A group of $10$ friends attend an amusement park. Each has visited three different attractions . Leaving the park and talking to each other, they found that any pair of friends visited at least one attraction in common. Determine what could be the minimum number of friends who could walk in the most visited attraction.

A company has a secret safe box that is locked by six locks. Several copies of the keys are distributed among the directors of the company. Each key can unlock exactly one lock. Each director has three keys for three different locks. No two directors can unlock the same three locks. No two directors together can unlock the safe. What is the maximum possible number of directors in the company?

Let $n$ be a positive integer and let $$1 = d_1 < d_2 < d_3 < d_4$$ the four smallest divisors of $n$. Find all$ n$ such that $$n^2 = d_1 + d_2^2+d_3^3 +d_4^4.$$

For any given pair of positive integers $m>n$ find all $a\in\mathbb R$ for which the polynomial $x^m-ax^n+1$ can be expressed as a quotient of two nonzero polynomials with real nonnegative coefficients.

The line $\ell$ passes through vertex$ B$ and the interior of regular hexagon $ABCDEF$. If the distances from $\ell$ to the vertices $A$ and $C$ are $7$ and $4$, respectively, compute the area of hexagon $ABCDEF$.

Incircle $\omega$ of triangle $ABC$ is tangent to sides $CB$ and $CA$ at $D$ and $E$, respectively. Point $X$ is the reflection of $D$ with respect to $B$. Suppose that the line $DE$ is tangent to the $A$-excircle at $Z$. Let the circumcircle of triangle $XZE$ intersect $\omega$ for the second time at $K$. Prove that the intersection of $BK$ and $AZ$ lies on $\omega$. Proposed by Mahdi Etesamifard and Alireza Dadgarnia

$BL $ is the bisector of an isosceles triangle $ABC $. A point $D $ is chosen on the Base $BC $ and a point $E $ is chosen on the lateral side $AB $ so that $AE=\frac {1}{2}AL=CD $. Prove that $LE=LD $. Tuymaada 2017 Q5 Juniors

$(SWE 1)$ Six points $P_1, . . . , P_6$ are given in $3-$dimensional space such that no four of them lie in the same plane. Each of the line segments $P_jP_k$ is colored black or white. Prove that there exists one triangle $P_jP_kP_l$ whose edges are of the same color.

For any positive integer $n$, let $c(n)$ be the largest divisor of $n$ not greater than $\sqrt{n}$ and let $s(n)$ be the least integer $x$ such that $n < x$ and the product $nx$ is divisible by an integer $y$ where $n < y < x$. Prove that, for every $n$, $s(n) = (c(n) + 1) \cdot \left( \frac{n}{c(n)}+1\right)$

Given triangle $\triangle ABC $ with $AC\neq BC $,and let $D $ be a point inside triangle such that $\measuredangle ADB=90^{\circ} + \frac {1}{2}\measuredangle ACB $.Tangents from $C $ to the circumcircles of $\triangle ABC $ and $\triangle ADC $ intersect $AB $ and $AD $ at $P $ and $Q $ , respectively.Prove that $PQ $ bisects the angle $\measuredangle BPC $.

For which of the following integers $n$, there is at least one integer $x$ such that $x^2 \equiv -1 \pmod{n}$? $ \textbf{(A)}\ 97 \qquad\textbf{(B)}\ 98 \qquad\textbf{(C)}\ 99 \qquad\textbf{(D)}\ 100 \qquad\textbf{(E)}\ \text{None of the preceding} $

Prove that if a,b,c are integers with $a+b+c=0$, then $2a^4+2b^4+2c^4$ is a perfect square.

Vincent has a fair die with sides labeled $1$ to $6$. He first rolls the die and records it on a piece of paper. Then, every second thereafter, he re-rolls the die. If Vincent rolls a different value than his previous roll, he records the value and continues rolling. If Vincent rolls the same value, he stops, does \emph{not} record his final roll, and computes the average of his previously recorded rolls. Given that Vincent first rolled a $1$, let $E$ be the expected value of his result. There exist rational numbers $r,s,t > 0$ such that $E = r-s\ln t$ and $t$ is not a perfect power. If $r+s+t = \frac mn$ for relatively prime positive integers $m$ and $n$, compute $100m+n$. [i]Proposed by Sean Li[/i]

Let the set $A=(a_{1},a_{2},a_{3},a_{4})$ . If the sum of elements in every 3-element subset of $A$ makes up the set $B=(-1,5,3,8)$ , then find the set $A$.

