Let $k$ be a positive integer. Show that there are infinitely many positive integer solutions $(m, n)$ to $(m - n)^2 = kmn + m + n$.

The equation of line $\ell_1$ is $24x-7y = 319$ and the equation of line $\ell_2$ is $12x-5y = 125$. Let $a$ be the number of positive integer values $n$ less than $2023$ such that for both $\ell_1$ and $\ell_2$ there exists a lattice point on that line that is a distance of $n$ from the point $(20,23)$. Determine $a$. [i]Proposed by Christopher Cheng[/i] [hide=Solution][i]Solution. [/i] $\boxed{6}$ Note that $(20,23)$ is the intersection of the lines $\ell_1$ and $\ell_2$. Thus, we only care about lattice points on the the two lines that are an integer distance away from $(20,23)$. Notice that $7$ and $24$ are part of the Pythagorean triple $(7,24,25)$ and $5$ and $12$ are part of the Pythagorean triple $(5,12,13)$. Thus, points on $\ell_1$ only satisfy the conditions when $n$ is divisible by $25$ and points on $\ell_2$ only satisfy the conditions when $n$ is divisible by $13$. Therefore, $a$ is just the number of positive integers less than $2023$ that are divisible by both $25$ and $13$. The LCM of $25$ and $13$ is $325$, so the answer is $\boxed{6}$.[/hide]

Allen plays a game on a tree with $2n$ vertices, each of whose vertices can be red or blue. Initially, all of the vertices of the tree are colored red. In one move, Allen is allowed to take two vertices of the same color which are connected by an edge and change both of them to the opposite color. He wins if at any time, all of the verices of the tree are colored blue. (a) Show that Allen can win if and only if the vertices can be split up into two groups $V_1$ and $V_2$ to size $n$, such that each edge in the tree has one endpoint in $V_1$ and one endpoint in $V_2$. (b) Let $V_1 = \left\{ a_1, \ldots, a_n \right\}$ and $V_2 = \left\{ b_1, \ldots, b_n \right\}$ from part (a). Let $M$ be the minimum over all permutations $\sigma$ of $\left\{ 1, \ldots, n \right\}$ of the quantity \[ \sum\limits_{i = 1}^{n} d(a_i, b_{\sigma(i)}), \] where $d(v, w)$ denotes the number of edges along the shortest path between vertices $v$ and $w$ in the tree. Show that if Allen can win, then the minimum number of moves that it can take for Allen to win is equal to $M$.

Positive integers $x$ and $y$ satisfy the equation $\sqrt{x}+\sqrt{y}=\sqrt{1183}.$ What is the minimum possible value of $x+y?$ $\textbf{(A) }585 \qquad\textbf{(B) }595\qquad\textbf{(C) }623\qquad\textbf{(D) }700\qquad\textbf{(E) }791$

Tessa has a unit cube, on which each vertex is labeled by a distinct integer between 1 and 8 inclusive. She also has a deck of 8 cards, 4 of which are black and 4 of which are white. At each step she draws a card from the deck, and[list][*]if the card is black, she simultaneously replaces the number on each vertex by the sum of the three numbers on vertices that are distance 1 away from the vertex;[*]if the card is white, she simultaneously replaces the number on each vertex by the sum of the three numbers on vertices that are distance $\sqrt2$ away from the vertex.[/list]When Tessa finishes drawing all cards of the deck, what is the maximum possible value of a number that is on the cube?

Determine all $6$-tuples $(p,q,r,x,y,z)$ where $p,q,r$ are prime, and $x,y,z$ natural numbers such that $p^{2x}=q^yr^z+1$.

Let $f : [0,1] \to [0,1]$ be a continuous, strictly increasing function such that $f(0) = 0$ and $f(1) = 1$. Prove that $$f\left(\frac{1}{10}\right) + f\left(\frac{2}{10}\right) +...+f\left(\frac{9}{10}\right) +f^{-1}\left(\frac{1}{10}\right) +...+f^{-1}\left(\frac{9}{10}\right) \le \frac{99}{10}$$

A regular hexagon with side length $1$ is given. Using a ruler construct points in such a way that among the given and constructed points there are two such points that the distance between them is $\sqrt7$. Notes: ''Using a ruler construct points $\ldots$'' means: Newly constructed points arise only as the intersection of straight lines connecting two points that are given or already constructed. In particular, no length can be measured by the ruler.

