Find all functions $f:\mathbb{N}\rightarrow \mathbb{N}$ such that the inequality $$f(x)+yf(f(x))\le x(1+f(y))$$ holds for all positive integers $x, y$. Proposed by Adrian Beker.

[b]M[/b]ary has a sequence $m_2,m_3,m_4,...$ , such that for each $b \ge 2$, $m_b$ is the least positive integer m for which none of the base-$b$ logarithms $log_b(m),log_b(m+1),...,log_b(m+2017)$ are integers. Find the largest number in her sequence.

Let $[ABCD]$ be a convex quadrilateral with area $2014$, and let $P$ be a point on $[AB]$ and $Q$ a point on $[AD]$ such that triangles $[ABQ]$ and $[ADP]$ have area $1$. Let $R$ be the intersection of $[AC]$ and $[PQ]$. Determine $\frac{\overline{RC}}{\overline{RA}}$.

Let $n$ be a positive integer divisible by $4$. Find the number of permutations $\sigma$ of $(1,2,3,\cdots,n)$ which satisfy the condition $\sigma(j)+\sigma^{-1}(j)=n+1$ for all $j \in \{1,2,3,\cdots,n\}$.

At the round table, $10$ people are sitting, some of them are knights, and the rest are liars (knights always say pride, and liars always lie) . It is clear thath I have at least one knight and at least one liar. What is the largest number of those sitting at the table can say: ''Both of my neighbors are knights '' ? (A statement that is at least partially false is considered false.)

Andrew has three identical semicircular mooncake halves, each with radius $3,$ and uses them to construct the following shape, which contains an equilateral triangle in the center. Compute the perimeter around this shape, in bold below. [center] [img] https://cdn.artofproblemsolving.com/attachments/7/2/2314ac2d34cd0706f47bace3eedbb87a91582a.png [/img] [/center]

Let $C$ be the coefficient of $x^2$ in the expansion of the product \[(1 - x)(1 + 2x)(1 - 3x)\cdots(1 + 14x)(1 - 15x).\] Find $|C|$.

a. Does there exist 4 distinct positive integers such that the sum of any 3 of them is prime? b. Does there exist 5 distinct positive integers such that the sum of any 3 of them is prime?

A real number $x$ is chosen uniformly at random from the interval $(0,10).$ Compute the probability that $\sqrt{x}, \sqrt{x+7},$ and $\sqrt{10-x}$ are the side lengths of a non-degenerate triangle.

Denote by $l(n)$ the largest prime divisor of $n$. Let $a_{n+1} = a_n + l(a_n)$ be a recursively defined sequence of integers with $a_1 = 2$. Determine all natural numbers $m$ such that there exists some $i \in \mathbb{N}$ with $a_i = m^2$. [i]Proposed by Nikola Velov, North Macedonia[/i]

If $y$ varies directly as $x$, and if $y=8$ when $x=4$, the value of $y$ when $x=-8$ is: ${{ \textbf{(A)}\ -16} \qquad\textbf{(B)}\ -4 \qquad\textbf{(C)}\ -2 \qquad\textbf{(D)}\ 4k, k= \pm1, \pm2, \dots}$ ${\qquad\textbf{(E)}\ 16k, k=\pm1,\pm2,\dots } $

The radius of the circle inscribed in triangle $ABC$ is equal to $r$, and the radius of the circle tangent to the segment $BC$ and the extensions of sides $AB$ and $AC$ (the exscribed circle corresponding to angle $A$) is equal to $R$. A circle with radius $x < r$ is inscribed in angle $\angle BAC$. Tangents to this circles passing through points $B$ and $C$ and different from $BA$ and $AC$ intersect at point $A'$. Let $y$ be the radius of the circle inscribed in triangle $BCK$. Find the greatest value of the sum $x + y$ as x changes from $0$ to $r$. (In this case, it is necessary to prove that this largest value is the same in any triangle with given $r$ and $R$).

In the triangle $ABC$ we have $\angle A = 2\angle C$ and $2\angle B = \angle A + \angle C$. The angle bisector of $\angle C$ intersects the segment $AB$ in $E$, let $F$ be the midpoint of $AE$, let $AD$ be the altitude of the triangle $ABC$. The perpendicular bisector of $DF$ intersects $AC$ in $M$. Prove that $AM = CM$.

