A box contains $28$ red balls, $20$ green balls, $19$ yellow balls, $13$ blue balls, $11$ white balls, and $9$ black balls. What is the minimum number of balls that must be drawn from the box without replacement to guarantee that at least $15$ balls of a single color will be drawn$?$ $\textbf{(A) } 75 \qquad\textbf{(B) } 76 \qquad\textbf{(C) } 79 \qquad\textbf{(D) } 84 \qquad\textbf{(E) } 91$

Let $A$ be a finite set of positive reals, let $B = \{x/y\mid x,y\in A\}$ and let $C = \{xy\mid x,y\in A\}$. Show that $|A|\cdot|B|\le|C|^2$. [i](Proposed by Gerhard Woeginger, Austria)[/i]

Let $ ABCD $ be a convex quadrilateral, and $ P $ be a point that isn't found on any of the lines formed by the sides of the quadrilateral. Prove that the centers of mass of the triangles $ PAB, PBC, PCD $ and $ PDA, $ form a parallelogram, and calculate the legths of its sides in terms of its diagonals.

Let $ABC$ be an acute triangle with altitude $AD$ ($D \in BC$). The line through $C$ parallel to $AB$ meets the perpendicular bisector of $AD$ at $G$. Show that $AC = BC$ if and only if $\angle AGC = 90^{\circ}$.

Three operations $f,g$ and $h$ are defined on subsets of the natural numbers $\mathbb{N}$ as follows: $f(n)=10n$, if $n$ is a positive integer; $g(n)=10n+4$, if $n$ is a positive integer; $h(n)=\frac{n}{2}$, if $n$ is an [i]even[/i] positive integer. Prove that, starting from $4$, every natural number can be constructed by performing a finite number of operations $f$, $g$ and $h$ in some order. $[$For example: $35=h(f(h(g(h(h(4)))))).]$

Let $n\ge 2$ be a positive integer, and suppose $a_1,a_2,\cdots ,a_n$ are positive real numbers whose sum is $1$ and whose squares add up to $S$. Prove that if $b_i=\tfrac{a^2_i}{S} \;(i=1,\cdots ,n)$, then for every $r>0$, we have $$\sum_{i=1}^n \frac{a_i}{{(1-a_i)}^r}\le \sum_{i=1}^n \frac{b_i}{{(1-b_i)}^r}.$$

Find the largest positive integer $n <30$ such that $\frac{1}{2}(n^8 + 3n^4 -4)$ is not divisible by the square of any prime number.

Let $ABC$ be a triangle with $AB = 13$, $BC = 14$, and $CA = 15$. Let $D$ be the foot of the altitude from $A$ to $BC$. The inscribed circles of triangles $ABD$ and $ACD$ are tangent to $AD$ at $P$ and $Q$, respectively, and are tangent to $BC$ at $X$ and $Y$ , respectively. Let $PX$ and $QY$ meet at $Z$. Determine the area of triangle $XY Z$.

Let $a,b,c$ be positive real numbers such that $a+b+c=10$ and $ab+bc+ca=25$. Let $m=\min\{ab,bc,ca\}$. Find the largest possible value of $m$.

In equilateral triangle $ABC$ with side length $2$, let the parabola with focus $A$ and directrix $BC$ intersect sides $AB$ and $AC$ at $A_1$ and $A_2$, respectively. Similarly, let the parabola with focus $B$ and directrix $CA$ intersect sides $BC$ and $BA$ at $B_1$ and $B_2$, respectively. Finally, let the parabola with focus $C$ and directrix $AB$ intersect sides $CA$ and $C_B$ at $C_1$ and $C_2$, respectively. Find the perimeter of the triangle formed by lines $A_1A_2$, $B_1B_2$, $C_1C_2$.

Maker and Breaker are building a wall. Maker has a supply of green cubical building blocks, and Breaker has a supply of red ones, all of the same size. On the ground, a row of $m$ squares has been marked in chalk as place-holders. Maker and Breaker now take turns in placing a block either directly on one of these squares, or on top of another block already in place, in such a way that the height of each column never exceeds $n$. Maker places the first block. Maker bets that he can form a green row, i.e. all $m$ blocks at a certain height are green. Breaker bets that he can prevent Maker from achieving this. Determine all pairs $(m,n)$ of positive integers for which Maker can make sure he wins the bet.

