The National Foundation of Happiness (NFoH) wants to estimate the happiness of people of country. NFoH selected $n$ random persons, and on every morning asked from each of them whether she is happy or not. On any two distinct days, exactly half of the persons gave the same answer. Show that after $k$ days, there were at most $n-\frac{n}{k}$ persons whose “yes” answers equals their “no” answers.

Find the least prime factor of $2024^{2024}-1$.

Two magicians are performing the following game. Initially the first magician encloses the second magician in a cabin where he can neither see nor hear anything. To start the game, the first magician invites Daniel, from the audience, to put on each square of a chessboard $n \times n$, at his (Daniel's) discretion, a token black or white. Then the first magician asks Daniel to show him a square $C$ of his own choice. At this point, the first magician chooses a square $D$ (not necessarily different from $C$) and replaces the token that is on $D$ with other color token (white with black or black with white). Then he opens the cabin in which the second magician was held. Looking at the chessboard, the second magician guesses what is the square $C$. For what value of $n$, the two magicians have a strategy such that the second magician makes a successful guess.

Find all functions $f : Z \to Z$ such that $x f (2f (y) - x) + y^2 f (2x - f (y)) = \frac{(f (x))^2}{x} + f (y f (y))$ , for all $x, y \in Z$, $x \ne 0$.

Given a matrix $\{a_{ij}\}_{i,j=0}^{9}$, $a_{ij}=10i+j+1$. Andrei is going to cover its entries by $50$ rectangles $1\times 2$ (each such rectangle contains two adjacent entries) so that the sum of $50$ products in these rectangles is minimal possible. Help him. [i]A. Badzyan[/i]

Find all prime numbers $p$, for which the number $p + 1$ is equal to the product of all the prime numbers which are smaller than $p$.

Each square of a chessboard is painted either blue or red . Prove that the squares of one colour possess the property that the chess queen can perform a tour of all of them. The rules are that the queen may visit the squares of this colour not necessarily only once each , and may not be placed on squares of the other colour, although she may pass over them ; the queen moves along any horizontal , vertical or diagonal file over any distance. (A . K . Tolpugo , Kiev)

Are there any rational numbers $x,y$ with $x^2 + y^2 = 2015$?

Consider an equilateral triangle $\vartriangle ABC$ of side length $4$. In the zeroth iteration, draw a circle $\Omega_0$ tangent to all three sides of the triangle. In the first iteration, draw circles $\Omega_{1A}$,$ \Omega_{1B}$, $\Omega_{1C}$ such that circle $\Omega_{1v}$ is externally tangent to $\Omega_0$ and tangent to the two sides that meet at vertex $v$ (for example, $\Omega_{1A}$ would be tangent to $\Omega_0$ and sides $AB$, $AC$). In the nth iteration, draw circle $\Omega_{n,v}$ externally tangent to $\Omega_{n-1,v}$ and the two sides that meet at vertex $v$. Compute the total area of all the drawn circles as the number of iterations approaches infinity.

Find all triples $(a, b, c)$ of positive integers for which $a + [a, b] = b + [b, c] = c + [c, a]$. Here $[a, b]$ denotes the least common multiple of integers $a, b$. [i](Proposed by Mykhailo Shtandenko)[/i]

Let $ P(x) \equal{} (x \minus{} 1)(x \minus{} 2)(x \minus{} 3)$. For how many polynomials $ Q(x)$ does there exist a polynomial $ R(x)$ of degree 3 such that $ P(Q(x)) \equal{} P(x) \cdot R(x)$? $ \textbf{(A)}\ 19\qquad \textbf{(B)}\ 22\qquad \textbf{(C)}\ 24\qquad \textbf{(D)}\ 27\qquad \textbf{(E)}\ 32$

$ABCD$ is a cyclic quadrilateral in which $AB=4$, $BC=3$, $CD=2$, and $AD=5$. Diagonals $AC$ and $BD$ intersect at $X$. A circle $\omega$ passes through $A$ and is tangent to $BD$ at $X$. $\omega$ intersects $AB$ and $AD$ at $Y$ and $Z$ respectively. Compute $YZ/BD$.

