In the overlapping triangles $ \triangle{ABC}$ and $ \triangle{ABE}$ sharing common side $ AB$, $ \angle{EAB}$ and $ \angle{ABC}$ are right angles, $ AB \equal{} 4$, $ BC \equal{} 6$, $ AE \equal{} 8$, and $ \overline{AC}$ and $ \overline{BE}$ intersect at $ D$. What is the difference between the areas of $ \triangle{ADE}$ and $ \triangle{BDC}$? [asy] defaultpen(linewidth(0.8)+fontsize(10));size(200); unitsize(5mm) ; pair A=(0,0), B=(4,0), C=(4,6), D=(0,8), H=intersectionpoint(C--A, D--B); draw(A--B--C--cycle) ; draw(A--B--D--cycle) ; label("E",(0,8), N) ; label("8",(0,4),W) ; label("A",A,S) ; label("B",B,SE) ; label("C",C,NE) ; label("6",(4,3),E) ; label("4",(2,0),S) ; label("D",H,2*dir(85)) ;[/asy] $ \textbf{(A)}\ 2 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 8 \qquad \textbf{(E)}\ 9$

200 students participated in a math contest. They had 6 problems to solve. Each problem was correctly solved by at least 120 participants. Prove that there must be 2 participants such that every problem was solved by at least one of these two students.

Prove that $\dfrac {x^2+1}{(x+y)^2+4 (z+1)}+\dfrac {y^2+1}{(y+z)^2+4 (x+1)}+\dfrac {z^2+1}{(z+x)^2+4 (y+1)} \ge \dfrac{1}{2} $ for all positive reals $x,y,z$

Let $\omega=e^{2\pi i/20}$ and let $S$ be the set $\{1, \omega, \ldots, \omega^{19}\}.$ How many subsets of $S$ sum to $0$? Include both $S$ and the empty set in your count.

A sequence $\{a_n\}$ is defined such that $a_i=i$ for $i=1,2,3\ldots,2020$ and for $i>2020$, $a_i$ is the average of the previous $2020$ terms. What is the largest integer less than or equal to $\displaystyle\lim_{n\to\infty}a_n$? [i]2020 CCA Math Bonanza Lightning Round #4.4[/i]

Let $ a,b,c\in \mathbb R$ such that $ a\ge b\ge c > 1$. Prove the inequality: $ \log_c\log_c b \plus{} \log_b\log_b a \plus{} \log_a\log_a c\geq 0$

Given an $8\times 8$ chess board, in how many ways can we select $56$ squares on the board while satisfying both of the following requirements: (a) All black squares are selected. (b) Exactly seven squares are selected in each column and in each row.

Compute $$\sum_{n=0}^\infty\frac{n+1}{2^n}.$$

Evan has $66000$ omons, particles that can cluster into groups of a perfect square number of omons. An omon in a cluster of $n^2$ omons has a potential energy of $\frac1n$. Evan accurately computes the sum of the potential energies of all the omons. Compute the smallest possible value of his result. [i]Proposed by Michael Ren and Luke Robitaille[/i]

Given the polynomials $P(x)=px^4+qx^3+rx^2+sx+t,\ Q(x)=\frac{d}{dx}P(x)$, find the real numbers $p,\ q,\ r,\ s,\ t$ such that $P(\sqrt{-5})=0,\ Q(\sqrt{-2})=0$ and $\int_0^1 P(x)dx=-\frac{52}{5}.$

Two is $ 10 \%$ of $ x$ and $ 20 \%$ of $ y$. What is $ x \minus{} y$? $ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 10 \qquad \textbf{(E)}\ 20$

Let $\omega$ be a circle and let $A,B,C,D,E$ be five points on $\omega$ in this order. Define $F=BC\cap DE$, such that the points $F$ and $A$ are on opposite sides, with regard to the line $BE$ and the line $AE$ is tangent to the circumcircle of the triangle $BFE$. a) Prove that the lines $AC$ and $DE$ are parallel b) Prove that $AE=CD$

Find the number of triples $(x,a,b)$ where $x$ is a real number and $a,b$ belong to the set $\{1,2,3,4,5,6,7,8,9\}$ such that $$x^2-a\{x\}+b=0.$$ where $\{x\}$ denotes the fractional part of the real number $x$.

