In $\vartriangle ABC$, let $D$ and $E$ be points on the angle bisector of $\angle BAC$ such that $\angle ABD = \angle ACE =90^o$ . Furthermore, let $F$ be the intersection of $AE$ and $BC$, and let $O$ be the circumcenter of $\vartriangle AF C$. If $\frac{AB}{AC} =\frac{3}{4}$, $AE = 40$, and $BD$ bisects $EF$, compute the perpendicular distance from $A$ to $OF$.

Let $ ABC$ be a non-isosceles triangle, in which $ X,Y,$ and $ Z$ are the tangency points of the incircle of center $ I$ with sides $ BC,CA$ and $ AB$ respectively. Denoting by $ O$ the circumcircle of $ \triangle{ABC}$, line $ OI$ meets $ BC$ at a point $ D.$ The perpendicular dropped from $ X$ to $ YZ$ intersects $ AD$ at $ E$. Prove that $ YZ$ is the perpendicular bisector of $ [EX]$.

Find all pairs $(p,q)$ of prime numbers such that $$ p(p^2 - p - 1) = q(2q + 3) .$$

Find the last digit of $2023^{2023}$. [i]Proposed by Yifan Kang[/i]

In equilateral triangle $ABC$, the midpoint of $\overline{BC}$ is $M$. If the circumcircle of triangle $MAB$ has area $36\pi$, then find the perimeter of the triangle. [i]Proposed by Isabella Grabski [/i]

A rectangle is said to be inscribed in a parallelogram if its vertices lie one on each side of the parallelogram. On the larger side $AB$ of the parallelogram $ABCD$, find all those points $K$ that are the vertices of the rectangles inscribed in $ABCD$.

In rectangle $ ABCD$, $ AD \equal{} 1$, $ P$ is on $ \overline{AB}$, and $ \overline{DB}$ and $ \overline{DP}$ trisect $ \angle ADC$. What is the perimeter of $ \triangle BDP$? [asy]unitsize(2cm); defaultpen(linewidth(.8pt)); dotfactor=4; pair D=(0,0), C=(sqrt(3),0), B=(sqrt(3),1), A=(0,1), P=(sqrt(3)/3,1); pair[] dotted={A,B,C,D,P}; draw(A--B--C--D--cycle); draw(B--D--P); dot(dotted); label("$A$",A,NW); label("$B$",B,NE); label("$C$",C,SE); label("$D$",D,SW); label("$P$",P,N);[/asy]$ \textbf{(A)}\ 3 \plus{} \frac {\sqrt3}{3} \qquad\textbf{(B)}\ 2 \plus{} \frac {4\sqrt3}{3}\qquad\textbf{(C)}\ 2 \plus{} 2\sqrt2\qquad\textbf{(D)}\ \frac {3 \plus{} 3\sqrt5}{2} \qquad\textbf{(E)}\ 2 \plus{} \frac {5\sqrt3}{3}$

Let $x,y,z$ be positive integers such that $x\neq y\neq z \neq x$ .Prove that $$(x+y+z)(xy+yz+zx-2)\geq 9xyz.$$ When does the equality hold? [i]Proposed by Dorlir Ahmeti, Albania[/i]

The sequence of integers $ a_1 $, $ a_2 $, $ \dots $ is defined as follows: $ a_1 = 1 $ and $ n> 1 $, $ a_ {n + 1} $ is the smallest integer greater than $ a_n $ and such, that $ a_i + a_j \neq 3a_k $ for any $ i, j $ and $ k $ from $ \{1, 2, \dots, n + 1 \} $ are not necessarily different. Define $ a_ {2004} $.

Determine all pairs of prime numbers $p$ and $q$ greater than $1$ and less than $100$, such that the following five numbers: $$p+6,p+10,q+4,q+10,p+q+1,$$ are all prime numbers.

There is a set of $N\geq 3$ points in the plane, such that no three of them are collinear. Three points $A$, $B$, $C$ in the set are said to form a [i]Baltic triangle[/i] if no other point in the set lies on the circumcircle of triangle $ABC$. Assume that there exists at least one Baltic triangle. Show that there exist at least $\displaystyle\frac{N}{3}$ Baltic triangles.

