Find all polynomials \(P(x)\) with real coefficients such that for all nonzero real numbers \(x\), \[P(x)+P\left(\frac1x\right) =\frac{P\left(x+\frac1x\right) +P\left(x-\frac1x\right)}2.\] Proposed by Holden Mui

Let \(\mathbb R_{>0}\) denote the set of positive real numbers. Find all functions \(f:\mathbb R_{>0}\to\mathbb R_{>0}\) such that for all positive real numbers \(x\) and \(y\), \[f(xy+1)=f(x)f\left(\frac1x+f\left(\frac1y\right)\right).\] Proposed by Luke Robitaille

Does there exist an infinite sequence of integers \(a_0\), \(a_1\), \(a_2\), \(\ldots\) such that \(a_0\ne0\) and, for any integer \(n\ge0\), the polynomial \[P_n(x)=\sum_{k=0}^na_kx^k\]has \(n\) distinct real roots? Proposed by Amol Rama and Espen Slettnes

Let \(f:\mathbb R\to\mathbb R\) be a function such that for all real numbers \(x\neq1\), \[f(x-f(x))+f(x)=\frac{x^2-x+1}{x-1}.\]Find all possible values of \(f(2023)\). Proposed by Linus Tang

Find the least positive integer \(M\) for which there exist a positive integer \(n\) and polynomials \(P_1(x)\), \(P_2(x)\), \(\ldots\), \(P_n(x)\) with integer coefficients satisfying \[Mx=P_1(x)^3+P_2(x)^3+\cdots+P_n(x)^3.\] Proposed by Karthik Vedula

Let \(\mathbb R_{>0}\) denote the set of positive real numbers and \(\mathbb R_{\ge0}\) the set of nonnegative real numbers. Find all functions \(f:\mathbb R\times \mathbb R_{>0}\to \mathbb R_{\ge0}\) such that for all real numbers \(a\), \(b\), \(x\), \(y\) with \(x,y>0\), we have \[f(a,x)+f(b,y)=f(a+b,x+y)+f(ay-bx,xy(x+y)).\] Proposed by Luke Robitaille

Elmo has 2023 cookie jars, all initially empty. Every day, he chooses two distinct jars and places a cookie in each. Every night, Cookie Monster finds a jar with the most cookies and eats all of them. If this process continues indefinitely, what is the maximum possible number of cookies that the Cookie Monster could eat in one night? Proposed by Espen Slettnes

Alice is performing a magic trick. She has a standard deck of 52 cards, which she may order beforehand. She invites a volunteer to pick an integer \(0\le n\le 52\), and cuts the deck into a pile with the top \(n\) cards and a pile with the remaining \(52-n\). She then gives both piles to the volunteer, who riffles them together and hands the deck back to her face down. (Thus, in the resulting deck, the cards that were in the deck of size \(n\) appear in order, as do the cards that were in the deck of size \(52-n\).) Alice then flips the cards over one-by-one from the top. Before flipping over each card, she may choose to guess the color of the card she is about to flip over. She stops if she guesses incorrectly. What is the maximum number of correct guesses she can guarantee? Proposed by Espen Slettnes

Find all pairs of positive integers \((a,b)\) with the following property: there exists an integer \(N\) such that for any integers \(m\ge N\) and \(n\ge N\), every \(m\times n\) grid of unit squares may be partitioned into \(a\times b\) rectangles and fewer than \(ab\) unit squares. Proposed by Holden Mui

Let \(n\) be a positive integer and consider an \(n\times n\) square grid. For \(1\le k\le n\), a python of length \(k\) is a snake that occupies \(k\) consecutive cells in a single row, and no other cells. Similarly, an anaconda of length \(k\) is a snake that occupies \(k\) consecutive cells in a single column, and no other cells. The grid contains at least one python or anaconda, and it satisfies the following properties: No cell is occupied by multiple snakes. If a cell in the grid is immediately to the left or immediately to the right of a python, then that cell must be occupied by an anaconda. If a cell in the grid is immediately to above or immediately below an anaconda, then that cell must be occupied by a python. Prove that the sum of the squares of the lengths of the snakes is at least \(n^2\). Proposed by Linus Tang

