For any prime $p$, prove that there exists integer $x_0$ such that $p | (x^2_0 - x_0 + 3)$ $\Leftrightarrow$ there exists integer $y_0$ such that $p | (y^2_0 - y_0 + 25).$

Let $ABCDEF$ be a convex hexagon such that $\angle A = \angle C = \angle E$ and $\angle B = \angle D = \angle F$ and the (interior) angle bisectors of $\angle A, ~\angle C,$ and $\angle E$ are concurrent. Prove that the (interior) angle bisectors of $\angle B, ~\angle D, $ and $\angle F$ must also be concurrent. [i]Note that $\angle A = \angle FAB$. The other interior angles of the hexagon are similarly described.[/i]

Define a figure which is constructed by unit squares "cross star" if it satisfies the following conditions: $(1)$Square bar $AB$ is bisected by square bar $CD$ $(2)$At least one square of $AB$ lay on both sides of $CD$ $(3)$At least one square of $CD$ lay on both sides of $AB$ There is a rectangular grid sheet composed of $38\times 53=2014$ squares,find the number of such cross star in this rectangle sheet

Find the maximum value of \[\sin^2 \theta_1+\sin^2 \theta_2+\cdots+\sin^2 \theta_n\] subject to the restrictions $0 \leq \theta_i , \theta_1+\theta_2+\cdots+\theta_n=\pi.$

Carlos went to a sports store to buy running shoes. Running shoes were on sale, with prices reduced by 20% on every pair of shoes. Carlos also knew that he had to pay a 7.5% sales tax on the discounted price. He had 43 dollars. What is the original (before discount) price of the most expensive shoes he could afford to buy? A)$46$ B)$50$ C)$48$ D)$47$ E)$49$

Prove that the polynomial $P_n(x)=1+x+\frac{x^2}{2!}+\cdots +\frac{x^n}{n!}$ has no real zeros if $n$ is even and has exatly one real zero if $n$ is odd

Two circles $\omega_1$ and $\omega_2$ intersect at points $A$ and $B$. Line $\ell$ is tangent to $\omega_1$ at $P$ and to $\omega_2$ at $Q$ so that $A$ is closer to $\ell$ than $B$. Let $X$ and $Y$ be points on major arcs $\overarc{PA}$ (on $\omega_1$) and $AQ$ (on $\omega_2$), respectively, such that $AX/PX = AY/QY = c$. Extend segments $PA$ and $QA$ through $A$ to $R$ and $S$, respectively, such that $AR = AS = c\cdot PQ$. Given that the circumcenter of triangle $ARS$ lies on line $XY$, prove that $\angle XPA = \angle AQY$.

Prove that there exist infinitely many integers $n$ such that $n, n + 1, n + 2$ are each the sum of two squares of integers.

[b]a)[/b] Prove that $ \{ a \} +\{ 1/a \} <3/2, $ for any positive real number $ a. $ [b]b)[/b] Give an example of a number $ b $ satisfying $ \{ b \} +\{ 1/b \} =1. $ [i]{} means fractional part[/i]

The Olcommunity consists of the next seven people: Christopher Took, Humberto Brandybuck, German son of Isildur, Leogolas, Argimli, Samzamora and Shago Baggins. This community needs to travel from the Olcomashire to Olcomordor to save the world. Each person can take with them a total of $4$ day-provisions, that can be transferred to other people that are on the same day of traveling, as long as nobody holds more than $4$ day-provisions at the time. If a person returns to Olcomashire, they will be too tired to go out again. What is the farthest away Olcomordor can be from Olcomashire, so that Shago Baggins can get to Olcomordor, and the rest of the Olcommunity can return save to Olcomashire? Note: All the members of the Olcommunity should eat exactly one day-provision while they are away from Olcomashire. The members only travel an integer number of days on each direction. Members of the Olcommunity may leave Olcomashire on different days.

Let $ABC$ be an acute scalene triangle, $H$ its orthocenter and $G$ its geocenter. The circumference with diameter $AH$ cuts the circumcircle of $BHC$ in $A'$ ($A' \neq H$). Points $B'$ and $C'$ are defined similarly. Show that the points $A'$, $B'$, $C'$, and $G$ lie in one circumference.

Let $I$ be the incenter of a nonisosceles triangle $ABC$. Prove that there exists a unique pair of points $M$, $N$ lying on the sides $AC$, $BC$ respectively, such that $\angle AIM = \angle BIN$ and $MN|| AB$.