A beam of length $ a $ is suspended horizontally with its ends on two parallel ropes equal $ b $. We turn the beam through an angle $ \varphi $ around a vertical axis passing through the center of the beam. By how much will the beam rise?

The plane is tiled by congruent squares and congruent pentagons as indicated. The percent of the plane that is enclosed by the pentagons is closest to $ \textbf{(A)} \ 50 \qquad \textbf{(B)} \ 52 \qquad \textbf{(C)} \ 54 \qquad \textbf{(D)} \ 56 \qquad \textbf{(E)} \ 58 \qquad$ [asy]unitsize(3mm); defaultpen(linewidth(0.8pt)); path p1=(0,0)--(3,0)--(3,3)--(0,3)--(0,0); path p2=(0,1)--(1,1)--(1,0); path p3=(2,0)--(2,1)--(3,1); path p4=(3,2)--(2,2)--(2,3); path p5=(1,3)--(1,2)--(0,2); path p6=(1,1)--(2,2); path p7=(2,1)--(1,2); path[] p=p1^^p2^^p3^^p4^^p5^^p6^^p7; for(int i=0; i<3; ++i) { for(int j=0; j<3; ++j) { draw(shift(3*i,3*j)*p); } }[/asy]

Let $ABC$ be an acute triangle, and let $M$ and $N$ be two points on the line $AC$ such that the vectors $MN$ and $AC$ are identical. Let $X$ be the orthogonal projection of $M$ on $BC$, and let $Y$ be the orthogonal projection of $N$ on $AB$. Finally, let $H$ be the orthocenter of triangle $ABC$. Show that the points $B$, $X$, $H$, $Y$ lie on one circle.

Triangle $DEF$ is acute. Circle $c_1$ is drawn with $DF$ as its diameter and circle $c_2$ is drawn with $DE$ as its diameter. Points $Y$ and $Z$ are on $DF$ and $DE$ respectively so that $EY$ and $FZ$ are altitudes of triangle $DEF$ . $EY$ intersects $c_1$ at $P$, and $FZ$ intersects $c_2$ at $Q$. $EY$ extended intersects $c_1$ at $R$, and $FZ$ extended intersects $c_2$ at $S$. Prove that $P$, $Q$, $R$, and $S$ are concyclic points.

In an isosceles right-angled triangle shaped billiards table , a ball starts moving from one of the vertices adjacent to hypotenuse. When it reaches to one side then it will reflect its path. Prove that if we reach to a vertex then it is not the vertex at initial position [i]By Sam Nariman[/i]

All contestants at one contest are sitting in $n$ columns and are forming a "good" configuration. (We define one configuration as "good" when we don't have 2 friends sitting in the same column). It's impossible for all the students to sit in $n-1$ columns in a "good" configuration. Prove that we can always choose contestants $M_1,M_2,...,M_n$ such that $M_i$ is sitting in the $i-th$ column, for each $i=1,2,...,n$ and $M_i$ is friend of $M_{i+1}$ for each $i=1,2,...,n-1$.

Prove the identity $\sum_{k=2}^{n} k(k-1)\binom{n}{k} =\binom{n}{2} 2^{n-1}$ for all $n=2,3,4,...$

Number of solutions of the equation $3^x+4^x=8^x$ in reals is [list=1] [*] 0 [*] 1 [*] 2 [*] $\infty$ [/list]

How many $3$ digit positive integers are not divided by $5$ neither by $7$?

Find all functions $f\colon \mathbb{R} \rightarrow \mathbb{R}$ such that for all $x,y \in \mathbb{R}$, \[f(x+yf(x))+y = xy + f(x+y).\] [i]Proposed by Giannis Galamatis, Greece[/i]

Given integer $n\geq 2$. Find the minimum value of $\lambda {}$, satisfy that for any real numbers $a_1$, $a_2$, $\cdots$, ${a_n}$ and ${b}$, $$\lambda\sum\limits_{i=1}^n\sqrt{|a_i-b|}+\sqrt{n\left|\sum\limits_{i=1}^na_i\right|}\geqslant\sum\limits_{i=1}^n\sqrt{|a_i|}.$$