Prove that \[\frac{1995}{2}-\frac{1994}{3}+\frac{1993}{4}-\ldots -\frac{2}{1995}+\frac{1}{1996}=\frac{1}{999}+\frac{3}{1000}+\ldots +\frac{1995}{1996}\]

Let $ ABCD $ be a rectangle that has $ M $ on its $ BD $ diagonal. If $ N,P $ are the projections of $ M $ on $ AB, $ respectively, $ AD, $ what's the locus of the intersection between $ CP $ and $ DN? $

There is a polynomial $P(x)$ with integer coefficients such that $$P(x)=\frac{(x^{2310}-1)^6}{(x^{105}-1)(x^{70}-1)(x^{42}-1)(x^{30}-1)}$$ holds for every $0<x<1.$ Find the coefficient of $x^{2022}$ in $P(x)$

Brandon wants to split his orchestra of $20$ violins, $15$ violas, $10$ cellos, and $5$ basses into three distinguishable groups, where all of the players of each instrument are indistinguishable. He wants each group to have at least one of each instrument and for each group to have more violins than violas, more violas than cellos, and more cellos than basses. How many ways are there for Brandon to split his orchestra following these conditions?

A non-Hookian spring has force $F = -kx^2$ where $k$ is the spring constant and $x$ is the displacement from its unstretched position. For the system shown of a mass $m$ connected to an unstretched spring initially at rest, how far does the spring extend before the system momentarily comes to rest? Assume that all surfaces are frictionless and that the pulley is frictionless as well. [asy] size(250); pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); draw((0,0)--(0,-1)--(2,-1)--(2+sqrt(3),-2)); draw((2.5,-2)--(4.5,-2),dashed); draw(circle((2.2,-0.8),0.2)); draw((2.2,-0.8)--(1.8,-1.2)); draw((0,-0.6)--(0.6,-0.6)--(0.75,-0.4)--(0.9,-0.8)--(1.05,-0.4)--(1.2,-0.8)--(1.35,-0.4)--(1.5,-0.8)--(1.65,-0.4)--(1.8,-0.8)--(1.95,-0.6)--(2.2,-0.6)); draw((2+0.3*sqrt(3),-1.3)--(2+0.3*sqrt(3)+0.6/2,-1.3+sqrt(3)*0.6/2)--(2+0.3*sqrt(3)+0.6/2+0.2*sqrt(3),-1.3+sqrt(3)*0.6/2-0.2)--(2+0.3*sqrt(3)+0.2*sqrt(3),-1.3-0.2)); //super complex Asymptote code gg draw((2+0.3*sqrt(3)+0.3/2,-1.3+sqrt(3)*0.3/2)--(2.35,-0.6677)); draw(anglemark((2,-1),(2+sqrt(3),-2),(2.5,-2))); label("$30^\circ$",(3.5,-2),NW); [/asy] $ \textbf{(A)}\ \left(\frac{3mg}{2k}\right)^{1/2} $ $ \textbf{(B)}\ \left(\frac{mg}{k}\right)^{1/2} $ $ \textbf{(C)}\ \left(\frac{2mg}{k}\right)^{1/2} $ $ \textbf{(D)}\ \left(\frac{\sqrt{3}mg}{k}\right)^{1/3} $ $ \textbf{(E)}\ \left(\frac{3\sqrt{3}mg}{2k}\right)^{1/3} $

Show that there exist infinitely many composite positive integers $n$ such that $n$ divides $2^{\frac{n-1}{2}}+1$

Let $\gamma$ be the circumcircle of the acute triangle $ABC$. Let $P$ be the midpoint of the minor arc $BC$. The parallel to $AB$ through $P$ cuts $BC, AC$ and $\gamma$ at points $R,S$ and $T$, respectively. Let $K \equiv AP \cap BT$ and $L \equiv BS \cap AR$. Show that $KL$ passes through the midpoint of $AB$ if and only if $CS = PR$.

Peter and Paul play the following game. First, Peter chooses some positive integer $a$ with the sum of its digits equal to $2012$. Paul wants to determine this number, he knows only that the sum of the digits of Peter’s number is $2012$. On each of his moves Paul chooses a positive integer $x$ and Peter tells him the sum of the digits of $|x - a|$. What is the minimal number of moves in which Paul can determine Peter’s number for sure?

