Let $ABCD$ be a convex quadrilateral such that the circle with diameter $AB$ is tangent to the line $CD$, and the circle with diameter $CD$ is tangent to the line $AB$. Prove that the two intersection points of these circles and the point $AC \cap BD$ are collinear.

Find all triplets $(a,b,c)$ of complex numbers such that the equation \[x^4-ax^3-bx+c=0\] has $a,b,c$ as roots.

The sequence $(p_n)$ is defined as follows: $p_1=2$ and for all $n$ greater than or equal to $2$, $p_n$ is the largest prime divisor of the expression $p_1p_2p_3\ldots p_{n-1}+1$. Prove that every $p_n$ is different from $5$.

What is the largest possible value of $|...| |a_1 - a_2| - a_3| - ... - a_{1990}|$, where $a_1, a_2, ... , a_{1990}$ is a permutation of $1, 2, 3, ... , 1990$?

[b]a)[/b] Among $9$ apparently identical coins, one is false and lighter than the others. How can you discover the fake coin by making $2$ weighing in a two-course balance? [b]b)[/b] Find the least necessary number of weighing that must be done to cover a false currency between $27$ coins if all the others are true.

Let $(I),(O)$ be the incircle, and, respectiely, circumcircle of $ABC$. $(I)$ touches $BC,CA,AB$ in $D,E,F$ respectively. We are also given three circles $\omega_a,\omega_b,\omega_c$, tangent to $(I),(O)$ in $D,K$ (for $\omega_a$), $E,M$ (for $\omega_b$), and $F,N$ (for $\omega_c$). [b]a)[/b] Show that $DK,EM,FN$ are concurrent in a point $P$; [b]b)[/b] Show that the orthocenter of $DEF$ lies on $OP$.

Let $ ABC$ is a triangle, let $ H$ is orthocenter of $ \triangle ABC$, let $ M$ is midpoint of $ BC$. Let $ (d)$ is a line perpendicular with $ HM$ at point $ H$. Let $ (d)$ meet $ AB, AC$ at $ E, F$ respectively. Prove that $ HE \equal{}HF$.

Let $a_1,a_2,a_3,\ldots$ be an infinite sequence of positive integers such that $a_{n+2m}$ divides $a_{n}+a_{n+m}$ for all positive integers $n$ and $m.$ Prove that this sequence is eventually periodic, i.e. there exist positive integers $N$ and $d$ such that $a_n=a_{n+d}$ for all $n>N.$

The fraction halfway between $ \frac{1}{5}$ and $ \frac{1}{3}$ (on the number line) is \[ \textbf{(A)}\ \frac{1}{4} \qquad \textbf{(B)}\ \frac{2}{15} \qquad \textbf{(C)}\ \frac{4}{15} \qquad \textbf{(D)}\ \frac{53}{200} \qquad \textbf{(E)}\ \frac{8}{15} \]

Let $ABC$ be an acute-angled triangle with $AB<AC$. Let $D$ be the midpoint of side $BC$ and $BE,CZ$ be the altitudes of the triangle $ABC$. Line $ZE$ intersects line $BC$ at point $O$. (i) Find all the angles of the triangle $ZDE$ in terms of angle $\angle A$ of the triangle $ABC$ (ii) Find the angle $\angle BOZ$ in terms of angles $\angle B$ and $\angle C$ of the triangle $ABC$

Given a quadrangle $ABCD$ and a point $M$ inside it such that $ABMD$ is a parallelogram. $ \angle CBM = \angle CDM$. Prove that the $ \angle ACD = \angle BCM$.

Two circles $\mathcal{C}_1$ and $\mathcal{C}_2$ intersect in $P$ and $Q$. A line through $P$ intersects $\mathcal{C}_1$ and $\mathcal{C}_2$ again at $A$ and $B$, respectively, and $X$ is the midpoint of $AB$. The line through $Q$ and $X$ intersects $C_1$ and $C_2$ again at $Y$ and $Z$, respectively. Prove that $X$ is the midpoint of $YZ$.

For positive reals $x,y,z\ge \frac{1}{2}$ with $x^2+y^2+z^2=1$, prove this inequality holds $$(\frac{1}{x}+\frac{1}{y}-\frac{1}{z})(\frac{1}{x}-\frac{1}{y}+\frac{1}{z})\ge 2$$

Let $a$ be a real number such that $\left(a + \frac{1}{a}\right)^2=11$. For which $n\in N$ is $a^n + \frac{1}{a^n}$ an integer? Does this depend on the exact value of $a$?

