The lengths of the altitudes of a triangle are positive integers, and the length of the radius of the incircle is a prime number. Find the lengths of the sides of the triangle.

If rose bushes are spaced about $1$ foot apart, approximately how many bushes are needed to surround a circular patio whose radius is $12$ feet? $ \text{(A)}\ 12\qquad\text{(B)}\ 38\qquad\text{(C)}\ 48\qquad\text{(D)}\ 75\qquad\text{(E)}\ 450 $

The positive numbers $a_1, a_2, \ldots , a_{2024}$ are placed on a circle clockwise in this order. Let $A_i$ be the arithmetic mean of the number $a_i$ and one or several following it clockwise. Prove that the largest of the numbers $A_1, A_2, \ldots , A_{2024}$ is not less than the arithmetic mean of all numbers $a_1, a_2, \ldots , a_{2024}$.

Do there exist $4$ vectors in the plane so that none is a multiple of another, but the sum of each pair is perpendicular to the sum of the other two? Do there exist $91$ non-zero vectors in the plane such that the sum of any $19$ is perpendicular to the sum of the others?

At each of the sixteen circles in the network below stands a student. A total of 3360 coins are distributed among the sixteen students. All at once, all students give away all their coins by passing an equal number of coins to each of their neighbors in the network. After the trade, all students have the same number of coins as they started with. Find the number of coins the student standing at the center circle had originally. [asy] import graph; unitsize(1 cm); pair[] O; O[1] = (0,0); O[2] = 0.6*dir(270); O[3] = 0.6*dir(270 + 360/5); O[4] = 0.6*dir(270 + 2*360/5); O[5] = 0.6*dir(270 + 3*360/5); O[6] = 0.6*dir(270 + 4*360/5); O[7] = 1.2*dir(90); O[8] = 1.2*dir(90 + 360/5); O[9] = 1.2*dir(90 + 2*360/5); O[10] = 1.2*dir(90 + 3*360/5); O[11] = 1.2*dir(90 + 4*360/5); O[12] = 2*dir(270); O[13] = 2*dir(270 + 360/5); O[14] = 2*dir(270 + 2*360/5); O[15] = 2*dir(270 + 3*360/5); O[16] = 2*dir(270 + 4*360/5); draw(O[1]--O[2]); draw(O[1]--O[3]); draw(O[1]--O[4]); draw(O[1]--O[5]); draw(O[1]--O[6]); draw(O[7]--O[5]--O[8]--O[6]--O[9]--O[2]--O[10]--O[3]--O[11]--O[4]--cycle); draw(O[12]--O[10]--O[13]--O[11]--O[14]--O[7]--O[15]--O[8]--O[16]--O[9]--cycle); draw(O[12]--O[13]--O[14]--O[15]--O[16]--cycle); for(int i = 1; i <= 16; ++i) { filldraw(Circle(O[i],0.2),white,black); } [/asy]

Circle $\omega$ is inscribed in quadrilateral $ABCD$ and is tangent to segments $BC, AD$ at $E,F$ , respectively.$DE$ intersects $\omega$ for the second time at $X$. if the circumcircle of triangle $DFX$ is tangent to lines $AB$ and $CD$ , prove that quadrilateral $AFXC$ is cyclic.

There are $15$ people, including Petruk, Gareng, and Bagong, which will be partitioned into $6$ groups, randomly, that consists of $3, 3, 3, 2, 2$, and $2$ people (orders are ignored). Determine the probability that Petruk, Gareng, and Bagong are in a group.

Let $AD$ be the altitude of an acute-angled triangle $ABC$ and $A'$ be the point on its circumcircle opposite to $A$. A point $P$ lies on the segment $AD$, and points $X$, $Y$ lie on the segments $AB$, $AC$ respectively in such a way that $\angle CBP = \angle ADY$, $\angle BCP = \angle ADX$. Let $PA'$ meet $BC$ at point $T$. Prove that $D$, $X$, $Y$, $T$ are concyclic.

The integers $n>1$ is given . The positive integer $a_1,a_2,\cdots,a_n$ satisfing condition : (1) $a_1<a_2<\cdots<a_n$; (2) $\frac{a^2_1+a^2_2}{2},\frac{a^2_2+a^2_3}{2},\cdots,\frac{a^2_{n-1}+a^2_n}{2}$ are all perfect squares . Prove that :$a_n\ge 2n^2-1.$

Let $x$ and $y$ be positive integers and assume that $z=\frac{4xy}{x+y}$ is an odd integer. Prove that at least one divisor of $z$ can be expressed in the form $4n-1$ where $n$ is a positive integer.

Let \(ABC\) be a triangle. Let \(E,F\) be the feet of the altitudes from \(B,C\) respectively. Let \(P,Q\) be the projections of \(B,C\) onto line \(EF\). Show that \(PE=QF\).

