It is known that odd prime numbers $x, y z$ satisfy $$x|(y^5+1),y|(z^5+1),z|(x^5+1).$$ Find the minimum value of the product $xyz$.

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.

Set $S = \{1, 2, 3, ..., 2005\}$. If among any $n$ pairwise coprime numbers in $S$ there exists at least a prime number, find the minimum of $n$.

A big circle is inscribed in a rhombus, each of two smaller circles touches two sides of the rhombus and the big circle as shown in the figure on the right. Prove that the four dashed lines spanning the points where the circles touch the rhombus as shown in the figure make up a square.

In how many ways, can we draw $n-3$ diagonals of a $n$-gon with equal sides and equal angles such that: $i)$ none of them intersect each other in the polygonal. $ii)$ each of the produced triangles has at least one common side with the polygonal.

Are there integers $a$ and $b$ such that $a^5b+3$ and $ab^5+3$ are both perfect cubes of integers?

Anna and Bob play the following game. In the beginning, Bob writes down the numbers $1, 2, ... , 2022$ on a piece of paper, such that half of the numbers are on the left and half on the right. Furthermore, we assume that the $1011$ numbers on both sides are written in some order. After Bob does this, Anna has the opportunity to swap the positions of the two numbers lying on different sides of the paper if they have different parity. Anna wins if, after finitely many moves, all odd numbers end up on the left, in increasing order, and all even ones end up on the right, in increasing order. Can Bob write down a arrangement of numbers for which Anna cannot win? For example, Bob could write down numbers in the following way: $$4, 2, 5, 7, 9, ... , 2021\,\,\,\,\,\,\,\,\,\,,\, \,\,\,\,\,\,\,\,\,\,,\, 3, 1, 6, 8, 10, ... , 2022$$ Then Anna could swap the numbers $1, 4$ and then swap $2, 3$ to win. However, if Anna swapped the pairs $3, 4$ and $1, 2$, the resulting numbers on the left and on the right would not be in increasing order, and hence Anna would not win.

Call a set of integers [i]unpredictable[/i] if no four elements in the set form an arithmetic sequence. How many unordered [i]unpredictable[/i] sets of five distinct positive integers $\{a, b, c, d, e\}$ exist such that all elements are strictly less than $12$? [i]Proposed by Anthony Yang[/i]

A concrete sewer pipe fitting is shaped like a cylinder with diameter $48$ with a cone on top. A cylindrical hole of diameter $30$ is bored all the way through the center of the fitting as shown. The cylindrical portion has height $60$ while the conical top portion has height $20$. Find $N$ such that the volume of the concrete is $N \pi$. [asy] import three; size(250); defaultpen(linewidth(0.7)+fontsize(10)); pen dashes = linewidth(0.7) + linetype("2 2"); currentprojection = orthographic(0,-15,5); draw(circle((0,0,0), 15),dashes); draw(circle((0,0,80), 15)); draw(scale3(24)*((-1,0,0)..(0,-1,0)..(1,0,0))); draw(shift((0,0,60))*scale3(24)*((-1,0,0)..(0,-1,0)..(1,0,0))); draw((-24,0,0)--(-24,0,60)--(-15,0,80)); draw((24,0,0)--(24,0,60)--(15,0,80)); draw((-15,0,0)--(-15,0,80),dashes); draw((15,0,0)--(15,0,80),dashes); draw("48", (-24,0,-20)--(24,0,-20)); draw((-15,0,-20)--(-15,0,-17)); draw((15,0,-20)--(15,0,-17)); label("30", (0,0,-15)); draw("60", (50,0,0)--(50,0,60)); draw("20", (50,0,60)--(50,0,80)); draw((50,0,60)--(47,0,60));[/asy]

Given is a trihedral angle $ OABC $ with a vertex $ O $ and a point $ P $ in its interior. Let $ V $ be the volume of a parallelepiped with two vertices at points $ O $ and $ P $, whose three edges are contained in the rays $ \overrightarrow{OA} $, $ \overrightarrow{OB} $, $ \overrightarrow{OC} $. Calculate the minimum volume of a tetrahedron whose three faces are contained in the faces of the trihedral angle $OABC$ and the fourth face contains the point $P$.

Let $a,b,c$ be non-zero real numbers.Prove that if function $f,g:\mathbb{R}\to\mathbb{R}$ satisfy $af(x+y)+bf(x-y)=cf(x)+g(y)$ for all real number $x,y$ that $y>2018$ then there exists a function $h:\mathbb{R}\to\mathbb{R}$ such that $f(x+y)+f(x-y)=2f(x)+h(y)$ for all real number $x,y$.

Let be two nonzero real numbers $ a,b $ such that $ |a|\neq |b| $ and let $ f:\mathbb{R}\longrightarrow\mathbb{R} $ be a function satisfying the functional relation $$ af(x)+bf(-x)=(x^3+x)^5+\sin^5 x . $$ Calculate $ \int_{-2019}^{2019}f(x)dx . $ [i]Constantin Rusu[/i]

Find the least number which is divisible by 2009 and its sum of digits is 2009.