On a circle are given the points $A_1, B_1, A_2, B_2, \cdots, A_9, B_9$ in this order. All the segments $A_iB_j (i, j=1, 2, \cdots, 9$ must be colored in one of $k$ colors, so that no two segments from the same color intersect (inside the circle) and for every $i$ there is a color, such that no segments with an end $A_i$, nor $B_i$ is colored such. What is the least possible $k$?

For how many ordered pairs of positive integers $(a,b)$ with $a,b<1000$ is it true that $a$ times $b$ is equal to $b^2$ divided by $a$? For example, $3$ times $9$ is equal to $9^2$ divided by $3$. [i]Ray Li[/i]

Find all functions $f$ from the real numbers to the real numbers such that $f(xy) \le \frac12 \left(f(x) + f(y) \right)$ for all real numbers $x$ and $y$.

Let $ m$, $ n$ be natural numbers such that $ m\plus{}n\plus{}1$ is prime and divides $ 2(m^2\plus{}n^2)\minus{}1$. Prove that $ m\equal{}n$.

A set of $n$ people participate in an online video basketball tournament. Each person may be a member of any number of $5$-player teams, but no two teams may have exactly the same $5$ members. The site statistics show a curious fact: The average, over all subsets of size $9$ of the set of $n$ participants, of the number of complete teams whose members are among those $9$ people is equal to the reciprocal of the average, over all subsets of size $8$ of the set of $n$ participants, of the number of complete teams whose members are among those $8$ people. How many values $n$, $9\leq n\leq 2017$, can be the number of participants? $\textbf{(A) } 477 \qquad \textbf{(B) } 482 \qquad \textbf{(C) } 487 \qquad \textbf{(D) } 557 \qquad \textbf{(E) } 562$

A circle $\omega_1$ with centre $O_1$ passes through the centre $O_2$ of a second circle $\omega_2$. The tangent lines to $\omega_2$ from a point $C$ on $\omega_1$ intersect $\omega_1$ again at points $A$ and $B$ respectively. Prove that $AB$ is perpendicular to $O_1O_2$. [i](5 points)[/i]

Determine the least real number $a > 1$ such that for any point $P$ in the interior of a square $ABCD$, the ratio of the areas of some two triangle $PAB, PBC, PCD, PDA$ lies in the interval $[1/a, a]$.

For how many three-digit whole numbers does the sum of the digits equal $25$? $\text{(A)}\ 2 \qquad \text{(B)}\ 4 \qquad \text{(C)}\ 6 \qquad \text{(D)}\ 8 \qquad \text{(E)}\ 10$

$(HUN 2)$ Prove that $1+\frac{1}{2^3}+\frac{1}{3^3}+\cdots+\frac{1}{n^3}<\frac{5}{4}$

Let there be $2004$ be bicolor tiles, white on one side and black on the other, placed in a circle. A move consists of choosing a black piece and turning over three pieces: the chosen one, the one on its left and the one on its right. If at the beginning there is only one black piece, will it be possible, repeating the movement described, to make all the pieces have the white face up?

Alex and Jack have $1000$ sheets each. Each of them writes the numbers $1,\ldots,2000$ on his sheets in an arbitrary order, with one number on each side of a sheet. The sheets are to be placed on the floor so that one side of each sheet is visible. Prove that they can do so in such a way that each of the numbers from $1$ to $2000$ is visible.

We are given an arbitrary $\bigtriangleup ABC$. a) Can we dissect $\bigtriangleup ABC$ in $4$ pieces, from which we can make two triangle similar to $\bigtriangleup ABC$ (each piece can be used only once)? Justify your answer! b) Is it possible that for every positive integer $n \ge 2$ , we are able to dissect $\bigtriangleup ABC$ in $2n$ pieces, from which we can make two triangles similar to $\bigtriangleup ABC$ (each piece can be used only once)? Justify your answer!