Define the mexth of \(k\) sets as the \(k\)th smallest positive integer that none of them contain, if it exists. Does there exist a family \(\mathcal F\) of sets of positive integers such that for any nonempty finite subset \(\mathcal G\) of \(\mathcal F\), the mexth of \(\mathcal G\) exists, and for any positive integer \(n\), there is exactly one nonempty finite subset \(\mathcal G\) of \(\mathcal F\) such that \(n\) is the mexth of \(\mathcal G\). Proposed by Espen Slettnes

For a set \(S\) of positive integers and a positive integer \(n\), consider the game of \((n,S)\)-nim, which is as follows. A pile starts with \(n\) watermelons. Two players, Deric and Erek, alternate turns eating watermelons from the pile, with Deric going first. On any turn, the number of watermelons eaten must be an element of \(S\). The last player to move wins. Let \(f(S)\) denote the set of positive integers \(n\) for which Deric has a winning strategy in \((n,S)\)-nim. Let \(T\) be a set of positive integers. Must the sequence \[T, \; f(T), \; f(f(T)), \;\ldots\]be eventually constant? Proposed by Brandon Wang and Edward Wan

A discrete hexagon with center \((a,b,c)\) \emph{(where \(a\), \(b\), \(c\) are integers) and radius \(r\) (a nonnegative integer)} is the set of lattice points \((x,y,z)\) such that \(x+y+z=a+b+c\) and \(\max(|x-a|,|y-b|,|z-c|)\le r\). Let \(n\) be a nonnegative integer and \(S\) be the set of triples \((x,y,z)\) of nonnegative integers such that \(x+y+z=n\). If \(S\) is partitioned into discrete hexagons, show that at least \(n+1\) hexagons are needed. Proposed by Linus Tang

Let \(n\ge3\) be a fixed integer, and let \(\alpha\) be a fixed positive real number. There are \(n\) numbers written around a circle such that there is exactly one \(1\) and the rest are \(0\)'s. An operation consists of picking a number \(a\) in the circle, subtracting some positive real \(x\le a\) from it, and adding \(\alpha x\) to each of its neighbors. Find all pairs \((n,\alpha)\) such that all the numbers in the circle can be made equal after a finite number of operations. Proposed by Anthony Wang

Let \(ABCDE\) be a regular pentagon. Let \(P\) be a variable point on the interior of segment \(AB\) such that \(PA\ne PB\). The circumcircles of \(\triangle PAE\) and \(\triangle PBC\) meet again at \(Q\). Let \(R\) be the circumcenter of \(\triangle DPQ\). Show that as \(P\) varies, \(R\) lies on a fixed line. Proposed by Karthik Vedula

Let \(ABC\) be an acute scalene triangle with orthocenter \(H\). Line \(BH\) intersects \(\overline{AC}\) at \(E\) and line \(CH\) intersects \(\overline{AB}\) at \(F\). Let \(X\) be the foot of the perpendicular from \(H\) to the line through \(A\) parallel to \(\overline{EF}\). Point \(B_1\) lies on line \(XF\) such that \(\overline{BB_1}\) is parallel to \(\overline{AC}\), and point \(C_1\) lies on line \(XE\) such that \(\overline{CC_1}\) is parallel to \(\overline{AB}\). Prove that points \(B\), \(C\), \(B_1\), \(C_1\) are concyclic. Proposed by Luke Robitaille

Two triangles intersect to form seven finite disjoint regions, six of which are triangles with area 1. The last region is a hexagon with area \(A\). Compute the minimum possible value of \(A\). Proposed by Karthik Vedula

Let \(D\) be a point on segment \(PQ\). Let \(\omega\) be a fixed circle passing through \(D\), and let \(A\) be a variable point on \(\omega\). Let \(X\) be the intersection of the tangent to the circumcircle of \(\triangle ADP\) at \(P\) and the tangent to the circumcircle of \(\triangle ADQ\) at \(Q\). Show that as \(A\) varies, \(X\) lies on a fixed line. Proposed by Elliott Liu and Anthony Wang