Determine all functions $f:\mathbb{R}\to\mathbb{R}$ satisfying \[f(xy+f(x))+f(y)=xf(y)+f(x+y),\]for all real numbers $x,y$.

There are six empty purses on the table. How many ways are there to put 12 two-euro coins in purses in such a way that at most one purse remains empty?

Among any four people at a party there is one who has met the three others before the party. Show that among any four people at the party there must be one who has met everyone at the party before the party

Find all real numbers $a > 1$ such that there exists an integer $k \ge 1$ such that the sequence $\{x_n\}_{n\ge 1}$ formed with the first $k$ digits of the number $\lfloor a^n\rfloor$ is periodical.

Let $H$ be the orthocenter of an acute triangle $ABC,$ and let $A_1, \: B_1, \: C_1$ be the feet of the altitudes belonging to the vertices $A, \: B, \: C,$ respectively. Let $K$ be a point on the smaller $AB_1$ arc of the circle with diameter $AB$ satisfying the condition $\angle HKB = \angle C_1KB.$ Let $M$ be the point of intersection of the line segment $AA_1$ and the circle with center $C$ and radius $CL$ where $KB \cap CC_1=\{L\}.$ Let $P$ and $Q$ be the points of intersection of the line $CC_1$ and the circle with center $B$ and radius $BM.$ Show that $A, \: K, \: P, \: Q$ are concyclic.

For a real number $x$, let $\lfloor x\rfloor$ stand for the largest integer that is less than or equal to $x$. Prove that \[ \left\lfloor{(n-1)!\over n(n+1)}\right\rfloor \] is even for every positive integer $n$.

A capacitor is made with two square plates, each with side length $L$, of negligible thickness, and capacitance $C$. The two-plate capacitor is put in a microwave which increases the side length of each square plate by $ 1 \% $. By what percent does the voltage between the two plates in the capacitor change? $ \textbf {(A) } \text {decreases by } 2\% \\ \textbf {(B) } \text {decreases by } 1\% \\ \textbf {(C) } \text {it does not change} \\ \textbf {(D) } \text {increases by } 1\% \\ \textbf {(E) } \text {increases by } 2\% $ [i]Problem proposed by Ahaan Rungta[/i]

Let $m$ and $n$ be positive integers such that $m>n$. Define $x_k=\frac{m+k}{n+k}$ for $k=1,2,\ldots,n+1$. Prove that if all the numbers $x_1,x_2,\ldots,x_{n+1}$ are integers, then $x_1x_2\ldots x_{n+1}-1$ is divisible by an odd prime.

Define a "hook" to be a figure made up of six unit squares as shown below in the picture, or any of the figures obtained by applying rotations and reflections to this figure. [asy] unitsize(0.5 cm); draw((0,0)--(1,0)); draw((0,1)--(1,1)); draw((2,1)--(3,1)); draw((0,2)--(3,2)); draw((0,3)--(3,3)); draw((0,0)--(0,3)); draw((1,0)--(1,3)); draw((2,1)--(2,3)); draw((3,1)--(3,3)); [/asy] Determine all $ m\times n$ rectangles that can be covered without gaps and without overlaps with hooks such that - the rectangle is covered without gaps and without overlaps - no part of a hook covers area outside the rectangle.

Consider integers $ a_i,i\equal{}\overline{1,2002}$ such that $ a_1^{ \minus{} 3} \plus{} a_2^{ \minus{} 3} \plus{} \ldots \plus{} a_{2002}^{ \minus{} 3} \equal{} \frac {1}{2}$ Prove that at least 3 of the numbers are equal.

The Princeton University Band plays a setlist of 8 distinct songs, 3 of which are tiring to play. If the Band can't play any two tiring songs in a row, how many ways can the band play its 8 songs?

Patty is standing on a line of planks playing a game. Define a block to be a sequence of adjacent planks, such that both ends are not adjacent to any planks. Every minute, a plank chosen uniformly at random from the block that Patty is standing on disappears, and if Patty is standing on the plank, the game is over. Otherwise, Patty moves to a plank chosen uniformly at random within the block she is in; note that she could end up at the same plank from which she started. If the line of planks begins with $n$ planks, then for sufficiently large n, the expected number of minutes Patty lasts until the game ends (where the first plank disappears a minute after the game starts) can be written as $P(1/n)f(n) + Q(1/n)$, where $P,Q$ are polynomials and $f(n) =\sum^n_{i=1}\frac{1}{i}$ . Find $P(2023) + Q(2023)$.