Evaluate \[ \log_{10}(\tan 1^{\circ})+ \log_{10}(\tan 2^{\circ})+ \log_{10}(\tan 3^{\circ})+ \cdots + \log_{10}(\tan 88^{\circ})+\log_{10}(\tan 89^{\circ}). \] $ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ \frac{1}{2}\log_{10}(\frac{\sqrt{3}}{2}) \qquad\textbf{(C)}\ \frac{1}{2}\log_{10}2 \qquad\textbf{(D)}\ 1 \qquad\textbf{(E)}\ \text{none of these} $

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 $S(n+1)$? $\textbf{(A)}\ 1 \qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 12\qquad\textbf{(D)}\ 1239\qquad\textbf{(E)}\ 1265$

A $6\times6$ square is given, and a quadratic trinomial with a positive leading coefficient is placed in each of its cells. There are $108$ coefficents in total, and these coefficents are chosen from the set $[-66;47]$, and each coefficient is different from each other. Prove that there exists at least one column such that the polynomial you get by summing the six trinomials in that column has a real root.

Let a triangle $ABC, A_1$ be the midpoint of the segment $[BC], B_1 \in (AC)$ ¸and $C_1 \in (AB)$ such that $[A_1B_1$ is the bisector of the angle $AA_1C$ and $A_1C_1$ is perpendicular to $AB$. Show that the lines $AA_1, BB_1$ and $CC_1$ are concurrent if and only if $ \angle BAC = 90^o$

Prove that the number $$2^{2^k-1}-2^k-1$$is composite (not prime) for all positive integers $k>2$.

Two equal-radii circles with centres $ O_1$ and $ O_2$ intersect each other at $ P$ and $ Q$, $ O$ is the midpoint of the common chord $ PQ$. Two lines $ AB$ and $ CD$ are drawn through $ P$ ( $ AB$ and $ CD$ are not coincide with $ PQ$ ) such that $ A$ and $ C$ lie on circle $ O_1$ and $ B$ and $ D$ lie on circle $ O_2$. $ M$ and $ N$ are the mipoints of segments $ AD$ and $ BC$ respectively. Knowing that $ O_1$ and $ O_2$ are not in the common part of the two circles, and $ M$, $ N$ are not coincide with $ O$. Prove that $ M$, $ N$, $ O$ are collinear.

Find all functions $f:\mathbb{R}\to\mathbb{R}$ which satisfy for all $x$: $(i)$ $f(x)=-f(-x);$ $(ii)$ $f(x+1)=f(x)+1;$ $(iii)$ $f\left( \frac{1}{x}\right)=\frac{1}{x^2}f(x)$ for $x\ne 0$

Let $k$, $n$ be fixed positive integers. In a circular table, there are placed pins numbered successively with the numbers $1, 2 \dots, n$, with $1$ and $n$ neighbors. It is known that pin $1$ is golden and the others are white. Arnaldo and Bernaldo play a game, in which a ring is placed initially on one of the pins and at each step it changes position. The game begins with Bernaldo choosing a starting pin for the ring, and the first step consists of the following: Arnaldo chooses a positive integer $d$ any and Bernaldo moves the ring $d$ pins clockwise or counterclockwise (positions are considered modulo $n$, i.e., pins $x$, $y$ equal if and only if $n$ divides $x-y$). After that, the ring changes its position according to one of the following rules, to be chosen at every step by Arnaldo: [b]Rule 1:[/b] Arnaldo chooses a positive integer $d$ and Bernaldo moves the ring $d$ pins clockwise or counterclockwise. [b]Rule 2:[/b] Arnaldo chooses a direction (clockwise or counterclockwise), and Bernaldo moves the ring in the chosen direction in $d$ or $kd$ pins, where $d$ is the size of the last displacement performed. Arnaldo wins if, after a finite number of steps, the ring is moved to the golden pin. Determine, as a function of $k$, the values of $n$ for which Arnaldo has a strategy that guarantees his victory, no matter how Bernaldo plays.

In a regular square room of side length $2\sqrt{2}$ ft, two cats that can see $2$ feet ahead of them are randomly placed into the four corners such that they do not share the same corner. If the probability that they don't see the mouse, also placed randomly into the room can be expressed as $\frac{a-b\pi}{c},$ where $a,b,c$ are positive integers with a greatest common factor of $1,$ then find $a+b+c.$ [i]Proposed by Ada Tsui[/i]

Any five points are taken inside or on a square with side length $1$. Let $a$ be the [i]smallest[/i] possible number with the property that it is always possible to select one pair of points from these five such that the distance between them is equal to or less than $a$. Then $a$ is: ${{ \textbf{(A)}\ \sqrt{3}/3 \qquad\textbf{(B)}\ \sqrt{2}/2 \qquad\textbf{(C)}\ 2\sqrt{2}/3 \qquad\textbf{(D)}\ 1 }\qquad\textbf{(E)}\ \sqrt{2} } $

