Given $n\ge 4$ real numbers $a_{n}>...>a_{1} > 0$. For $r > 0$, let $f_{n}(r)$ be the number of triples $(i,j,k)$ with $1\leq i<j<k\leq n$ such that $\frac{a_{j}-a_{i}}{a_{k}-a_{j}}=r$. Prove that ${f_{n}(r)}<\frac{n^{2}}{4}$.

There are $100$ numbers from $(0,1)$ on the board. On every move we replace two numbers $a,b$ with roots of $x^2-ax+b=0$(if it has two roots). Prove that process is not endless.

Let $ABCD$ be a convex quadrilateral whose diagonals $AC$ and $BD$ meet at $P$. Let the area of triangle $APB$ be $24$ and let the area of triangle $CPD$ be $25$. What is the minimum possible area of quadrilateral $ABCD$?

How many pairs $(a, b)$ for integers $a, b \ge 2$ which exist the sequence $x_1, x_2, . . . , x_{1000}$ which satisfy conditions as below? 1.Terms $x_1, x_2, . . . , x_{1000}$ are sorting of $1, 2, . . . , 1000$. 2.For each integers $1 \le i < 1000$, the sequence forms $x_{i+1} = x_i + a$ or $x_{i+1} = x_i - b$.

Determine all the functions $f : \mathbb{R}\rightarrow\mathbb{R}$ that satisfies the following. $f(xf(y)+yf(z)+zf(x))=yf(x)+zf(y)+xf(z)$

Jim has two fair $6$-sided dice, one whose faces are labelled from $1$ to $6$, and the second whose faces are labelled from $3$ to $8$. Twice, he randomly picks one of the dice (each die equally likely) and rolls it. Given the sum of the resulting two rolls is $9$, if $\frac{m}{n}$ is the probability he rolled the same die twice where $m, n$ are relatively prime positive integers, then what is $m + n$?

Let $A_0A_1A_2$ be a scalene triangle. Find the locus of the centres of the equilateral triangles $X_0X_1X_2$ , such that $A_k$ lies on the line $X_{k+1}X_{k+2}$ for each $k=0,1,2$ (with indices taken modulo $3$).

Let $n$ be a positive integer and let $A_n$ respectively $B_n$ be the set of nonnegative integers $k<n$ such that the number of distinct prime factors of $\gcd(n,k)$ is even (respectively odd). Show that $|A_n|=|B_n|$ if $n$ is even and $|A_n|>|B_n|$ if $n$ is odd. Example: $A_{10} = \left\{ 0,1,3,7,9 \right\}$, $B_{10} = \left\{ 2,4,5,6,8 \right\}$.

Consider the integral $$\int_{-1}^1 x^nf(x) \; dx$$ for every $n$-th degree polynomial $f$ with integer coefficients. Let $\alpha_n$ denote the smallest positive real number that such an integral can give. Determine the limit value $$\lim_{n\to \infty} \frac{\log \alpha_n}n.$$

There are relatively prime positive integers $p$ and $q$ such that $\dfrac{p}{q}=\displaystyle\sum_{n=3}^{\infty} \dfrac{1}{n^5-5n^3+4n}$. Find $p+q$.

Let a sequence of integers $a_0, a_1, a_2, \cdots, a_{2010}$ such that $a_0 = 1$ and $2011$ divides $a_{k-1}a_k - k$ for all $k = 1, 2, \cdots, 2010$. Prove that $2011$ divides $a_{2010} + 1$.

How many sets of two or more consecutive positive integers have a sum of 15? $ \textbf{(A) } 1\qquad \textbf{(B) } 2\qquad \textbf{(C) } 3\qquad \textbf{(D) } 4\qquad \textbf{(E) } 5$

Find all integers $n\geq 3$ for which every convex equilateral $n$-gon of side length $1$ contains an equilateral triangle of side length $1$. (Here, polygons contain their boundaries.)

