Let a,b, and c be positive reals. Prove: $\left(\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\right)^{2}\ge (a+b+c)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)$

Find, with proof, all triples of real numbers $(a, b, c)$ such that all four roots of the polynomial $f(x) = x^4 +ax^3 +bx^2 +cx+b$ are positive integers. (The four roots need not be distinct.)

Find all positive integers $n$ such that: $$n = a^2 + b^2 + c^2 + d^2,$$ where $a < b < c < d$ are the smallest divisors of $n$.

Let $A$ be a set with at least $3$ integers, and let $M$ be the maximum element in $A$ and $m$ the minimum element in $A$. it is known that there exist a polynomial $P$ such that: $m<P(a)<M$ for all $a$ in $A$. And also $p(m)<p(a)$ for all $a$ in $A-(m,M)$. Prove that $n<6$ and there exist integers $b$ and $c$ such that $p(x)+x^2+bx+c$ is cero in $A$.

Prove that: \[\det(A)=\frac{1}{n!}\left| \begin{array}{llllll}\mbox{tr}(A) & 1 & 0 & \ldots & \ldots & 0 \\ \mbox{tr}(A^{2}) & \mbox{tr}(A) & 2 & 0 & \ldots & 0 \\ \mbox{tr}(A^{3}) & \mbox{tr}(A^{2}) & \mbox{tr}(A) & 3 & & \vdots \\ \vdots & & & & & n-1 \\ \mbox{tr}(A^{n}) & \mbox{tr}(A^{n-1}) & \mbox{tr}(A^{n-2}) & \ldots & \ldots & \mbox{tr}(A) \end{array}\right|\]

Let $ABC$ be an isosceles triangle with $AB = AC$. Let $D$ and $E$ be the midpoints of segments $AB$ and $AC$, respectively. Suppose that there exists a point$ F$ on ray $\overrightarrow{DE}$ outside of $ABC$ such that triangle $BFA$ is similar to triangle $ABC$. Compute $\frac{AB}{BC}$

Let $n,m$ be positive integers. Let $A_1,A_2,A_3, ... ,A_m$ be sets such that $A_i \subseteq \{1, 2, 3, . . . , n\}$ and $|A_i| = 3$ for all $i$ (i.e. $A_i$ consists of three different positive integers each at most $n$). Suppose for all $i < j$ we have $|A_i \cap A_j | \le 1$ (i.e. $A_i$ and $A_j$ have at most one element in common). (a) Prove that $m \le \frac{n(n-1)}{ 6}$ . (b) Show that for all $n \ge3$ it is possible to have $m \ge \frac{(n-1)(n-2)}{ 6}$ .

Let $ABC$ be a triangle. Construct three circles $k_1$, $k_2$, and $k_3$ with the same radius such that they intersect each other at a common point $O$ inside the triangle $ABC$ and $k_1\cap k_2=\{A,O\}$, $k_2 \cap k_3=\{B,O\}$, $k_3\cap k_1=\{C,O\}$. Let $t_a$ be a common tangent of circles $k_1$ and $k_2$ such that $A$ is closer to $t_a$ than $O$. Define $t_b$ and $t_c$ similarly. Those three tangents determine a triangle $MNP$ such that the triangle $ABC$ is inside the triangle $MNP$. Prove that the area of $MNP$ is at least $9$ times the area of $ABC$.

If $5$ times a number is $2$, then $100$ times the reciprocal of the number is $\text{(A)}\ 2.5 \qquad \text{(B)}\ 40 \qquad \text{(C)}\ 50 \qquad \text{(D)}\ 250 \qquad \text{(E)}\ 500$

$ ABCD$ is a quadrilateral. $ A'BCD'$ is the reflection of $ ABCD$ in $ BC,$ $ A''B'CD'$ is the reflection of $ A'BCD'$ in $ CD'$ and $ A''B''C'D'$ is the reflection of $ A''B'CD'$ in $ D'A''.$ Show that; if the lines $ AA''$ and $ BB''$ are parallel, then ABCD is a cyclic quadrilateral.

Let $\triangle ABC$ a triangle, and $D$ the intersection of the angle bisector of $\angle BAC$ and the perpendicular bisector of $AC$. the line parallel to $AC$ passing by the point $B$, intersect the line $AD$ at $X$. the line parallel to $CX$ passing by the point $B$, intersect $AC$ at $Y$. $E = (AYB) \cap BX$ . prove that $C$ , $D$ and $E$ collinear.

In $\vartriangle ABC$, $AB = 13$, $BC = 14$, and $AC = 15$. Draw the circumcircle of $\vartriangle ABC$, and suppose that the circumcircle has center $O$. Extend $AO$ past $O$ to a point $D$, $BO$ past $O$ to a point $E$, and $CO$ past $O$ to a point $F$ such that $D, E, F$ also lie on the circumcircle. Compute the area of the hexagon $AF BDCE$.

