC^2.$ [i]Proposed by Kyle Lee[/i]

Find all positive integers $k$ for which there are infinitely many positive integers $n$ such that $\binom{(2025+k)n}{2025n}$ is not divisible by $kn+1$.

Prove that $ \frac{\pi}{2}\minus{}1<\int_{0}^{1}e^{\minus{}2x^{2}}\ dx$.

Let $ABCD$ be a cyclic quadrilateral. Prove that \[ |AB - CD| + |AD - BC| \geq 2|AC - BD|. \]

In an acute triangle $ABC$, points $D$ and $E$ lie on side $BC$ with $BD<BE$. Let $O_1, O_2, O_3, O_4, O_5, O_6$ be the circumcenters of triangles $ABD, ADE, AEC, ABE, ADC, ABC$, respectively. Prove that $O_1, O_3, O_4, O_5$ are con-cyclic if and only if $A, O_2, O_6$ are collinear.

Find the necessary and sufficient conditions that the real numbers $ a $, $ b $, $ c $ should satisfy so that the equation $$x^3 + ax^2 + bx + c = 0$$ has three real roots creating an arithmetic progression.

Daniel has a (mostly) standard deck of 54 cards, consisting of 4 suits each containing the ranks 1 to 13 as well as 2 jokers. Daniel plays the following game: He shuffles the deck uniformly randomly and then takes all of the cards that end up strictly between the two jokers. He then sums up the ranks of all the cards he has taken and calls that his score. Let $p$ be the probability that his score is a multiple of 13. There exists relatively prime positive integers $a$ and $b,$ with $b$ as small as possible, such that $|p - a/b| < 10^{-10}.$ What is $a/b?$ [i]Proposed by Dilhan Salgado, Daniel Li[/i]

Let $n$ be a positive integer and $d(n)$ is the number of positive integer divisors of $n$. For every two positive integer divisor $x,y$ of $n$, the remainders when $x,y$ divided by $d(n)+1$ are pairwise distinct. Show that either $d(n)+1$ is equal to prime or $4$.

In the game of Flow, a path is drawn through a $3\times3$ grid of squares obeying the following rules: i A path is continuous with no breaks (it can be drawn without lifting a pencil). ii A path that spans multiple squares can only be drawn between colored squares that share a side. iii A path cannot go through a square more than once. Compute the number of ways to color a positive number of squares on the grid such that a valid path can be drawn. An example of one such coloring and a valid path is shown below. [Insert Diagram] [i]Proposed by Alex Li[/i]

Let $a_1, a_2, a_3, a_4, a_5$ be non-negative real numbers satisfied \[\sum_{k = 1}^{5} a_k = 20 \ \ \ \ \text{and} \ \ \ \ \sum_{k=1}^{5} a_k^2 = 100\] Find the minimum and maximum of $\text{max} \{a_1, a_2, a_3, a_4, a_5\}$

Let $\{\varepsilon_i\}_{i\ge 1}, \{\theta_i\}_{i\ge 0}$ be two infinite sequences of real numbers, such that $\varepsilon_i \in \{-1,1\}$ for all $i$, and the numbers $\theta_i$ obey$$\tan \theta_{n+1} = \tan \theta_{n}+\varepsilon_n \sec(\theta_{n})-\tan \theta_{n-1} , \qquad n \ge 1$$and $\theta_0 = \frac{\pi}{4}, \theta_1 = \frac{2\pi}{3}$. Compute the sum of all possible values of $$\lim_{m \to \infty} \left(\sum_{n=1}^m \frac{1}{\tan \theta_{n+1} + \tan \theta_{n-1}} + \tan \theta_m - \tan \theta_{m+1}\right)$$ [i]Proposed by Grant Yu[/i]

The edges of a cube are labeled from $1$ to $12$. Show that there must exist at least eight triples $(i, j, k)$ with $1 \leq i < j < k \leq 12$ so that the edges $i, j, k$ are consecutive edges of a path. Also show that there exists labeling in which we cannot find nine such triples.

