Determine all $ f:R\rightarrow R $ such that $$ f(xf(y)+y^3)=yf(x)+f(y)^3 $$

Find all real numbers $ a,b,c,d$ such that \[ \left\{\begin{array}{cc}a \plus{} b \plus{} c \plus{} d \equal{} 20, \\ ab \plus{} ac \plus{} ad \plus{} bc \plus{} bd \plus{} cd \equal{} 150. \end{array} \right.\]

An $n\times n\times n$ cube is divided into unit cubes. We are given a closed non-self-intersecting polygon (in space), each of whose sides joins the centers of two unit cubes sharing a common face. The faces of unit cubes which intersect the polygon are said to be distinguished. Prove that the edges of the unit cubes may be colored in two colors so that each distinguished face has an odd number of edges of each color, while each nondistinguished face has an even number of edges of each color. [i]M. Smurov[/i]

Consider a function $f$ on nonnegative integers such that $f(0)=1, f(1)=0$ and $f(n)+f(n-1)=nf(n-1)+(n-1)f(n-2)$ for $n \ge 2$. Show that \[\frac{f(n)}{n!}=\sum_{k=0}^n \frac{(-1)^k}{k!}\]

Given is a chessboard 8x8. We have to place $n$ black queens and $n$ white queens, so that no two queens attack. Find the maximal possible $n$. (Two queens attack each other when they have different colors. The queens of the same color don't attack each other)

Let $a$ and $b$ be integers. Define $a$ to be a power of $b$ if there exists a positive integer $n$ such that $a = b^n$. Define $a$ to be a multiple of $b$ if there exists an integer $n$ such that $a = bn$. Let $x$, $y$ and $z$ be positive integer such that $z$ is a power of both $x$ and $y$. Decide for each of the following statements whether it is true or false. Prove your answers. (a) The number $x + y$ is even. (b) One of $x$ and $y$ is a multiple of the other one. (c) One of $x$ and $y$ is a power of the other one. (d) There exist an integer $v$ such that both $x$ and $y$ are powers of $v$ (e) For each power of $x$ and for each power of $y$, an integer $w$ can be found such that $w$ is a power of each of these powers. (f) There exists a positive integer $k$ such that $x^k > y$.

Show that \[(-1)^{\frac{p-1}{2}}{p-1 \choose{\frac{p-1}{2}}}\equiv 4^{p-1}\pmod{p^{3}}\] for all prime numbers $p$ with $p \ge 5$.

Let $A_1A_2 \dots A_{13}$ and $B_1B_2 \dots B_{13}$ be two regular $13$-gons in the plane such that the points $B_1$ and $A_{13}$ coincide and lie on the segment $A_1B_{13}$, and both polygons lie in the same semiplane with respect to this segment. Prove that the lines $A_1A_9, B_{13}B_8$ and $A_8B_9$ are concurrent.

Consider a container of a hollow cube $ABGCDEPF$ (where $ABGC$, $DEPF$ are squares and $AD\parallel BE\parallel GP\parallel CF$). The cube is placed on a table in a way that the space diagonal $AP=1$ is perpendicular to the table. Then, water is poured into the cube. Denote $x$ the length of part of $AP$ submerged in water. Determine the volume of water $y$ in terms of $x$ when a) $0 < x \leq\frac13$, b) $\frac13 < x \leq\frac12$.

Compute the sum of the series $\sum_{k=0}^{\infty} \frac{1}{(4k+1)(4k+2)(4k+3)(4k+4)} = \frac{1}{1\cdot2\cdot3\cdot4} + \frac{1}{5\cdot6\cdot7\cdot8} + ...$

Prove that $102^{1991} + 103^{1991}$ is not a proper power of an integer.

For x > 1, let $f(x) = log_2(x + log_2(x + log_2(x +...)))$. Compute $\Sigma_{k=2}^{10} f^{-1}(k)$

In triangle $ABC$, let $D$ be the midpoint of $BC$, and $BE$, $CF$ are the altitudes. Prove that $DE$ and $DF$ are both tangents to the circumcircle of triangle $AEF$

a) There is an infinite sequence of $0,1$, like $\dots,a_{-1},a_{0},a_{1},\dots$ (i.e. an element of $\{0,1\}^{\mathbb Z}$). At each step we make a new sequence. There is a function $f$ such that for each $i$, $\mbox{new }a_{i}=f(a_{i-100},a_{i-99},\dots,a_{i+100})$. This operation is mapping $F: \{0,1\}^{\mathbb Z}\longrightarrow\{0,1\}^{\mathbb Z}$. Prove that if $F$ is 1-1, then it is surjective. b) Is the statement correct if we have an $f_{i}$ for each $i$?

