Ms. Blackwell gives an exam to two classes. The mean of the scores of the students in the morning class is $84$, and the afternoon class’s mean score is $70$. The ratio of the number of students in the morning class to the number of students in the afternoon class is $\frac{3}{4}$. What is the mean of the scores of all the students? $\textbf{(A) }74 \qquad \textbf{(B) }75 \qquad \textbf{(C) }76 \qquad \textbf{(D) }77 \qquad \textbf{(E) }78$

For a positive integer $m$, let $f(m)$ denote the smallest power of $2024$ not less than $m$ (e.g. $f(1)=1, f(2023)=f(2024)=2024,$ and $f(2025)=2024^2$). Find all positive real numbers $c$ for which there exists a sequence $x_1,x_2,\cdots$ of real numbers in $[0,1]$ such that $$|x_m-x_n|\geq\frac{c}{f(m)}$$ for all positive integers $m>n\geq1$. Proposed by Shantanu Nene

The Magician and his Assistant show trick. The Viewer writes on the board the sequence of $N$ digits. Then the Assistant covers some pair of adjacent digits so that they become invisible. Finally, the Magician enters the show, looks at the board and guesses the covered digits and their order. Find the minimal $N$ such that the Magician and his Assistant can agree in advance so that the Magician always guesses right

Circle $B$, which has radius 2008, is tangent to horizontal line $A$ at point $P$. Circle $C_1$ has radius 1 and is tangent both to circle $B$ and to line $A$ at a point to the right of point $P$. Circle $C_2$ has radius larger than 1 and is tangent to line $A$ and both circles B and $C_1$. For $n>1$, circle $C_n$ is tangent to line $A$ and both circles $B$ and $C_{n-1}$. Find the largest value of n such that this sequence of circles can be constructed through circle $C_n$ where the n circles are all tangent to line $A$ at points to the right of $P$. [asy] size(300); draw((-10,0)--(10,0)); draw(arc((0,10),10,210,330)); label("$P$",(0,0),S); pair C=(0,10),X=(12,3); for(int kk=0;kk<6;++kk) { pair Y=(X.x-X.y,X.y); for(int k=0;k<20;++k) Y+=(abs(Y-X)-X.y-Y.y,abs(Y-C)-10-Y.y)/3; draw(circle(Y,Y.y)); X=Y; }[/asy]

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 $25$? $\text{(A)}\ 2 \qquad \text{(B)}\ 4 \qquad \text{(C)}\ 6 \qquad \text{(D)}\ 8 \qquad \text{(E)}\ 10$