Let $a_1, a_2, ...,a_n$ be positive real numbers such that $$a_1 + a_2 + ... + a_n = a_1^2 + a_2^2 + ... + a_n^2$$ Prove that $$\sum_{1\le i<j\le n} a_ia_j(1 - a_ia_j) \ge 0$$ Võ Quốc Bá Cẩn.

In the triangle $ABC$, it is known that $AB = BC = 20$ cm, and $AC = 24$ cm. The point $M$ lies on the side $BC$ and is equidistant from sides $AB$ and $AC$. Find this distance. (Alexander Shkolny)

Determine all natural numbers $ \overline{ab} $ with $ a<b $ which are equal with the sum of all the natural numbers between $ a $ and $ b, $ inclusively.

$N$ unit squares on the infinite sheet of cross-lined paper are painted with black colour. Prove that you can cut out the finite number of square pieces and satisfy two conditions all the black squares are contained in those pieces the area of black squares is not less than $1/5$ and not greater than $4/5$ of every piece area.

Let $ABC$ be an acute-angled triangle. The tangents to its circumcircle at $A, B, C$ form a triangle $PQR$ with $C \in PQ$ and $B \in PR$. Let $C_{1}$ be the foot of the altitude from $C$ in $\Delta ABC$ . Prove that $CC_{1}$ bisects $\widehat{QC_{1}P}$ .

Prove that the equation $$\sqrt{2-x^2}+\sqrt[3]{3-x^3}=0$$ has no real solutions.

An integer $a$ is given. Find all real-valued functions $f (x)$ defined on integers $x \ge a$, satisfying the equation $f (x+y) = f (x) f (y)$ for all $x,y \ge a$ with $x + y \ge a$.

Let $f$ be a differentiable real-valued function defined on the positive real numbers. The tangent lines to the graph of $f$ always meet the $y$-axis 1 unit lower than where they meet the function. If $f(1)=0$, what is $f(2)$?

Using each of the first eight primes exactly once and several algebraic operations, obtain the result $2010$.

We are given a right-angled triangle $MNP$ with right angle in $P$. Let $k_M$ be the circle with center $M$ and radius $MP$, and let $k_N$ be the circle with center $N$ and radius $NP$. Let $A$ and $B$ be the common points of $k_M$ and the line $MN$, and let $C$ and $D$ be the common points of $k_N$ and the line $MN$ with with $C$ between $A$ and $B$. Prove that the line $PC$ bisects the angle $\angle APB$.

In isosceles trapezoid $ABCD$ with $AB < CD$ and $BC = AD$, the angle bisectors of $\angle A$ and $\angle B$ intersect $CD$ at $E$ and $F$ respectively, and intersect each other outside the trapezoid at $G$. Given that $AD = 8$, $EF = 3$, and $EG = 4$, the area of $ABCD$ can be expressed as $\frac{a\sqrt{b}}{c}$ for positive integers $a, b$, and $c$, with $a$ and $c$ relatively prime and $b$ squarefree. Find $10000a +100b +c$.

Prove the inequality [b]a.)[/b] $ \left( a_{1}+a_{2}+...+a_{k}\right) ^{2}\leq k\left( a_{1}^{2}+a_{2}^{2}+...+a_{k}^{2}\right) , $ where $k\geq 1$ is a natural number and $a_{1},$ $a_{2},$ $...,$ $a_{k}$ are arbitrary real numbers. [b]b.)[/b] Using the inequality (1), show that if the real numbers $a_{1},$ $a_{2},$ $...,$ $a_{n}$ satisfy the inequality \[ a_{1}+a_{2}+...+a_{n}\geq \sqrt{\left( n-1\right) \left( a_{1}^{2}+a_{2}^{2}+...+a_{n}^{2}\right) }, \] then all of these numbers $a_{1},$ $a_{2},$ $\ldots,$ $a_{n}$ are non-negative.