This is an ARML Super Relay! I'm sure you know how this works! You start from #1 and #15 and meet in the middle. We are going to require you to solve all $15$ problems, though -- so for the entire task, submit the sum of all the answers, rather than just the answer to #8. Also, uhh, we can't actually find the slip for #1. Sorry about that. Have fun anyways! Problem 2. Let $T = TNYWR$. Find the number of way to distribute $6$ indistinguishable pieces of candy to $T$ hungry (and distinguishable) schoolchildren, such that each child gets at most one piece of candy. Problem 3. Let $T = TNYWR$. If $d$ is the largest proper divisor of $T$, compute $\frac12 d$. Problem 4. Let $T = TNYWR$ and flip $4$ fair coins. Suppose the probability that at most $T$ heads appear is $\frac mn$, where $m$ and $n$ are coprime positive integers. Compute $m+n$. Problem 5. Let $T = TNYWR$. Compute the last digit of $T^T$ in base $10$. Problem 6. Let $T = TNYWR$ and flip $6$ fair coins. Suppose the probability that at most $T$ heads appear is $\frac mn$, where $m$ and $n$ are coprime positive integers. Compute $m+n$. Problem 7. Let $T = TNYWR$. Compute the smallest prime $p$ for which $n^T \not\equiv n \pmod{p}$ for some integer $n$. Problem 8. Let $M$ and $N$ be the two answers received, with $M \le N$. Compute the number of integer quadruples $(w,x,y,z)$ with $w+x+y+z = M \sqrt{wxyz}$ and $1 \le w,x,y,z \le N$. Problem 9. Let $T = TNYWR$. Compute the smallest integer $n$ with $n \ge 2$ such that $n$ is coprime to $T+1$, and there exists positive integers $a$, $b$, $c$ with $a^2+b^2+c^2 = n(ab+bc+ca)$. Problem 10. Let $T = TNYWR$ and flip $10$ fair coins. Suppose the probability that at most $T$ heads appear is $\frac mn$, where $m$ and $n$ are coprime positive integers. Compute $m+n$. Problem 11. Let $T = TNYWR$. Compute the last digit of $T^T$ in base $10$. Problem 12. Let $T = TNYWR$ and flip $12$ fair coins. Suppose the probability that at most $T$ heads appear is $\frac mn$, where $m$ and $n$ are coprime positive integers. Compute $m+n$. Problem 13. Let $T = TNYWR$. If $d$ is the largest proper divisor of $T$, compute $\frac12 d$. Problem 14. Let $T = TNYWR$. Compute the number of way to distribute $6$ indistinguishable pieces of candy to $T$ hungry (and distinguishable) schoolchildren, such that each child gets at most one piece of candy. Also, we can't find the slip for #15, either. We think the SFBA coaches stole it to prevent us from winning the Super Relay, but that's not going to stop us, is it? We have another #15 slip that produces an equivalent answer. Here you go! Problem 15. Let $A$, $B$, $C$ be the answers to #8, #9, #10. Compute $\gcd(A,C) \cdot B$.

Find the second smallest prime factor of $18!+1.$ [i]Proposed by Kaylee Ji[/i]

Find all lists $(x_1, x_2, \ldots, x_{2020})$ of non-negative real numbers such that the following three conditions are all satisfied: [list] [*] $x_1 \le x_2 \le \ldots \le x_{2020}$; [*] $x_{2020} \le x_1 + 1$; [*] there is a permutation $(y_1, y_2, \ldots, y_{2020})$ of $(x_1, x_2, \ldots, x_{2020})$ such that $$\sum_{i = 1}^{2020} ((x_i + 1)(y_i + 1))^2 = 8 \sum_{i = 1}^{2020} x_i^3.$$ [/list] [i]A permutation of a list is a list of the same length, with the same entries, but the entries are allowed to be in any order. For example, $(2, 1, 2)$ is a permutation of $(1, 2, 2)$, and they are both permutations of $(2, 2, 1)$. Note that any list is a permutation of itself.[/i]

Find all number sets $(a,b,c,d)$ s.t. $1 < a \le b \le c \le d$, $a,b,c,d \in \mathbb{N}$, and $a^2+b+c+d$, $a+b^2+c+d$, $a+b+c^2+d$, and $a+b+c+d^2$ are all square numbers. Sum the value of $d$ across all solution set(s).