Let \(ABC\) be an acute triangle with circumcircle \(\omega\). Let \(P\) be a variable point on the arc \(BC\) of \(\omega\) not containing \(A\). Squares \(BPDE\) and \(PCFG\) are constructed such that \(A\), \(D\), \(E\) lie on the same side of line \(BP\) and \(A\), \(F\), \(G\) lie on the same side of line \(CP\). Let \(H\) be the intersection of lines \(DE\) and \(FG\). Show that as \(P\) varies, \(H\) lies on a fixed circle. Proposed by Karthik Vedula

Let \(ABCDEF\) be a convex cyclic hexagon such that quadrilateral \(ABDF\) is a square, and the incenter of \(\triangle ACE\) lines on \(\overline{BF}\). Diagonal \(CE\) intersects diagonals \(BD\) and \(DF\) at points \(P\) and \(Q\), respectively. Prove that the circumcircle of \(\triangle DPQ\) is tangent to \(\overline{BF}\). Proposed by Elliott Liu

Let \(\mathcal E\) be an ellipse with foci \(F_1\) and \(F_2\), and let \(P\) be a point on \(\mathcal E\). Suppose lines \(PF_1\) and \(PF_2\) intersect \(\mathcal E\) again at distinct points \(A\) and \(B\), and the tangents to \(\mathcal E\) at \(A\) and \(B\) intersect at point \(Q\). Show that the midpoint of \(\overline{PQ}\) lies on the circumcircle of \(\triangle PF_1F_2\). Proposed by Karthik Vedula

Convex quadrilaterals \(ABCD\), \(A_1B_1C_1D_1\), and \(A_2B_2C_2D_2\) are similar with vertices in order. Points \(A\), \(A_1\), \(B_2\), \(B\) are collinear in order, points \(B\), \(B_1\), \(C_2\), \(C\) are collinear in order, points \(C\), \(C_1\), \(D_2\), \(D\) are collinear in order, and points \(D\), \(D_1\), \(A_2\), \(A\) are collinear in order. Diagonals \(AC\) and \(BD\) intersect at \(P\), diagonals \(A_1C_1\) and \(B_1D_1\) intersect at \(P_1\), and diagonals \(A_2C_2\) and \(B_2D_2\) intersect at \(P_2\). Prove that points \(P\), \(P_1\), and \(P_2\) are collinear. Proposed by Holden Mui

Let \(m\) be a positive integer. Find, in terms of \(m\), all polynomials \(P(x)\) with integer coefficients such that for every integer \(n\), there exists an integer \(k\) such that \(P(k)=n^m\). Proposed by Raymond Feng

Determine the greatest positive integer \(n\) for which there exists a sequence of distinct positive integers \(s_1\), \(s_2\), \(\ldots\), \(s_n\) satisfying \[s_1^{s_2}=s_2^{s_3}=\cdots=s_{n-1}^{s_n}.\] Proposed by Holden Mui

Let \(a\), \(b\), and \(n\) be positive integers. A lemonade stand owns \(n\) cups, all of which are initially empty. The lemonade stand has a filling machine and an emptying machine, which operate according to the following rules: If at any moment, \(a\) completely empty cups are available, the filling machine spends the next \(a\) minutes filling those \(a\) cups simultaneously and doing nothing else. If at any moment, \(b\) completely full cups are available, the emptying machine spends the next \(b\) minutes emptying those \(b\) cups simultaneously and doing nothing else. Suppose that after a sufficiently long time has passed, both the filling machine and emptying machine work without pausing. Find, in terms of \(a\) and \(b\), the least possible value of \(n\). Proposed by Raymond Feng

Let \(d(n)\) denote the number of positive divisors of \(n\). The sequence \(a_0\), \(a_1\), \(a_2\), \(\ldots\) is defined as follows: \(a_0=1\), and for all integers \(n\ge1\), \[a_n=d(a_{n-1})+d(d(a_{n-2}))+\cdots+ {\underbrace{d(d(\ldots d(a_0)\ldots))}_{n\text{ times}}}.\]Show that for all integers \(n\ge1\), we have \(a_n\le3n\). Proposed by Karthik Vedula

An ordered pair \((k,n)\) of positive integers is good if there exists an ordered quadruple \((a,b,c,d)\) of positive integers such that \(a^3+b^k=c^3+d^k\) and \(abcd=n\). Prove that there exist infinitely many positive integers \(n\) such that \((2022,n)\) is not good but \((2023,n)\) is good. Proposed by Luke Robitaille