Scalene triangle $ABC$ has incenter $I$ and circumcircle $\Omega$ with center $O$. $H$ is the orthocenter of triangle $BIC$, and $T$ is a point on $\Omega$ for which $\angle ATI=90^\circ$. Circle $(AIO)$ intersects line $IH$ again at $X$. Show that the lines $AX, HT$ intersect on $\Omega$.

Let $ABCD$ be a cyclic quadrilateral whose center of the circumscribed circle is inside this quadrilateral, and its diagonals intersect in point $S{}$. Let $P{}$ and $Q{}$ be the centers of the curcimuscribed circles of triangles $ABS$ and $BCS$. The lines through the points $P{}$ and $Q{}$, which are parallel to the sides $AD$ and $CD$, respectively, intersect at the point $R$. Prove that the point $R$ lies on the line $BD$.

At a certain grocery store, cookies may be bought in boxes of $10$ or $21.$ What is the minimum positive number of cookies that must be bought so that the cookies may be split evenly among $13$ people? [i]Author: Ray Li[/i]

Can you make $2015$ positive integers $1,2, \ldots , 2015$ to be a certain permutation which can be ordered in the circle such that the sum of any two adjacent numbers is a multiple of $4$ or a multiple of $7$?

Let $k$ be a circle centered at $M$ and let $t$ be a tangentline to $k$ through some point $T\in k$. Let $P$ be a point on $t$ and let $g\neq t$ be a line through $P$ intersecting $k$ at $U$ and $V$. Let $S$ be the point on $k$ bisecting the arc $UV$ not containing $T$ and let $Q$ be the the image of $P$ under a reflection over $ST$. Prove that $Q$, $T$, $U$ and $V$ are vertices of a trapezoid.

Let $ABC$ be a triangle such that such that $AB=14, BC=13$, and $AC=15$. Let $X$ be a point inside triangle $ABC$. Compute the minimum possible value of $(\sqrt{2}AX+BX+CX)^2$.

Does there exist a right angled triangle, which hypotenuse is $2016^{2017}$ and two other sides positive integers.

Each square in a $5 \times 5$ grid is either filled or empty, and has up to eight adjacent neighboring squares, where neighboring squares share either a side or a corner. The grid is transformed by the following rules: [list] [*] Any filled square with two or three filled neighbors remains filled. [*] Any empty square with exactly three filled neighbors becomes a filled square. [*] All other squares remain empty or become empty. [/list] A sample transformation is shown in the figure below. [asy] import geometry; unitsize(0.6cm); void ds(pair x) { filldraw(x -- (1,0) + x -- (1,1) + x -- (0,1)+x -- cycle,gray+opacity(0.5),invisible); } ds((1,1)); ds((2,1)); ds((3,1)); ds((1,3)); for (int i = 0; i <= 5; ++i) { draw((0,i)--(5,i)); draw((i,0)--(i,5)); } label("Initial", (2.5,-1)); draw((6,2.5)--(8,2.5),Arrow); ds((10,2)); ds((11,1)); ds((11,0)); for (int i = 0; i <= 5; ++i) { draw((9,i)--(14,i)); draw((i+9,0)--(i+9,5)); } label("Transformed", (11.5,-1)); [/asy] Suppose the $5 \times 5$ grid has a border of empty squares surrounding a $3 \times 3$ subgrid. How many initial configurations will lead to a transformed grid consisting of a single filled square in the center after a single transformation? (Rotations and reflections of the same configuration are considered different.) [asy] import geometry; unitsize(0.6cm); void ds(pair x) { filldraw(x -- (1,0) + x -- (1,1) + x -- (0,1)+x -- cycle,gray+opacity(0.5),invisible); } for (int i = 1; i < 4; ++ i) { for (int j = 1; j < 4; ++j) { label("?",(i + 0.5, j + 0.5)); } } for (int i = 0; i <= 5; ++i) { draw((0,i)--(5,i)); draw((i,0)--(i,5)); } label("Initial", (2.5,-1)); draw((6,2.5)--(8,2.5),Arrow); ds((11,2)); for (int i = 0; i <= 5; ++i) { draw((9,i)--(14,i)); draw((i+9,0)--(i+9,5)); } label("Transformed", (11.5,-1)); [/asy] $$\textbf{(A) 14}~\textbf{(B) 18}~\textbf{(C) 22}~\textbf{(D) 26}~\textbf{(E) 30}$$

