Determine all functions $f:\mathbb{N}\to \mathbb{N}$ such that for all $a, b \in \mathbb{N}$ the following conditions hold: $(i)$ $f(f(a)+b) \mid b^a-1$; $(ii)$ $f(f(a))\geq f(a)-1$.

The average of the five numbers in a list is $54$. The average of the first two numbers is $48$. What is the average of the last three numbers? $\textbf{(A)}\ 55\qquad \textbf{(B)}\ 56\qquad \textbf{(C)}\ 57\qquad \textbf{(D)}\ 58 \qquad \textbf{(E)}\ 59$

Let $n$ and $k$ be positive integers with $k$ $\le$ $n$. In a group of $n$ people, each one or always speak the truth or always lie. Arnaldo can ask questions for any of these people provided these questions are of the type: “In set $A$, what is the parity of people who speak to true? ”, where $A$ is a subset of size $ k$ of the set of $n$ people. The answer can only be “$even$” or “$odd$”. a) For which values of $n$ and $k$ is it possible to determine which people speak the truth and which people always lie? b) What is the minimum number of questions required to determine which people speak the truth and which people always lie, when that number is finite?

In a triangle $ABC$ with $BC = CA + \frac 12 AB$, point $P$ is given on side $AB$ such that $BP : PA = 1 : 3$. Prove that $\angle CAP = 2 \angle CPA.$

Let $P(x) = x^3 - 3x^2 + 3$. For how many positive integers $n < 1000$ does there not exist a pair $(a, b)$ of positive integers such that the equation \[ \underbrace{P(P(\dots P}_{a \text{ times}}(x)\dots))=\underbrace{P(P(\dots P}_{b \text{ times}}(x)\dots))\] has exactly $n$ distinct real solutions?

Let $ABCD$ by a cyclic quadrilateral with circumcenter $O$. Let $P$ be the point of intersection of the diagonals $AC$ and $BD$, and $K, L, M, N$ the circumcenters of triangles $AOP, BOP$, $COP, DOP$, respectively. Prove that $KL = MN$.

Given an acute triangle $ABC$. The inscribed circle of triangle $ABC$ is tangent to $AB$ and $AC$ at $X$ and $Y$ respectively. Let $CH$ be the altitude. The perpendicular bisector of the segment $CH$ intersects the line $XY$ at $Z$. Prove that $\angle BZC = 90^o.$

Find all triangles whose side lengths are consecutive integers, and one of whose angles is twice another.

Which one of the following bar graphs could represent the data from the circle graph? [asy] unitsize(36); draw(circle((0,0),1),gray); fill((0,0)--arc((0,0),(0,-1),(1,0))--cycle,gray); fill((0,0)--arc((0,0),(1,0),(0,1))--cycle,black); [/asy] [asy] unitsize(4); fill((1,0)--(1,15)--(5,15)--(5,0)--cycle,gray); fill((6,0)--(6,15)--(10,15)--(10,0)--cycle,black); draw((11,0)--(11,20)--(15,20)--(15,0)); fill((26,0)--(26,15)--(30,15)--(30,0)--cycle,gray); fill((31,0)--(31,15)--(35,15)--(35,0)--cycle,black); draw((36,0)--(36,15)--(40,15)--(40,0)); fill((51,0)--(51,10)--(55,10)--(55,0)--cycle,gray); fill((56,0)--(56,10)--(60,10)--(60,0)--cycle,black); draw((61,0)--(61,20)--(65,20)--(65,0)); fill((76,0)--(76,10)--(80,10)--(80,0)--cycle,gray); fill((81,0)--(81,15)--(85,15)--(85,0)--cycle,black); draw((86,0)--(86,20)--(90,20)--(90,0)); fill((101,0)--(101,15)--(105,15)--(105,0)--cycle,gray); fill((106,0)--(106,10)--(110,10)--(110,0)--cycle,black); draw((111,0)--(111,20)--(115,20)--(115,0)); for(int a = 0; a < 5; ++a) { draw((25*a,21)--(25*a,0)--(25*a+16,0)); } label("(A)",(8,21),N); label("(B)",(33,21),N); label("(C)",(58,21),N); label("(D)",(83,21),N); label("(E)",(108,21),N); [/asy]

Let $ n$ be a composite natural number and $ p$ a proper divisor of $ n.$ Find the binary representation of the smallest natural number $ N$ such that \[ \frac{(1 \plus{} 2^p \plus{} 2^{n\minus{}p})N \minus{} 1}{2^n}\] is an integer.