A triangle $ABC$ is inscribed in a circle $\omega$. A variable line $\ell$ chosen parallel to $BC$ meets segments $AB$, $AC$ at points $D$, $E$ respectively, and meets $\omega$ at points $K$, $L$ (where $D$ lies between $K$ and $E$). Circle $\gamma_1$ is tangent to the segments $KD$ and $BD$ and also tangent to $\omega$, while circle $\gamma_2$ is tangent to the segments $LE$ and $CE$ and also tangent to $\omega$. Determine the locus, as $\ell$ varies, of the meeting point of the common inner tangents to $\gamma_1$ and $\gamma_2$. [i](Russia) Vasily Mokin and Fedor Ivlev[/i]

Determine all three-digit numbers $N$ having the property that $N$ is divisible by $11,$ and $\frac{N}{11}$ is equal to the sum of the squares of the digits of $N.$

Determine all triples $(x, y, z)$ of nonnegative real numbers that verify the following system of equations: $$x^2 - y = (z -1)^2 $$ $$y^2 - z = (x -1)^2$$ $$z^2 - x = (y - 1)^2$$

Positive real numbers $a,b,c,d$ satisfy the equalities \[a^3+b^3+c^3=3d^3\\ b^4+c^4+d^4=3a^4\\ c^5+d^5+a^5=3b^5. \] Prove that $a=b=c=d$.

Given triangle $ABC$. Point $A_1$ is symmetric to vertex $A$ wrt line $BC$, and point $C_1$ is symmetric to vertex $C$ wrt line $AB$. Prove that if points $A_1$, $B$ and $C_1$ lie on the same line and $C_1B = 2A_1B$, then angle $\angle CA_1B$ is right.

Solve the system of equations: $$ 2(3-2\cos y)^2+2(4-2\sin y)^2=2(3-x)^2+32=(x-2\cos y)^2+4\sin^2y$$

Let $ABCD$ be a rectangle. Two perpendicular lines pass through point $B$. One of them meets segment $AD$ at point $K$, and the second one meets the extension of side $CD$ at point $L$. Let $F$ be the common point of $KL$ and $AC$. Prove that $BF\perp KL$.

Given $2020*2020$ chessboard, what is the maximum number of warriors you can put on its cells such that no two warriors attack each other. Warrior is a special chess piece which can move either $3$ steps forward and one step sideward and $2$ step forward and $2$ step sideward in any direction.

Positive real numbers $a,b,c$ satisfy $a+b+c=1$. Prove that $$(a+1)\sqrt{2a(1-a)} + (b+1)\sqrt{2b(1-b)} + (c+1)\sqrt{2c(1-c)} \geq 8(ab+bc+ca).$$ Also, find the values of $a,b,c$ for which the equality happens.

Let $(a_n)_n\geq 0$ and $a_{m+n}+a_{m-n}=\frac{1}{2}(a_{2m}+a_{2n})$ for every $m\geq n\geq0.$ If $a_1=1,$ then find the value of $a_{2007}.$

Six small circles of radius $1$ are drawn so that they are all tangent to a larger circle, and two of them are tangent to sides $BC$ and $AD$ of a rectangle $ABCD$ at their midpoints $K$ and $L$. Each of the remaining four small circles is tangent to two sides of the rectangle. The large circle is tangent to sides $AB$ and $CD$ of the rectangle and cuts the other two sides. Find the radius of the large circle. [img]https://cdn.artofproblemsolving.com/attachments/b/4/a134da78d709fe7162c48d6b5c40bd1016c355.png[/img]

Let $f(x)$ be a function defined in $0<x<\frac{\pi}{2}$ satisfying: (i) $f\left(\frac{\pi}{6}\right)=0$ (ii) $f'(x)\tan x=\int_{\frac{\pi}{6}}^x \frac{2\cos t}{\sin t}dt$. Find $f(x)$. [i]1987 Sapporo Medical University entrance exam[/i]

Ivan claims that for all positive integers $n$, $$\left\lfloor\sqrt[2]{\frac{n}{1^3}}\right\rfloor + \left\lfloor\sqrt[2]{\frac{n}{2^3}}\right\rfloor + \left\lfloor\sqrt[2]{\frac{n}{3^3}}\right\rfloor + \cdots = \left\lfloor\sqrt[3]{\frac{n}{1^2}}\right\rfloor + \left\lfloor\sqrt[3]{\frac{n}{2^2}}\right\rfloor + \left\lfloor\sqrt[3]{\frac{n}{3^2}}\right\rfloor + \cdots$$ Why is he correct? (Note: $\lfloor x \rfloor$ denotes the floor function.)