In the triangle $ABC$, let $B_1$ be on $AC, E$ on $AB, G$ on $BC$, and let $EG$ be parallel to $AC$. Furthermore, let $EG$ be tangent to the inscribed circle of the triangle $ABB_1$ and intersect $BB_1$ at $F$. Let $r, r_1$, and $r_2$ be the inradii of the triangles $ABC, ABB_1$, and $BFG$, respectively. Prove that $r = r_1 + r_2.$

Find the area enclosed by the graph of $a^2x^4=b^2x^2-y^2\ (a>0,\ b>0).$

For a science project, Sammy observed a chipmunk and a squirrel stashing acorns in holes. The chipmunk hid $3$ acorns in each of the holes it dug. The squirrel hid $4$ acorns in each of the holes it dug. They each hid the same number of acorns, although the squirrel needed $4$ fewer holes. How many acorns did the chipmunk hide? ${{ \textbf{(A)}\ 30\qquad\textbf{(B)}\ 36\qquad\textbf{(C)}\ 42\qquad\textbf{(D)}\ 48}\qquad\textbf{(E)}\ 54} $

Determine, with proof, the smallest positive integer $c$ such that for any positive integer $n$, the decimal representation of the number $c^n+2014$ has digits all less than $5$. [i]Proposed by Evan Chen[/i]

Let $n$ be a positive integer, $1<n<2018$. For each $i=1, 2, \ldots ,n$ we define the polynomial $S_i(x)=x^2-2018x+l_i$, where $l_1, l_2, \ldots, l_n$ are distinct positive integers. If the polynomial $S_1(x)+S_2(x)+\cdots+S_n(x)$ has at least an integer root, prove that at least one of the $l_i$ is greater or equal than $2018$.

Let $b$ and $c$ be integers chosen randomly (uniformly and independently) from the set \[ \{ -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6 \} . \] (Note that $b$ and $c$ can be equal.) What is the probability that the two roots of the quadratic $x^2 + bx + c$ are consecutive integers?

