On the plane circles $k$ and $\ell$ are intersected at points $C$ and $D$, where circle $k$ passes through the center $L$ of circle $\ell$. The straight line passing through point $D$ intersects circles $k$ and $\ell$ for the second time at points $A$ and $B$ respectively in such a way that $D$ is the interior point of segment $AB$. Show that $AB = AC$.

[b]Randy:[/b] "Hi Rachel, that's an interesting quadratic equation you have written down. What are its roots?'' [b]Rachel:[/b] "The roots are two positive integers. One of the roots is my age, and the other root is the age of my younger brother, Jimmy.'' [b]Randy:[/b] "That is very neat! Let me see if I can figure out how old you and Jimmy are. That shouldn't be too difficult since all of your coefficients are integers. By the way, I notice that the sum of the three coefficients is a prime number.'' [b]Rachel:[/b] "Interesting. Now figure out how old I am.'' [b]Randy:[/b] "Instead, I will guess your age and substitute it for $x$ in your quadratic equation $\dots$ darn, that gives me $-55$, and not $0$.'' [b]Rachel:[/b] "Oh, leave me alone!'' (1) Prove that Jimmy is two years old. (2) Determine Rachel's age.

Let $p$ be a prime number. How many solutions does the congruence $x^2+y^2+z^2+1\equiv 0\pmod{p}$ have among the modulo $p$ remainder classes? [i]Proposed by: Zoltán Gyenes, Budapest[/i]

Find all quadruplets of positive integers $(m,n,k,l)$ such that $3^m = 2^k + 7^n$ and $m^k = 1 + k + k^2 + k^3 + \ldots + k^l$.

How many pairs of integers $x, y$ are there between $1$ and $1000$ such that $x^2 + y^2$ is divisible by $7$?

For a positive integer $n$, define $f(x)$ to be the smallest positive integer $x$ satisfying the following conditions: there exists a positive integer $k$ and $k$ distinct positive integers $n=a_0<a_1<a_2<\cdots<a_{k-1}=x$ such that $a_0a_1\cdots a_{k-1}$ is a perfect square. Find the smallest real number $c$ such that there exists a positive integer $N$ such that for all $n>N$ we have $f(n)\leq cn$. [i]Proposed by Fysty and amano_hina[/i]

Let $n$ be a positive integer of the form $4k+1$, $k\in \mathbb N$ and $A = \{ a^2 + nb^2 \mid a,b \in \mathbb Z\}$. Prove that there exist integers $x,y$ such that $x^n+y^n \in A$ and $x+y \notin A$.

(F.Nilov, A.Zaslavsky) Let $ CC_0$ be a median of triangle $ ABC$; the perpendicular bisectors to $ AC$ and $ BC$ intersect $ CC_0$ in points $ A_c$, $ B_c$; $ C_1$ is the common point of $ AA_c$ and $ BB_c$. Points $ A_1$, $ B_1$ are defined similarly. Prove that circle $ A_1B_1C_1$ passes through the circumcenter of triangle $ ABC$.

A point $D$ lies inside triangle $ABC$. Let $A_1, B_1, C_1$ be the second intersection points of the lines $AD$, $BD$, and $CD$ with the circumcircles of $BDC$, $CDA$, and $ADB$, respectively. Prove that $$\frac{AD}{AA_1} + \frac{BD}{BA_1} + \frac{CD}{CC_1} = 1.$$

Suppose $a,b,c\in[0,2]$ and $a+b+c=3$. Find the maximal and minimal value of the expression $$\sqrt{a(b+1)}+\sqrt{b(c+1)}+\sqrt{c(a+1)}.$$

Find all polynomials $P$ with integer coefficients such that $$s(x)=s(y) \implies s(|P(x)|)=s(|P(y)|).$$ for all $x,y\in \mathbb{N}$. Note: $s(x)$ denotes the sum of digits of $x$. [i]Proposed by Şevket Onur YILMAZ[/i]

The numerical value of the following integral $$\int^1_0 (-x^2 + x)^{2017} \lfloor 2017x \rfloor dx$$ can be expressed in the form $a\frac{m!^2}{ n!}$ where a is minimized. Find $a + m + n$. (Note $\lfloor x\rfloor$ is the largest integer less than or equal to x.)

Construct a triangle $ABC$ given angle $A$ and the medians drawn from vertices $B$ and $C$.

Find the number of sequences $a_1,a_2,\dots,a_{100}$ such that $\text{(i)}$ There exists $i\in\{1,2,\dots,100\}$ such that $a_i=3$, and $\text{(ii)}$ $|a_i-a_{i+1}|\leq 1$ for all $1\leq i<100$.

Let $s(n)$ be the number of positive integer divisors of $n$. Find the all positive values of $k$ that is providing $k=s(a)=s(b)=s(2a+3b)$.

