For positive $x, y$, and $z$ that satisfy the condition $xyz = 1$, prove the inequality $$\sqrt[3]{\frac{x+y}{2z}}+\sqrt[3]{\frac{y+z}{2x}}+\sqrt[3]{\frac{z+x}{2y}}\le \frac{5(x+y+z)+9}{8}$$

A convex polygon $\mathcal{P}$ with a center of symmetry $O{}$ is drawn in the plane. Prove that it is possible to place a rhombus in $\mathcal{P}$ whose image following a homothety of factor two centered at $O$ contains $\mathcal{P}$. [i]Proposed by I. Bogdanov, S. Gerdzhikov and N. Nikolov[/i]

Let $\Delta ABC$ be an isosceles triangle with $AB=AC$. The incircle $\Gamma$ of $\Delta ABC$ has centre $I$, and it is tangent to the sides $AB$ and $AC$ at $F$ and $E$ respectively. Let $\Omega$ be the circumcircle of $\Delta AFE$. The two external common tangents of $\Gamma$ and $\Omega$ intersect at a point $P$. If one of these external common tangents is parallel to $AC$, prove that $\angle PBI=90^{\circ}$.

For each pair of integers \( j, k \geq 2 \), define the function \( f_{jk} : \mathbb{R} \to \mathbb{R} \) given by \[ f_{jk}(x) = 1 - (1 - x^j)^k. \] (a) Prove that for any integers \( j, k \geq 2 \), there exists a unique real number \( p_{jk} \in (0, 1) \) such that \( f_{jk}(p_{jk}) = p_{jk} \). Furthermore, defining \( \lambda_{jk} := f'_{jk}(p_{jk}) \), prove that \( \lambda_{jk} > 1 \). (b) Prove that \( p^j_{jk} = 1 - p_{kj} \) for any integers \( j, k \geq 2 \). (c) Prove that \( \lambda_{jk} = \lambda_{kj} \) for any integers \( j, k \geq 2 \).

$(GDR 2)^{IMO1}$ Prove that there exist infinitely many natural numbers $a$ with the following property: The number $z = n^4 + a$ is not prime for any natural number $n.$

Given the line $3x+5y=15$ and a point on this line equidistant from the coordinate axes. Such a point exists in: $ \textbf{(A)}\ \text{none of the quadrants} \qquad\textbf{(B)}\ \text{quadrant I only} \qquad\textbf{(C)}\ \text{quadrants I, II only} \qquad$ $\textbf{(D)}\ \text{quadrants I, II, III only} \qquad\textbf{(E)}\ \text{each of the quadrants} $

Let $M$ be the point of intersection of the diagonals, and $K$ be the point of intersection of the bisectors of the angles $B$ and $C$ of the convex quadrilateral $ABCD$. Prove that points $A, B, M, K$ lie on the same circle if the following relation holds: $|AB|=|BC|=|CD|$

The altitudes of a tetrahedron meet at a single point. Prove that this point, the centroid of one face of the tetrahedron, the foot of the altitude on that face, and the three points dividing the other three altitudes in ratio $2:1$ (closer to the feet) all lie on a sphere.

The function $f: \mathbb{Z} \rightarrow \mathbb{Z}$ is called [i]“Sozopolian”[/i], if it satisfies the following two properties: For each two $x,y\in \mathbb{Z}$ which aren’t multiples of 17 the number $f(xy)-f(x)-f(y)$ is divisible by 8; For $\forall x\in \mathbb{Z}$ the number $f(x+17)-f(x)$ is divisible by 8. Does there exist a [i]Sozopolian[/i] function for which a) $f(2)=1; \quad$ b) $f(3)=1$?

The figure below shows a hexagonal net formed by many congruent equilateral triangles. Taking turns, Gabriel and Arnaldo play a game as follows. On his turn, the player colors in a segment, including the endpoints, following these three rules: i) The endpoints must coincide with vertices of the marked equilateral triangles. ii) The segment must be made up of one or more of the sides of the triangles. iii) The segment cannot contain any point (endpoints included) of a previously colored segment. Gabriel plays first, and the player that cannot make a legal move loses. Find a winning strategy and describe it.

Let $n \ne 0$ be a natural number. A sequence of numbers is briefly called a sequence “$F_n$” if $n$ different numbers $z_1$, $z_2$, $...$, $z_n$ exist so that the following conditions are fulfilled: (1) Each term of the sequence is one of the numbers $z_1$, $z_2$, $...$, $z_n$. (2) Each of the numbers $z_1$, $z_2$, $...$, $z_n$ occurs at least once in the sequence. (3) Any two immediately consecutive members of the sequence are different numbers. (4) No subsequence of the sequence has the form $\{a, b, a, b\}$ with $a \ne b$. Note: A subsequence of a given sequence $\{x_1, x_2, x_3, ...\}$ or $\{x_1, x_2, x_3, ..., x_s\}$ is called any sequence of the form $\{x_{m1}, x_{m2}, x_{m3}, ...\}$ or $\{x_{m1}, x_{m2}, x_{m3}, ..., x_{mt}\}$ with natural numbers $m_1 < m_2 < m_3 < ...$ Answer the following questions: a) Given $n$, are there sequences $F_n$ of arbitrarily long length? b) If question (a) is answered in the negative for an $n$: What is the largest possible number of terms that a sequence $F_n$ can have (given $n$)?

Let $ABC$ be a triangle with $AB < BC$. The bisector of angle $C$ meets the line parallel to $AC$ and passing through $B$, at point $P$. The tangent at $B$ to the circumcircle of $ABC$ meets this bisector at point $R$. Let $R'$ be the reflection of $R$ with respect to $AB$. Prove that $\angle R'P B = \angle RPA$.

