Let $o$, $r$, $g$, $t$, $n$, $i$, $z$, $e$, and $d$ be positive reals. Show that \[ \sqrt{(d+o+t+t+e+d)(o+r+z+i+n+g)} > \sqrt{ti} + \sqrt{go} + \sqrt[6]{orz}. \]when $d^2e \geq \tfrac{2}{1434}$. Proposed by David Fox

Prove that for any convex quadrilateral there exist an inellipse and circumellipse which are homothetic. Proposed by Benny Wang + Oron Wang

Fix a positive integer $n$. Define sequences $a, b, c \in \mathbb{Q}^{n+1}$ by $(a_0, b_0, c_0) = (0, 0, 1)$ and \[ a_k = (n-k+1) \cdot c_{k-1}, \quad b_k = \binom nk - c_k - a_k, \quad \text{and} \quad c_k = \frac{b_{k-1}}{k} \]for each integer $1 \leq k \leq n$. $ $ $ $ $ $ $ $ $ $ Determine for which $n$ it happens that $a, b, c \in \mathbb{Z}^{n+1}$. Proposed by Jonathan Du

Determine all triples of positive integers $(A,B,C)$ for which some function $f \colon \mathbb Z_{\geq 0} \to \mathbb Z_{\geq 0}$ satisfies \[ f^{f(y)} (y + f(2x)) + f^{f(y)} (2y) = (Ax+By)^{C} \]for all nonnegative integers $x$ and $y$, where $f^k$ as usual denotes $f$ composed $k$ times. Proposed by Benny Wang

Inscribe three mutually tangent pink disks of radii $450$, $450$, and $720$ in an uncolored circle $\Omega$ of radius $1200$. In one move, Elmo selects an uncolored region inside $\Omega$ and draws in it the largest possible pink disk. Can Elmo ever draw a disk with a radius that is a perfect square of a rational? Proposed by Ritwin Narra

Bob and Cob are playing a game on an infinite grid of hexagons. On Bob's turn, he chooses one hexagon that has not yet been chosen, and draws a segment from the center of the hexagon to the midpoints of three of its sides. On Cob's turn, he erases one of Bob's edges made on the previous turn. Bob wins if his edges form a closed loop. Can Bob guarantee to win in a finite amount of time? (Note that Bob may win before Cob can play his next turn.) Proposed by Jonathan He

A scalene triangle $ABC$ was drawn, and Elmo marked its incenter $I$, Feuerbach point $X$, and Nagel point $N$. Sadly, after taking the abcdEfghijkLMnOpqrstuvwxyz, Elmo lost the triangle $ABC$. Can Elmo use only a ruler and compass to reconstruct the triangle? Proposed by Karn Chutinan

In $\triangle ABC$ let $D$ and $E$ be points on $AB$ and $AC$ respectively. The circumcircle of $\triangle CDE$ meets $AB$ again at $F$, and the circumcircle of $\triangle ACD$ meets $BC$ again at $G$. Show that if the circumcircles of $DFG$ and $ADE$ meet at $H$, then the three lines $AG$, $BE$, and $DH$ concur. Proposed by Oron Wang inspired by Tiger Zhang

Find all solutions to \[ (abcde)^2 = a^2+b^2+c^2+d^2+e^2+f^2. \]in integers. Proposed by Seongjin Shim

In triangle $ABC$ let the $A$-foot be $E$ and the $B$-excenter be $L$. Suppose the incircle of $ABC$ is tangent to $AC$ at $I$. Construct a hyperbola $\mathcal H$ through $A$ with $B$ and $C$ as the foci such that $A$ lies on the branch of the $\mathcal H$ closer to $C$. Construct an ellipse $\mathcal E$ passing through $I$ with $B$ and $C$ as the foci. Suppose $\mathcal E$ meets $\overline{AB}$ again at point $H$. Let $\overline{CH}$ and $\overline{BI}$ intersect the $C$-branch of $\mathcal H$ at points $M$ and $O$ respectively. Prove $E$, $L$, $M$, $O$ are concyclic. Proposed by Alex Wang

