Solve in the real numbers the equation $ |\sin 3x+\cos (7\pi /2 -5x)|=2. $

An equilateral triangle has side length $ 6$. What is the area of the region containing all points that are outside the triangle and not more than $ 3$ units from a point of the triangle? $ \textbf{(A)}\ 36\plus{}24\sqrt{3} \qquad \textbf{(B)}\ 54\plus{}9\pi \qquad \textbf{(C)}\ 54\plus{}18\sqrt{3}\plus{}6\pi \qquad \textbf{(D)}\ \left(2\sqrt{3}\plus{}3\right)^2\pi \\ \textbf{(E)}\ 9\left(\sqrt{3}\plus{}1\right)^2\pi$

Let $N$ be the sum of all binomial coefficients $\binom{a}{b}$ such that $a$ and $b$ are nonnegative integers and $a+b$ is an even integer less than 100. Find the remainder when $N$ is divided by 144. (Note: $\binom{a}{b} = 0$ if $a<b$, and $\binom{0}{0} = 1$.)

We have an $n \times n$ board, filled with $n$ rectangles aligned to the grid. The $n$ rectangles cover all the board and are never superposed. Find, in terms of $n$, the smallest value the sum of the $n$ diagonals of the rectangles can take.

The three squares $ACC_{1}A''$, $ABB_{1}'A'$, $BCDE$ are constructed externally on the sides of a triangle $ABC$. Let $P$ be the center of the square $BCDE$. Prove that the lines $A'C$, $A''B$, $PA$ are concurrent.

Let $\{x_n\}$ be a sequence of natural numbers such that \[(a) 1 = x_1 < x_2 < x_3 < \ldots; \quad (b) x_{2n+1} \leq 2n \quad \forall n.\] Prove that, for every natural number $k$, there exist terms $x_r$ and $x_s$ such that $x_r - x_s = k.$

Initially given $31$ tuplets $$(1,0,0,\dots,0),(0,1,0,\dots,0),\dots, (0,0,0,\dots,1)$$ were written on the blackboard. At every move we choose two written $31$ tuplets as $(a_1,a_2,a_3,\dots, a_{31})$ and $(b_1,b_2,b_3,\dots,b_{31})$, then write the $31$ tuplet $(a_1+b_1,a_2+b_2,a_3+b_3,\dots, a_{31}+b_{31})$ to the blackboard too. Find the least possible value of the moves such that one can write the $31$ tuplets $$(0,1,1,\dots,1),(1,0,1,\dots,1),\dots, (1,1,1,\dots,0)$$ to the blackboard by using those moves.

[b]p1.[/b] Sometimes one finds in an old park a tetrahedral pile of cannon balls, that is, a pile each layer of which is a tightly packed triangular layer of balls. A. How many cannon balls are in a tetrahedral pile of cannon balls of $N$ layers? B. How high is a tetrahedral pile of cannon balls of $N$ layers? (Assume each cannon ball is a sphere of radius $R$.) [b]p2.[/b] A prime is an integer greater than $1$ whose only positive integer divisors are itself and $1$. A. Find a triple of primes $(p, q, r)$ such that $p = q + 2$ and $q = r + 2$ . B. Prove that there is only one triple $(p, q, r)$ of primes such that $p = q + 2$ and $q = r + 2$ . [b]p3.[/b] The function $g$ is defined recursively on the positive integers by $g(1) =1$, and for $n>1$ , $g(n)= 1+g(n-g(n-1))$ . A. Find $g(1)$ , $g(2)$ , $g(3)$ and $g(4)$ . B. Describe the pattern formed by the entire sequence $g(1) , g(2 ), g(3), ...$ C. Prove your answer to Part B. [b]p4.[/b] Let $x$ , $y$ and $z$ be real numbers such that $x + y + z = 1$ and $xyz = 3$ . A. Prove that none of $x$ , $y$ , nor $z$ can equal $1$. B. Determine all values of $x$ that can occur in a simultaneous solution to these two equations (where $x , y , z$ are real numbers). [b]p5.[/b] A round robin tournament was played among thirteen teams. Each team played every other team exactly once. At the conclusion of the tournament, it happened that each team had won six games and lost six games. A. How many games were played in this tournament? B. Define a [i]circular triangle[/i] in a round robin tournament to be a set of three different teams in which none of the three teams beat both of the other two teams. How many circular triangles are there in this tournament? C. Prove your answer to Part B. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all functions $ f : \mathbb{R} \rightarrow \mathbb{R} $ such that for all reals $ x, y $,$$ f(x^2+f(x+y))=y+xf(x+1) $$

Prove that for any natural number $n{}$ the numbers $1,2,\ldots,n$ can be divided into several groups so that the sum of the numbers in each group is equal to a power of three. [i]Proposed by V. Novikov[/i]

In the game of Flow, a path is drawn through a $3\times3$ grid of squares obeying the following rules: i A path is continuous with no breaks (it can be drawn without lifting a pencil). ii A path that spans multiple squares can only be drawn between colored squares that share a side. iii A path cannot go through a square more than once. Compute the number of ways to color a positive number of squares on the grid such that a valid path can be drawn. An example of one such coloring and a valid path is shown below. [Insert Diagram] [i]Proposed by Alex Li[/i]

In triangle $\triangle ABC$ we have $BC=a,CA=b,AB=c$ and $\angle B=4\angle A$ Show that \[ab^2c^3=(b^2-a^2-ac)((a^2-b^2)^2-a^2c^2)\]

A tetrahedron $ABCD$ is given. The incircles of triangles $ ABC$ and $ABD$ with centers $O_1, O_2$, touch $AB$ at points $T_1, T_2$. The plane $\pi_{AB}$ passing through the midpoint of $T_1T_2$ is perpendicular to $O_1O_2$. The planes $\pi_{AC},\pi_{BC}, \pi_{AD}, \pi_{BD}, \pi_{CD}$ are defined similarly. Prove that these six planes have a common point.

