Let $A,B,P$ be points on a line $\ell$, with $P$ outside the segment $AB$. Lines $a$ and $b$ pass through $A$ and $B$ and are perpendicular to $\ell$. A line $m$ through $P$, which is neither parallel nor perpendicular to $\ell$, intersects $a$ and $b$ at $Q$ and $R$, respectively. The perpendicular from $B$ to $AR$ meets $a$ and $AR$ at $S$ and $U$, and the perpendicular from $A$ to $BQ$ meets $b$ and $BQ$ at $T$ and $V$, respectively. (a) Prove that $P,S,T$ are collinear. (b) Prove that $P,U,V$ are collinear.

How many real numbers $x$ satisfy the equation $\frac{1}{5}\log_2 x = \sin (5\pi x)$?

Antonette gets $ 70\%$ on a 10-problem test, $ 80\%$ on a 20-problem test and $ 90\%$ on a 30-problem test. If the three tests are combined into one 60-problem test, which percent is closest to her overall score? $ \textbf{(A)}\ 40 \qquad \textbf{(B)}\ 77 \qquad \textbf{(C)}\ 80 \qquad \textbf{(D)}\ 83 \qquad \textbf{(E)}\ 87$

Four regular hexagons surround a square with a side length $1$, each one sharing an edge with the square, as shown in the figure below. The area of the resulting 12-sided outer nonconvex polygon can be written as $m\sqrt{n} + p$, where $m$, $n$, and $p$ are integers and $n$ is not divisible by the square of any prime. What is $m + n + p$? [asy] import geometry; unitsize(3cm); draw((0,0) -- (1,0) -- (1,1) -- (0,1) -- cycle); draw(shift((1/2,1-sqrt(3)/2))*polygon(6)); draw(shift((1/2,sqrt(3)/2))*polygon(6)); draw(shift((sqrt(3)/2,1/2))*rotate(90)*polygon(6)); draw(shift((1-sqrt(3)/2,1/2))*rotate(90)*polygon(6)); draw((0,1-sqrt(3))--(1,1-sqrt(3))--(3-sqrt(3),sqrt(3)-2)--(sqrt(3),0)--(sqrt(3),1)--(3-sqrt(3),3-sqrt(3))--(1,sqrt(3))--(0,sqrt(3))--(sqrt(3)-2,3-sqrt(3))--(1-sqrt(3),1)--(1-sqrt(3),0)--(sqrt(3)-2,sqrt(3)-2)--cycle,linewidth(2)); [/asy] $\textbf{(A)}-12~\textbf{(B)}-4~\textbf{(C)} 4~\textbf{(D)}24~\textbf{(E)}32$

a) Replace each letter in the following sum by a digit from $0$ to $9$, in such a way that the sum is correct. $\tab$ $\tab$ $ABC$ $\tab$ $\tab$ $DEF$ [u]$+GHI$[/u] $\tab$ $\tab$ $\tab$ $J J J$ Different letters must be replaced by different digits, and equal letters must be replaced by equal digits. Numbers $ABC$, $DEF$, $GHI$ and $JJJ$ cannot begin by $0$. b) Determine how many triples of numbers $(ABC,DEF,GHI)$ can be formed under the conditions given in a).

Assume that $a_1, a_2, a_3$ are three given positive integers consider the following sequence: $a_{n+1}=\text{lcm}[a_n, a_{n-1}]-\text{lcm}[a_{n-1}, a_{n-2}]$ for $n\ge 3$ Prove that there exist a positive integer $k$ such that $k\le a_3+4$ and $a_k\le 0$. ($[a, b]$ means the least positive integer such that$ a\mid[a,b], b\mid[a, b]$ also because $\text{lcm}[a, b]$ takes only nonzero integers this sequence is defined until we find a zero number in the sequence)

Circle $\Gamma$ has radius $10$, center $O$, and diameter $AB$. Point $C$ lies on $\Gamma$ such that $AC = 12$. Let $P$ be the circumcenter of $\vartriangle AOC$. Line $AP$ intersects $\Gamma$ at $Q$, where $Q$ is different from $A$. Then the value of $\frac{AP}{AQ}$ can be expressed in the form $\frac{m}{n}$, where m and n are relatively prime positive integers. Compute $m + n$.