In the mathematical talent show called “The $X^2$-factor” contestants are scored by a a panel of $8$ judges. Each judge awards a score of $0$ (‘fail’), $X$ (‘pass’), or $X^2$ (‘pass with distinction’). Three of the contestants were Ann, Barbara and David. Ann was awarded the same score as Barbara by exactly $4$ of the judges. David declares that he obtained different scores to Ann from at least $4$ of the judges, and also that he obtained different scores to Barbara from at least $4$ judges. In how many ways could scores have been allocated to David, assuming he is telling the truth?

Let $A, B$, and $F$ be positive integers, and assume $A < B < 2A$. A flea is at the number $0$ on the number line. The flea can move by jumping to the right by $A$ or by $B$. Before the flea starts jumping, Lavaman chooses finitely many intervals $\{m+1, m+2, \ldots, m+A\}$ consisting of $A$ consecutive positive integers, and places lava at all of the integers in the intervals. The intervals must be chosen so that: ([i]i[/i]) any two distinct intervals are disjoint and not adjacent; ([i]ii[/i]) there are at least $F$ positive integers with no lava between any two intervals; and ([i]iii[/i]) no lava is placed at any integer less than $F$. Prove that the smallest $F$ for which the flea can jump over all the intervals and avoid all the lava, regardless of what Lavaman does, is $F = (n-1)A + B$, where $n$ is the positive integer such that $\frac{A}{n+1} \le B-A < \frac{A}{n}$.

Jacob rolls two fair six-sided dice. If the outcomes of these dice rolls are the same, he rolls a third fair six-sided die. Compute the probability that the sum of the outcomes of all the dice he rolls is even.

Prove that a tetrahedron with just one edge length greater than $1$ has volume at most $ \frac{1}{8}.$

Find all pairs $(m, n)$ of natural numbers for which the numbers $m^2 - 4n$ and $n^2 - 4m$ are both perfect squares.

Let the incircle $\omega $ of $\triangle ABC $ touch $AC $ and $AB $ at points $E $ and $F $ respectively. Points $X $, $Y $ of $\omega $ are such that $\angle BXC=\angle BYC=90^{\circ} $. Prove that $EF $ and $XY $ meet on the medial line of $ABC $.

In the triangle $ABC$, the angle $A$ is equal to $60^o$, and the median $BD$ is equal to the altitude $CH$. Prove that this triangle is equilateral.

Find all real numbers $a$ such that $3 < a < 4$ and $a(a-3\{a\})$ is an integer. (Here $\{a\}$ denotes the fractional part of $a$.)

Let $ABC$ be a triangle with circumcenter $O$. Let $\omega (O_1)$ be the circumference which passes through $A$ and $B$ and is tangent to $BC$ at $B$. $\omega (O_2)$ the circle that passes through $A$ and $C$ and is tangent to $BC$ at $C$. Let $M$ the midpoint of $O_1O_2$ and $D$ the symmetric point of $O$ with respect to $A$. Prove that $\angle O_1DM = \angle ODO_2$.

In $\triangle ABC$, $AB<AC$. $D$ and $P$ are the feet of the internal and external angle bisectors of $\angle BAC$, respectively. $M$ is the midpoint of segment $BC$, and $\omega$ is the circumcircle of $\triangle APD$. Suppose $Q$ is on the minor arc $AD$ of $\omega$ such that $MQ$ is tangent to $\omega$. $QB$ meets $\omega$ again at $R$, and the line through $R$ perpendicular to $BC$ meets $PQ$ at $S$. Prove $SD$ is tangent to the circumcircle of $\triangle QDM$. [i]Proposed by Ray Li[/i]

Let $P_n(x)=1+2x+3x^2+\cdots+nx^{n-1}.$ Prove that the polynomials $P_j(x)$ and $P_k(x)$ are relatively prime for all positive integers $j$ and $k$ with $j\ne k.$

A box contains exactly five chips, three red and two white. Chips are randomly removed one at a time without replacement until all the red chips are drawn or all the white chips are drawn. What is the probability that the last chip drawn is white? $ \displaystyle \textbf{(A)} \ \frac {3}{10} \qquad \textbf{(B)} \ \frac {2}{5} \qquad \textbf{(C)} \ \frac {1}{2} \qquad \textbf{(D)} \ \frac {3}{5} \qquad \textbf{(E)} \ \frac {7}{10}$

Given that \[ \frac{((3!)!)!}{3!} = k \cdot n!, \] where $k$ and $n$ are positive integers and $n$ is as large as possible, find $k + n$.