Prove that in any set consisting of $117$ pairwise distinct three-digit numbers, you can choose $4$ pairwise disjoint subsets in which the sums of numbers are equal.

Each field of the table $(mn + 1) \times (mn + 1)$ contains a real number from the interval $[0, 1]$. The sum the numbers in each square section of the table with dimensions $n x n$ is equal to $n$. Determine how big it can be sum of all numbers in the table.

A palindrome is a number that is the same when its digits are reversed. Find all numbers that have at least one multiple that is a palindrome.

Each vertex of a convex hexagon is the center of a circle whose radius is equal to the shorter side of the hexagon that contains the vertex. Show that if the intersection of all six circles (including their boundaries) is not empty, then the hexagon is regular.

Let $p$ be a prime and let $f(x) = ax^2 + bx + c$ be a quadratic polynomial with integer coefficients such that $0 < a, b, c \le p$. Suppose $f(x)$ is divisible by $p$ whenever $x$ is a positive integer. Find all possible values of $a + b + c$.

Let $ABC$ be a triangle inscribed in circle $C$. Circles $C_1, C_2, C_3$ are tangent internally with circle $C$ in $A_1, B_1, C_1$ and tangent to sides $[BC], [CA], [AB]$ in points $A_2, B_2, C_2$ respectively, so that $A, A_1$ are on one side of $BC$ and so on. Lines $A_1A_2, B_1B_2$ and $C_1C_2$ intersect the circle $C$ for second time at points $A’,B’$ and $C’$, respectively. If $ M = BB’ \cap CC’$, prove that $m (\angle MAA’) = 90^\circ$ .

For all positive reals $a$ and $b$, show that $$\frac{a^2+b^2}{2a^5b^5} + \frac{81a^2b^2}{4} + 9ab > 18.$$

Let $A(n)$ denote the number of sequences $a_1\ge a_2\ge\cdots{}\ge a_k$ of positive integers for which $a_1+\cdots{}+a_k = n$ and each $a_i +1$ is a power of two $(i = 1,2,\cdots{},k)$. Let $B(n)$ denote the number of sequences $b_1\ge b_2\ge \cdots{}\ge b_m$ of positive integers for which $b_1+\cdots{}+b_m =n$ and each inequality $b_j\ge 2b_{j+1}$ holds $(j=1,2,\cdots{}, m-1)$. Prove that $A(n) = B(n)$ for every positive integer $n$. [i]Senior Problems Committee of the Australian Mathematical Olympiad Committee[/i]

For positive integers $n,$ let $f(n)$ equal the number of subsets of the first $13$ positive integers whose members sum to $n.$ Compute \[ \sum_{n=46}^{86} f(n). \]

A sphere with center $O$ has radius $6$. A triangle with sides of length $15, 15,$ and $24$ is situated in space so that each of its sides is tangent to the sphere. What is the distance between $O$ and the plane determined by the triangle? $ \textbf{(A) }2\sqrt{3}\qquad \textbf{(B) }4\qquad \textbf{(C) }3\sqrt{2}\qquad \textbf{(D) }2\sqrt{5}\qquad \textbf{(E) }5\qquad $

$\lim _{x \rightarrow 0^{+}} \frac{[x]}{\tan x}$ where $[x]$ is the greatest integer function [list=1] [*] -1 [*] 0 [*] 1 [*] Does not exists [/list]

$x,y,z$ are positive real numbers such that: $$xyz+x+y+z=6$$ $$xyz+2xy+yz+zx+z=10$$ Find the maximum value of: $$(xy+1)(yz+1)(zx+1)$$

A number $N$ is in base 10, $503$ in base $b$ and $305$ in base $b+2$ find product of digits of $N$

A circle passes through the points $A,C$ of triangle $ABC$ intersects with the sides $AB,BC$ at points $D,E$ respectively. Let $ \frac{BD}{CE}=\frac{3}{2}$, $BE=4$, $AD=5$ and $AC=2\sqrt{7} $. Find the angle $ \angle BDC$.