Determine all real numbers $a, b, c, d$ for which $$ab + c + d = 3$$ $$bc + d + a = 5$$ $$cd + a + b = 2$$ $$da + b + c = 6$$

Let $1 = d_0 < d_1 < \dots < d_m = 4k$ be all positive divisors of $4k$, where $k$ is a positive integer. Prove that there exists $i \in \{1, \dots, m\}$ such that $d_i - d_{i-1} = 2$. [i]Ivan Mitrofanov[/i]

Jo and Blair take turns counting from $1$ to one more than the last number said by the other person. Jo starts by saying "$1$", so Blair follows by saying "$1$, $2$". Jo then says "$1$, $2$, $3$", and so on. What is the $53$rd number said? $ \textbf{(A) }2\qquad\textbf{(B) }3\qquad\textbf{(C) }5\qquad\textbf{(D) }6\qquad\textbf{(E) }8 $

Circles $C_1,C_2,$ and $C_3$ have radii $2,3,$ and $6$ respectively. If the fourth circle $C_4$ is the sum of the areas of $C_1,C_2,$ and $C_3,$ compute the radius of $C_4.$ [i]Proposed by Alex Li[/i]

Georg writes a positive integer $a$ on a blackboard. As long as there is a number on the blackboard, he does the following each day: $\bullet$ If the last digit in the number on the blackboard is less than or equal to $5$, he erases that last digit. (If there is only this digit, the blackboard thus becomes empty.) $\bullet$ Otherwise he erases the entire number and writes $9$ times the number. Can Georg choose $a$ in such a way that the blackboard never becomes empty?

Steven draws a line segment between every two of the points \[A(2,2), B(-2,2), C(-2,-2), D(2,-2), E(1,0), F(0,1), G(-1,0), H(0,-1).\] How many regions does he divide the square $ABCD$ into? [i]Proposed by Michael Ren

Let $ABCD$ be an isosceles trapezoid with bases $AD> BC$. Let $X$ be the intersection of the bisectors of $\angle BAC$ and $BC$. Let $E$ be the intersection of$ DB$ with the parallel to the bisector of $\angle CBD$ through $X$ and let $F$ be the intersection of $DC$ with the parallel to the bisector of $\angle DCB$ through $X$. Show that quadrilateral $AEFD$ is cyclic.

A box contains $p$ white balls and $q$ black balls. Beside the box there is a pile of black balls. Two balls are taken out of the box. If they have the same color, a black ball from the pile is put into the box. If they have different colors, the white ball is put back into the box. This procedure is repeated until the last two balls are removed from the box and one last ball is put in. What is the probability that this last ball is white?

Call a real-valued function $ f$ [i]very convex[/i] if \[ \frac {f(x) \plus{} f(y)}{2} \ge f\left(\frac {x \plus{} y}{2}\right) \plus{} |x \minus{} y| \] holds for all real numbers $ x$ and $ y$. Prove that no very convex function exists.

Prove the identity \[\sum\limits_{k=0}^n\binom{n}{k}\left(\tan\frac{x}{2}\right)^{2k}\left(1+\frac{2^k}{\left(1-\tan^2\frac{x}{2}\right)^k}\right)=\sec^{2n}\frac{x}{2}+\sec^n x\] for any natural number $n$ and any angle $x.$

Find $c$ if $a$, $b$, and $c$ are positive integers which satisfy $c=(a + bi)^3 - 107i$, where $i^2 = -1$.

A finite non-empty set of integers is called $3$-[i]good[/i] if the sum of its elements is divisible by $3$. Find the number of $3$-good subsets of $\{0,1,2,\ldots,9\}$.