Let $n$ be a fixed natural number. Find all $n$ tuples of natural pairwise distinct and coprime numbers like $a_1,a_2,\ldots,a_n$ such that for $1\leq i\leq n$ we have \[ a_1+a_2+\ldots+a_n|a_1^i+a_2^i+\ldots+a_n^i \]

For all rational $x$ satisfying $0 \leq x < 1$, the functions $f$ is defined by \[f(x)=\begin{cases}\frac{f(2x)}{4},&\mbox{for }0 \leq x < \frac 12,\\ \frac 34+ \frac{f(2x - 1)}{4}, & \mbox{for } \frac 12 \leq x < 1.\end{cases}\] Given that $x = 0.b_1b_2b_3 \cdots $ is the binary representation of $x$, find, with proof, $f(x)$.

When two stars are very far apart their gravitational potential energy is zero; when they are separated by a distance $d$ the gravitational potential energy of the system is $U$. If the stars are separated by a distance $2d$ the gravitational potential energy of the system is $ \textbf{(A)}\ U/4\qquad\textbf{(B)}\ U/2 \qquad\textbf{(C)}\ U \qquad\textbf{(D)}\ 2U\qquad\textbf{(E)}\ 4U $

Let $n$ be a positive integer. Define a chameleon to be any sequence of $3n$ letters, with exactly $n$ occurrences of each of the letters $a, b,$ and $c$. Define a swap to be the transposition of two adjacent letters in a chameleon. Prove that for any chameleon $X$ , there exists a chameleon $Y$ such that $X$ cannot be changed to $Y$ using fewer than $3n^2/2$ swaps.

For every positive integer $n$ let $\tau (n)$ denote the number of its positive factors. Determine all $n \in N$ that satisfy the equality $\tau (n) = \frac{n}{3}$

Let $M$ be a positive integer. At a party with 120 people, 30 wear red hats, 40 wear blue hats, and 50 wear green hats. Before the party begins, $M$ pairs of people are friends. (Friendship is mutual.) Suppose also that no two friends wear the same colored hat to the party. During the party, $X$ and $Y$ can become friends if and only if the following two conditions hold: [list] [*] There exists a person $Z$ such that $X$ and $Y$ are both friends with $Z$. (The friendship(s) between $Z,X$ and $Z,Y$ could have been formed during the party.) [*] $X$ and $Y$ are not wearing the same colored hat. [/list] Suppose the party lasts long enough so that all possible friendships are formed. Let $M_1$ be the largest value of $M$ such that regardless of which $M$ pairs of people are friends before the party, there will always be at least one pair of people $X$ and $Y$ with different colored hats who are not friends after the party. Let $M_2$ be the smallest value of $M$ such that regardless of which $M$ pairs of people are friends before the party, every pair of people $X$ and $Y$ with different colored hats are friends after the party. Find $M_1+M_2$. [hide="Clarifications"] [list] [*] The definition of $M_2$ should read, ``Let $M_2$ be the [i]smallest[/i] value of $M$ such that...''. An earlier version of the test read ``largest value of $M$''.[/list][/hide] [i]Victor Wang[/i]

Prove the inequalities \[\frac{u}{v}\le \frac{\sin u}{\sin v}\le \frac{\pi}{2}\times\frac{u}{v},\text{ for }0 \le u < v \le \frac{\pi}{2}\]

Let $ABCD$ be a square with side $20$ and $T_1, T_2, ..., T_{2000}$ are points in $ABCD$ such that no $3$ points in the set $S = \{A, B, C, D, T_1, T_2, ..., T_{2000}\}$ are collinear. Prove that there exists a triangle with vertices in $S$, such that the area is less than $1/10$.

The [i]minimal polynomial[/i] of a complex number $r$ is the unique polynomial with rational coefficients of minimal degree with leading coefficient $1$ that has $r$ as a root. If $f$ is the minimal polynomial of $\cos\frac\pi7$, what is $f(-1)$?

Evaluate $ \int_1^e \frac{\sqrt[3]{x}}{x(\sqrt{x}\plus{}\sqrt[3]{x})}\ dx$.