The diagram below shows a triangle divided into sections by three horizontal lines which divide the altitude of the triangle into four equal parts, and three lines connecting the top vertex with points that divide the opposite side into four equal parts. If the shaded region has area $100$, find the area of the entire triangle. [asy] import graph; size(5cm); pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); pen dotstyle = black; filldraw((-1,2.5)--(-1,1.75)--(0.5,1.75)--(0,2.5)--cycle,grey); draw((-1,4)--(-2,1)); draw((-1,4)--(2,1)); draw((-2,1)--(2,1)); draw((-1,4)--(-1,1)); draw((-1,4)--(-0.5,2.5)); draw((-0.25,1.75)--(0,1)); draw((-1,2.5)--(-1,1.75)); draw((-1,1.75)--(0.5,1.75)); draw((0.5,1.75)--(0,2.5)); draw((0,2.5)--(-1,2.5)); draw((-1.25,3.25)--(-0.25,3.25)); draw((-1.5,2.5)--(0.5,2.5)); draw((1.25,1.75)--(-1.75,1.75)); draw((-1,4)--(0,2.5)); draw((0.47,1.79)--(1,1)); dot((-1,1),dotstyle); dot((0,1),dotstyle); dot((1,1),dotstyle); dot((-1.25,3.25),dotstyle); dot((-1.5,2.5),dotstyle); dot((-1.75,1.75),dotstyle); dot((1.25,1.75),dotstyle); dot((0.5,2.5),dotstyle); dot((-0.25,3.25),dotstyle); [/asy]

For each positive integer $n$ determine the maximum number of points in space creating the set $A$ which has the following properties: $1)$ the coordinates of every point from the set $A$ are integers from the range $[0, n]$ $2)$ for each pair of different points $(x_1,x_2,x_3), (y_1,y_2,y_3)$ belonging to the set $A$ it is satisfied at least one of the following inequalities $x_1< y_1, x_2<y_2, x_3<y_3$ and at least one of the following inequalities $x_1>y_1, x_2>y_2,x_3>y_3$.

Let $O$ be the circumcenter and $I$ be the incenter of $\vartriangle$, $P$ is the reflection from $I$ through $O$, the foot of perpendicular from $P$ to $BC,CA,AB$ is $X,Y,Z$, respectively. Prove that $AP^2+PX^2=BP^2+PY^2=CP^2+PZ^2$.

Let $(x_j,y_j)$, $1\le j\le 2n$, be $2n$ points on the half-circle in the upper half-plane. Suppose $\sum_{j=1}^{2n}x_j$ is an odd integer. Prove that $\displaystyle{\sum_{j=1}^{2n}y_j \ge 1}$.

Let $ABC$ be a triangle with $AC = 28$, $BC = 33$, and $\angle ABC = 2\angle ACB$. Compute the length of side $AB$. [i]2016 CCA Math Bonanza #10[/i]

Given two distinct points $A,B$ in the plane, for each point $C$ not on the line $AB$, we denote by $G$ and $I$ the centroid and incenter of the triangle $ABC$, respectively. (a) For $0<\alpha<\pi$, let $\Gamma$ be the set of points $C$ in the plane such that $\angle\left(\overrightarrow{CA},\overrightarrow{CB}\right)=\alpha+2k\pi$ as an oriented angle, where $k\in\mathbb Z$. If $C$ describes $\Gamma$, show that points $G$ and $I$ also descibre arcs of circles, and determine these circles. (b) Suppose that in addition $\frac\pi3<\alpha<\pi$. For which positions of $C$ in $\Gamma$ is $GI$ minimal? (c) Let $f(\alpha)$ denote the minimal $GI$ from the part (b). Give $f(\alpha)$ explicitly in terms of $a=AB$ and $\alpha$. Find the minimum value of $f(\alpha)$ for $\alpha\in\left(\frac\pi3,\pi\right)$.

$240$ students are participating a big performance show. They stand in a row and face to their coach. The coach askes them to count numbers from left to right, starting from $1$. (Of course their counts be like $1,2,3,...$)The coach askes them to remember their number and do the following action: First, if your number is divisible by $3$ then turn around. Then, if your number is divisible by $5$ then turn around. Finally, if your number is divisible by $7$ then turn around. (a) How many students are face to coach now? (b) What is the number of the $66^{\text{th}}$ student counting from left who is face to coach?

Let $ f(x)$ be a nonnegative, integrable function on $ (0,2\pi)$ whose Fourier series is $ f(x)\equal{}a_0\plus{}\sum_{k\equal{}1}^{\infty} a_k \cos (n_k x)$, where none of the positive integers $ n_k$ divides another. Prove that $ |a_k| \leq a_0$. [i]G. Halasz[/i]

A candy company makes $5$ colors of jellybeans, which come in equal proportions. If I grab a random sample of $5$ jellybeans, what is the probability that I get exactly $2$ distinct colors?

Let $[z]$ denote the greatest integer not exceeding $z$. Let $x$ and $y$ satisfy the simultaneous equations \[ \begin{array}{c} y=2[x]+3, \\ y=3[x-2]+5. \end{array} \]If $x$ is not an integer, then $x+y$ is $\textbf {(A) } \text{an integer} \qquad \textbf {(B) } \text{between 4 and 5} \qquad \textbf {(C) } \text{between -4 and 4} \qquad \textbf {(D) } \text{between 15 and 16} \qquad \textbf {(E) } 16.5$

A cube is cut by a plane such that the cross section is a pentagon. Show there is a side of the pentagon of length $\ell$ such that the inequality holds: \[ |\ell-1|>\frac{1}{5} \]

Eric has assembled a convex polygon $P$ from finitely many centrally symmetric (not necessarily congruent or convex) polygonal tiles. Prove that $P$ is centrally symmetric. [i]Proposed by Josef Tkadlec - Czech Republic[/i]

Exactly $4n$ numbers in set $A= \{ 1,2,3,...,6n \} $ of natural numbers painted in red, all other in blue. Proved that exist $3n$ consecutive natural numbers from $A$, exactly $2n$ of which numbers is red.