A rectangular sheet of paper is folded so that two diagonally opposite corners come together. If the crease formed is the same length as the longer side of the sheet, what is the ratio of the longer side of the sheet to the shorter side?

Let $p$ and $q$ be prime numbers. The sequence $(x_n)$ is defined by $x_1 = 1$, $x_2 = p$ and $x_{n+1} = px_n - qx_{n-1}$ for all $n \geq 2$. Given that there is some $k$ such that $x_{3k} = -3$, find $p$ and $q$.

Let $ABC$ be a triangle with $AB=13$, $BC=14$, and $CA=15$. Points $P$, $Q$, and $R$ are chosen on segments $BC$, $CA$, and $AB$, respectively, such that triangles $AQR$, $BPR$, $CPQ$ have the same perimeter, which is $\frac{4}{5}$ of the perimeter of $PQR$. What is the perimeter of $PQR$? [i]2021 CCA Math Bonanza Individual Round #2[/i]

Let $n \geq 2$ be an integer. Let $S$ be a subset of $\{1,2,\dots,n\}$ such that $S$ neither contains two elements one of which divides the other, nor contains two elements which are coprime. What is the maximal possible number of elements of such a set $S$?

In an $8$-square board -like the one in the figure- there is initially one checker in each square. $ \begin{tabular}{ | l | c | c |c | c| c | c | c | r| } \hline & & & & & & & \\ \hline \end{tabular} $ A move consists of choosing two tokens and moving one of them one square to the right and the other one one square to the left. If after $4$ moves the $8$ checkers are distributed in only $2$ boxes, determine what those boxes can be and how many checkers are in each one.

Let $ a, b, c $ be the side length of a triangle, with $ ab + bc + ca = 18 $ and $ a, b, c > 1 $. Prove that $$ \sum_{cyc}\frac{1}{(a - 1)^3} > \frac{1}{a + b + c - 3} $$

Let $ABCD$ be a trapezoid with bases $AB$ and $CD$, inscribed in a circle of center $O$. Let $P$ be the intersection of the lines $BC$ and $AD$. A circle through $O$ and $P$ intersects the segments $BC$ and $AD$ at interior points $F$ and $G$, respectively. Show that $BF=DG$.

Find the values of $p,\ q,\ r\ (-1<p<q<r<1)$ such that for any polynomials with degree$\leq 2$, the following equation holds: \[\int_{-1}^p f(x)\ dx-\int_p^q f(x)\ dx+\int_q^r f(x)\ dx-\int_r^1 f(x)\ dx=0.\] [i]1995 Hitotsubashi University entrance exam/Law, Economics etc.[/i]

Is it possible to tile an $8\times8$ board with dominoes ($2\times1$ tiles) so that no two dominoes form a $2\times2$ square?

The positive integers are colored with black and white such that: - There exists a bijection from the black numbers to the white numbers, - The sum of three black numbers is a black number, and - The sum of three white numbers is a white number. Find the number of possible colorings that satisfies the above conditions.

Determine all solutions in real numbers $x,y,z,w$ of the system $$x+y+z=w, \frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{1}{w}.$$

Let $ a$, $ b$, $ c$, $ d$ be positive real numbers such that $ abcd \equal{} 1$ and $ a \plus{} b \plus{} c \plus{} d > \dfrac{a}{b} \plus{} \dfrac{b}{c} \plus{} \dfrac{c}{d} \plus{} \dfrac{d}{a}$. Prove that \[ a \plus{} b \plus{} c \plus{} d < \dfrac{b}{a} \plus{} \dfrac{c}{b} \plus{} \dfrac{d}{c} \plus{} \dfrac{a}{d}\] [i]Proposed by Pavel Novotný, Slovakia[/i]

Find all functions $ f : Z\rightarrow Z$ for which we have $ f (0) \equal{} 1$ and $ f ( f (n)) \equal{} f ( f (n\plus{}2)\plus{}2) \equal{} n$, for every natural number $ n$.