Let $f,g$ are defined in $(a,b)$ such that $f(x),g(x)\in\mathcal{C}^2$ and non-decreasing in an interval $(a,b)$ . Also suppose $f^{\prime \prime}(x)=g(x),g^{\prime \prime}(x)=f(x)$. Also it is given that $f(x)g(x)$ is linear in $(a,b)$. Show that $f\equiv 0 \text{ and } g\equiv 0$ in $(a,b)$.

Given a $\triangle ABC$, $\angle A=45^\circ$, $O$ is the circumcenter and $H$ is the orthocenter of $\triangle ABC$. $M$ is the midpoint of $\overline{BC}$, and $N$ is the midpoint of $\overline{OH}$. Prove that $\angle BAM=\angle CAN$.

When dividing an integer $m$ by a positive integer $n$, $(0< n\leq 100)$, a student finds $\frac{m}{n}= 0,167a_{1}a_{2}...$. Prove that the student made a mistake while computing.

[b]p1.[/b] Let $X = 2022 + 022 + 22 + 2$. When $X$ is divided by $22$, there is a remainder of $R$. What is the value of $R$? [b]p2.[/b] When Amy makes paper airplanes, her airplanes fly $75\%$ of the time. If her airplane flies, there is a $\frac56$ chance that it won’t fly straight. Given that she makes $80$ airplanes, what is the expected number airplanes that will fly straight? [b]p3.[/b] It takes Joshua working alone $24$ minutes to build a birdhouse, and his son working alone takes $16$ minutes to build one. The effective rate at which they work together is the sum of their individual working rates. How long in seconds will it take them to make one birdhouse together? [b]p4.[/b] If Katherine’s school is located exactly $5$ miles southwest of her house, and her soccer tournament is located exactly $12$ miles northwest of her house, how long, in hours, will it take Katherine to bike to her tournament right after school given she bikes at $0.5$ miles per hour? Assume she takes the shortest path possible. [b]p5.[/b] What is the largest possible integer value of $n$ such that $\frac{4n+2022}{n+1}$ is an integer? [b]p6.[/b] A caterpillar wants to go from the park situated at $(8, 5)$ back home, located at $(4, 10)$. He wants to avoid routes through $(6, 7)$ and $(7, 10)$. How many possible routes are there if the caterpillar can move in the north and west directions, one unit at a time? [b]p7.[/b] Let $\vartriangle ABC$ be a triangle with $AB = 2\sqrt{13}$, $BC = 6\sqrt2$. Construct square $BCDE$ such that $\vartriangle ABC$ is not contained in square $BCDE$. Given that $ACDB$ is a trapezoid with parallel bases $\overline{AC}$, $\overline{BD}$, find $AC$. [b]p8.[/b] How many integers $a$ with $1 \le a \le 1000$ satisfy $2^a \equiv 1$ (mod $25$) and $3^a \equiv 1$ (mod $29$)? [b]p9.[/b] Let $\vartriangle ABC$ be a right triangle with right angle at $B$ and $AB < BC$. Construct rectangle $ADEC$ such that $\overline{AC}$,$\overline{DE}$ are opposite sides of the rectangle, and $B$ lies on $\overline{DE}$. Let $\overline{DC}$ intersect $\overline{AB}$ at $M$ and let $\overline{AE}$ intersect $\overline{BC}$ at $N$. Given $CN = 6$, $BN = 4$, find the $m+n$ if $MN^2$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m, n$. [b]p10.[/b] An elimination-style rock-paper-scissors tournament occurs with $16$ players. The $16$ players are all ranked from $1$ to $16$ based on their rock-paper-scissor abilities where $1$ is the best and $16$ is the worst. When a higher ranked player and a lower ranked player play a round, the higher ranked player always beats the lower ranked player and moves on to the next round of the tournament. If the initial order of players are arranged randomly, and the expected value of the rank of the $2$nd place player of the tournament can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m, n$ what is the value of $m+n$? [b]p11.[/b] Estimation (Tiebreaker) Estimate the number of twin primes (pairs of primes that differ by $2$) where both primes in the pair are less than $220022$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ABCD$ be a rectangle. A line $r$ moves parallel to $AB$ and intersects diagonal $AC$ , forming two triangles opposite the vertex, inside the rectangle. Prove that the sum of the areas of these triangles is minimal when $r$ passes through the midpoint of segment $AD$ .

Find all functions $f$ which are defined on all non-negative real numbers, take nonnegative real values only, and satisfy the condition $x \cdot f(y) + y\cdot f(x) = f(x) \cdot f(y) \cdot (f(x) + f(y))$ for all non-negative real numbers $x, y$.