Prove for all $a_1, ..., a_n > 0$ the following inequality and determine all cases in where the equaloty holds: $$\sum_{k=1}^{n}ka_k\le {n \choose 2}+\sum_{k=1}^{n}a_k^k.$$

Let \( ABC \) be a triangle with \( AB < AC \). Let \( M \) be the midpoint of \( AC \), and let \( D \) be a point on segment \( AC \) such that \( DB = DC \). Let \( E \) be the point of intersection, different from \( B \), of the circumcircle of triangle \( ABM \) and line \( BD \). Define \( P \) and \( Q \) as the points of intersection of line \( BC \) with \( EM \) and \( AE \), respectively. Prove that \( P \) is the midpoint of \( BQ \).

Let $ x_1$, $ x_2$, $ \dots$, $ x_n$ be a sequence of integers such that (i) $ \minus{}1 \le x_i \le 2$, for $ i \equal{} 1,2,3,\dots,n$; (ii) $ x_1 \plus{} x_2 \plus{} \cdots \plus{} x_n \equal{} 19$; and (iii) $ x_1^2 \plus{} x_2^2 \plus{} \cdots \plus{} x_n^2 \equal{} 99$. Let $ m$ and $ M$ be the minimal and maximal possible values of $ x_1^3 \plus{} x_2^3 \plus{} \cdots \plus{} x_n^3$, respectively. Then $ \frac{M}{m} \equal{}$ $ \textbf{(A)}\ 3\qquad \textbf{(B)}\ 4\qquad \textbf{(C)}\ 5\qquad \textbf{(D)}\ 6\qquad \textbf{(E)}\ 7$

Find all polynomials $p(x)$ satisfying the condition: $$p(x^2-2x)=p(x-2)^2.$$

Which figure can be the images of equations $mx+ny^2=0$ and $mx^2+ny^2=1$$(m,n\neq0)$? [img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvZC9kLzUzZTAwZjU1YzEyN2I3ZDJjNjcwNDQ2ZmQ5MDBmYWZlODAwNGU0LnBuZw==&rn=YWFhYWFhYWFhYWFhYWFhYWFhYWFhYWFhYWFhYWFhYWFhYWFhYS5wbmc=[/img]

Let $f(x) = x^2 + 6x + 7$. Determine the smallest possible value of $f(f(f(f(x))))$ over all real numbers $x.$

$O$ is the point of intersection of the diagonals of the convex quadrilateral $ABCD$. It is known that the areas of triangles $AOB, BOC, COD$ and $DOA$ are expressed in natural numbers. Prove that the product of these areas cannot end in $1985$.

For a positive integer $p$, define the positive integer $n$ to be $p$-safe if $n$ differs in absolute value by more than $2$ from all multiples of $p$. For example, the set of $10$-safe numbers is $\{3, 4, 5, 6, 7, 13, 14, 15, 16, 17,23, \ldots \}$. Find the number of positive integers less than or equal to $10,000$ which are simultaneously $7$-safe, $11$-safe, and $13$-safe.·

In triangle $ABC, AH$ is an altitude ($H$ is on $BC$) and $BE$ is a bisector ($E$ is on $AC$) . We are given that angle $BEA$ equals $45^o$ .Prove that angle $EHC$ equals $45^o$ . (I. Sharygin , Moscow)

Let $P(x) \in \mathbb{R}[x]$ be a monic, non-constant polynomial. Determine all continuous functions $f: \mathbb{R} \to \mathbb{R}$ such that $$f(f(P(x))+y+2023f(y))=P(x)+2024f(y),$$ for all reals $x,y$.

On each of the cards written in $2013$ by number, all of these $2013$ numbers are different. The cards are turned down by numbers. In a single move is allowed to point out the ten cards and in return will report one of the numbers written on them (do not know what). For what most $w$ guaranteed to be able to find $w$ cards for which we know what numbers are written on each of them?