Let $n$ a positive integer. In a $2n\times 2n$ board, $1\times n$ and $n\times 1$ pieces are arranged without overlap. Call an arrangement [b]maximal[/b] if it is impossible to put a new piece in the board without overlapping the previous ones. Find the least $k$ such that there is a [b]maximal[/b] arrangement that uses $k$ pieces.

Let $ABCD$ be a convex quadrilateral with $\angle ABC = 120^{\circ}$ and $\angle BCD = 90^{\circ}$, and let $M$ and $N$ denote the midpoints of $\overline{BC}$ and $\overline{CD}$. Suppose there exists a point $P$ on the circumcircle of $\triangle CMN$ such that ray $MP$ bisects $\overline{AD}$ and ray $NP$ bisects $\overline{AB}$. If $AB + BC = 444$, $CD = 256$ and $BC = \frac mn$ for some relatively prime positive integers $m$ and $n$, compute $100m+n$. [i]Proposed by Michael Ren[/i]

For $n\in\mathbb{N}$, $n\geq 2$, $a_{i}, b_{i}\in\mathbb{R}$, $1\leq i\leq n$, such that \[\sum_{i=1}^{n}a_{i}^{2}=\sum_{i=1}^{n}b_{i}^{2}=1, \sum_{i=1}^{n}a_{i}b_{i}=0. \] Prove that \[\left(\sum_{i=1}^{n}a_{i}\right)^{2}+\left(\sum_{i=1}^{n}b_{i}\right)^{2}\leq n. \] [i]Cezar Lupu & Tudorel Lupu[/i]

Let $(O_1), (O_2)$ be given two circles intersecting at $A$ and $B$. The tangent lines of $(O_1)$ at $A, B$ intersect at $O$. Let $I$ be a point on the circle $(O_1)$ but outside the circle $(O_2)$. The lines $IA, IB$ intersect circle $(O_2)$ at $C, D$. Denote by $M$ the midpoint of $C D$. Prove that $I, M, O$ are collinear.

The sequence of $a_n$ is determined by the relation $$a_{n+1}=\frac{k+a_n}{1-a_n}$$ where $k > 0$. It is known that $a_{13} = a_1$. What values can $k$ take?

Given $n$ chairs around a circle which are marked with numbers from 1 to $n$ .There are $k$, $k \leq 4 \cdot n$ students sitting on those chairs .Two students are called neighbours if there is no student sitting between them. Between two neighbours students ,there are at less 3 chairs. Find the number of choices of $k$ chairs so that $k$ students can sit on those and the condition is satisfied.

Coins are placed in some squares on a $n\times n$ board. Each coin can be moved towards the square symmetrical with respect to either of the two diagonals, as long as that square is empty. The initial coin setup is said to be [i]good [/i], if any coin can make the first move. (a) Determine the maximum number of coins $M$ that can be placed on the $n\times n$ board, such that the configuration is good. (b) Calculate the total number of good configurations that have exactly $M$ coins.

On a round table, $n$ bowls are arranged in a circle. Anja walks around the table clockwise, placing marbles in the bowls according to the following rule: She places a marble in any first bowl, then goes one bowl further and puts a marble in there. Then she goes two shells before putting another marble, then she goes three shells, etc. If there is at least one marble in each shell, she stops. For which $n$ does this occur?

Let $\Gamma$ be a circle of radius $1$ centered at $O$. A circle $\Omega$ is said to be \emph{friendly} if there exist distinct circles $\omega_1$, $\omega_2$, $\ldots$, $\omega_{2020}$, such that for all $1\le i\le2020$, $\omega_i$ is tangent to $\Gamma$, $\Omega$, and $\omega_{i+1}$. (Here, $\omega_{2021} = \omega_1$.) For each point $P$ in the plane, let $f(P)$ denote the sum of the areas of all friendly circles centered at $P$. If $A$ and $B$ are points such that $OA=\frac12$ and $OB=\frac13$, determine $f(A)-f(B)$. [i]Proposed by Michael Ren.[/i]

Kelvin the Frog was bored in math class one day, so he wrote all ordered triples $(a, b, c)$ of positive integers such that $abc=2310$ on a sheet of paper. Find the sum of all the integers he wrote down. In other words, compute $$\sum_{\stackrel{abc=2310}{a,b,c\in \mathbb{N}}} (a+b+c),$$ where $\mathbb{N}$ denotes the positive integers.

