For $n \ge 3$, let $S=a_1+a_2+\cdots+a_n$ and $T=b_1b_2\cdots b_n$ for positive real numbers $a_1,a_2,\ldots,a_n, b_1,b_2 ,\ldots,b_n$, where the numbers $b_i$ are pairwise distinct. (a) Find the number of distinct real zeroes of the polynomial \[f(x)=(x-b_1)(x-b_2)\cdots(x-b_n)\sum_{j=1}^n \frac{a_j}{x-b_j}\] (b) Prove the inequality \[\frac1{n-1}\sum_{j=1}^n \left(1-\frac{a_j}{S}\right)b_j > \left(\frac{T}{S}\sum_{j=1}^{n} \frac{a_j}{b_j}\right)^{\frac1{n-1}}\]

Prove that there exist infinitely many collections of positive integers $ (a,b,c,d,e,f)$ such that $ a < b < c$ and the equalities $ ab \minus{} c \equal{} de,$ $ bc \minus{} a \equal{} ef$ and $ ac \minus{} b \equal{} df$ hold.

Points $A,B,C$ with $AB = BC$ are given on a circle with radius $r$, and $D$ is a point inside the circle such that the triangle $BCD$ is equilateral. The line $AD$ meets the circle again at $E$. Show that $DE = r$.

$200 \times 200$ square is colored in chess order. In one move we can take every $2 \times 3$ rectangle and change color of all its cells. Can we make all cells of square in same color ?

Let $ABCD$ be a square of side length $10.$ Point $E$ is on ray $\overrightarrow{AB}$ such that $AE=17,$ and point $F$ is on ray $\overrightarrow{AD}$ such that $AF=14.$ The line through $B$ parallel to $CE$ and the line through $D$ parallel to $CF$ meet at $P.$ Compute the area of quadrilateral $AEPF.$

In a certain land, all Arogs are Brafs, all Crups are Brafs, all Dramps are Arogs, and all Crups are Dramps. Which of the following statements is implied by these facts? $ \textbf{(A)}\ \text{All Dramps are Brafs and are Crups.}\qquad \\ \textbf{(B)}\ \text{All Brafs are Crups and are Dramps.}\qquad \\ \textbf{(C)}\ \text{All Arogs are Crups and are Dramps.}\qquad \\ \textbf{(D)}\ \text{All Crups are Arogs and are Brafs.}\qquad \\ \textbf{(E)}\ \text{All Arogs are Dramps and some Arogs may not be Crups.}$

Given the equation \[ x^4 \plus{} px^3 \plus{} qx^2 \plus{} rx \plus{} s \equal{} 0\] has four real, positive roots, prove that (a) $ pr \minus{} 16s \geq 0$ (b) $ q^2 \minus{} 36s \geq 0$ with equality in each case holding if and only if the four roots are equal.

Solve in $ \mathbb{Z} $ the system: $$ \left\{ \begin{matrix} 2^x+\log_3 x=y^2 \\ 2^y+\log_3 y=x^2 \end{matrix} \right. . $$

Inside the triangle $ABC$ a point $M$ is given . Find the points $P,Q$ and $R$ lying on the sides $AB,BC$ and $AC$ respectively and such so that the sum $MP+PQ+QR+RM$ is the smallest.

Find all family $\mathcal{F}$ of subsets of $[n]$ such that for any nonempty subset $X\subseteq [n]$, exactly half of the elements $A\in \mathcal{F}$ satisfies that $|A\cap X|$ is even.

Find all triples $ (x,y,z)$ of positive real numbers which satisfy $ 2x^3 \equal{} 2y(x^2 \plus{} 1) \minus{} (z^2 \plus{} 1)$; $ 2y^4 \equal{} 3z(y^2 \plus{} 1) \minus{} 2(x^2 \plus{} 1)$; $ 2z^5 \equal{} 4x(z^2 \plus{} 1) \minus{} 3(y^2 \plus{} 1)$.