Let $ S$ be a finite set of points in the plane such that no three of them are on a line. For each convex polygon $ P$ whose vertices are in $ S$, let $ a(P)$ be the number of vertices of $ P$, and let $ b(P)$ be the number of points of $ S$ which are outside $ P$. A line segment, a point, and the empty set are considered as convex polygons of $ 2$, $ 1$, and $ 0$ vertices respectively. Prove that for every real number $ x$ \[\sum_{P}{x^{a(P)}(1 \minus{} x)^{b(P)}} \equal{} 1,\] where the sum is taken over all convex polygons with vertices in $ S$. [i]Alternative formulation[/i]: Let $ M$ be a finite point set in the plane and no three points are collinear. A subset $ A$ of $ M$ will be called round if its elements is the set of vertices of a convex $ A \minus{}$gon $ V(A).$ For each round subset let $ r(A)$ be the number of points from $ M$ which are exterior from the convex $ A \minus{}$gon $ V(A).$ Subsets with $ 0,1$ and 2 elements are always round, its corresponding polygons are the empty set, a point or a segment, respectively (for which all other points that are not vertices of the polygon are exterior). For each round subset $ A$ of $ M$ construct the polynomial \[ P_A(x) \equal{} x^{|A|}(1 \minus{} x)^{r(A)}. \] Show that the sum of polynomials for all round subsets is exactly the polynomial $ P(x) \equal{} 1.$ [i]Proposed by Federico Ardila, Colombia[/i]

Point $P$ lies in the interior of rectangle $ABCD$ such that $AP + CP = 27$, $BP - DP = 17$, and $\angle DAP \cong \angle DCP$. Compute the area of rectangle $ABCD$. [i]Proposed by Aaron Lin[/i]

Let $P$ and $Q$ be fixed points in the Euclidean plane. Consider another point $O_0$. Define $O_{i+1}$ as the center of the unique circle passing through $O_i$, $P$ and $Q$. (Assume that $O_i,P,Q$ are never collinear.) How many possible positions of $O_0$ satisfy that $O_{2021}=O_{0}$? [i]Proposed by Fei Peng[/i]

If $f$ is a monic cubic polynomial with $f(0)=-64$, and all roots of $f$ are non-negative real numbers, what is the largest possible value of $f(-1)$? (A polynomial is monic if it has a leading coefficient of $1$.)

Suppose that $a,b,c$ are positive integers such that $a^b$ divides $b^c$, and $a^c$ divides $c^b$. Prove that $a^2$ divides $bc$.

We consider the two sequences $(a_n)_{n\ge 0}$ and $(b_n) _{n\ge 0}$ of integers, which are given by $a_0 = b_0 = 2$ and $a_1= b_1 = 14$ and for $n\ge 2$ they are defined as $a_n = 14a_{n-1} + a_{n-2}$ , $b_n = 6b_{n-1}-b_{n-2}$. Determine whether there are infinite numbers that occur in both sequences

Pete's research shows that the number of nuts collected by the squirrels in any park is proportional to the square of the number of squirrels in that park. If Pete notes that four squirrels in a park collect $60$ nuts, how many nuts are collected by $20$ squirrels in a park?

Determine which of the following is larger: \[ \sqrt{2+\sqrt[3]{5}}\qquad \text{or}\qquad \sqrt[3]{5+\sqrt{2}}.\] Fully explain your reasoning.

Shuffle a deck of $71$ playing cards which contains $6$ aces. Then turn up cards from the top until you see an ace. What is the average number of cards required to be turned up to find the first ace?

