Given the polygons $P$ and $Q$ as shown in the grid below, cut $P$ into two polygons $P_1$ and $P_2$ such that, when pasted together differently, they form $Q$. [asy] import graph; size(16cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-2.05,xmax=15.10,ymin=-1.87,ymax=9.74; pen cqcqcq=rgb(0.75,0.75,0.75), zzttqq=rgb(0.6,0.2,0); draw((7,5)--(12,5)--(12,2)--(7,2)--cycle,zzttqq); draw((2,2)--(2,5)--(3,6)--(6,6)--(6,3)--(5,2)--cycle,zzttqq); /*grid*/ pen gs=linewidth(0.7)+cqcqcq+linetype("2 2"); real gx=1,gy=1; for(real i=ceil(xmin/gx)*gx;i<=floor(xmax/gx)*gx;i+=gx) draw((i,ymin)--(i,ymax),gs); for(real i=ceil(ymin/gy)*gy;i<=floor(ymax/gy)*gy;i+=gy) draw((xmin,i)--(xmax,i),gs); draw((0,8)--(0,0)); draw((0,0)--(13,0)); draw((13,0)--(13,8)); draw((13,8)--(0,8)); draw((7,5)--(12,5),zzttqq); draw((12,5)--(12,2),zzttqq); draw((12,2)--(7,2),zzttqq); draw((7,2)--(7,5),zzttqq); draw((2,2)--(2,5),zzttqq); draw((2,5)--(3,6),zzttqq); draw((3,6)--(6,6),zzttqq); draw((6,6)--(6,3),zzttqq); draw((6,3)--(5,2),zzttqq); draw((5,2)--(2,2),zzttqq); dot((0,0),linewidth(1pt)+ds); dot((13,0),linewidth(1pt)+ds); dot((0,8),linewidth(1pt)+ds); dot((2,2),linewidth(1pt)+ds); dot((6,6),linewidth(1pt)+ds); dot((13,8),linewidth(1pt)+ds); dot((7,2),linewidth(1pt)+ds); dot((7,5),linewidth(1pt)+ds); dot((12,2),linewidth(1pt)+ds); dot((12,5),linewidth(1pt)+ds); label("$Q$",(8.42,2.56),NE*lsf,zzttqq); dot((5,2),linewidth(1pt)+ds); dot((6,3),linewidth(1pt)+ds); dot((2,5),linewidth(1pt)+ds); dot((3,6),linewidth(1pt)+ds); label("$P$",(4.65,2.74),NE*lsf,zzttqq); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); [/asy]

Does there exist a positive integer $ n$ such that $ n$ has exactly 2000 prime divisors and $ n$ divides $ 2^n \plus{} 1$?

Given $n$ positive real numbers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n \ge 0$ and $x_1^2+x_2^2+\cdots+x_n^2=1$, prove that \[\frac{x_1}{\sqrt{1}}+\frac{x_2}{\sqrt{2}}+\cdots+\frac{x_n}{\sqrt{n}}\ge 1.\]

Theo and Wendy are commuting to school from their houses. Theo travels at $x$ miles per hour, while Wendy travels at $x + 5$ miles per hour. The school is $4$ miles from Theo’s house and $10$ miles from Wendy’s house. If Wendy’s commute takes double the amount of time that Theo’s commute takes, how many minutes does it take Wendy to get to school?

Let $v_{0}$ be the zero ector and let $v_{1},...,v_{n+1}\in\mathbb{R}^{n}$ such that the Euclidian norm $|v_{i}-v_{j}|$ is rational for all $0\le i,j\le n+1$. Prove that $v_{1},...,v_{n+1}$ are linearly dependent over the rationals.

Prove the equality: $\sin 54^o -\sin 18^o = 0.5$

A week ago, Sandy’s seasonal Little League batting average was $360$. After five more at bats this week, Sandy’s batting average is up to $400$. What is the smallest number of hits that Sandy could have had this season?

A splitting of a planar polygon is a finite set of triangles whose interiors are pairwise disjoint, and whose union is the polygon in question. Given an integer $n \ge 3$, determine the largest integer $m$ such that no planar $n$-gon splits into less than $m$ triangles.

