Points $B_1$ and $C_1$ are marked respectively on the sides $AB$ and $AC$ of an acute isosceles triangle $ABC$( $AB=AC$) such that $BB_1=AC_1$. The points $B,C$ and $S$ lie in the same half-plane with respect to the line $B_1C_1$ so that $\angle SB_1C_1=\angle SC_1B_1 = \angle BAC$ Prove that $B,C,S$ are colinear if and only if the triangle $ABC$ is equilateral.

Let $k$ be a fixed real number. Find all functions $f: R \to R$ such that $f(x)+ (f(y))^2 = kf(x + y^2)$ for all real numbers $x$ and $y$.

Let $n$ and $k$ be positive integers. A baby uses $n^2$ blocks to form a $n\times n$ grid, with each of the blocks having a positive integer no greater than $k$ on it. The father passes by and notice that: 1. each row on the grid can be viewed as an arithmetic sequence with the left most number being its leading term, with all of them having distinct common differences; 2. each column on the grid can be viewed as an arithmetic sequence with the top most number being its leading term, with all of them having distinct common differences, Find the smallest possible value of $k$ (as a function of $n$.) Note: The common differences might not be positive. Proposed by Chu-Lan Kao

Find the number of ordered pairs $(a, b, c, d)$ of positive integers satisfying the equation: \[a + 2b + 3c + 1000d = 2024.\] [i]Proposed by Irakli Khutsishvili, Georgia [/i]

Given $n$ positive real numbers $a_1, a_2, \ldots , a_n$ such that $a_1a_2 \cdots a_n = 1$, prove that \[(1 + a_1)(1 + a_2) \cdots (1 + a_n) \geq 2^n.\]

Let $m,n$ be positive integers, greater than 2.Find the number of polynomials of degree $2n-1$ with distinct coefficients from the set $\left\{ 1,2,\ldots,m\right\}$ which are divisible by $x^{n-1}+x^{n-2}+\ldots+1.$

Given is a cyclic quadrilateral $ABCD$ and a point $E$ lies on the segment $DA$ such that $2\angle EBD = \angle ABC$. Prove that $DE= \frac {AC.BD}{AB+BC}$.

Let $A\subset \{1,2,\ldots,4014\}$, $|A|=2007$, such that $a$ does not divide $b$ for all distinct elements $a,b\in A$. For a set $X$ as above let us denote with $m_{X}$ the smallest element in $X$. Find $\min m_{A}$ (for all $A$ with the above properties).

Let $ n > 1$ be an integer, and $ n$ can divide $ 2^{\phi(n)} \plus{} 3^{\phi(n)} \plus{} \cdots \plus{} n^{\phi(n)},$ let $ p_{1},p_{2},\cdots,p_{k}$ be all distinct prime divisors of $ n$. Show that $ \frac {1}{p_{1}} \plus{} \frac {1}{p_{2}} \plus{} \cdots \plus{} \frac {1}{p_{k}} \plus{} \frac {1}{p_{1}p_{2}\cdots p_{k}}$ is an integer. ( where $ \phi(n)$ is defined as the number of positive integers $ \leq n$ that are relatively prime to $ n$.)

Let f:N→N be a function satisfying $ kf(n)\le f(kn)\le kf(n) \plus{} k \minus{} 1$ for all $ k, n\in N$. (a)Prove that $ f(a) \plus{} f(b)\le f(a \plus{} b)\le f(a) \plus{} f(b) \plus{} 1$ for all $ a, b\in N$. (b)If $ f$ satisfies $ f(2007n)\le 2007f(n) \plus{} 200$ for every $ n\in N$, show that there exists $ c\in N$ such that $ f(2007c) \equal{} 2007f(c)$.

In the space, there is a convex polyhedron $D$ such that for every vertex of $D$, there are an even number of edges passing through that vertex. We choose a face $F$ of $D$. Then we assign each edge of $D$ a positive integer such that for all faces of $D$ different from $F$, the sum of the numbers assigned on the edges of that face is a positive integer divisible by $2024$. Prove that the sum of the numbers assigned on the edges of $F$ is also a positive integer divisible by $2024$.

On a circle of center $O$, let $A$ and $B$ be points on the circle such that $\angle AOB = 120^o$. Point $C$ lies on the small arc $AB$ and point $D$ lies on the segment $AB$. Let also $AD = 2, BD = 1$ and $CD = \sqrt2$. Calculate the area of triangle $ABC$.

Suppose $1, 2, 3$ are the roots of the equation $x^4 + ax^2 + bx = c$. Find the value of $c$.