In order to estimate the value of $x-y$ where $x$ and $y$ are real numbers with $x>y>0$, Xiaoli rounded $x$ up by a small amount, rounded $y$ down by the same amount, and then subtracted her values. Which of the following statements is necessarily correct? $ \textbf{(A)}\ \text{Her estimate is larger than }x-y\\ \textbf{(B)}\ \text{Her estimate is smaller than }x-y\\ \textbf{(C)}\ \text{Her estimate equals }x-y\\ \textbf{(D)}\ \text{Her estimate equals }y-x\\ \textbf{(E)}\ \text{Her estimate is }0 $

Find all positive integers $n$ for which there exist positive integers $x_1, x_2, \dots, x_n$ such that $$ \frac{1}{x_1^2}+\frac{2}{x_2^2}+\frac{2^2}{x_3^2}+\cdots +\frac{2^{n-1}}{x_n^2}=1.$$

Determine the greatest positive integer $k$ that satisfies the following property: The set of positive integers can be partitioned into $k$ subsets $A_1, A_2, \ldots, A_k$ such that for all integers $n \geq 15$ and all $i \in \{1, 2, \ldots, k\}$ there exist two distinct elements of $A_i$ whose sum is $n.$ [i]Proposed by Igor Voronovich, Belarus[/i]

$\Delta ABC$ is such that $AC+BC=2$ and the sum of its altitude through $C$ and its base $AB$ is $CD+AB=\sqrt{5}$. Find the sides of the triangle.

Let $ABCD$ be a rectangle. Two perpendicular lines pass through point $B$. One of them meets segment $AD$ at point $K$, and the second one meets the extension of side $CD$ at point $L$. Let $F$ be the common point of $KL$ and $AC$. Prove that $BF\perp KL$.

Two straight lines are drawn on a plane, intersecting at an angle of $40^o$. A circle with center at point $O$ touches these lines. Let's consider a line, different from the given ones, tangent to the same circle and intersecting the given lines at points $B$ and $C$. What can the angle $\angle BOC$ be equal to?

The product of several distinct positive integers is divisible by ${2006}^{2}$. Determine the minimum value the sum of such numbers can take.

How many non-overlapping $2$ by $2$ squares will fit into a circle with radius $8$?

You have to divide a square paper into three parts, by two straight cuts, so that by locating these parts properly, without gaps or overlaps, an obtuse triangle is formed. Indicate how to cut the square and how to assemble the triangle with the three parts.

Given two positive integers $r > s$, and let $F$ be an infinite family of sets, each of size $r$, no two of which share fewer than $s$ elements. Prove that there exists a set of size $r -1$ that shares at least $s$ elements with each set in $F$.

Let $a,b,c,d$ be positive real numbers satisfying $a^2+b^2+c^2+d^2=1$. Prove that \[a^3+b^3+c^3+d^3+abcd\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}\right)\leqslant 1.\]

Let $p$ and $q$ be two given positive integers. A set of $p+q$ real numbers $a_1<a_2<\cdots <a_{p+q}$ is said to be balanced iff $a_1,\ldots,a_p$ were an arithmetic progression with common difference $q$ and $a_p,\ldots,a_{p+q}$ where an arithmetic progression with common difference $p$. Find the maximum possible number of balanced sets, so that any two of them have nonempty intersection. Comment: The intended problem also had "$p$ and $q$ are coprime" in the hypothesis. A typo when the problems where written made it appear like that in the exam (as if it were the only typo in the olympiad). Fortunately, the problem can be solved even if we didn't suppose that and it can be further generalized: we may suppose that a balanced set has $m+n$ reals $a_1<\cdots <a_{m+n-1}$ so that $a_1,\ldots,a_m$ is an arithmetic progression with common difference $p$ and $a_m,\ldots,a_{m+n-1}$ is an arithmetic progression with common difference $q$.

Consider a grid of black and white squares with $3$ rows and $n$ columns. If there is a non-empty sequence of white squares $s_1$, $...$, $s_m$ such that $s_1$ is in the top row and $s_m$ is in the bottom row and consecutive squares in the sequence share an edge, then we say that the grid percolates. Let $T_n$ be the number of grids which do not percolate. There exists constants a, b such that $\frac{T_n}{ab^n}$ approaches $ 1$ as $n$ approaches $\infty$. Then $b$ is expressible as $(x+ \sqrt{y})/z$ for squarefree $y$ and coprime $x, z$. Find $x + y + z$.

Zadam Heng bets Saniel Dun that he can win in a free throw contest. Zadam shoots until he has made $5$ shots. He wins if this takes him $9$ or fewer attempts. The probability that Zadam makes any given attempt is $\frac{1}{2}$. What is the probability that Zadam Heng wins the bet? [i]2018 CCA Math Bonanza Individual Round #4[/i]

On the sides of the right triangle, outside are constructed regular nonagons, which are constructed on one of the catheti and on the hypotenuse, with areas equal to $1602$ $cm^2$ and $2019$ $cm^2$, respectively. What is the area of the nonagon that is constructed on the other cathetus of this triangle? (Vladislav Kirilyuk)

Which atom has the highest electronegativity? ${ \textbf{(A)}\ \text{Na}\qquad\textbf{(B)}\ \text{P}\qquad\textbf{(C)}\ \text{Cl}\qquad\textbf{(D)}}\ \text{Br}\qquad $

Let $a,b,c$ be real numbers that satisfy $\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{a+c} =1$ and $ab+bc+ac >0$. Show that \[ a+b+c - \frac{abc}{ab+bc+ac} \ge 4 \]