Tetrahedron $ABCD$ has $AD=BC=28$, $AC=BD=44$, and $AB=CD=52$. For any point $X$ in space, define $f(X)=AX+BX+CX+DX$. The least possible value of $f(X)$ can be expressed as $m\sqrt{n}$, where $m$ and $n$ are positive integers, and $n$ is not divisible by the square of any prime. Find $m+n$.

Let $a_0,a_1,a_2,\dots$ be a sequence of nonnegative integers such that $a_2=5$, $a_{2014}=2015$, and $a_n=a_{a_{n-1}}$ for all positive integers $n$. Find all possible values of $a_{2015}$.

Suppose two regular pyramids with the same base $ABC$: $P-ABC$ and $Q-ABC$ are circumscribed by the same sphere. If the angle formed by one of the lateral face and the base of pyramid $P-ABC$ is $\frac{\pi}{4}$, find the tangent value of the angle formed by one of the lateral face and the base of the pyramid $Q-ABC$.

Find all real solutions of the system of equations $$\begin{cases} (x_3 + x_4 + x_5)^5 = 3x_1 \\ (x_4 + x_5 + x_1)^5 = 3x_2\\ (x_5 + x _1 + x_2)^5 = 3x_3\\ (x_1 + x_2 + x_3)^5 = 3x_4\\ (x_2 + x_3 + x_4)^5 = 3x_5 \end{cases}$$ (L. Tumescu , Romania)

We are given a non-obtuse $\Delta ABC$ $(BC>AC)$ with an altitude $CD$ $(D\in AB)$, center $O$ of its circumscribed circle, and a middle point $M$ of its side $AB$. Point $E$ lies on the ray $\overrightarrow{BA}$ in such way that $AE.BE=DE.ME$. If the line $OE$ bisects the area of $\Delta ABC$ and $CO=CD.cos\angle ACB$, determine the angles of $\Delta ABC$.

What number is nine more than four times the answer to this question?

Let $f_1, f_2, \ldots$ be continuous real functions on the real line. Is it true that if the series $\sum_{n=1}^{\infty} f_n(x)$ is divergent for every $x$, then this holds also true for any typical choice of the signs in the sum (i.e. the set of those $\{ \epsilon _n\}_{n=1}^{\infty} \in \{ +1, -1\}^{\mathbb{N}}$ sequences, for which there series $\sum_{n=1}^{\infty} \epsilon_nf_n(x)$ is convergent at least at one point $x$, forms a subset of first category within the set $\{+1,-1\}^{\mathbb{N}} $)? (translated by L. Erdős)

[b]p1. [/b] Evaluate $S$. $$S =\frac{10000^2 - 1}{\sqrt{10000^2 - 19999}}$$ [b]p2. [/b] Starting on a triangular face of a right triangular prism and allowing moves to only adjacent faces, how many ways can you pass through each of the other four faces and return to the first face in five moves? [b]p3.[/b] Given that $$(a + b) + (b + c) + (c + a) = 18$$ $$\frac{1}{a + b}+\frac{1}{b + c}+ \frac{1}{c + a}=\frac59,$$ determine $$\frac{c}{a + b}+\frac{a}{b + c}+\frac{b}{c + a}.$$ [b]p4.[/b] Find all primes $p$ such that $2^{p+1} + p^3 - p^2 - p$ is prime. [b]p5.[/b] In right triangle $ABC$ with the right angle at $A$, $AF$ is the median, $AH$ is the altitude, and $AE$ is the angle bisector. If $\angle EAF = 30^o$ , find $\angle BAH$ in degrees. [b]p6.[/b] For which integers $a$ does the equation $(1 - a)(a - x)(x- 1) = ax$ not have two distinct real roots of $x$? [b]p7. [/b]Given that $a^2 + b^2 - ab - b +\frac13 = 0$, solve for all $(a, b)$. [b]p8. [/b] Point $E$ is on side $\overline{AB}$ of the unit square $ABCD$. $F$ is chosen on $\overline{BC}$ so that $AE = BF$, and $G$ is the intersection of $\overline{DE}$ and $\overline{AF}$. As the location of $E$ varies along side $\overline{AB}$, what is the minimum length of $\overline{BG}$? [b]p9.[/b] Sam and Susan are taking turns shooting a basketball. Sam goes first and has probability $P$ of missing any shot, while Susan has probability $P$ of making any shot. What must $P$ be so that Susan has a $50\%$ chance of making the first shot? [b]p10.[/b] Quadrilateral $ABCD$ has $AB = BC = CD = 7$, $AD = 13$, $\angle BCD = 2\angle DAB$, and $\angle ABC = 2\angle CDA$. Find its area. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $\mathbb{F}_p$ be the set of integers modulo $p$. Call a function $f : \mathbb{F}_p^2 \to \mathbb{F}_p$ [i]quasiperiodic[/i] if there exist $a,b \in \mathbb{F}_p$, not both zero, so that $f(x + a, y + b) = f(x, y)$ for all $x,y \in \mathbb{F}_p$. Find the number of functions $\mathbb{F}_p^2 \to \mathbb{F}_p$ that can be written as the sum of some number of quasiperiodic functions.

