Let $a, b, c, d > 0$ satisfying $abcd = 1$. Prove that $$\frac{1}{a + b + 2}+\frac{1}{b + c + 2}+\frac{1}{c + d + 2}+\frac{1}{d + a + 2} \le 1$$

Weights of $1$ g, $2$ g,$ ...$ , $200$ g are placed on the two pans of a balance such that on each pan there are $100$ weights and the balance is in equilibrium. Prove that one can swap $50$ weights from one pan with $50$ weights from the other pan such that the balance remains in equilibrium. Kvant Magazine

Consider a circle centered at $O$ with radius $r$ and a line $\ell$ not passing through $O$. A grasshopper is jumping to and fro between the points of the circle and the line, the length of each jump being $r$. Prove that there are at most $8$ points for the grasshopper to reach.

In the acute-angled triangle $ABC$, with $AB \ne AC$, $D$ is the foot of the angle bisector of angle $A$, and $E, F$ are the feet of the altitudes from $B$ and $C$, respectively. The circumcircles of triangles $DBF$ and $DCE$ intersect for the second time at $M$. Prove that $ME = MF$. Leonard Giugiuc

a) Prove that for every positive integer n, there exist $a, b \in R - Z$ such that the set $A_n = \{a - b, a^2 - b^2, a^3 - b^3,...,a^n - b^n\}$ contains only positive integers. b) Let $a$ and $b$ be two real numbers such that the set $A = \{a^k - b^k | k \in N*\}$ contains only positive integers. Prove that $a$ and $b$ are integers.

If $a, b, c > 0$ satisfy $a + b + c = 3$, then prove that $$\frac{a^2(b + 1)}{ ab + a + b} + \frac{b^2(c + 1)}{ bc + b + c} + \frac{c^2(a + 1)}{ ca + c + a} \ge 2$$ Mathematical Excalibur P322/Vol.14, no.2

Call the number $\overline{a_1a_2... a_m}$ ($a_1 \ne 0,a_m \ne 0$) the reverse of the number $\overline{a_m...a_2a_1}$. Prove that the sum between a number $n$ and its reverse is a multiple of $81$ if and only if the sum of the digits of $n$ is a multiple of $81$.

The three-element subsets of a seven-element set are colored. If the intersection of two sets is empty then they have different colors. What is the minimum number of colors needed?

Let $H$ be the orthocenter of an acute-angled triangle $ABC$ and $P$ a point on the circumcenter of triangle $ABC$. Prove that the Simson line of $P$ bisects the segment $[P H]$.

Let $n$ be a positive integer. Determine all positive integers $p$ for which there exist positive integers $x_1 < x_2 <...< x_n$ such that $\frac{1}{x_1}+\frac{2}{x_2}+ ... +\frac{n}{x_n}= p$ Irish Mathematical Olympiad

Find all positive integers $x,y,z$ such that $7^x + 13^y = 8^z$

Let $ABCD$ be a cyclic quadrilateral and $\omega_1, \omega_2$ the incircles of triangles $ABC$ and $BCD$. Show that the common external tangent line of $\omega_1$ and $\omega_2$, the other one than $BC$, is parallel with $AD$

Find all integers $n \ge 2$ with the property: there is a permutation $(a_1,a2,..., a_n)$ of the set $\{1, 2,...,n\}$ so that the numbers $a_1 + a_2 +...+ a_k, k = 1, 2,...,n$ have diffferent remainders when divided by $n$

Let $a, b, c$ be positive real numbers such that $a + b + c = 1$. Show that $$\frac{1 - a^2}{a + bc} + \frac{1 - b^2}{b + ca} + \frac{1 - c^2}{c + ab} \ge 6$$

Let $D$ be the midpoint of the side $[BC]$ of the triangle $ABC$ with $AB \ne AC$ and $E$ the foot of the altitude from $BC$. If $P$ is the intersection point of the perpendicular bisector of the segment line $[DE]$ with the perpendicular from $D$ onto the the angle bisector of $BAC$, prove that $P$ is on the Euler circle of triangle $ABC$.

For any sequence ($a_1,a_2,...,a_{2013}$) of integers, we call a triple ($i,j, k$) satisfying $1 \le i < j < k \le 2013$ to be progressive if $a_k-a_j = a_j -a_i = 1$. Determine the maximum number of progressive triples that a sequence of $2013$ integers could have.

Find all pairs of integers $(x,y)$ satisfying the following condition: each of the numbers $x^3 + y$ and $x + y^3$ is divisible by $x^2 + y^2$ Tournament of Towns

Let $M$ be the set of integer coordinate points situated on the line $d$ of real numbers. We color the elements of M in black or white. Show that at least one of the following statements is true: (a) there exists a finite subset $F \subset M$ and a point $M \in d$ so that the elements of the set $M - F$ that are lying on one of the rays determined by $M$ on $d$ are all white, and the elements of $M - F$ that are situated on the opposite ray are all black, (b) there exists an infinite subset $S \subset M$ and a point $T \in d$ so that for each $A \in S$ the reflection of A about $T$ belongs to $S$ and has the same color as $A$

Find the minimum and the maximum value of the expression $\sqrt{4 -a^2} +\sqrt{4 -b^2} +\sqrt{4 -c^2}$ where $a,b, c$ are positive real numbers satisfying the condition $a^2 + b^2 + c^2=6$

Consider acute triangles $ABC$ and $BCD$, with $\angle BAC = \angle BDC$, such that $A$ and $D$ are on opposite sides of line $BC$. Denote by $E$ the foot of the perpendicular line to $AC$ through $B$ and by $F$ the foot of the perpendicular line to $BD$ through $C$. Let $H_1$ be the orthocenter of triangle $ABC$ and $H_2$ be the orthocenter of $BCD$. Show that lines $AD, EF$ and $H_1H_2$ are concurrent.