Consider any rectangular table having finitely many rows and columns, with a real number $a(r, c)$ in the cell in row $r$ and column $c$. A pair $(R, C)$, where $R$ is a set of rows and $C$ a set of columns, is called a [i]saddle pair[/i] if the following two conditions are satisfied: [list] [*] $(i)$ For each row $r^{\prime}$, there is $r \in R$ such that $a(r, c) \geqslant a\left(r^{\prime}, c\right)$ for all $c \in C$; [*] $(ii)$ For each column $c^{\prime}$, there is $c \in C$ such that $a(r, c) \leqslant a\left(r, c^{\prime}\right)$ for all $r \in R$. [/list] A saddle pair $(R, C)$ is called a [i]minimal pair[/i] if for each saddle pair $\left(R^{\prime}, C^{\prime}\right)$ with $R^{\prime} \subseteq R$ and $C^{\prime} \subseteq C$, we have $R^{\prime}=R$ and $C^{\prime}=C$. Prove that any two minimal pairs contain the same number of rows.

A chord divides the interior of a circle $k$ into two parts. Variable circles $k_1$ and $k_2$ are inscribed in these two parts, touching the chord at the same point. Show that the ratio of the radii of circles $k_1$ and $k_2$ is constant, i.e. independent of the tangency point with the chord.

A cyclic $n$-gon is given. The midpoints of all its sides are concyclic. The sides of the $n$-gon cut $n$ arcs of this circle lying outside the $n$-gon. Prove that these arcs can be coloured red and blue in such a way that the sum of the lengths of the red arcs is equal to the sum of the lengths of the blue arcs.

For how many positive integers $ n$ is $ n^2\minus{}3n\plus{}2$ a prime number? $ \textbf{(A)}\ \text{none} \qquad \textbf{(B)}\ \text{one} \qquad \textbf{(C)}\ \text{two} \qquad \textbf{(D)}\ \text{more than two, but finitely many}\\ \textbf{(E)}\ \text{infinitely many}$

Let $k,l$ be two given integers. Prove that there exist infinite many integers $m\ge k$ such that $\gcd\left(\binom{m}{k},l\right)=1$.

Let $ {\mathbb Q}^ \plus{}$ be the set of positive rational numbers. Construct a function $ f : {\mathbb Q}^ \plus{} \rightarrow {\mathbb Q}^ \plus{}$ such that \[ f(xf(y)) \equal{} \frac {f(x)}{y} \] for all $ x$, $ y$ in $ {\mathbb Q}^ \plus{}$.

In the following diagram, points $E, F, G, H, I$, and $J$ lie on a circle. The triangle $ABC$ has side lengths $AB = 6$, $BC = 7$, and $CA = 9$. The three chords have lengths $EF = 12$, $GH = 15$, and $IJ = 16$. Compute $6 \cdot AE + 7 \cdot BG + 9 \cdot CI$. [img]https://cdn.artofproblemsolving.com/attachments/2/7/661b3d6a0f0baac0cd3b8d57c4cd4c62eeab46.png[/img]

Let $ABCD$ be a convex quadrilateral, and let $P$, $Q$, $R$, and $S$ be points on the sides $AB$, $BC$, $CD$, and $DA$, respectively. Let the line segment $PR$ and $QS$ meet at $O$. Suppose that each of the quadrilaterals $APOS$, $BQOP$, $CROQ$, and $DSOR$ has an incircle. Prove that the lines $AC$, $PQ$, and $RS$ are either concurrent or parallel to each other.

For a rational number $r$, its *period* is the length of the smallest repeating block in its decimal expansion. for example, the number $r=0.123123123...$ has period $3$. If $S$ denotes the set of all rational numbers of the form $r=\overline{abcdefgh}$ having period $8$, find the sum of all elements in $S$.

The incircle of the triangle $ABC$ touches the sides $BC, CA$ and $AB$ in points $A_1, B_1$ and $C_1$ respectively. $C_1D$ is a diameter of the incircle. Finally, let $E$ be the intersection of the lines $B_1C_1$ and $A_1D$. Prove that the segments $CE$ and $CB_1$ have equal length.

Evaluate $ \int_0^1 \frac{(1\minus{}2x)e^{x}\plus{}(1\plus{}2x)e^{\minus{}x}}{(e^x\plus{}e^{\minus{}x})^3}\ dx.$

Baron Minchausen talked to a mathematician. Baron said that in his country from any town one can reach any other town by a road. Also, if one makes a circular trip from any town, one passes through an odd number of other towns. By this, as an answer to the mathematician’s question, Baron said that each town is counted as many times as it is passed through. Baron also added that the same number of roads start at each town in his country, except for the town where he was born, at which a smaller number of roads start. Then the mathematician said that baron lied. How did he conclude that?

Find all triples $(x, y, z)$ of nonnegative integers such that $$ x^5+x^4+1=3^y7^z $$