Determine the set of all pairs (a,b) of positive integers for which the set of positive integers can be decomposed into 2 sets A and B so that $a\cdot A=b\cdot B$.

At the start of the day, the four numbers $(a_0,b_0,c_0,d_0)$ were written on the board. Every minute, Danny replaces the four numbers written on the board with new ones according to the following rule: if the numbers written on the board are $(a,b,c,d)$, then Danny first calculates the numbers \begin{align*} a'&=a+4b+16c+64d\\ b'&=b+4c+16d+64a\\ c'&=c+4d+16a+64b\\ d'&=d+4a+16b+64c \end{align*} and replaces the numbers $(a,b,c,d)$ with the numbers $(a'd',c'd',c'b',b'a')$. For which initial quadruples $(a_0,b_0,c_0,d_0)$, will Danny write at some point a quadruple of numbers all of which are divisible by $5780^{5780}$?

Define $ n!!$ to be $ n(n\minus{}2)(n\minus{}4)\ldots3\cdot1$ for $ n$ odd and $ n(n\minus{}2)(n\minus{}4)\ldots4\cdot2$ for $ n$ even. When $ \displaystyle \sum_{i\equal{}1}^{2009} \frac{(2i\minus{}1)!!}{(2i)!!}$ is expressed as a fraction in lowest terms, its denominator is $ 2^ab$ with $ b$ odd. Find $ \displaystyle \frac{ab}{10}$.

Let $n?$ denote the product of all primes smaller than $n$. Prove that $n? > n$ holds for any natural number $n > 3$.

Let $ \mathbb{R}$ denote the set of all real numbers and $ \mathbb{R}^\plus{}$ the subset of all positive ones. Let $ \alpha$ and $ \beta$ be given elements in $ \mathbb{R},$ not necessarily distinct. Find all functions $ f: \mathbb{R}^\plus{} \mapsto \mathbb{R}$ such that \[ f(x)f(y) \equal{} y^{\alpha} f \left( \frac{x}{2} \right) \plus{} x^{\beta} f \left( \frac{y}{2} \right) \forall x,y \in \mathbb{R}^\plus{}.\]

Determine all pairs $(a, b)$ of real numbers for which there exists a unique symmetric $2\times 2$ matrix $M$ with real entries satisfying $\mathrm{trace}(M)=a$ and $\mathrm{det}(M)=b$. (Proposed by Stephan Wagner, Stellenbosch University)

Take any 26 distinct numbers from {1, 2, ... , 100}. Show that there must be a non-empty subset of the $ 26$ whose product is a square. [hide] I think that the upper limit for such subset is 37.[/hide]

In whatever follows $f$ denotes a differentiable function from $\mathbb{R}$ to $\mathbb{R}$. $f \circ f$ denotes the composition of $f(x)$. $\textbf{(a)}$ If $f\circ f(x) = f(x) \forall x \in \mathbb{R}$ then for all $x$, $f'(x) =$ or $f'(f(x)) =$. Fill in the blank and justify. $\textbf{(b)}$Assume that the range of $f$ is of the form $ \left(-\infty , +\infty \right), [a, \infty ),(- \infty , b], [a, b] $. Show that if $f \circ f = f$, then the range of $f$ is $\mathbb{R}$. [hide=Hint](Hint: Consider a maximal element in the range of f)[/hide] $\textbf{(c)}$ If $g$ satisfies $g \circ g \circ g = g$, then $g$ is onto. Prove that $g$ is either strictly increasing or strictly decreasing. Furthermore show that if $g$ is strictly increasing, then $g$ is unique.

In rectangle $ABCD$, points $E$ and $F$ are on sides $\overline{BC}$ and $\overline{AD}$, respectively. Two congruent semicircles are drawn with centers $E$ and $F$ such that they both lie entirely on or inside the rectangle, the semicircle with center $E$ passes through $C$, and the semicircle with center $F$ passes through $A$. Given that $AB=8$, $CE=5$, and the semicircles are tangent, find the length $BC$. [i]Proposed by Ada Tsui[/i]

Show that, if the real numbers $a, b, c, A, B, C$ satisfy $$aC -2bB + cA = 0 \ \ and \ \ ac - b^2 > 0,$$ then $$AC - B^2 \le 0.$$

Mother baked $15$ pasties. She placed them on a round plate in a circular way: $7$ with cabbage, $7$ with meat and one with cherries in that exact order and put the plate into a microwave. All pasties look the same but Olga knows the order. However she doesn't know how the plate has been rotated in the microwave. She wants to eat a pasty with cherries. Can Olga eat her favourite pasty for sure if she is not allowed to try more than three other pasties?

