Cagney can frost a cupcake every $20$ seconds and Lacey can frost a cupcake every $30$ seconds. Working together, how many cupcakes can they frost in $5$ minutes? $ \textbf{(A)}\ 10 \qquad\textbf{(B)}\ 15 \qquad\textbf{(C)}\ 20 \qquad\textbf{(D)}\ 25 \qquad\textbf{(E)}\ 30 $

Suppose for real numbers $a, b, c$ we know $a + \dfrac 1b = 3,$ and $b + \dfrac 3c = \dfrac 13.$ What is the value of $c + \dfrac{27}a?$ $$ \mathrm a. ~ 1\qquad \mathrm b.~3\qquad \mathrm c. ~8 \qquad \mathrm d. ~9 \qquad \mathrm e. ~21 $$

Let $S_n$ denote the sum of the first $n$ prime numbers. Prove that for any $n$ there exists the square of an integer between $S_n$ and $S_{n+1}$.

Let $a_n$ be the number of sets $S$ of positive integers for which \[ \sum_{k\in S}F_k=n,\] where the Fibonacci sequence $(F_k)_{k\ge 1}$ satisfies $F_{k+2}=F_{k+1}+F_k$ and begins $F_1=1$, $F_2=1$, $F_3=2$, $F_4=3$. Find the largest number $n$ such that $a_n=2020$.

Let $p$ be an odd prime. A degree $d$ polynomial $f$ with non-negative integer coefficients less than $p$ is called $p-floppy$ if the coefficients of $f(x)f(-x) - f(x^2)$ are all divisible by $p$ and if exactly $d$ entries in the sequence $(f(0), f(1), f(2), \dots , f(p-1))$ are divisible by $p$. How many non-constant $61$-floppy polynomials are there?

A set of $n$ lines are said to be in [i]standard form[/i] if no two are parallel and no three are concurrent. Does there exist a value of $k$ such that given any $n$ lines in [i]standard form[/i], it is possible to colour the regions bounded by the $n$ lines using $k$ colours in such a way that no two regions of the same colour share a common intersection point of the $n$ lines?

In an arcade game, the "monster" is the shaded sector of a circle of radius $ 1$ cm, as shown in the figure. The missing piece (the mouth) has central angle $ 60^{\circ}$. What is the perimeter of the monster in cm? [asy]size(100); defaultpen(linewidth(0.7)); filldraw(Arc(origin,1,30,330)--dir(330)--origin--dir(30)--cycle, yellow, black); label("1", (sqrt(3)/4, 1/4), NW); label("$60^\circ$", (1,0)); [/asy] $ \textbf{(A)}\ \pi \plus{} 2 \qquad \textbf{(B)}\ 2\pi \qquad \textbf{(C)}\ \frac53 \pi \qquad \textbf{(D)}\ \frac56 \pi \plus{} 2 \qquad \textbf{(E)}\ \frac53 \pi \plus{} 2$

Let $A$ be a set containing $4k$ consecutive positive integers, where $k \geq 1$ is an integer. Find the smallest $k$ for which the set A can be partitioned into two subsets having the same number of elements, the same sum of elements, the same sum of the squares of elements, and the same sum of the cubes of elements.

Let $A=\{-2562,-2561,...,2561,2562\}$. Prove that for any bijection (1-1, onto function) $f:A\to A$, $$\sum_{k=1}^{2562}\left\lvert f(k)-f(-k)\right\rvert\text{ is maximized if and only if } f(k)f(-k)<0\text{ for any } k=1,2,...,2562.$$

$12.$ In a rectangle $ABCD$ $AB=8$ and $BC=20.$ Let $P$ be a point on $AD$ such that $\angle{BPC}=90^o.$ If $r_1,r_2,r_3.$ are the radii of the incircles of triangles $APB,$ $BPC,$ and $CPD.$ what is the value of $r_1+r_2+r_3 ?$

Two circles, $\omega_1$ and $\omega_2$, centered at $O_1$ and $O_2$, respectively, meet at points $A$ and $B$. A line through $B$ meet $\omega_1$ again at $C$, and $\omega_2$ again at $D$. The tangents to $\omega_1$ and $\omega_2$ at $C$ and $D$, respectively, meet at $E$, and the line $AE$ meets the circle $\omega$ through $A, O_1,O_2$ again at $F$. Prove that the length of the segment $EF$ is equal to the diameter of $\omega$.