Let ABC be a triangle with angles $\alpha, \beta, \gamma$ commensurable with $\pi$. Starting from a point $P$ interior to the triangle, a ball reflects on the sides of $ABC$, respecting the law of reflection that the angle of incidence is equal to the angle of reflection. Prove that, supposing that the ball never reaches any of the vertices $A,B,C$, the set of all directions in which the ball will move through time is finite. In other words, its path from the moment $0$ to infinity consists of segments parallel to a finite set of lines.

Given is a cyclic quadrilateral $ABCD$ with $|AB| = |BC|$. Point $E$ is on the arc $CD$ where $A$ and $B$ are not on. Let $P$ be the intersection point of $BE$ and $CD$ , let $Q$ be the intersection point of $AE$ and $BD$ . Prove that $PQ \parallel AC$.

A triangle $ABC$ with $AB<AC<BC$ is given. The point $P$ is the center of an excircle touching the line segment $AB$ at $D$. The point $Q$ is the center of an excircle touching the line segment $AC$ at $E$. The circumcircle of the triangle $ADE$ intersects $\overline{PE}$ and $\overline{QD}$ again at $G$ and $H$ respectively. The line perpendicular to $\overline{AG}$ at $G$ intersects the side $AB$ at $R$. The line perpendicular to $\overline{AH}$ at $H$ intersects the side $AC$ at $S$. Prove that $\overline{DE}$ and $\overline{RS}$ are parallel.

Let $P$ be a point interior to triangle $ABC$ (with $CA \neq CB$). The lines $AP$, $BP$ and $CP$ meet again its circumcircle $\Gamma$ at $K$, $L$, respectively $M$. The tangent line at $C$ to $\Gamma$ meets the line $AB$ at $S$. Show that from $SC = SP$ follows $MK = ML$. [i]Proposed by Marcin E. Kuczma, Poland[/i]

Two polynomials with real coefficients have the leading coefficients equal to 1 . Each polynomial has an odd degree that is equal to the number of its distinct real roots. The product of the values of the first polynomial at the roots of the second polynomial is equal to 2024. Find the product of the values of the second polynomial at the roots of the first one. Sergey Yanzhinov

There are two triangles $ABC$ and $BKL$ on the plane so that the segment $AK$ is divided into three equal parts by the point of intersection of the medians $\vartriangle ABC$ and the point of intersection of the bisectors $ \vartriangle BKL $ ($AK $ - median $ \vartriangle ABC$, $KA$ - bisector $\vartriangle BKL $) and quadrilateral $KALC $ is trapezoid. Find the angles of the triangle $BKL$. (Bogdan Rublev)

Find all triples of primes $(p,q,r)$ satisfying $3p^{4}-5q^{4}-4r^{2}=26$.

[u]Set 1[/u] [b]p1.[/b] Assume the speed of sound is $343$ m/s. Anastasia and Bananastasia are standing in a field in front of you. When they both yell at the same time, you hear Anastasia’s yell $5$ seconds before Bananastasia’s yell. If Bananastasia yells first, and then Anastasia yells when she hears Bananastasia yell, you hear Anastasia’s yell $5$ seconds after Bananastasia’s yell. What is the distance between Anastasia and Bananastasia in meters? [b]p2.[/b] Michelle picks a five digit number with distinct digits. She then reverses the digits of her number and adds that to her original number. What is the largest possible sum she can get? [b]p3.[/b] Twain is trying to crack a $4$-digit number combination lock. They know that the second digit must be even, the third must be odd, and the fourth must be different from the previous three. If it takes Twain $10$ seconds to enter a combination, how many hours would it take them to try every possible combination that satisfies these rules? PS. You should use hide for answers.

Triangle $A_1 A_2 A_3$ has a right angle at $A_3$. A sequence of points is now defined by the following iterative process, where $n$ is a positive integer. From $A_n$ ($n \geq 3$), a perpendicular line is drawn to meet $A_{n-2}A_{n-1}$ at $A_{n+1}$. (a) Prove that if this process is continued indefinitely, then one and only one point $P$ is interior to every triangle $A_{n-2} A_{n-1} A_{n}$, $n \geq 3$. (b) Let $A_1$ and $A_3$ be fixed points. By considering all possible locations of $A_2$ on the plane, find the locus of $P$.

Find all pairs $(a, b)$ of real numbers such that the roots of polynomials $6x^2 -24x -4a$ and $x^3 + ax^2 + bx - 8$ are all non-negative real numbers.

Let $M\subset \{1, 2, 3, . . . , 2007\}$ a set with the following property: Among every three numbers one can always choose two from $M$ such that one is divisible by the other. How many numbers can $M$ contain at most?

For every $ n\in\mathbb{N}$ let $ d(n)$ denote the number of (positive) divisors of $ n$. Find all functions $ f: \mathbb{N}\to\mathbb{N}$ with the following properties: [list][*] $ d\left(f(x)\right) \equal{} x$ for all $ x\in\mathbb{N}$. [*] $ f(xy)$ divides $ (x \minus{} 1)y^{xy \minus{} 1}f(x)$ for all $ x$, $ y\in\mathbb{N}$.[/list] [i]Proposed by Bruno Le Floch, France[/i]