Let $(A,+,\cdot)$ be a finite unit ring, with $n\geq 3$ elements in which there exist [b]exactly[/b] $\dfrac {n+1}2$ perfect squares (e.g. a number $b\in A$ is called a perfect square if and only if there exists an $a\in A$ such that $b=a^2$). Prove that a) $1+1$ is invertible; b) $(A,+,\cdot)$ is a field. [i]Proposed by Marian Andronache[/i]

Let $m \leq n$ be positive integers and $p$ be a prime. Let $p-$expansions of $m$ and $n$ be \[m = a_0 + a_1p + \dots + a_rp^r\]\[n = b_0 + b_1p + \dots + b_sp^s\] respectively, where $a_r, b_s \neq 0$, for all $i \in \{0,1,\dots,r\}$ and for all $j \in \{0,1,\dots,s\}$, we have $0 \leq a_i, b_j \leq p-1$ . If $a_i \leq b_i$ for all $i \in \{0,1,\dots,r\}$, we write $ m \prec_p n$. Prove that \[p \nmid {{n}\choose{m}} \Leftrightarrow m \prec_p n\].

In the interior of the square $ABCD$, the point $P$ lies in such a way that $\angle DCP = \angle CAP=25^{\circ}$. Find all possible values of $\angle PBA$.

A convex quadrilateral has equal diagonals. An equilateral triangle is constructed on the outside of each side of the quadrilateral. The centers of the triangles on opposite sides are joined. Show that the two joining lines are perpendicular. [i]Alternative formulation.[/i] Given a convex quadrilateral $ ABCD$ with congruent diagonals $ AC \equal{} BD.$ Four regular triangles are errected externally on its sides. Prove that the segments joining the centroids of the triangles on the opposite sides are perpendicular to each other. [i]Original formulation:[/i] Let $ ABCD$ be a convex quadrilateral such that $ AC \equal{} BD.$ Equilateral triangles are constructed on the sides of the quadrilateral. Let $ O_1,O_2,O_3,O_4$ be the centers of the triangles constructed on $ AB,BC,CD,DA$ respectively. Show that $ O_1O_3$ is perpendicular to $ O_2O_4.$

Chitoge is painting a cube; she can paint each face either black or white, but she wants no vertex of the cube to be touching three faces of the same color. In how many ways can Chitoge paint the cube? Two paintings of a cube are considered to be the same if you can rotate one cube so that it looks like the other cube.

Let $a_1, a_2, \dots, a_6$ be a permutation of $1, 2, 2, 3, 4, 4$. Let $b_i = 5 - a_i$. Find the minimum value of \[\sum_{i=1}^7{\left(\prod_{j=0}^{i-1}{b_j}\right)\left(\prod_{j=i}^{6}{a_j}\right)}.\] [i]Lightning 4.1[/i]

Consider a right triangle $\triangle ABC$ with right angle at $A$. Let $CD$ be the bisector of angle $\angle ACB$, where $D$ lies on segment $AB$. The perpendicular line from $B$ to $BC$ intersects $CD$ at $E$. Let $F$ be the reflection of $E$ over $B$, and let $P$ be the intersection of $DF$ with $BC$. Prove that lines $EP$ and $CF$ are perpendicular.

Prove such $a$, that all the numbers $\cos a, \cos 2a, \cos 4a, ... , \cos (2^na)$ are negative.

Given a convex quadrilateral with $\angle B+\angle D<180$.Let $P$ be an arbitrary point on the plane,define $f(P)=PA*BC+PD*CA+PC*AB$. (1)Prove that $P,A,B,C$ are concyclic when $f(P)$ attains its minimum. (2)Suppose that $E$ is a point on the minor arc $AB$ of the circumcircle $O$ of $ABC$,such that$AE=\frac{\sqrt 3}{2}AB,BC=(\sqrt 3-1)EC,\angle ECA=2\angle ECB$.Knowing that $DA,DC$ are tangent to circle $O$,$AC=\sqrt 2$,find the minimum of $f(P)$.

Prove that in an isosceles trapezoid circumscibed around a circle, the segments connecting the points of tangency of opposite sides with the circle pass through the point of intersection of the diagonals.

\[ a\plus{}4b\plus{}9c\plus{}16d\plus{}25e\plus{}36f\plus{}49g\equal{}1\] \[ 4a\plus{}9b\plus{}16c\plus{}25d\plus{}36e\plus{}49f\plus{}64g\equal{}12\] \[ 9a\plus{}16b\plus{}25c\plus{}36d\plus{}49e\plus{}64f\plus{}81g\equal{}123\] Determine the value of $ 16a\plus{}25b\plus{}36c\plus{}49d\plus{}64e\plus{}81f\plus{}100g$.

Let a triangle $ABC$ inscribed in circle $\Gamma$ be given. Circle $\Theta$ lies in angle $Â$ of triangle and touches sides $AB, AC$ at $M_1, N_1$ and touches internally $\Gamma$ at $P_1$. The points $M_2, N_2, P_2$ and $M_3, N_3, P_3$ are defined similarly to angles $B$ and $C$ respectively. Show that $M_1N_1, M_2N_2$ and $M_3N_3$ intersect each other at their midpoints.