Given a polynomial $P$, assume that $L = \{z \in \mathbb{C}: |P(z)| = 1\}$ is a Jordan curve. Show that the zeros of $P'$ are in the interior of $L$.

From the set of all permutations $f$ of $\{1, 2, ... , n\}$ that satisfy the condition: $f(i) \geq i-1$ $i=1,...,n$ one is chosen uniformly at random. Let $p_n$ be the probability that the chosen permutation $f$ satisfies $f(i) \leq i+1$ $i=1,...,n$ Find all natural numbers $n$ such that $p_n > \frac{1}{3}$.

Find all pairs of primes $(p, q)$ for which $p-q$ and $pq-q$ are both perfect squares.

Let $a, b,c$ satises the conditions $\begin{cases} 5 \ge a \ge b \ge c \ge 0 \\ a + b \le 8 \\ a + b + c = 10 \end{cases}$ Prove that $a^2 + b^2 + c^2 \le 38$

A knight is modified so that it moves $p$ fields horizontally or vertically and $q$ fields in the perpendicular direction. It is placed on an infinite chessboard. If the knight returns to the initial field after $n$ moves, show that $n$ must be even.

$ a)$ Two players play a cooperative game. They can discuss a strategy prior to the game, however, they cannot communicate and have no information about the other player during the game. The game master chooses one of the players in each round. The player on turn has to guess the number of the current round. Players keep note of the number of rounds they were chosen, however, they have no information about the other player's rounds. If the player's guess is correct, the players are awarded a point. Player's are not notified whether they've scored or not. The players win the game upon collecting 100 points. Does there exist a strategy with which they can surely win the game in a finite number of rounds? $b)$ How does this game change, if in each round the player on turn has two guesses instead of one, and they are awarded a point if one of the guesses is correct (while keeping all the other rules of the game the same)? [i]Proposed by Gábor Szűcs, Budapest[/i]

Prove that:$a+b+c+d \geq \frac{2}{3}(ab+bc+ca+ad+ac+bd)$ where $a;b;c;d$ are positive real numbers satisfying $2(ab+bc+cd+da+ac+bd)+abc+bcd+cda+dab=16$

Anne multiplies each two-digit number by $588$ in turn, and writes down the so-obtained products. How many perfect squares does she write down?

Let $m$ be equal to $1$ or $2$ and $n<10799$ be a positive integer. Determine all such $n$ for which $\sum_{k=1}^{n}\frac{1}{\sin{k}\sin{(k+1)}}=m\frac{\sin{n}}{\sin^{2}{1}}$.

An integer $n>1$ is given . Find the smallest positive number $m$ satisfying the following conditions： for any set $\{a,b\}$ $\subset \{1,2,\cdots,2n-1\}$ ,there are non-negative integers $ x, y$ ( not all zero) such that $2n|ax+by$ and $x+y\leq m.$

Find the integer represented by $\left[ \sum_{n=1}^{10^9} n^{-2/3} \right] $. Here $[x]$ denotes the greatest integer less than or equal to $x.$

$12$ distinct points are equally spaced around a circle. How many ways can Bryan choose $3$ points (not in any order) out of these $12$ points such that they form an acute triangle (Rotations of a set of points are considered distinct). [i]Proposed by Bryan Guo [/i]

Given three non-negative real numbers $a$, $b$ and $c$. The sum of the modules of their pairwise differences is equal to $1$, i.e. $|a- b| + |b -c| + |c -a| = 1$. What can the sum $a + b + c$ be equal to?

Let $O$ be the circumcenter of triangle $ABC$. Define $O_{A0} = O_{B0} = O_{C0} = O$. Recursively, define $O_{An}$ to be the circumcenter of $\vartriangle BO_{A(n-1)}C$. Similarly define $O_{Bn}, O_{Cn}$. Find all $n \ge 1$ so that for any triangle $ABC$ such that $O_{An}, O_{Bn}, O_{Cn}$ all exist, it is true that $AO_{An}, BO_{Bn}, CO_{Cn}$ are concurrent. (Li4)