Prove that for all non negative real numbers $a,b,c$ we have \[a^2+b^2+c^2\leq\sqrt{(b^2-bc+c^2)(c^2-ca+a^2)}+\sqrt{(c^2-ca+a^2)(a^2-ab+b^2)}+\sqrt{(a^2-ab+b^2)(b^2-bx+c^2)} \]

$(MOR 1)$ Denote by $[x]$ the greatest integer not exceeding $x$. For all real $k > 1$, define two sequences: \[a_n(k) = [nk]\text{ and } b_n(k) =\left[\frac{nk}{k - 1}\right]\] If $A(k) = \{a_n(k) : n \in\mathbb{N}\}$ and $B(k) = \{b_n(k) : n \in \mathbb{N}\}$, prove that $A(k)$ and $B(k)$ form a partition of $\mathbb{N}$ if and only if $k$ is irrational.

Suppose that $X\sim\operatorname{Uniform}(0,1)$ and $Y\sim\operatorname{Bernoulli}\left(\frac14\right)$, independently of each other. Let $Z=X+Y$. Then which of the following is true? $\textbf{(A)}~\text{The distribution of }Z\text{ is symmetric about }1$ $\textbf{(B)}~Z\text{ has a probability density function}$ $\textbf{(C)}~E(Z)=\frac54$ $\textbf{(D)}~P(Z\le1)=\frac14$

Let $x$ and $y$ be real numbers satisfying $9x^2+16y^2=144$. What is the maximum possible value of $xy$?

A large collection of congruent right triangles is given, each with side length 3,4,5. Find the maximal number of such triangles you can place inside a 20x20 square, with no two triangles intersecting (in their interiors).

[b]p1.[/b] What is the maximum amount of a $12\%$ acid solution that can be obtained from $1$ liter of $5\%$, $10\%$ and $15\%$ solutions? [b]p2.[/b] Which number is greater: $199,719,971,997^2$ or $199,719,971,996 * 19,9719,971,998$ ? [b]p3.[/b] Is there a convex $1998$-gon whose angles are all integer degrees? [b]p4.[/b] Is there a ten-digit number divisible by $11$ that uses all the digits from$ 0$ to $9$? [b]p5.[/b] There are $20$ numbers written in a circle, each of which is equal to the sum of its two neighbors. Prove that the sum of all numbers is $0$. [b]p6.[/b] Is there a convex polygon that has neither an axis of symmetry nor a center of symmetry, but which transforms into itself when rotated around some point through some angle less than $180$ degrees? [b]p7.[/b] In a convex heptagon, draw as many diagonals as possible so that no three of them are sides of the same triangle, the vertices of which are at the vertices of the original heptagon. [b]p8.[/b] Give an example of a natural number that is divisible by $30$ and has exactly $105$ different natural factors, including $1$ and the number itself. [b]p9.[/b] In the writing of the antipodes, numbers are also written with the digits $0, ..., 9$, but each of the numbers has different meanings for them and for us. It turned out that the equalities are also true for the antipodes $5 * 8 + 7 + 1 = 48$ $2 * 2 * 6 = 24$ $5* 6 = 30$ a) How will the equality $2^3 = ...$ in the writing of the antipodes be continued? b) What does the number$ 9$ mean among the Antipodes? Clarifications: a) It asks to convert $2^3$ in antipodes language, and write with what number it is equal and find a valid equality in both numerical systems. b) What does the digit $9$ mean among the antipodes, i.e. with which digit is it equal in our number system? [b]p10.[/b] Is there a convex quadrilateral that can be cut along a straight line into two parts of the same size and shape, but neither the diagonal nor the straight line passing through the midpoints of opposite sides divides it into two equal parts? PS.1. There was typo in problem $9$, it asks for $2^3$ and not $23$. PS.2. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics]here.[/url]

Suppose that three prime numbers $p,q,$ and $r$ satisfy the equations $pq + qr + rp = 191$ and $p + q = r - 1$. Find $p + q + r$. [i]Proposed by Andrew Wu[/i]

Two cirlces $ C_1$ and $ C_2$, with center $ O_1$ and $ O_2$ respectively, intersect at $ A$ and $ B$. Let $ O_1$ lies on $ C_2$. A line $ l$ passes through $ O_1$ but does not pass through $ O_2$. Let $ P$ and $ Q$ be the projection of $ A$ and $ B$ respectively on the line $ l$ and let $ M$ be the midpoint of $ \overline{AB}$. Prove that $ MPQ$ is an isoceles triangle.

The Lucas numbers $L_n$ are defined recursively as follows: $L_0=2,L_1=1,L_n=L_{n-1}+L_{n-2}$ for $n\geq2$. Let $r=0.21347\dots$, whose digits form the pattern of the Lucas numbers. When the numbers have multiple digits, they will "overlap," so $r=0.2134830\dots$, [b]not[/b] $0.213471118\dots$. Express $r$ as a rational number $\frac{p}{q}$, where $p$ and $q$ are relatively prime.

Solve the given equation in primes \begin{align*} xyz=1 +2^{y^2+1} \end{align*}