Let $m>1$ be a fixed positive integer. For a nonempty string of base-ten digits $S$, let $c(S)$ be the number of ways to split $S$ into contiguous nonempty strings of digits such that the base-ten number represented by each string is divisible by $m$. These strings are allowed to have leading zeroes. In terms of $m$, what are the possible values that $c(S)$ can take? For example, if $m=2$, then $c(1234)=2$ as the splits $1234$ and $12|34$ are valid, while the other six splits are invalid.

Ted's grandfather used his treadmill on 3 days this week. He went 2 miles each day. On Monday he jogged at a speed of 5 miles per hour. He walked at the rate of 3 miles per hour on Wednesday and at 4 miles per hour on Friday. If Grandfather had always walked at 4 miles per hour, he would have spent less time on the treadmill. How many minutes less? $\textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 5$

The points $A_1,A_2, \ldots , A_{1983}$ are set on the circumference of a circle and each is given one of the values $\pm 1$. Show that if the number of points with the value $+1$ is greater than $1789$, then at least $1207$ of the points will have the property that the partial sums that can be formed by taking the numbers from them to any other point, in either direction, are strictly positive.

In a square $10^{2019} \times 10^{2019}, 10^{4038}$ points are marked. Prove that there is such a rectangle with sides parallel to the sides of a square whose area differs from the number of points located in it by at least $6$.

A square grid $6 \times 6$ is tiled with $18$ dominoes. Prove that there is a line, intersecting no dominoes. Can this line be unique?

Let $D$ and $E$ be the midpoints of the sides $BC$ and $AC$ of a right triangle $ABC$. Prove that if $\angle CAD=\angle ABE$, then $$\frac{5}{6} \le \frac{AD}{AB}\le \frac{\sqrt{73}}{10}.$$

(1) Find all positive integer $n$ such that for any odd integer $a$, we have $4\mid a^n-1$ (2) Find all positive integer $n$ such that for any odd integer $a$, we have $2^{2017}\mid a^n-1$

A, B and C are three persons among a set P of n (n[u]>[/u]3) persons. It is known that A, B and C are friends of one another, and that every one of the three persons has already made friends with more than half the total number of people in P. Given that every three persons who are friends of one another form a [i]friendly group[/i], what is the minimum number of friendly groups that may exist in P?

Find the number of elements in the set $$ \left\{(a, b) \in \mathbb{N}: 2 \leq a, b \leq 2023, \log _a(b)+6 \log _b(a)=5\right\} $$

Each voter in an election puts $n$ names of candidates on the ballot. There are $n + 1$ at the polling station urn. After the elections it turned out that each ballot box contained at least at least one ballot, for every choice of the $(n + 1)$-th ballot, one from each ballot box, there is a candidate whose surname appears in each of the selected ballots. Prove that in at least one ballot box all ballots contain the name of the same candidate.

Quadratic equation $ x^2\plus{}ax\plus{}b\plus{}1\equal{}0$ have 2 positive integer roots, for integers $ a,b$. Show that $ a^2\plus{}b^2$ is not a prime.

Find all positive integers $ n$ such that there exists sequence consisting of $ 1$ and $ - 1: a_{1},a_{2},\cdots,a_{n}$ satisfying $ a_{1}\cdot1^2 + a_{2}\cdot2^2 + \cdots + a_{n}\cdot n^2 = 0.$

Does there exist a triangle $ABC$ whose sides are rational numbers and $BC$ equals to the altitude from $A$?

A circle contains the points $(0, 11)$ and $(0, -11)$ on its circumference and contains all points $(x, y)$ with $x^2+y^2<1$ in its interior. Compute the largest possible radius of the circle.