Find all real solutions of the equation: $$x=\frac{2z^2}{1+z^2}$$ $$y=\frac{2x^2}{1+x^2}$$ $$z=\frac{2y^2}{1+y^2}$$

A positive integer is written on each corner of a square such that numbers on opposite vertices are relatively prime while numbers on adjacent vertices are not relatively prime. What is the smallest possible value of the sum of these $4$ numbers?

Let any point $D$ be chosen on the side $BC$ of the triangle $ABC$. Let the radii of the incircles of the triangles $ABC, ABD$ and $ACD$ be $r_1, r_2$ and $r_3$. Prove that $r_1 <r_2 + r_3$.

Consider an acute-angled triangle $ABC$, with $AC>AB$, and let $\Gamma$ be its circumcircle. Let $E$ and $F$ be the midpoints of the sides $AC$ and $AB$, respectively. The circumcircle of the triangle $CEF$ and $\Gamma$ meet at $X$ and $C$, with $X\neq C$. The line $BX$ and the tangent to $\Gamma$ through $A$ meet at $Y$. Let $P$ be the point on segment $AB$ so that $YP = YA$, with $P\neq A$, and let $Q$ be the point where $AB$ and the parallel to $BC$ through $Y$ meet each other. Show that $F$ is the midpoint of $PQ$.

Find the number of distinct real roots of the following equation $x^2 +\frac{9x^2}{(x + 3)^2} = 40$. A. $0$ B. $1$ C. $2$ D. $3$ E. $4$

Let $ABC$ be an acute triangle with orthocentre $H$. Let $D$ be a point outside the circumcircle of triangle $ABC$ such that $\angle ABD=\angle DCA$. The reflection of $AB$ in $BD$ intersects $CD$ at $X$. The reflection of $AC$ in $CD$ intersects $BD$ at $Y$. The lines through $X$ and $Y$ perpendicular to $AC$ and $AB$, respectively, intersect at $P$. Prove that points $D$, $P$ and $H$ are collinear.

We call a graph symmetric, if we can put its vertices on the plane such that if the edges are segments, the graph has a reflectional symmetry with respect to a line not passing through its vertices. Find the least value of $K$ such that the edges of every graph with $100$ vertices, can be divided into $K$ symmetric subgraphs.

Amanda has the list of even numbers $2, 4, 6, \dots 100$ and Billy has the list of odd numbers $1, 3, 5, \dots 99$. Carlos creates a list by adding the square of each number in Amanda's list to the square of the corresponding number in Billy's list. Daisy creates a list by taking twice the product of corresponding numbers in Amanda's list and Billy's list. What is the positive difference between the sum of the numbers in Carlos's list and the sum of the numbers in Daisy's list? [i]2016 CCA Math Bonanza Individual #3[/i]

For any positive integer $n$, and $i=1,2$, let $f_{i}(n)$ denote the number of divisors of $n$ of the form $3 k+i$ (including $1$ and $n$ ). Define, for any positive integer $n$, $$f(n)=f_{1}(n)-f_{2}(n)$$ Find the value of $f\left(5^{2022}\right)$ and $f\left(21^{2022}\right)$.

Given is an acute angled triangle $ \triangle ABC$ with side lengths $ a$, $ b$ and $ c$ (in an usual way) and circumcenter $ O$. Angle bisector of angle $ \angle BAC$ intersects circumcircle at points $ A$ and $ A_{1}$. Let $ D$ be projection of point $ A_{1}$ onto line $ AB$, $ L$ and $ M$ be midpoints of $ AC$ and $ AB$ , respectively. (i) Prove that $ AD\equal{}\frac{1}{2}(b\plus{}c)$ (ii) If triangle $ \triangle ABC$ is an acute angled prove that $ A_{1}D\equal{}OM\plus{}OL$

Let $x,y,z$ be three distinct real positive numbers, Determine whether or not the three real numbers \[ \left| \frac{x}{y} - \frac{y}{x}\right| ,\left| \frac{y}{z} - \frac{z}{y}\right |, \left| \frac{z}{x} - \frac{x}{z}\right| \] can be the lengths of the sides of a triangle.

Consider a right-angled triangle whose legs are $1$ cm long. Suppose that each point of the triangle was assigned a color from the set of Brown, Blue, Green and Orange colors. It proves that, whatever way this was done, there is at least one pair of points of the same color at a distance equal to or greater than $2-\sqrt 2$ cm from each other.

Let $p, q$, and $r$ be the angles of a triangle, and let $a = \sin2p, b = \sin2q$, and $c = \sin2r$. If $s = \frac{(a + b + c)}2$, show that \[s(s - a)(s - b)(s -c) \geq 0.\] When does equality hold?

