A nonzero polynomial $f(x)$ with real coefficients has the property that $f(x)=f^\prime(x)f^{\prime\prime}(x)$. What is the leading coefficient of $f(x)$?

Chandler the Octopus is at a tentacle party! At this party, there is $1$ creature with $2$ tentacles, $2$ creatures with $3$ tentacles, $3$ creatures with $4$ tentacles, all the way up to $14$ creatures with $15$ tentacles. Each tentacle is distinguishable from all other tentacles. For some $2\le m < n \le 15$, a creature with m tentacles “meets” a creature with n tentacles; “meeting” another creature consists of shaking exactly 1 tentacle with each other. Find the number of ways there are to pick distinct $m < n$ between $2$ and $15$, inclusive, and then to pick a creature with $m$ tentacles to “meet” a selected creature with $n$ tentacles. [i]Proposed by Armaan Tipirneni, Richard Chen, and Denise the Octopus[/i]

Given a positive integer $n$, it can be shown that every complex number of the form $r+si$, where $r$ and $s$ are integers, can be uniquely expressed in the base $-n+i$ using the integers $1,2,\ldots,n^2$ as digits. That is, the equation\[ r+si=a_m(-n+i)^m+a_{m-1}(-n+i)^{m-1}+\cdots +a_1(-n+i)+a_0 \]is true for a unique choice of non-negative integer $m$ and digits $a_0,a_1,\ldots,a_m$ chosen from the set $\{0,1,2,\ldots,n^2\}$, with $a_m\ne 0$. We write \[ r+si=(a_ma_{m-1}\ldots a_1a_0)_{-n+i} \]to denote the base $-n+i$ expansion of $r+si$. There are only finitely many integers $k+0i$ that have four-digit expansions \[ k=(a_3a_2a_1a_0)_{-3+i}~~~~a_3\ne 0. \]Find the sum of all such $k$.

Let $S(n)$ equal the sum of the digits of positive integer $n$. For example, $S(1507) = 13$. For a particular positive integer $n$, $S(n) = 1274$. Which of the following could be the value of $S(n+1)$? $\textbf{(A)}\ 1 \qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 12\qquad\textbf{(D)}\ 1239\qquad\textbf{(E)}\ 1265$

Let $ABC$ be a scalene acute triangle with incentre $I{}$ and circumcentre $O{}$. Let $AI$ cross $BC$ at $D$. On circle $ABC$, let $X$ and $Y$ be the mid-arc points of $ABC$ and $BCA$, respectively. Let $DX{}$ cross $CI{}$ at $E$ and let $DY{}$ cross $BI{}$ at $F{}$. Prove that the lines $FX, EY$ and $IO$ are concurrent on the external bisector of $\angle BAC$. [i]David-Andrei Anghel[/i]

Let $P,Q\in\mathbb{R}[x]$, such that $Q$ is a $2021$-degree polynomial and let $a_{1}, a_{2}, \ldots , a_{2022}, b_{1}, b_{2}, \ldots , b_{2022}$ be real numbers such that $a_{1}a_{2}\ldots a_{2022}\neq 0$. If for all real $x$ \[P(a_{1}Q(x) + b_{1}) + \ldots + P(a_{2021}Q(x) + b_{2021}) = P(a_{2022}Q(x) + b_{2022})\] prove that $P(x)$ has a real root.

$6$ indistinguishable coins are given, $4$ are authentic, all of the same weight, and $2$ are false, one is more light than the real ones and the other one, heavier than the real ones. The two false ones together weigh same as two authentic coins. Find two authentic coins using a balance scale twice only by two plates, no weights. Clarification: A two-pan scale only reports if the left pan weighs more, equal or less that right.

The shortest distance from an integral point to line $y=\frac{5}{3}x+\frac{4}{5}$ is $\text{(A)}\frac{\sqrt{34}}{170}\qquad\text{(B)}\frac{\sqrt{34}}{85}\qquad\text{(C)}\frac{1}{20}\qquad\text{(D)}\frac{1}{30}$

Find the area of the part enclosed by the following curve. \[x^2+2axy+y^2=1\ (-1<a<1)\]

A finite sequence of three-digit integers has the property that the tens and units digits of each terms are, respectively, the hundreds and tens digits of the next term, and the tens and units digits of the last term are, respectively, the hundreds and tens digits of the first term. For example, such a sequence might begin with terms $ 247$, $ 475$, and $ 756$ and end with the term $ 824$. Let $ S$ be the sum of all the terms in the sequence. What is the largest prime number that always divides $ S$? $ \textbf{(A)}\ 3 \qquad \textbf{(B)}\ 7 \qquad \textbf{(C)}\ 13 \qquad \textbf{(D)}\ 37 \qquad \textbf{(E)}\ 43$

The $64$ whole numbers from $1$ through $64$ are written, one per square, on a checkerboard (an $8$ by $8$ array of $64$ squares). The first $8$ numbers are written in order across the first row, the next $8$ across the second row, and so on. After all $64$ numbers are written, the sum of the numbers in the four corners will be $\text{(A)}\ 130 \qquad \text{(B)}\ 131 \qquad \text{(C)}\ 132 \qquad \text{(D)}\ 133 \qquad \text{(E)}\ 134$

Let $a$ and $b$ be perfect squares whose product exceeds their sum by $4844$. Compute the value of \[\left(\sqrt a + 1\right)\left(\sqrt b + 1\right)\left(\sqrt a - 1\right)\left(\sqrt b - 1\right) - \left(\sqrt{68} + 1\right)\left(\sqrt{63} + 1\right)\left(\sqrt{68} - 1\right)\left(\sqrt{63} - 1\right).\]

In triangle $ABC$ medians from $B$ and $C$ are perpendicular. Prove that $\frac{\sin(B+C)}{\sin B \cdot \sin C} \geq \frac{2}{3}.$

For each positive integer $k$, let $S(k)$ be the sum of the digits of $k$ in the decimal system. Find all positive integers N for which there exist positive integers $a$,$b$,$c$, coprime two by two, such that: $S(ab) = S(bc) = S(ca) = N$.

Let $ABC$ be a scalene triangle with $\angle A = 60^{\circ}$. Let $E$ and $F$ be the feet of the angle bisectors of $\angle ABC$ and $\angle ACB$, respectively, and let $I$ be the incenter of $\triangle ABC$. Let $P,Q$ be distinct points such that $\triangle PEF$ and $\triangle QEF$ are equilateral. If $O$ is the circumcenter of of $\triangle APQ$, show that $\overline{OI}\perp \overline{BC}$. [i]Proposed by Vincent Huang

$AH$ is the altitude of triangle $ABC$ and $H^\prime$ is the reflection of $H$ trough the midpoint of $BC$. If the tangent lines to the circumcircle of $ABC$ at $B$ and $C$, intersect each other at $X$ and the perpendicular line to $XH^\prime$ at $H^\prime$, intersects $AB$ and $AC$ at $Y$ and $Z$ respectively, prove that $\angle ZXC=\angle YXB$.

A circle center $O$ is inscribed in the quadrilateral $ABCD$. $AB$ is parallel to and longer than $CD$ and has midpoint $M$. The line $OM$ meets $CD$ at $F$. $CD$ touches the circle at $E$. Show that $DE = CF$ iff $AB = 2CD$.