Paula the painter and her two helpers each paint at constant, but different, rates. They always start at $\text{8:00 AM}$, and all three always take the same amount of time to eat lunch. On Monday the three of them painted $50\%$ of a house, quitting at $\text{4:00 PM}$. On Tuesday, when Paula wasn't there, the two helpers painted only $24\%$ of the house and quit at $\text{2:12 PM}$. On Wednesday Paula worked by herself and finished the house by working until $\text{7:12 PM}$. How long, in minutes, was each day's lunch break? $ \textbf{(A)}\ 30 \qquad\textbf{(B)}\ 36 \qquad\textbf{(C)}\ 42 \qquad\textbf{(D)}\ 48 \qquad\textbf{(E)}\ 60 $

What is the sum of all integer solutions to $1<(x-2)^2<25$? $ \textbf{(A)}\ 10\qquad\textbf{(B)}\ 12\qquad\textbf{(C)}\ 15\qquad\textbf{(D)}\ 19\qquad\textbf{(E)}\ 25 $

A cue ball is shot at a $45$ degree angle from the upper right corner of a billiard table with dimensions $4\text{ ft}$ by $5\text{ ft}$, as shown. How many times does the ball bounce before hitting another corner? Assume that when the ball bounces, its path is perfectly reflected. The final impact in the corner does not count as a bounce. [asy] size(120); draw((0,0)--(5,0)--(5,4)--(0,4)--cycle); label("$5$",(0,0)--(5,0),S); label("$4$",(0,0)--(0,4),W); filldraw(circle((0.4,3.6),0.4),black); draw((0,4)--(1.5,2.5),EndArrow); draw((1.5,2.5)--(4,0)--(5,1), dashed); draw(arc((0,4),1.25,315,270)); label(scale(0.8)*"$45^\circ$",(0.2,2.8),SE); [/asy] $\textbf{(A) }3\qquad\textbf{(B) }4\qquad\textbf{(C) }5\qquad\textbf{(D) }6\qquad\textbf{(E) }7$

Let $ABC$ be a scalene triangle. The incircle is tangent to lines $BC$, $AC$, and $AB$ at points $D$, $E$, and $F$, respectively, and the $A$-excircle is tangent to lines $BC$, $AC$, and $AB$ at points $D_1$, $E_1$, and $F_1$, respectively. Suppose that lines $AD$, $BE$, and $CF$ are concurrent at point $G$, and suppose that lines $AD_1$, $BE_1$, and $CF_1$ are concurrent at point $G_1$. Let line $GG_1$ intersect the internal bisector of angle $BAC$ at point $X$. Suppose that $AX=1$, $\cos{\angle BAC}=\sqrt{3}-1$, and $BC=8\sqrt[4]{3}$. Then $AB \cdot AC = \frac{j+k\sqrt{m}}{n}$ for positive integers $j$, $k$, $m$, and $n$ such that $\gcd(j,k,n)=1$ and $m$ is not divisible by the square of any integer greater than $1$. Compute $1000j+100k+10m+n$. [i]Proposed by Luke Robitaille and Brandon Wang[/i]

Let be the sequence $ \left( a_n \right)_{n\ge 0} $ of positive real numbers defined by $$ a_n=1+\frac{a_{n-1}}{n} ,\quad\forall n\ge 1. $$ Calculate $ \lim_{n\to\infty } a_n ^n . $ [i]Florian Dumitrel[/i]

Let $A$ be the set of all functions $f:\mathbb{R} \to \mathbb{R}$ such that $f(xy)=xf(y)$ for all $x,y \in \mathbb{R}$. (a) If $f \in A$ then show that $f(x+y)=f(x)+f(y)$ for all $x,y \in \mathbb{R}$ (b) For $g,h \in A$, define a function $g\circ h$ by $(g \circ h)(x)=g(h(x))$ for $x \in \mathbb{R}$. Prove that $g \circ h$ is in $A$ and is equal to $h \circ g$.

Prove that there are infinitely many different natural numbers of the form $k^2 + 1$, $k \in N$ that have no real divisor of this form.

In a country with $5$ distinct cities, there may or may not be a road between each pair of cities. It’s possible to get from any city to any other city through a series of roads, but there is no set of three cities $\{A,B,C\}$ such that there are roads between $A$ and $B$, $B$ and $C$, and $C$ and $A$. How many road systems between the five cities are possible?