Let $n \ge 2$ be a fixed even integer. We consider polynomials of the form \[P(x) = x^n + a_{n-1}x^{n-1} + \cdots + a_1x + 1\] with real coefficients, having at least one real roots. Find the least possible value of $a^2_1 + a^2_2 + \cdots + a^2_{n-1}$.

A positive integer $N$ is said to be $n-$ good if (i) $N$ has at least $n$ distinct prime divisors, and (ii) there exist distinct positive divisors $1, x_{2}, . . . , x_{n}$ whose sum is $N$ . Show that there exists an $n-$ good number for each $n\geq 6$.

Consider a game on an \( n \times n \) board, where each square starts with exactly one stone. A move consists of choosing $5$ consecutive squares in the same row or column of the board and toggling the state of each of those squares (removing the stone from squares with a stone and placing a stone in squares without a stone). For which positive integers \( n \geq 5 \) is it possible to end up with exactly one stone on the board after a finite number of moves?

Given a cyclic quadrilateral $ABCD$, let the diagonals $AC$ and $BD$ meet at $E$ and the lines $AD$ and $BC$ meet at $F$. The midpoints of $AB$ and $CD$ are $G$ and $H$, respectively. Show that $EF$ is tangent at $E$ to the circle through the points $E$, $G$ and $H$. [i]Proposed by David Monk, United Kingdom[/i]

Prove that if $|x| < 1$ and $|y| < 1$, then $\left|\frac{x - y}{1 -xy}\right|< 1$.

Let be $a,b$ real numbers such that $|a|\neq |b|$ and $\frac{a+b}{a-b}+\frac{a-b}{a+b}=6$ . Calculate: $\frac{a^3+b^3}{a^3-b^3}+\frac{a^3-b^3}{a^3+b^3}$

Given a triangle $ABC$ satisfying $AC+BC=3\cdot AB$. The incircle of triangle $ABC$ has center $I$ and touches the sides $BC$ and $CA$ at the points $D$ and $E$, respectively. Let $K$ and $L$ be the reflections of the points $D$ and $E$ with respect to $I$. Prove that the points $A$, $B$, $K$, $L$ lie on one circle. [i]Proposed by Dimitris Kontogiannis, Greece[/i]

George is taking a ten-question true-false exam, where the answer key has been selected uniformly at random; however, he doesn't know any of the answers! Luckily, a friend has helpfully hinted that no two consecutive questions have true as the correct answer. If George takes the exam and maximizes the expected number of questions he gets correct, how many of his answers are expected to be right?

Given $a$, $b$, and $c$ are complex numbers satisfying \[ a^2+ab+b^2=1+i \] \[ b^2+bc+c^2=-2 \] \[ c^2+ca+a^2=1, \] compute $(ab+bc+ca)^2$. (Here, $i=\sqrt{-1}$)

In the figure on the right, $ABCD$ is a con­vex quadrilateral, $K, L, M,$ and $N$ are the mid­points of its sides, and $PQRS$ is the quadrilateral formed by the intersections of $AK, BL, CM,$ and $DN$. Determine the area of quadrilateral $PQRS$ if the area of quadrilateral $ABCD$ is $3000$, and the areas of quadrilaterals $AMQP$ and $CKSR$ are $513$ and $388$, respectively. [asy] defaultpen(linewidth(0.7)+fontsize(10));size(200); pair A=origin, B=(14,0), C=(13,10), D=(2,9), K=midpoint(C--D), L=midpoint(D--A), M=midpoint(A--B), N=midpoint(B--C), P=intersectionpoint(B--L, A--K), Q=intersectionpoint(B--L, C--M), R=intersectionpoint(C--M, D--N), S=intersectionpoint(D--N, A--K); draw(K--A--B--C--D--A^^D--N^^B--L^^C--M); pair point=(7,6); label("$A$", A, dir(point--A)); label("$B$", B, dir(point--B)); label("$C$", C, dir(point--C)); label("$D$", D, dir(point--D)); label("$S$", S, dir(160)*dir(point--S)); label("$R$", R, dir(190)*dir(point--R)); label("$Q$", Q, dir(180)*dir(point--Q)); label("$P$", P, dir(180)*dir(point--P)); label("$K$", K, dir(point--K)); label("$L$", L, dir(point--L)); label("$M$", M, dir(point--M)); label("$N$", N, dir(point--N));[/asy]

Let $M$ be a positive integer, and $A = \{1, 2,... , M + 1\}$. Show that if $f$ is a bijection from $A$ to $A$ then $\sum_{n=1}^{M} \frac{1}{f(n) + f(n + 1)} > \frac{M}{M + 3}$

Pingu is given two positive integers $m$ and $n$ without any common factors greater than $1$. a) Help Pingu find positive integers $p, q$ such that $$\operatorname{gcd}(pm+q, n) \cdot \operatorname{gcd}(m, pn+q) = mn$$ b) Prove to Pingu that he can never find positive integers $r, s$ such that $$\operatorname{lcm}(rm+s, n) \cdot \operatorname{lcm}(m, rn+s) = mn$$ regardless of the choice of $m$ and $n$.

Given a finite set $S \subset \mathbb{R}^3$, define $f(S)$ to be the mininum integer $k$ such that there exist $k$ planes that divide $\mathbb{R}^3$ into a set of regions, where no region contains more than one point in $S$. Suppose that \[M(n) = \max\{f(S) : |S| = n\} \text{ and } m(n) = \min\{f(S) : |S| = n\}.\] Evaluate $M(200) \cdot m(200)$.