Find all functions $f : \mathbb{R} \to \mathbb{R}$ such that $$f(f(x)^2 + |y|) = x^2 + f(y)$$

Find all triples $(a, b, c)$ of positive integers for which $$\begin{cases} a + bc=2010 \\ b + ca = 250\end{cases}$$

Let $n$ be a positive integer. If $x_1, x_2, \ldots, x_n \geq 0$, $x_1+x_2+\ldots+x_n=1$ and, assuming $x_{n+1}=x_1$, find the maximal value of $$\sum_{k=1}^n \frac{1+x_k^2+x_k^4}{1+x_{k+1}+x_{k+1}^2+x_{k+1}^3+x_{k+1}^4}.$$

[u]Round 1[/u] [b]p1.[/b] A party is attended by ten people (men and women). Among them is Pat, who always lies to people of the opposite gender and tells the truth to people of the same gender. Pat tells five of the guests: “There are more men than women at the party.” Pat tells four of the guests: “There are more women than men at the party.” Is Pat a man or a woman? [b]p2.[/b] Once upon a time in a land far, far away there lived $100$ knights, $99$ princesses, and $101$ dragons. Over time, knights beheaded dragons, dragons ate princesses, and princesses poisoned knights. But they always obeyed an ancient law that prohibits killing any creature who has killed an odd number of others. Eventually only one creature remained alive. Could it have been a knight? A dragon? A princess? [b]p3.[/b] The numbers $1 \circ 2 \circ 3 \circ 4 \circ 5 \circ 6 \circ 7 \circ 8 \circ 9 \circ 10$ are written in a line. Alex and Vicky play a game, taking turns inserting either an addition or a multiplication symbol between adjacent numbers. The last player to place a symbol wins if the resulting expression is odd and loses if it is even. Alex moves first. Who wins? (Remember that multiplication is performed before addition.) [b]p4.[/b] A chess tournament had $8$ participants. Each participant played each other participant once. The winner of a game got $1$ point, the loser $0$ points, and in the case of a draw each got $1/2$ a point. Each participant scored a different number of points, and the person who got $2$nd prize scored the same number of points as the $5$th, $6$th, $7$th and $8$th place participants combined. Can you determine the result of the game between the $3$rd place player and the $5$th place player? [b]p5.[/b] One hundred gnomes sit in a circle. Each gnome gets a card with a number written on one side and a different number written on the other side. Prove that it is possible for all the gnomes to lay down their cards so that no two neighbors have the same numbers facing up. [u]Round 2[/u] [b]p6.[/b] A casino machine accepts tokens of $32$ different colors, one at a time. For each color, the player can choose between two fixed rewards. Each reward is up to $\$10$ cash, plus maybe another token. For example, a blue token always gives the player a choice of getting either $\$5$ plus a red token or $\$3$ plus a yellow token; a black token can always be exchanged either for $\$10$ (but no token) or for a brown token (but no cash). A player may keep playing as long as he has a token. Rob and Bob each have one white token. Rob watches Bob play and win $\$500$. Prove that Rob can win at least $\$1000$. [img]https://cdn.artofproblemsolving.com/attachments/6/6/e55614bae92233c9b2e7d66f5f425a18e6475a.png[/img] [b]p7.[/b] Each of the $100$ residents of Pleasantville has at least $30$ friends in town. A resident votes in the mayoral election only if one of her friends is a candidate. Prove that it is possible to nominate two candidates for mayor so that at least half of the residents will vote. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Consider an $n$-by-$n$ board of unit squares for some odd positive integer $n$. We say that a collection $C$ of identical dominoes is a [i]maximal grid-aligned configuration[/i] on the board if $C$ consists of $(n^2-1)/2$ dominoes where each domino covers exactly two neighboring squares and the dominoes don't overlap: $C$ then covers all but one square on the board. We are allowed to slide (but not rotate) a domino on the board to cover the uncovered square, resulting in a new maximal grid-aligned configuration with another square uncovered. Let $k(C)$ be the number of distinct maximal grid-aligned configurations obtainable from $C$ by repeatedly sliding dominoes. Find the maximum value of $k(C)$ as a function of $n$. [i]Proposed by Holden Mui[/i]

Find the smallest positive integer $j$ such that for every polynomial $p(x)$ with integer coefficients and for every integer $k,$ the integer \[p^{(j)}(k)=\left. \frac{d^j}{dx^j}p(x) \right|_{x=k}\] (the $j$-th derivative of $p(x)$ at $k$) is divisible by $2016.$

In the country of Óxisz the minister of finance observed at the end of the tax census that the sum of properties of any two neighboring city counted in dinar is divisible by $1000$, and she also observed that the sum of properties of all cities is also divisible by $1000$. What is the least sum of properties of all cities if the map of the cities looks as follows? [img]https://cdn.artofproblemsolving.com/attachments/0/5/274730ebfdd52d0c3642dfbd0596fe587eb211.png[/img] [i]Remark: The cities may have non-integer properties, but it is also positive. On the map the points are the cities, and two cities are neighboring if there is a direct connection between them.[/i]

The slope of the line $\frac{x}{3} + \frac{y}{2} = 1$ is $ \textbf{(A)}\ -\frac{3}{2}\qquad\textbf{(B)}\ -\frac{2}{3}\qquad\textbf{(C)}\ \frac{1}{3}\qquad\textbf{(D)}\ \frac{2}{3}\qquad\textbf{(E)}\ \frac{3}{2} $