We will say that a triangle is multiplicative if the product of the heights of two of its sides is equal to the length of the third side. Given $ABC \dots XYZ$ is a regular polygon with $n$ sides of length $1$. The $n-3$ diagonals that go out from vertex $A$ divide the triangle $ZAB$ in $n-2$ smaller triangles. Prove that each one of these triangles is multiplicative.

Define the [i]factorial function[/i] of $n,$ denoted $\partial (n),$ as the sum of the factorials of the digits of $n.$ For example, $\partial(2024)=2!+0!+2!+4!=29.$ There are four positive integers such that $\partial(\partial(n))=n$ and $\partial(n) \neq n.$ Given that $n=871$ is one of them, compute the sum of the other three.

Let $ABC$ be a triangle, and $M$ be the midpoint of $BC$, Let $N$ be the point on the segment $AM$ with $AN = 3NM$, and $P$ be the intersection point of the lines $BN$ and $AC$. What is the area in cm$^2$ of the triangle $ANP$ if the area of the triangle $ABC$ is $40$ cm$^2$?

Given are $50$ distinct sets of positive integers, each of size $30$, such that every $30$ of them have a common element. Prove that all of them have a common element.

[b]p1.[/b] There are eight different dominoes in the box (fig.), but the boundaries between them are not visible. Draw the boundaries. [img]https://cdn.artofproblemsolving.com/attachments/6/f/6352b18c25478d68a23820e32a7f237c9f2ba9.png[/img] [b]p2.[/b] The teacher drew a quadrilateral $ABCD$ on the board. Vanya and Vitya marked points $X$ and $Y$ inside it, from which all sides of the quadrilateral are visible at equal angles. What is the distance between points $X$ and $Y$? (From point $X$, side $AB$ is visible at angle $AXB$.) [b]pЗ.[/b] Several identical black squares, perhaps partially overlapping, were placed on a white plane. The result was a black polygonal figure, possibly with holes or from several pieces. Could it be that this figure does not have a single right angle? [b]p4.[/b] The bus ticket number consists of six digits (the first digits may be zeros). A ticket is called [i]lucky [/i] if the sum of the first three digits is equal to the sum of the last three. Prove that the sum of the numbers of all lucky tickets is divisible by $13$. [b]p5.[/b] The Meandrovka River, which has many bends, crosses a straight highway under thirteen bridges. Prove that there are two neighboring bridges along both the highway and the river. (Bridges are called river neighbors if there are no other bridges between them on the river section; bridges are called highway neighbors if there are no other bridges between them on the highway section.) PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics]here.[/url]

Given two circles intersecting at points $P$ and $Q$. Let C be an arbitrary point distinct from $P$ and $Q$ on the former circle. Let lines $CP$ and $CQ$ intersect again the latter circle at points A and B, respectively. Determine the locus of the circumcenters of triangles $ABC$.

Prove that for any integer the number $2n^3+3n^2+7n$ is divisible by $6$.

Find all triples of natural numbers $ (x, y, z) $ that satisfy the condition $ (x + 1) ^ {y + 1} + 1 = (x + 2) ^ {z + 1}. $

2) Let $a, b, c$ be real numbers such that $$a+\frac{b}{c}=b+\frac{c}{a}=c+\frac{a}{b}=1$$a) Prove that $ab+bc+ca=0$ and $a+b+c=3$. b) Prove that $|a|+|b|+|c|< 5$

Let $ABCD$ be a quadrilateral with $\angle ADC = 70^{\circ}$, $\angle ACD = 70^{\circ}$, $\angle ACB = 10^{\circ}$ and $\angle BAD = 110^{\circ}$. The measure of $\angle CAB$ (in degrees) is:

Let $ABC$ be a non-isosceles triangle. The altitude from $A$, the bisector from $B$ and the median from $C$ concur at point $K$. a) Which of the sidelengths of the triangle is medial (intermediate in length)? b) Which of the lengths of segments $AK, BK, CK$ is medial (intermediate in length)?

Find for which natural numbers $n$ one can color the sides and diagonals of a regular $n$-gon with $n$ colors in such a way that for each triplet in pairs of different colors, a triangle can be found, the sides of which are sides or diagonals of $n$-gon and which is colored with exactly these three colors.