Let $a,b,c,d$ be four real numbers such that $a+b+c+d=20$ and $ab+bc+cd+da=16$. Find the maximum possible value of $abc+bcd+cda+dab$. [i]Proposed by Yannick Yao[/i]

Let $a_1,a_2,a_3,\cdots$ be a non-decreasing sequence of positive integers. For $m\ge1$, define $b_m=\min\{n: a_n \ge m\}$, that is, $b_m$ is the minimum value of $n$ such that $a_n\ge m$. If $a_{19}=85$, determine the maximum value of \[a_1+a_2+\cdots+a_{19}+b_1+b_2+\cdots+b_{85}.\]

A polynomial $x^n + a_1x^{n-1} + a_2x^{n-2} +... + a_{n-2}x^2 + a_{n-1}x + a_n$ has $n$ distinct real roots $x_1, x_2,...,x_n$, where $n > 1$. The polynomial $nx^{n-1}+ (n - 1)a_1x^{n-2} + (n - 2)a_2x^{n-3} + ...+ 2a_{n-2}x + a_{n-1}$ has roots $y_1, y_2,..., y_{n_1}$. Prove that $\frac{x^2_1+ x^2_2+ ...+ x^2_n}{n}>\frac{y^2_1 + y^2_2 + ...+ y^2_{n-1}}{n - 1}$

A simplified model of a bicycle of mass $M$ has two tires that each comes into contact with the ground at a point. The wheelbase of this bicycle (the distance between the points of contact with the ground) is $w$, and the center of mass of the bicycle is located midway between the tires and a height h above the ground. The bicycle is moving to the right, but slowing down at a constant rate. The acceleration has a magnitude $a$. Air resistance may be ignored. [asy] size(175); pen dps = linewidth(0.7) + fontsize(4); defaultpen(dps); draw(circle((0,0),1),black+linewidth(2.5)); draw(circle((3,0),1),black+linewidth(2.5)); draw((1.5,0)--(0,0)--(1,1.5)--(2.5,1.5)--(1.5,0)--(1,1.5),black+linewidth(1)); draw((3,0)--(2.4,1.8),black+linewidth(1)); filldraw(circle((1.5,2/3),0.05),gray); draw((1.3,1.6)--(0.7,1.6)--(0.7,1.75)--cycle,black+linewidth(1)); label("center of mass of bicycle",(2.5,1.9)); draw((1.55,0.85)--(1.8,1.8),BeginArrow); draw((4.5,-1)--(4.5,2/3),BeginArrow,EndArrow); label("$h$",(4.5,-1/6),E); draw((1.5,2/3)--(4.5,2/3),dotted); draw((0,-1)--(4.5,-1),dotted); draw((0,-5/4)--(3,-5/4),BeginArrow,EndArrow); label("$w$",(3/2,-5/4),S); draw((0,-1)--(0,-6/4),dotted); draw((3,-1)--(3,-6/4),dotted); [/asy] Case 2 ([b][u]Question 30[/u][/b]): Assume, instead, that the coefficient of sliding friction between each tire and the ground is different: $\mu_1$ for the front tire and $\mu_2$ for the rear tire. Let $\mu_1 = 2\mu_2$. Assume that both tires are skidding: sliding without rotating. What is the maximum value of $a$ so that both tires remain in contact with the ground? $ \textbf{(A)}\ \frac{wg}{h} $ $ \textbf{(B)}\ \frac{wg}{3h} $ $ \textbf{(C)}\ \frac{2wg}{3h} $ $ \textbf{(D)}\ \frac{hg}{2w}$ $ \textbf{(E)}\ \text{none of the above} $

Sphere $S$ touches a plane. Let $A,B,C,D$ be four points on the plane such that no three of them are collinear. Consider the point $A'$ such that $S$ in tangent to the faces of tetrahedron $A'BCD$. Points $B',C',D'$ are defined similarly. Prove that $A',B',C',D'$ are coplanar and the plane $A'B'C'D'$ touches $S$. [i]Proposed by Alexey Zaslavsky (Russia)[/i]