Let $ABCD$ be a convex quadrilateral such that $AC = BD$. Equilateral triangles are constructed on the sides of the quadrilateral. Let $O_1,O_2,O_3,O_4$ be the centers of the triangles constructed on $AB,BC,CD,DA$ respectively. Show that $O_1O_3$ is perpendicular to $O_2O_4.$

Find all the real numbers $x,y,z$ which satisfy the following conditions: $$ \begin{cases} 3(x^2+y^2+z^2)=1\\ x^2y^2+y^2z^2+z^2x^2=xyz(x+y+z)^3\\ \end{cases} $$

Compute $$\prod^{2019}_{i=1} (2^{2^i}- 2^{2^{i-1}} + 1).$$

If $a, b, c$ are real numbers greater than $1$, prove that for any real number $r$ \[(\log_{a}bc)^{r}+(\log_{b}ca)^{r}+(\log_{c}ab)^{r}\geq 3 \cdot 2^{r}. \]

Given a natural prime $ p, $ find the number of integer solutions of the equation $ p+xy=p(x+y). $

A soccer ball is usually made from a polyhedral fugure model, with two types of faces, hexagons and pentagons, and in every vertex incide three faces - two hexagons and one pentagon. We call a polyhedron [i]soccer-ball[/i] if it is similar to the traditional soccer ball, in the following sense: its faces are \(m\)-gons or \(n\)-gons, \(m \not= n\), and in every vertex incide three faces, two of them being \(m\)-gons and the other one being an \(n\)-gon. [list='i'] [*] Show that \(m\) needs to be even. [*] Find all soccer-ball polyhedra. [/list]

A frog is jumping between lattice points on the coordinate plane in the following way: On each jump, the frog randomly goes to one of the $8$ closest lattice points to it, such that the frog never goes in the same direction on consecutive jumps. If the frog starts at $(20, 19)$ and jumps to $(20, 20)$, then what is the expected value of the frog’s position after it jumps for an infinitely long time?

Let $n \geq 5$ be a positive integer. $a_1, b_1, a_2, b_2, \ldots, a_n, b_n$ are integers. $( a_i, b_i)$ are pairwisely distinct for $i = 1, 2, \ldots, n$, and $|a_1b_2 - a_2b_1| = |a_2b_3 -a_3b_2| = \cdots = |a_{n-1}b_n -a_nb_{n-1}| = 1$. Prove that there exists a pair of indexes $i, j$ satisfying $2 \leq |i - j| \leq n - 2$ and $|a_ib_j -a_jb_i| = 1.$

Let $s(m)$ denote the sum of the digits of the positive integer $m$. Find the largest positive integer that has no digits equal to zero and satisfies the equation \[2^{s(n)} = s(n^2).\]

In a wagon, every $m \geq 3$ people have exactly one common friend. (When $A$ is $B$'s friend, $B$ is also $A$'s friend. No one was considered as his own friend.) Find the number of friends of the person who has the most friends.

The pie charts below indicate the percent of students who prefer golf, bowling, or tennis at East Junior High School and West Middle School. The total number of students at East is $2000$ and at West, $2500$. In the two schools combined, the percent of students who prefer tennis is [asy] unitsize(18); draw(circle((0,0),4)); draw(circle((9,0),4)); draw((-4,0)--(0,0)--4*dir(352.8)); draw((0,0)--4*dir(100.8)); draw((5,0)--(9,0)--(4*dir(324)+(9,0))); draw((9,0)--(4*dir(50.4)+(9,0))); label("$48\%$",(0,-1),S); label("bowling",(0,-2),S); label("$30\%$",(1.5,1.5),N); label("golf",(1.5,0.5),N); label("$22\%$",(-2,1.5),N); label("tennis",(-2,0.5),N); label("$40\%$",(8.5,-1),S); label("tennis",(8.5,-2),S); label("$24\%$",(10.5,0.5),E); label("golf",(10.5,-0.5),E); label("$36\%$",(7.8,1.7),N); label("bowling",(7.8,0.7),N); label("$\textbf{East JHS}$",(0,-4),S); label("$\textbf{2000 students}$",(0,-5),S); label("$\textbf{West MS}$",(9,-4),S); label("$\textbf{2500 students}$",(9,-5),S); [/asy] $\text{(A)}\ 30\% \qquad \text{(B)}\ 31\% \qquad \text{(C)}\ 32\% \qquad \text{(D)}\ 33\% \qquad \text{(E)}\ 34\%$

Find all polynomials with real coefficients, for which the equality \[ P(2P(x)) \equal{} 2P(P(x)) \plus{} 2(P(x))^{2}\] holds for any real number $ x$.