$40$ people, numbered $1$ through $40$ counterclockwise, sit around a circular table. They begin playing a game. Each person is initially considered "alive". Starting with person $1$, the first person eliminates the closest "alive" person to their right (so Person $1$ eliminates Person $2$). Then the next "alive" person, moving counterclockwise, eliminates the closest "alive" person to their right (so since Person $2$ is eliminated, Person $3$ eliminates Person $4$). This process continues until there is only $1$ "alive" person remaining. What is the number of the last "alive" person? [asy] usepackage("cancel", "makeroom, thicklines"); usepackage("bm"); size(15cm); picture p; draw(p, circle((0,0), 5)); for(int i = 0; i < 4; ++i) { label(p, "$" + string(40 - i) + "$", 5 * dir(-20 * i - 100), 2 * dir(-20 * i - 100)); label(p, "$" + string(i + 1) + "$", 5 * dir(20 * i - 80), 2 * dir(20 * i - 80)); } int n = 20; for(int i = 0; i <= n; ++i) { label(p, scale(2)*"$\cdot$", 6 *dir(180 / n * i)); } draw(p, arc((0,0), 8 * dir(-80), 8 * dir(0)), EndArrow); add(shift(-20, 0) * p); draw((-11, 0)--(-8,0), EndArrow); picture q; draw(q, circle((0,0), 5)); for(int i = 0; i < 4; ++i) { label(q, "$" + string(40 - i) + "$", 5 * dir(-20 * i - 100), 2 * dir(-20 * i - 100)); if(i != 1) label(q, "$" + string(i + 1) + "$", 5 * dir(20 * i - 80), 2 * dir(20 * i - 80)); } int n = 20; for(int i = 0; i <= n; ++i) { label(q, scale(2)*"$\cdot$", 6 *dir(180 / n * i)); } draw(q, arc((0,0), 8 * dir(-80), 8 * dir(0)), EndArrow); for(int i = 0; i < 1; i+=2) { //label(q, "\bm\xcancel{~}", 5 * dir(-20 * i - 100), 2 * dir(-20 * i - 100)); label(q, "\xcancel{2}", 5 * dir(20 * (i + 1) - 80), 2 * dir(20 * (i + 1) - 80)); } add(q); draw((9,0)--(12,0), EndArrow); picture r; draw(r, circle((0,0), 5)); for(int i = 0; i < 4; ++i) { if(i % 2 == 1) label(r, "$" + string(40 - i) + "$", 5 * dir(-20 * i - 100), 2 * dir(-20 * i - 100)); if(i % 2 != 1) label(r, "$" + string(i + 1) + "$", 5 * dir(20 * i - 80), 2 * dir(20 * i - 80)); } int n = 20; for(int i = 0; i <= n; ++i) { label(r, scale(2)*"$\cdot$", 6 *dir(180 / n * i)); } draw(r, arc((0,0), 8 * dir(-80), 8 * dir(0)), EndArrow); for(int i = 0; i < 4; i+=2) { label(r, "\xcancel{" + string(40 - i) +"}", 5 * dir(-20 * i - 100), 2 * dir(-20 * i - 100)); label(r, "\xcancel{" + string(i + 1) + "}", 5 * dir(20 * (i + 1) - 80), 2 * dir(20 * (i + 1) - 80)); } add(shift(20, 0) * r); [/asy] [center]In the last step here, Person $39$ eliminates Person $40$. Next turn, Person $1$ eliminates the closest person to his right, Person $3$.[/center]

Let $S$ be a circle with centre $O$. A chord $AB$, not a diameter, divides $S$ into two regions $R_1$ and $R_2$ such that $O$ belongs to $R_2$. Let $S_1$ be a circle with centre in $R_1$, touching $AB$ at $X$ and $S$ internally. Let $S_2$ be a circle with centre in $R_2$, touching $AB$ at $Y$, the circle $S$ internally and passing through the centre of $S$. The point $X$ lies on the diameter passing through the centre of $S_2$ and $\angle YXO=30^o$. If the radius of $S_2$ is $100 $ then what is the radius of $S_1$?

For each positive integer $k$, define $a_k$ as the number obtained from adding $k$ zeroes and a $1$ to the right of $2024$, all written in base $10$. Determine wether there's a $k$ such that $a_k$ has at least $2024^{2024}$ distinct prime divisors.

Lily has an unfair coin that has $\tfrac23$ probability of showing heads and $\tfrac13$ probability of showing tails. She flips the coin twice. What is the probability that the first flip is heads while the second is tails? $\textbf{(A) }0\qquad\textbf{(B) }1/9\qquad\textbf{(C) }2/9\qquad\textbf{(D) }4/9\qquad\textbf{(E) }1$

a) Prove that it is possible to choose one number out of any 39 consecutive positive integers, having the sum of its digits divisible by 11; b) Find the first 38 consecutive positive integers none of which have the sum of its digits divisible by 11.