In a faraway place in the Universe, a villain has a medal with special powers and wants to hide it so that no one else can use it. For this, the villain hides it in a vertex of a regular polygon with $2019$ sides. Olcoman, the savior of the Olcomita people, wants to get the medal to restore peace in the Universe, for which you have to pay $1000$ olcolones for each time he makes the following move: on each turn he chooses a vertex of the polygon, which turns green if the medal is on it or in one of the four vertices closest to it, or otherwise red. Find the fewest olcolones Olcoman needs to determine with certainty the position of the medal.

At the nodes of graph paper, gardeners live; everywhere around them grow flowers. Each flower is to be taken care of by the three gardeners nearest to it. One of the gardeners wishes to know which are the flowers (s)he has to take care of. Sketch the plot of these gardeners. Proposed by I. F. Sharygin

a) Show that $\forall n \in \mathbb{N}_0, \exists A \in \mathbb{R}^{n\times n}: A^3=A+I$. b) Show that $\det(A)>0, \forall A$ fulfilling the above condition.

There are $n$ students in a class, and some pairs of these students are friends. Among any six students, there are two of them that are not friends, and for any pair of students that are not friends there is a student among the remaining four that is friends with both of them. Find the maximum value of $n$.

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 $RN=RL$

Consider the set $S = \{1,2,3, ...,1441\}$. 1. Nora counts thoses subsets of $S$ having exactly two elements, tbe sum of which is even. Rania counts those subsets of $S$ having exactly two elements, the sum of which is odd. Determine the numbers counted by Nora and Rania. 2. Let $t$ be the number of subsets of $S$ which have at least two elements and the product of the elements is even. Determine the greatest power of $2$ which divides $t$. 3. Ahmad counts the subsets of $S$ having $77$ elements such that in each subset the sum of the elements is even. Bushra counts the subsets of $S$ having $77$ elements such that in each subset the sum of the elements is odd. Whose number is bigger? Determine the difference between the numbers found by Ahmad and Bushra.

Let $N$ be the number of polynomials $P(x_1, x_2, \dots, x_{2016})$ of degree at most $2015$ with coefficients in the set $\{0, 1, 2 \}$ such that $P(a_1,a_2,\cdots ,a_{2016}) \equiv 1 \pmod{3}$ for all $(a_1,a_2,\cdots ,a_{2016}) \in \{0, 1\}^{2016}.$ Compute the remainder when $v_3(N)$ is divided by $2011$, where $v_3(N)$ denotes the largest integer $k$ such that $3^k | N.$ [i]Proposed by Yang Liu[/i]

Let $ABCD$ be a cyclic quadrilateral such that $AB \cdot BC=2 \cdot AD \cdot DC$. Prove that its diagonals $AC$ and $BD$ satisfy the inequality $8BD^2 \le 9AC^2$. [color=#FF0000] Moderator says: Do not double post [url]http://www.artofproblemsolving.com/Forum/viewtopic.php?f=46&t=590175[/url][/color]

Let $ABC$ be a triangle. Consider the circle $\omega_B$ internally tangent to the sides $BC$ and $BA$, and to the circumcircle of the triangle $ABC$, let $P$ be the point of contact of the two circles. Similarly, consider the circle $\omega_C$ internally tangent to the sides $CB$ and $CA$, and to the circumcircle of the triangle $ABC$, let $Q$ be the point of contact of the two circles. Show that the incentre of the triangle $ABC$ lies on the segment $PQ$ if and only if $AB + AC = 3BC$. proposed by Luis Eduardo Garcia Hernandez, Mexico

Find the smallest positive integer $n$ such that for any $n$ distinct real numbers $b_1, b_2,\ldots ,b_n$ in the interval $[ 1, 1000 ]$ there always exist $b_i$ and $b_j$ such that: $$0<b_i-b_j<1+3\sqrt[3]{b_ib_j}$$

Three circles are pair-wise tangent to each other. Prove that the circle passing through the three tangent points is perpendicular to each of the initial three circles.

We are given $m\times n$ table tiled with $3\times 1$ stripes and we are given that $6 | mn$. Prove that there exists a tiling of the table with $2\times 1$ dominoes such that each of these stripes contains one whole domino.

Two circles each have radius 1. No point is inside both circles. The circles are contained in a square. What is the area of the smallest such square?