Unit square $ABCD$ is divided into four rectangles by $EF$ and $GH$, with $BF = \frac14$. $EF$ is parallel to $AB$ and $GH$ parallel to $BC$. $EF$ and $GH$ meet at point $P$. Suppose $BF + DH = FH$, calculate the nearest integer to the degree of $\angle FAH$. [asy] size(100); defaultpen(linewidth(0.7)+fontsize(10)); pair D2(pair P) { dot(P,linewidth(3)); return P; } // NOTE: I've tampered with the angles to make the diagram not-to-scale. The correct numbers should be 72 instead of 76, and 45 instead of 55. pair A=(0,1), B=(0,0), C=(1,0), D=(1,1), F=intersectionpoints(A--A+2*dir(-76),B--C)[0], H=intersectionpoints(A--A+2*dir(-76+55),D--C)[0], E=F+(0,1), G=H-(1,0), P=intersectionpoints(E--F,G--H)[0]; draw(A--B--C--D--cycle); draw(F--A--H); draw(E--F); draw(G--H); label("$A$",D2(A),NW); label("$B$",D2(B),SW); label("$C$",D2(C),SE); label("$D$",D2(D),NE); label("$E$",D2(E),plain.N); label("$F$",D2(F),S); label("$G$",D2(G),W); label("$H$",D2(H),plain.E); label("$P$",D2(P),SE); [/asy]

Given a finite sequence of real numbers $a_1,a_2,...,a_n$ such that $$a_1 + a_2 + ... + a_n > 0.$$ Prove that there is at least one index $ i$ such that $$a_i > 0, a_i + a_{i+1} > 0, ..., a_i + a_{i+1} + ...+ a_n > 0.$$

In a convex $n$-gon, several diagonals are drawn. Among these diagonals, a diagonal is called [i]good[/i] if it intersects exactly one other diagonal drawn (in the interior of the $n$-gon). Find the maximum number of good diagonals.

The area of the ring between two concentric circles is $12\tfrac12\pi$ square inches. The length of a chord of the larger circle tangent to the smaller circle, in inches, is: $\textbf{(A) }\dfrac5{\sqrt2}\qquad \textbf{(B) }5\qquad \textbf{(C) }5\sqrt2\qquad \textbf{(D) }10\qquad \textbf{(E) }10\sqrt2$

Ana and Carlos entertain themselves with the next game. At the beginning of game in each vertex of the square there is an empty box. In each step, the corresponding player has two possibilities: either he adds a stone to an arbitrary box, or move each box clockwise to the next vertex of the square. Carlos starts and they take 2012 steps in turn (each player 1006). So Carlos marks one of the vertices of the square and allows Ana to make a more play. Carlos wins if after this last step the number ofstones in some box is greater than the number of stones in the box which is at the vertex marked by Carlos; otherwise Ana wins. Which of the two players has a winning strategy?

Given $a_1\ge 1$ and $a_{k+1}\ge a_k+1$ for all $k\ge 1,2,\dots,n$, show that $a_1^3+a_2^3+\dots+a_n^3\ge (a_1+a_2+\dots+a_n)^2$

Let a binary operation $*$ on ordered pairs of integers be defined by $(a,b)*(c,d)=(a-c,b+d)$. Then, if $(3,2)*(0,0)$ and $(x,y)*(3,2)$ represent idential pairs, $x$ equals: $\textbf{(A) }-3\qquad \textbf{(B) }0\qquad \textbf{(C) }2\qquad \textbf{(D) }3\qquad \textbf{(E) }6$

Given an acute triangle $ABC$ with circumcenter $O$. The circumcircle of $BCH$ and a circle with diameter of $AC$ intersect at $P (P \neq C)$. A point $Q$ on segment of $PC$ such that $PB = PQ$. Prove that $\angle ABC = \angle AQP$

Let $(a_n)_{n\ge0}$ be a sequence of positive numbers that satisfy the relations $a_{i-1}a_{i+1}\le a_i^2$ for all $i\in\mathbb N$. For any integer $n>1$, prove the inequality $$\frac{a_0+\ldots+a_n}{n+1}\cdot\frac{a_1+\ldots+a_{n-1}}{n-1}\ge\frac{a_0+\ldots+a_{n-1}}n\cdot\frac{a_1+\ldots+a_n}n.$$

A point $D$ lies inside a triangle $ABC$ on the bisector of angle $B$. Let $\omega_{1}$ and $\omega_{2}$ be the circles touching $AD$ and $CD$ at $D$ and passing through $B$; $P$ and $Q$ be the common points of $\omega_{1}$ and $\omega_{2}$ with the circumcircle of $ABC$ distinct from $B$. Prove that the circumcircles of the triangles $PQD$ and $ACD$ are tangent. Proposed by: L Shatunov

Let $f:\mathbb{R}_{+}^{*}$$\rightarrow$$\mathbb{R}_{+}^{*}$ such that $f'''(x)>0$ for all $x$ $\in$ $\mathbb{R}_{+}^{*}$. Prove that: $f(a^{2}+b^{2}+c^{2})+2f(ab+bc+ac)$ $\geq$ $f(a^{2}+2bc)+f(b^{2}+2ca)+f(c^{2}+2ab)$, for all $a,b,c$ $\in$ $\mathbb{R}_{+}^{*}$.

Prove that there exist infinitely many composite positive integers $n$ such that $n$ divides $3^{n-1}-2^{n-1}$.