Let $\mathbb{N}_0=\{0,1,2 \cdots \}$. Does there exist a function $f: \mathbb{N}__0 \to \mathbb{N}_0$ such that: \[ f^{2003}(n)=5n, \forall n \in \mathbb{N}_0 \] where we define: $f^1(n)=f(n)$ and $f^{k+1}(n)=f(f^k(n))$, $\forall k \in \mathbb{N}_0$?

Find all pairs $(a,b)$ of positive integers such that $(ab)^2 - 4(a+b)$ is the square of an integer.

Let $\Gamma$ be a circle and let $d$ be a line such that $\Gamma$ and $d$ have no common points. Further, let $AB$ be a diameter of the circle $\Gamma$; assume that this diameter $AB$ is perpendicular to the line $d$, and the point $B$ is nearer to the line $d$ than the point $A$. Let $C$ be an arbitrary point on the circle $\Gamma$, different from the points $A$ and $B$. Let $D$ be the point of intersection of the lines $AC$ and $d$. One of the two tangents from the point $D$ to the circle $\Gamma$ touches this circle $\Gamma$ at a point $E$; hereby, we assume that the points $B$ and $E$ lie in the same halfplane with respect to the line $AC$. Denote by $F$ the point of intersection of the lines $BE$ and $d$. Let the line $AF$ intersect the circle $\Gamma$ at a point $G$, different from $A$. Prove that the reflection of the point $G$ in the line $AB$ lies on the line $CF$.

The sequence $\{a_n\}$ is defined by $a_0 = 1, a_1 = 2,$ and for $n \geq 2,$ $$a_n = a_{n-1}^2 + (a_0a_1 \dots a_{n-2})^2.$$ Let $k$ be a positive integer, and let $p$ be a prime factor of $a_k.$ Show that $p > 4(k-1).$

Let $ABCD$ be circumscribed quadrilateral such that the midpoints of $AB$,$BC$,$CD$ and $DA$ are $K$, $L$, $M$, $N$ respectively. Let the reflections of the point $M$ wrt the lines $AD$ and $BC$ be $P$ and $Q$ respectively. Let the circumcenter of the triangle $KPQ$ be $R$. Prove that $RN=RL$

Let $n \geq 3$ be an integer. Let $a_1,a_2,\dots, a_n\in[0,1]$ satisfy $a_1 + a_2 + \cdots + a_n = 2$. Prove that $$\sqrt{1-\sqrt{a_1}}+\sqrt{1-\sqrt{a_2}}+\cdots+\sqrt{1-\sqrt{a_n}}\leq n-3+\sqrt{9-3\sqrt{6}}.$$

Let $n$ be an integer greater than or equal to $1$. Find, as a function of $n$, the smallest integer $k\ge 2$ such that, among any $k$ real numbers, there are necessarily two of which the difference, in absolute value, is either strictly less than $1 / n$, either strictly greater than $n$.

Find the smallest prime that is the fifth term of an increasing arithmetic sequence, all four preceding terms also being prime.

Suppose that $ s_1,s_2,s_3, \ldots$ is a strictly increasing sequence of positive integers such that the sub-sequences \[s_{s_1},\, s_{s_2},\, s_{s_3},\, \ldots\qquad\text{and}\qquad s_{s_1+1},\, s_{s_2+1},\, s_{s_3+1},\, \ldots\] are both arithmetic progressions. Prove that the sequence $ s_1, s_2, s_3, \ldots$ is itself an arithmetic progression. [i]Proposed by Gabriel Carroll, USA[/i]

Find the largest positive integer $k$ so that any binary string of length $2024$ contains a palindromic substring of length at least $k$.

The incircle of a triangle $ABC$ touches the sides $BC,CA,AB$ at $D,E,F$, respectively. Let $G$ be a point on the incircle such that $FG$ is a diameter. The lines $EG$ and $FD$ intersect at $H$. Prove that $CH\parallel AB$.