Fix a point $A$, a circle $\Omega$ centered at $O$, and reals $r$ and $\theta$. Let $X$ and $Y$ be variable points on $\Omega$ so that $\measuredangle XOY = \theta$. The tangents to $\Omega$ at $X$ and $Y$ meet at $T$, and a dilation at $T$ with scale factor $r$ sends $A$ to $A'$. Let $P$ be the foot from $A'$ to $TX$. $ $ $ $ $ $ $ $ $ $ Suppose that some point $P^*$ is the same for two different $X$. Show that $\measuredangle TXY = \measuredangle AP^\ast O$. (All angles are directed.) Proposed by Karn Chutinan

Prove that $\forall n\in\mathbb{Z}^+_0:(\exists b\in\mathbb{Z}^+_0:(\forall m\in\mathbb{Z}^+_0:((\exists x\in\mathbb{Z}^+_0:(x+m = b))\lor(\exists s\in\mathbb{Z}^+_0:(\exists p\in\mathbb{Z}^+_0:((\neg(\exists x\in\mathbb{Z}^+_0:(p+x = 1)))\land(\neg(\exists x\in\mathbb{Z}^+_0:(\exists y\in\mathbb{Z}^+_0:(p = (x+2) \cdot (y+2)))))\land(\exists x\in\mathbb{Z}^+_0:(p = m+x+1))\land(\exists r\in\mathbb{Z}^+_0:((\forall x\in\mathbb{Z}^+_0:(\forall y\in\mathbb{Z}^+_0:((\neg(x \cdot y = r))\lor(x = 1)\lor(\exists z\in\mathbb{Z}^+_0:(x = z \cdot p)))))\land(\exists x\in\mathbb{Z}^+_0:(\exists y\in\mathbb{Z}^+_0:((\exists z\in\mathbb{Z}^+_0:(r = y+z+1))\land(s = r \cdot (p \cdot x + m) + y))))))\land(\forall u\in\mathbb{Z}^+_0:((\exists x\in\mathbb{Z}^+_0:(u = p+x))\lor(u = 0)\lor(u = n+1)\lor(\neg(\exists r\in\mathbb{Z}^+_0:((\forall x\in\mathbb{Z}^+_0:(\forall y\in\mathbb{Z}^+_0:((\neg(x \cdot y = r))\lor(x = 1)\lor(\exists z\in\mathbb{Z}^+_0:(x = z \cdot p)))))\land(\exists x\in\mathbb{Z}^+_0:(\exists y\in\mathbb{Z}^+_0:((\exists z\in\mathbb{Z}^+_0:(r = y+z+1))\land(s = r \cdot (p \cdot x + u) + y)))))))\lor(\exists v\in\mathbb{Z}^+_0:(\exists k\in\mathbb{Z}^+_0:((\neg(v = 0))\land((u = v \cdot (k+2))\lor(u = v \cdot (k+2) + 1))\land(\exists r\in\mathbb{Z}^+_0:((\forall x\in\mathbb{Z}^+_0:(\forall y\in\mathbb{Z}^+_0:((\neg(x \cdot y = r))\lor(x = 1)\lor(\exists z\in\mathbb{Z}^+_0:(x = z \cdot p)))))\land(\exists x\in\mathbb{Z}^+_0:(\exists y\in\mathbb{Z}^+_0:((\exists z\in\mathbb{Z}^+_0:(r = y+z+1))\land(s = r \cdot (p \cdot x + v) + y)))))))))))))))))$. Proposed by Warren Bei

The RELMO is out! There has been a change to the format of the test; we formerly announced it of a single 4.5 hour, 7 problem test. However, to prioritize returners' mental sanity, we broke it up into two sections: a 1.5 hour section of 3 problems and a 3 hour section of 4 problems. Best of luck! We hope you enjoy our problems. Guide to the different testsBesides the RELMO, we have made the RELSMO, an olympiad in the same format as the RELMO but even more cursed. Several problems appear on both the RELMO and the RELSMO, so only take one. We have also produced some variants of the RELSMO with the same problems but improved flavortext. You choose! Links The RELMO The S variantsThe RELSMO The RELBMO The RELMORZ The RELSSMO The RELXMO Submission form