[b]p1.[/b] Compute $1 + 3 + 6 + 10 + 15 + 21 + 28 + 36 + 45 + 55$. [b]p2.[/b] Given that $a$, $b$, and $c$ are positive integers such that $a+b = 9$ and $bc = 30$, find the minimum possible value of $a + c$. [b]p3.[/b] Points $X$ and $Y$ lie outside regular pentagon $ABCDE$ such that $ABX$ and $DEY$ are equilateral triangles. Find the degree measure of $\angle XCY$ . [b]p4.[/b] Let $N$ be the product of the positive integer divisors of $8!$, including itself. The largest integer power of $2$ that divides $N$ is $2^k$. Compute $k$. [b]p5.[/b] Let $A=(-20, 22)$, $B = (k, 0)$, and $C = (202, 2)$ be points on the coordinate plane. Given that $\angle ABC = 90^o$, find the sum of all possible values of $k$. [b]p6.[/b] Tej is typing a string of $L$s and $O$s that consists of exactly $7$ $L$s and $4$ $O$s. How many different strings can he type that do not contain the substring ‘$LOL$’ anywhere? A substring is a sequence of consecutive letters contained within the original string. [b]p7.[/b] How many ordered triples of integers $(a, b, c)$ satisfy both $a+b-c = 12$ and $a^2+b^2-c^2 = 24$? [b]p8.[/b] For how many three-digit base-$7$ numbers $\overline{ABC}_7$ does $\overline{ABC}_7$ divide $\overline{ABC}_{10}$? (Note: $\overline{ABC}_D$ refers to the number whose digits in base $D$ are, from left to right, $A$, $B$, and $C$; for example, $\overline{123}_4$ equals $27$ in base ten). [b]p9.[/b] Natasha is sitting on one of the $35$ squares of a $5$-by-$7$ grid of squares. Wanda wants to walk through every square on the board exactly once except the one Natasha is on, starting and ending on any $2$ squares she chooses, such that from any square she can only go to an adjacent square (two squares are adjacent if they share an edge). How many squares can Natasha choose to sit on such that Wanda cannot go on her walk? [b]p10.[/b] In triangle $ABC$, $AB = 13$, $BC = 14$, and $CA = 15$. Point $P$ lies inside $ABC$ and points $D,E$, and $F$ lie on sides $BC$, $CA$, and $AB$, respectively, so that $PD \perp BC$, $PE \perp CA$, and $PF \perp AB$. Given that $PD$, $PE$, and $PF$ are all integers, find the sum of all possible distinct values of $PD \cdot PE \cdot PF$. [b]p11.[/b] A palindrome is a positive integer which is the same when read forwards or backwards. Find the sum of the two smallest palindromes that are multiples of $137$. [b]p12.[/b] Let $P(x) = x^2+px+q$ be a quadratic polynomial with positive integer coefficients. Compute the least possible value of p such that 220 divides p and the equation $P(x^3) = P(x)$ has at least four distinct integer solutions. [b]p13.[/b] Everyone at a math club is either a truth-teller, a liar, or a piggybacker. A truth-teller always tells the truth, a liar always lies, and a piggybacker will answer in the style of the previous person who spoke (i.e., if the person before told the truth, they will tell the truth, and if the person before lied, then they will lie). If a piggybacker is the first one to talk, they will randomly either tell the truth or lie. Four seniors in the math club were interviewed and here was their conversation: Neil: There are two liars among us. Lucy: Neil is a piggybacker. Kevin: Excluding me, there are more truth-tellers than liars here. Neil: Actually, there are more liars than truth-tellers if we exclude Kevin. Jacob: One plus one equals three. Define the base-$4$ number $M = \overline{NLKJ}_4$, where each digit is $1$ for a truth-teller, $2$ for a piggybacker, and $3$ for a liar ($N$ corresponds to Neil, $L$ to Lucy, $K$ corresponds to Kevin, and $J$ corresponds to Jacob). What is the sum of all possible values of $M$, expressed in base $10$? [b]p14.[/b] An equilateral triangle of side length $8$ is tiled by $64$ equilateral triangles of unit side length to form a triangular grid. Initially, each triangular cell is either living or dead. The grid evolves over time under the following rule: every minute, if a dead cell is edge-adjacent to at least two living cells, then that cell becomes living, and any living cell remains living. Given that every cell in the grid eventually evolves to be living, what is the minimum possible number of living cells in the initial grid? [b]p15.[/b] In triangle $ABC$, $AB = 7$, $BC = 11$, and $CA = 13$. Let $\Gamma$ be the circumcircle of $ABC$ and let $M$, $N$, and $P$ be the midpoints of minor arcs $BC$ , $CA$, and $AB$ of $\Gamma$, respectively. Given that $K$ denotes the area of $ABC$ and $L$ denotes the area of the intersection of $ABC$ and $MNP$, the ratio $L/K$ can be written as $a/b$ , where $a$ and $b$ are relatively prime positive integers. Compute $a + b$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].