Consider a triangle $ ABC $ having incenter $ I $ and inradius $ r. $ Let $ D $ be the tangency of $ ABC $ 's incircle with $ BC, $ and $ E $ on the line $ BC $ such that $ AE $ is perpendicular to $ BC, $ and $ M\neq E $ on the segment $ AE $ such that $ AM=r. $ [b]a)[/b] Give an idenity for $ \frac{BD}{DC} $ involving only the lengths of the sides of the triangle. [b]b)[/b] Prove that $ AB \cdot \overrightarrow{IC} +BC\cdot \overrightarrow{IA} +CA\cdot \overrightarrow{IB} =0. $ [b]c)[/b] Show that $ MI $ passes through the middle of the side $ BC. $ [i]Cătălin Zârnă[/i]

[b]p1.[/b] Find the smallest positive integer $N$ such that $2N -1$ and $2N +1$ are both composite. [b]p2.[/b] Compute the number of ordered pairs of integers $(a, b)$ with $1 \le a, b \le 5$ such that $ab - a - b$ is prime. [b]p3.[/b] Given a semicircle $\Omega$ with diameter $AB$, point $C$ is chosen on $\Omega$ such that $\angle CAB = 60^o$. Point $D$ lies on ray $BA$ such that $DC$ is tangent to $\Omega$. Find $\left(\frac{BD}{BC} \right)^2$. [b]p4.[/b] Let the roots of $x^2 + 7x + 11$ be $r$ and $s$. If $f(x)$ is the monic polynomial with roots $rs + r + s$ and $r^2 + s^2$, what is $f(3)$? [b]p5.[/b] Regular hexagon $ABCDEF$ has side length $3$. Circle $\omega$ is drawn with $AC$ as its diameter. $BC$ is extended to intersect $\omega$ at point $G$. If the area of triangle $BEG$ can be expressed as $\frac{a\sqrt{b}}{c}$ for positive integers $a, b, c$ with $b$ squarefree and $gcd(a, c) = 1$, find $a + b + c$. [b]p6.[/b] Suppose that $x$ and $y$ are positive real numbers such that $\log_2 x = \log_x y = \log_y 256$. Find $xy$. [b]p7.[/b] Call a positive three digit integer $\overline{ABC}$ fancy if $\overline{ABC} = (\overline{AB})^2 - 11 \cdot \overline{C}$. Find the sum of all fancy integers. [b]p8.[/b] Let $\vartriangle ABC$ be an equilateral triangle. Isosceles triangles $\vartriangle DBC$, $\vartriangle ECA$, and $\vartriangle FAB$, not overlapping $\vartriangle ABC$, are constructed such that each has area seven times the area of $\vartriangle ABC$. Compute the ratio of the area of $\vartriangle DEF$ to the area of $\vartriangle ABC$. [b]p9.[/b] Consider the sequence of polynomials an(x) with $a_0(x) = 0$, $a_1(x) = 1$, and $a_n(x) = a_{n-1}(x) + xa_{n-2}(x)$ for all $n \ge 2$. Suppose that $p_k = a_k(-1) \cdot a_k(1)$ for all nonnegative integers $k$. Find the number of positive integers $k$ between $10$ and $50$, inclusive, such that $p_{k-2} + p_{k-1} = p_{k+1} - p_{k+2}$. [b]p10.[/b] In triangle $ABC$, point $D$ and $E$ are on line segments $BC$ and $AC$, respectively, such that $AD$ and $BE$ intersect at $H$. Suppose that $AC = 12$, $BC = 30$, and $EC = 6$. Triangle BEC has area 45 and triangle $ADC$ has area $72$, and lines CH and AB meet at F. If $BF^2$ can be expressed as $\frac{a-b\sqrt{c}}{d}$ for positive integers $a$, $b$, $c$, $d$ with c squarefree and $gcd(a, b, d) = 1$, then find $a + b + c + d$. [b]p11.[/b] Find the minimum possible integer $y$ such that $y > 100$ and there exists a positive integer x such that $x^2 + 18x + y$ is a perfect fourth power. [b]p12.[/b] Let $ABCD$ be a quadrilateral such that $AB = 2$, $CD = 4$, $BC = AD$, and $\angle ADC + \angle BCD = 120^o$. If the sum of the maximum and minimum possible areas of quadrilateral $ABCD$ can be expressed as $a\sqrt{b}$ for positive integers $a$, $b$ with $b$ squarefree, then find $a + b$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

The height and radius of the base of the cylinder are equal to $1$. What is the smallest number of balls of radius $1$ that can cover the entire cylinder?