Circle $w$ with center $O$ meets circle $\Gamma$ at $X,Y,$ and $O$ is on $\Gamma$. Point $Z\in\Gamma$ lies outside $w$ such that $XZ=11$, $OZ=15$, and $YZ=13$. If the radius of circle $w$ is $r$, find $r^2$.

Prove that for any positive number $a,b,c$ and any nonnegative integer $p$ $$a^{p+2}+b^{p+2}+c^{p+2}\ge a^pbc+b^pca+c^pab.$$

Given a regular tetrahedron $ABCD$ with edge length $1$ and a point $P$ inside it. What is the maximum value of $\left|PA\right|+\left|PB\right|+\left|PC\right|+\left|PD\right|$.

You are given a pair of triangles for which two sides of one triangle are equal in length to two sides of the second triangle, and the triangles are similar, but not necessarily congruent. Prove that the ratio of the sides that correspond under the similarity is a number between $ \frac {1}{2}(\sqrt {5} \minus{} 1)$ and $ \frac {1}{2}(\sqrt {5} \plus{} 1)$.

$99$ identical balls lie on a table. $50$ balls are made of copper, and $49$ balls are made of zinc. The assistant numbered the balls. Once spectrometer test is applied to $2$ balls and allows to determine whether they are made of the same metal or not. However, the results of the test can be obtained only the next day. What minimum number of tests is required to determine the material of each ball if all the tests should be performed today? [i]Proposed by N. Vlasova, S. Berlov[/i]

On a circle, points $A,B,C,D$ lie counterclockwise in this order. Let the orthocenters of $ABC,BCD,CDA,DAB$ be $H,I,J,K$ respectively. Let $HI=2$, $IJ=3$, $JK=4$, $KH=5$. Find the value of $13(BD)^2$.

Prove that every nonzero coefficient of the Taylor series of $(1-x+x^2)e^x$ about $x=0$ is a rational number whose numerator (in lowest terms) is either $1$ or a prime number.

If the equation: $f(\frac{x-3}{x+1}) + f(\frac{3+x}{1-x}) = x$ holds true for all real x but $\pm 1$, find $f(x)$.

Let $ R $ be the circumradius of a triangle $ ABC. $ The points $ B,C, $ lie on a circle of radius $ \rho $ that intersects $ AB,AC $ at $ E,D, $ respectively. $ \rho' $ is the circumradius of $ ADE. $ Show that there exists a triangle with sides $ R,\rho ,\rho' , $ and having an angle whose value doesn't depend on $ \rho . $ [i]Laurențiu Panaitopol[/i]

Determine all possible values of $a_1$ for which there exists a sequence $a_1, a_2, \dots$ of rational numbers satisfying $$a_{n+1}^2-a_{n+1}=a_n$$ for all positive integers $n$.

A tournament is a directed graph for which every (unordered) pair of vertices has a single directed edge from one vertex to the other. Let us define a proper directed-edge-coloring to be an assignment of a color to every (directed) edge, so that for every pair of directed edges $\overrightarrow{uv}$ and $\overrightarrow{vw}$, those two edges are in different colors. Note that it is permissible for $\overrightarrow{uv}$ and $\overrightarrow{uw}$ to be the same color. The directed-edge-chromatic-number of a tournament is defined to be the minimum total number of colors that can be used in order to create a proper directed-edge-coloring. For each $n$, determine the minimum directed-edge-chromatic-number over all tournaments on $n$ vertices. [i]Proposed by Po-Shen Loh[/i]

Find the number of perfect squares of five digits whose last two digits are equal. [i]Gheorghe Iurea[/i]

Four cars participate in a rally on a circular racecourse. They start simultaneously from the same point and go with a constant (but different) speeds. It is known that any three of them meet at some point. Prove that all four of them will meet again at some point.

How many positive integers $n\leq100$ satisfy $\left\lfloor n\pi\right\rfloor=\left\lfloor\left(n-1\right)\pi\right\rfloor+3$? Here $\left\lfloor x\right\rfloor$ is the greatest integer less than or equal to $x$; for example, $\left\lfloor\pi\right\rfloor=3$. [i]2018 CCA Math Bonanza Lightning Round #3.2[/i]