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 \[ a+b+c - \frac{abc}{ab+bc+ac} \ge 4 \]

Suppose $A_1 A_2 \ldots A_n$ is an $n$-sided polygon with $n \geq 3$ and $\angle A_j \leq 180^{\circ}$ for each $j$ (in other words, the polygon is convex or has fewer than $n$ distinct sides). For each $i \leq n$, suppose $\alpha_i$ is the smallest possible value of $\angle{A_i A_j A_{i+1}}$ where $j$ is neither $i$ nor $i+1$. (Here, we define $A_{n+1} = A_1$.) Prove that \[ \alpha_1 + \alpha_2 + \cdots + \alpha_n \leq 180^{\circ} \] and determine all equality cases.

A point $P$ lies inside an equilateral triangle $ABC$ such that $AP=15$ and $BP=8$. Find the maximum possible value of the sum of areas of triangles $ABP$ and $BCP$.

In the $\textit{Fragmented Game of Spoons}$, eight players sit in a row, each with a hand of four cards. Each round, the first player in the row selects the top card from the stack of unplayed cards and either passes it to the second player, which occurs with probability $\tfrac12$, or swaps it with one of the four cards in his hand, each card having an equal chance of being chosen, and passes the new card to the second player. The second player then takes the card from the first player and chooses a card to pass to the third player in the same way. Play continues until the eighth player is passed a card, at which point the card he chooses to pass is removed from the game and the next round begins. To win, a player must hold four cards of the same number, one of each suit. During a game, David is the eighth player in the row and needs an Ace of Clubs to win. At the start of the round, the dealer picks up a Ace of Clubs from the deck. Suppose that Justin, the fifth player, also has a Ace of Clubs, and that all other Ace of Clubs cards have been removed. The probability that David is passed an Ace of Clubs during the round is $\tfrac mn$, where $m$ and $n$ are positive integers with $\gcd(m, n) = 1.$ Find $100m + n.$ [i]Proposed by David Altizio[/i]

Given a function $f(x)=x^{2}+ax+b$ with $a,b \in R$, if there exists a real number $m$ such that $\left| f(m) \right| \leq \frac{1}{4}$ and $\left| f(m+1) \right| \leq \frac{1}{4}$, then find the maximum and minimum of the value of $\Delta=a^{2}-4b$.

Let $n$ be a natural number and $L = Z^2$ the set of points on the plane with integer coordinates. Every point in $L$ is colored now in one of the colors red or green. Show that there are $n$ different points $x_1,...,x_n \in L$ all of which have the same color and whose center of gravity is also in $L$ and is of the same color.

Let $c\in \Big(0,\dfrac{\pi}{2}\Big) , a = \Big(\dfrac{1}{sin(c)}\Big)^{\dfrac{1}{cos^2 (c)}}, b = \Big(\dfrac{1}{cos(c)}\Big)^{\dfrac{1}{sin^2 (c)}}$. \\Prove that at least one of $a,b$ is bigger than $\sqrt[11]{2015}$.

Postman Pete has a pedometer to count his steps. The pedometer records up to $ 99999$ steps, then flips over to $ 00000$ on the next step. Pete plans to determine his mileage for a year. On January $ 1$ Pete sets the pedometer to $ 00000$. During the year, the pedometer flips from $ 99999$ to $ 00000$ forty-four times. On December $ 31$ the pedometer reads $ 50000$. Pete takes $ 1800$ steps per mile. Which of the following is closest to the number of miles Pete walked during the year? $ \textbf{(A)}\ 2500 \qquad \textbf{(B)}\ 3000 \qquad \textbf{(C)}\ 3500 \qquad \textbf{(D)}\ 4000 \qquad \textbf{(E)}\ 4500$

Berit has $11$ twenty kroner coins, $14$ ten kroner coins, and $12$ five kroner coins. An exchange machine can exchange three ten kroner coins into one twenty kroner coin and two five kroner coins, and the reverse. It can also exchange two twenty kroner coins into three ten kroner coins and two five kroner coins, and the reverse. (i) Can Berit get the same number of twenty kroner and ten kroner coins, but no five kroner coins? (ii) Can she get the same number each of twenty kroner, ten kroner, and five kroner coins?

We examine the following two sequences: The Fibonacci sequence: $F_{0}= 0, F_{1}= 1, F_{n}= F_{n-1}+F_{n-2 }$ for $n \geq 2$; The Lucas sequence: $L_{0}= 2, L_{1}= 1, L_{n}= L_{n-1}+L_{n-2}$ for $n \geq 2$. It is known that for all $n \geq 0$ \[F_{n}=\frac{\alpha^{n}-\beta^{n}}{\sqrt{5}},L_{n}=\alpha^{n}+\beta^{n},\] where $\alpha=\frac{1+\sqrt{5}}{2},\beta=\frac{1-\sqrt{5}}{2}$. These formulae can be used without proof. We call a nonnegative integer $r$-Fibonacci number if it is a sum of $r$ (not necessarily distinct) Fibonacci numbers. Show that there infinitely many positive integers that are not $r$-Fibonacci numbers for any $r, 1 \leq r\leq 5.$

Compute the sum of all positive integers less than $100$ that do not have consecutive $1$s in their binary representation.

Find all triples $(p,m,n)$ satisfying the equation $p^m-n^3=8$ where $p$ is a prime number and $m,n$ are nonnegative integers.

Prove that for every positive integer $m$, every prime $p$ and every positive integer $j \le p^{m-1}$, $p^m$ divides $${p^m \choose p^j }- {p^{m-1} \choose j}$$

If the real function $f(x)=\cos x+\sum_{i=1}^{n}\cos(a_ix)$ is periodic, prove that $a_i,i\in\{1,2,...,n\}$, are rational numbers.

Suppose you have a sphere tangent to the $xy$-plane with its center having positive $z$-coordinate. If it is projected from a point $P=(0,b,a)$ to the $xy$-plane, it gives the conic section $y=x^2$. If we write $a=\tfrac pq$ where $p,q$ are integers, find $p+q$.

The point $H$ is the orthocentre of a triangle $ABC$, and the segments $AD,BE,CF$ are its altitudes. The points $I_1,I_2,I_3$ are the incentres of the triangles $EHF,FHD,DHE$ respectively. Prove that the lines $AI_1,BI_2,CI_3$ intersect at a single point.

A cube with sides of length 3cm is painted red and then cut into 3 x 3 x 3 = 27 cubes with sides of length 1cm. If a denotes the number of small cubes (of 1cm x 1cm x 1cm) that are not painted at all, b the number painted on one sides, c the number painted on two sides, and d the number painted on three sides, determine the value a-b-c+d.

Let $ABC$ be a triangle. Suppose $D$ is on $BC$ such that $AD$ bisects $\angle BAC$. Suppose $M$ is on $AB$ such that $\angle MDA = \angle ABC$, and $N$ is on $AC$ such that $\angle NDA = \angle ACB$. If $AD$ and $MN$ intersect on $P$, prove that $AD^3 = AB \cdot AC \cdot AP$.