Determine all pairs of natural numbers $(m, r)$ with $2014 \ge m \ge r \ge 1$ that fulfill $\binom{2014}{m}+\binom{m}{r}=\binom{2014}{r}+\binom{2014-r}{m-r} $

Let $ABCD$ be a cyclic quadrilateral, and angle bisectors of $\angle BAD$ and $\angle BCD$ meet at point $I$. Show that if $\angle BIC = \angle IDC$, then $I$ is the incenter of triangle $ABD$.

Yuri and Vlad are playing a game on the table $4 \times 100$. Firstly, Yuri chooses $73$ squares $2 \times 2$ (squares can intersect, but cannot be equal). Then Vlad colours the cells of the table in $4$ colours such that in any row and in any column, and in any square chosen by Yuri, there were cells of all 4 colours. After that Vlad pays 2 rubles for every square $2 \times 2$, not chosen by Yuri, which cells of all 4 colours. What is the maximum possible number of rubles Yuri can get regardless of Vlad's actions [i]M. Shutro[/i]

Given positive integers $n$ and $k$, $n > k^2 >4.$ In a $n \times n$ grid, a $k$[i]-group[/i] is a set of $k$ unit squares lying in different rows and different columns. Determine the maximal possible $N$, such that one can choose $N$ unit squares in the grid and color them, with the following condition holds: in any $k$[i]-group[/i] from the colored $N$ unit squares, there are two squares with the same color, and there are also two squares with different colors.

For a positive integer that doesn’t end in $0$, define its reverse to be the number formed by reversing its digits. For instance, the reverse of $102304$ is $403201$. In terms of $n \geq 1$, how many numbers when added to its reverse give $10^{n}-1$, the number consisting of $n$ nines?

$30$ volleyball teams have participated in a league. Any two teams have played a match with each other exactly once. At the end of the league, a match is called [b]unusual[/b] if at the end of the league, the winner of the match have a smaller amount of wins than the loser of the match. A team is called [b]astonishing[/b] if all its matches are [b]unusual[/b] matches. Find the maximum number of [b]astonishing[/b] teams.

[b]Problem 1 [/b] Let $A$ and $B$ be two $3 \times 3$ matrices with real entries. Prove that $$ A-(A^{-1} +(B^{-1}-A)^{-1})^{-1} =ABA\ , $$ provided all the inverses appearing on the left-hand side of the equality exist.

Let \[ T \equal{} \frac1{3\minus{}\sqrt8} \minus{} \frac1{\sqrt8 \minus{} \sqrt7} \plus{} \frac1{\sqrt7\minus{}\sqrt6} \minus{} \frac1{\sqrt6\minus{}\sqrt5} \plus{} \frac1{\sqrt5\minus{}2}\] then $ \textbf{(A)}\ T<1 \qquad \textbf{(B)}\ T\equal{}1 \qquad \textbf{(C)}\ 1<T<2 \qquad \textbf{(D)}\ T>2 \qquad$ $ \textbf{(E)}\ T \equal{} \frac1{(3\minus{}\sqrt8)(\sqrt8\minus{}\sqrt7)(\sqrt7\minus{}\sqrt6)(\sqrt6\minus{}\sqrt5)(\sqrt5\minus{}2)}$

Does there exist a natural number $n$ such that $n!$ ends with exactly $2021$ zeros?

The shores of the Tvertsy River are two parallel straight lines. There are point-like villages on the shores in some order: 20 villages on the left shore and 15 villages on the right shore. We want to build a system of non-intersecting bridges, that is, segments connecting a couple of villages from different shores, so that from any village you can get to any other village only by bridges (you can't walk along the shore). In how many ways can such a bridge system be built?

The circles $k_1$ and $k_2$ intersect at points $D$ and $P$. The common tangent of the two circles on the side of $D$ touches $k_1$ at $A$ and $k_2$ at $B$. The straight line $AD$ intersects $k_2$ for a second time at $C$. Let $M$ be the center of the segment $BC$. Show that $ \angle DPM = \angle BDC$ .

What is the greatest number of regions into which four planes can divide three-dimensional space?