Let $ABCDEF$ be a convex hexagon inscribed in a circle . Prove that the diagonals $AD, BE$ and $CF$ intersect at one point if and only if $$\frac{AB}{BC} \cdot \frac{CD}{DE}\cdot \frac{EF}{FA}=1$$

Let $ABC$ be a triangle with $AB < AC$, incenter $I$, and $A$ excenter $I_{A}$. The incircle meets $BC$ at $D$. Define $E = AD\cap BI_{A}$, $F = AD\cap CI_{A}$. Show that the circumcircle of $\triangle AID$ and $\triangle I_{A}EF$ are tangent to each other

Let $n\ge k$ be positive integers, and let $\mathcal{F}$ be a family of finite sets with the following properties: (i) $\mathcal{F}$ contains at least $\binom{n}{k}+1$ distinct sets containing exactly $k$ elements; (ii) for any two sets $A, B\in \mathcal{F}$, their union $A\cup B$ also belongs to $\mathcal{F}$. Prove that $\mathcal{F}$ contains at least three sets with at least $n$ elements. (Proposed by Fedor Petrov, St. Petersburg State University)

A wall made of mirrors has the shape of $\triangle ABC$, where $AB = 13$, $BC = 16$, and $CA = 9$. A laser positioned at point $A$ is fired at the midpoint $M$ of $BC$. The shot reflects about $BC$ and then strikes point $P$ on $AB$. If $\tfrac{AM}{MP} = \tfrac{m}{n}$ for relatively prime positive integers $m, n$, compute $100m+n$. [i]Proposed by Michael Tang[/i]

The domain of a function $f$ is $\mathbb{N}$ (The set of natural numbers). The function is defined as follows : $$f(n)=n+\lfloor\sqrt{n}\rfloor$$ where $\lfloor k\rfloor$ denotes the nearest integer smaller than or equal to $k$. Prove that, for every natural number $m$, the following sequence contains at least one perfect square $$m,~f(m),~f^2(m),~f^3(m),\cdots$$ The notation $f^k$ denotes the function obtained by composing $f$ with itself $k$ times.

Find all triples of integers $(a, b, c)$ satisfying $a^2 + b^2 + c^2 =3(ab + bc + ca).$

In the Olympic school the exams are graded with whole numbers, the lowest possible grade is $0$, and the highest is $10$. In the arithmetic class the teacher takes two exams. This year he has $15$ students. When one of his students gets less than $3$ on the first exam and more than $7$ on the second exam, he calls him an overachieving student. The teacher, at the end of correcting the exams, averaged the $30$ grades and obtained $8$. What is the largest number of students who passed this class could have had?

In the following sum: $1 + 2 + 3 + 4 + 5 + 6$, if we remove the first two “+” signs, we obtain the new sum $123 + 4 + 5 + 6 = 138$. By removing three “$+$” signs, we can obtain $1 + 23 + 456 = 480$. Let us now consider the sum $1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13$, in which some “$+$” signs are to be removed. What are the three smallest multiples of $100$ that we can get in this way?