[i]25 problems for 30 minutes[/i] [b]p1.[/b] You and nine friends spend $4000$ dollars on tickets to attend the new Harry Styles concert. Unfortunately, six friends cancel last minute due to the u. You and your remaining friends still attend the concert and split the original cost of $4000$ dollars equally. What percent of the total cost does each remaining individual have to pay? [b]p2.[/b] Find the number distinct $4$ digit numbers that can be formed by arranging the digits of $2021$. [b]p3.[/b] On a plane, Darnay draws a triangle and a rectangle such that each side of the triangle intersects each side of the rectangle at no more than one point. What is the largest possible number of points of intersection of the two shapes? [b]p4.[/b] Joy is thinking of a two-digit number. Her hint is that her number is the sum of two $2$-digit perfect squares $x_1$ and $x_2$ such that exactly one of $x_i - 1$ and $x_i + 1$ is prime for each $i = 1, 2$. What is Joy's number? [b]p5.[/b] At the North Pole, ice tends to grow in parallelogram structures of area $60$. On the other hand, at the South Pole, ice grows in right triangular structures, in which each triangular and parallelogram structure have the same area. If every ice triangle $ABC$ has legs $\overline{AB}$ and $\overline{AC}$ that are integer lengths, how many distinct possible lengths are there for the hypotenuse $\overline{BC}$? [b]p6.[/b] Carlsen has some squares and equilateral triangles, all of side length $1$. When he adds up the interior angles of all shapes, he gets $1800^o$. When he adds up the perimeters of all shapes, he gets $24$. How many squares does he have? [b]p7.[/b] Vijay wants to hide his gold bars by melting and mixing them into a water bottle. He adds $100$ grams of liquid gold to $100$ grams of water. His liquefied gold bars have a density of $20$ g/ml and water has a density of $1$ g/ml. Given that the density of the mixture in g/mL can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$, compute the sum $m + n$. (Note: density is mass divided by volume, gram (g) is unit of mass and ml is unit of volume. Further, assume the volume of the mixture is the sum of the volumes of the components.) [b]p8.[/b] Julius Caesar has epilepsy. Specifically, if he sees $3$ or more flashes of light within a $0.1$ second time frame, he will have a seizure. His enemy Brutus has imprisoned him in a room with $4$ screens, which flash exactly every $4$, $5$, $6$, and $7$ seconds, respectively. The screens all flash at once, and $105$ seconds later, Caesar opens his eyes. How many seconds after he opened his eyes will Caesar first get a seizure? [b]p9.[/b] Angela has a large collection of glass statues. One day, she was bored and decided to use some of her statues to create an entirely new one. She melted a sphere with radius $12$ and a cone with height of 18 and base radius of $2$. If Angela wishes to create a new cone with a base radius $2$, what would the the height of the newly created cone be? [b]p10.[/b] Find the smallest positive integer $N$ satisfying these properties: (a) No perfect square besides $1$ divides $N$. (b) $N$ has exactly $16$ positive integer factors. [b]p11.[/b] The probability of a basketball player making a free throw is $\frac15$. The probability that she gets exactly $2$ out of $4$ free throws in her next game can be expressed as $\frac{m}{n}$ for relatively prime positive integers m and n. Find $m + n$. [b]p12.[/b] A new donut shop has $1000$ boxes of donuts and $1000$ customers arriving. The boxes are numbered $1$ to $1000$. Initially, all boxes are lined up by increasing numbering and closed. On the first day of opening, the first customer enters the shop and opens all the boxes for taste testing. On the second day of opening, the second customer enters and closes every box with an even number. The third customer then "reverses" (if closed, they open it and if open, they close it) every box numbered with a multiple of three, and so on, until all $1000$ customers get kicked out for having entered the shop and reversing their set of boxes. What is the number on the sixth box that is left open? [b]p13.[/b] For an assignment in his math class, Michael must stare at an analog clock for a period of $7$ hours. He must record the times at which the minute hand and hour hand form an angle of exactly $90^o$, and he will receive $1$ point for every time he records correctly. What is the maximum number of points Michael can earn on his assignment? [b]p14.[/b] The graphs of $y = x^3 +5x^2 +4x-3$ and $y = -\frac15 x+1$ intersect at three points in the Cartesian plane. Find the sum of the $y$-coordinates of these three points. [b]p15.[/b] In the quarterfinals of a single elimination countdown competition, the $8$ competitors are all of equal skill. When any $2$ of them compete, there is exactly a $50\%$ chance of either one winning. If the initial bracket is randomized, the probability that two of the competitors, Daniel and Anish, face off in one of the rounds can be expressed as $\frac{p}{q}$ for relatively prime positive integers $p$, $q$. Find $p + q$. [b]p16.[/b] How many positive integers less than or equal to $1000$ are not divisible by any of the numbers $2$, $3$, $5$ and $11$? [b]p17.[/b] A strictly increasing geometric sequence of positive integers $a_1, a_2, a_3,...$ satisfies the following properties: (a) Each term leaves a common remainder when divided by $7$ (b) The first term is an integer from $1$ to $6$ (c) The common ratio is an perfect square Let $N$ be the smallest possible value of $\frac{a_{2021}}{a_1}$. Find the remainder when $N$ is divided by $100$. [b]p18.[/b] Suppose $p(x) = x^3 - 11x^2 + 36x - 36$ has roots $r, s,t$. Find %\frac{r^2 + s^2}{t}+\frac{s^2 + t^2}{r}+\frac{t^2 + r^2}{s}%. [b]p19.[/b] Let $a, b \le 2021$ be positive integers. Given that $ab^2$ and $a^2b$ are both perfect squares, let $G = gcd(a, b)$. Find the sum of all possible values of $G$. [b]p20.[/b] Jessica rolls six fair standard six-sided dice at the same time. Given that she rolled at least four $2$'s and exactly one $3$, the probability that all six dice display prime numbers can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$, $n$. What is $m + n$? [b]p21.[/b] Let $a, b, c$ be numbers such $a + b + c$ is real and the following equations hold: $$a^3 + b^3 + c^3 = 25$$ $$\frac{1}{ab}+\frac{1}{bc}+\frac{1}{ac}= 1$$ $$\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{25}{9}$$ The value of $a + b + c$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$, $n$. Find $m + n$. [b]p22.[/b] Let $\omega$ be a circle and $P$ be a point outside $\omega$. Let line $\ell$ pass through $P$ and intersect $\omega$ at points $A,B$ and with $PA < PB$ and let $m$ be another line passing through $P$ intersecting $\omega$ at points $C,D$ with $PC < PD$. Let X be the intersection of $AD$ and $BC$. Given that $\frac{PC}{CD}=\frac23$, $\frac{PC}{PA}=\frac45$, and $\frac{[ABC]}{[ACD]}=\frac79$,the value of $\frac{[BXD]}{[BXA]}$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m, n$: Find $m + n$. [b]p23.[/b] Define the operation $a \circ b =\frac{a^2 + 2ab + a - 12}{b}$. Given that $1 \circ (2 \circ (3 \circ (... 2019 \circ (2020 \circ 2021)))...)$ can be expressed as $-\frac{a}{b}$ for some relatively prime positive integers $a,b$, compute $a + b$. [b]p24.[/b] Find the largest integer $n \le 2021$ for which $5^{n-3} | (n!)^4$ [b]p25.[/b] On the Cartesian plane, a line $\ell$ intersects a parabola with a vertical axis of symmetry at $(0, 5)$ and $(4, 4)$. The focus $F$ of the parabola lies below $\ell$, and the distance from $F$ to $\ell$ is $\frac{16}{\sqrt{17}}$. Let the vertex of the parabola be $(x, y)$. The sum of all possible values of $y$ can be expressed as $\frac{p}{q}$ for relatively prime positive integers $p, q$. Find $p + q$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $f(x)=\sqrt{\sin^4 x + 4\cos^2 x}-\sqrt{\cos^4x + 4\sin^2x}$. An equivalent form of $f(x)$ is $\textbf{(A) }1-\sqrt2\sin x\qquad\textbf{(B) }-1+\sqrt2\cos x\qquad\textbf{(C) }\cos\dfrac x2-\sin\dfrac x2$ $\textbf{(D) }\cos x-\sin x\qquad\textbf{(E) }\cos2x$