There are $n$ MOPpers $p_1,...,p_n$ designing a carpool system to attend their morning class. Each $p_i$'s car fits $\chi (p_i)$ people ($\chi : \{p_1,...,p_n\} \to \{1,2,...,n\}$). A $c$-fair carpool system is an assignment of one or more drivers on each of several days, such that each MOPper drives $c$ times, and all cars are full on each day. (More precisely, it is a sequence of sets $(S_1, ...,S_m)$ such that $|\{k: p_i\in S_k\}|=c$ and $\sum_{x\in S_j} \chi(x) = n$ for all $i,j$. ) Suppose it turns out that a $2$-fair carpool system is possible but not a $1$-fair carpool system. Must $n$ be even? [i]Proposed by Nathan Ramesh and Palmer Mebane

The board has a natural number greater than $1$. At each step, Igor writes the number $n +\frac{n}{p}$ instead of the number $n$ on the board , where $p$ is some prime divisor of $n$. Prove that if Igor continues to rewrite the number infinite times, then he will choose infinitely times the number $3$ as a prime divisor of $p$. [hide=original wording]На доске записано какое-то натуральное число, большее 1. На каждом шагу Игорь переписывает имеющееся на доске число n на число n +n/p, где p - это какой-нибудь простой делитель числа n. Доказать, что если Игорь будет продолжать переписывать число бесконечно долго, то он бесконечно много раз выберет в качестве простого делителя p число 3.[/hide]

Consider all $6$-digit numbers of the form $abccba$ where $b$ is odd. Determine the number of all such $6$-digit numbers that are divisible by $7$.

Let $\omega_1$ and $\omega_2$ be two circles with centers $O_1$ and $O_2$. The two circles intersect at $A$ and $B$. $\ell$ is the circles' common external tangent that is closer to $B$, and it meets $\omega_1$ at $T_1$ and $\omega_2$ at $T_2$. Let $C$ be the point on line $AB$ not equal to $A$ that is the same distance from $\ell$ as $A$ is. Given that $O_1O_2=15$, $AT_1=5$ and $AT_2=12$, find $AC^2+{T_1T_2}^2$. [i]Proposed by Zachary Perry[/i]

Let $ \mathbb{R} ^{+} $ denote the set of all positive real numbers. Find all functions $ \mathbb{R} ^{+} \to \mathbb{R} ^{+} $ such that \[ f(x+f(y)) = yf(xy+1)\] holds for all $ x, y \in \mathbb{R} ^{+} $.

Prove that there is no set of integers $m, n, p$ except $0, 0, 0$ for which \[m + n \sqrt2 + p \sqrt3 = 0.\]

The Rational Unit Jumping Frog starts at $(0, 0)$ on the Cartesian plane, and each minute jumps a distance of exactly $1$ unit to a point with rational coordinates. (a) Show that it is possible for the frog to reach the point $\left(\frac15,\frac{1}{17}\right)$ in a finite amount of time. (b) Show that the frog can never reach the point $\left(0,\frac14\right)$.

$1600$ bananas are distributed among $100$ monkeys (it is possible that some monkeys do not receive bananas). Prvove that at least four monkeys receive the same amount of bananas.

Let $\mathcal{H}$ be the region of points $(x, y)$, such that $(1, 0), (x, y), (-x, y)$, and $(-1,0)$ form an isosceles trapezoid whose legs are shorter than the base between $(x, y)$ and $(-x,y)$. Find the least possible positive slope that a line could have without intersecting $\mathcal{H}$.

Determine the polynomials P of two variables so that: [b]a.)[/b] for any real numbers $t,x,y$ we have $P(tx,ty) = t^n P(x,y)$ where $n$ is a positive integer, the same for all $t,x,y;$ [b]b.)[/b] for any real numbers $a,b,c$ we have $P(a + b,c) + P(b + c,a) + P(c + a,b) = 0;$ [b]c.)[/b] $P(1,0) =1.$

Let $X$ and $Y$ be two finite subsets of the half-open interval $[0, 1)$ such that $0 \in X \cap Y$ and $x + y = 1$ for no $x \in X$ and no $y \in Y$. Prove that the set $\{x + y - \lfloor x + y \rfloor : x \in X \textrm{ and } y \in Y\}$ has at least $|X| + |Y| - 1$ elements. [i]***[/i]

Let $S$ be a circle, and $\alpha =\{A_1,\ldots ,A_n\}$ a family of open arcs in $S$. Let $N(\alpha )=n$ denote the number of elements in $\alpha$. We say that $\alpha$ is a covering of $S$ if $\bigcup_{k=1}^n A_k\supset S$. Let $\alpha=\{A_1,\ldots ,A_n\}$ and $\beta =\{B_1,\ldots ,B_m\}$ be two coverings of $S$. Show that we can choose from the family of all sets $A_i\cap B_j,\ i=1,2,\ldots ,n,\ j=1, 2,\ldots ,m,$ a covering $\gamma$ of $S$ such that $N(\gamma )\le N(\alpha)+N(\beta)$.