Let $ M$ be the intersection of the diagonals of a convex quadrilateral $ ABCD$. Determine all such quadrilaterals for which there exists a line $ g$ that passes through $ M$ and intersects the side $ AB$ in $ P$ and the side $ CD$ in $ Q$, such that the four triangles $ APM$, $ BPM$, $ CQM$, $ DQM$ are similar.

Consider the acute triangle $ABC$. Let $C_1$ and $C_2$ be semicircles with diameters $AB$ and $AC$, respectively, positioned outside triangle $ABC$. The altitude passing through $C$ intersects $C_1$ at $P$, and similarly, $Q$ is the intersection of $C_2$ with the extension of the altitude passing through $B$. Prove that $AP = AQ$.

$77$ stones weighing $1,2,\dots, 77$ grams are divided into $k$ groups such that total weights of each group are different from each other and each group contains less stones than groups with smaller total weights. For how many $k\in \{9,10,11,12\}$, is such a division possible? $ \textbf{(A)}\ 4 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 1 \qquad\textbf{(E)}\ \text{None of above} $

Let us consider sequences of complex numbers that are infinite in both directions $c=(c_k) , k\in Z$ with finite norm $||c||= (\sum_{k \in Z} |c_k|^2)^{1/2}$ Let $T_m-$ this is a shift operation sequences on m ($(T_mc)_k=c_{k-m}$) Prove that: $\lim_{n \to \infty} \frac{\sum_{i=0}^{n-1} T_ic}{n} =0$ (Adding and multiplying a sequence by a number defined component by component)

A plane contains points $ A$ and $ B$ with $ AB \equal{} 1$. Let $ S$ be the union of all disks of radius $ 1$ in the plane that cover $ \overline{AB}$. What is the area of $ S$? $ \textbf{(A)}\ 2\pi \plus{} \sqrt3 \qquad \textbf{(B)}\ \frac {8\pi}{3} \qquad \textbf{(C)}\ 3\pi \minus{} \frac {\sqrt3}{2} \qquad \textbf{(D)}\ \frac {10\pi}{3} \minus{} \sqrt3 \qquad \textbf{(E)}\ 4\pi \minus{} 2\sqrt3$

Let $A',\,B',\,C'$ be points, in which excircles touch corresponding sides of triangle $ABC$. Circumcircles of triangles $A'B'C,\,AB'C',\,A'BC'$ intersect a circumcircle of $ABC$ in points $C_1\ne C,\,A_1\ne A,\,B_1\ne B$ respectively. Prove that a triangle $A_1B_1C_1$ is similar to a triangle, formed by points, in which incircle of $ABC$ touches its sides.

Let $X,Y,Z$ be the points outside the $\triangle ABC$ such that $\angle BAZ=\angle CAY,\angle CBX=\angle ABZ,\angle ACY=\angle BCX.$ Prove that the lines $AX, BY, CZ$ are concurrent.

Let $ABCD$ be a rectangle with $AB \ne BC$ and the center the point $O$. Perpendicular from $O$ on $BD$ intersects lines $AB$ and $BC$ in points $E$ and $F$ respectively. Points $M$ and $N$ are midpoints of segments $[CD]$ and $[AD]$ respectively. Prove that $FM \perp EN$ .

Find positive reals $a, b, c$ which maximizes the value of $a+ 2b+ 3c$ subject to the constraint that $9a^2 + 4b^2 + c^2 = 91$