Given $k\ge 2$, for which polynomials $P\in \mathbb{Z}[X]$ does there exist a function $h:\mathbb{N}\rightarrow\mathbb{N}$ with $h^{(k)}(n)=P(n)$?

Find all functions $ f: \mathbb{R}\longrightarrow \mathbb{R}$ such that \[f(x\plus{}y)\plus{}f(y\plus{}z)\plus{}f(z\plus{}x)\ge 3f(x\plus{}2y\plus{}3z)\] for all $x, y, z \in \mathbb R$.

Solve in integers the equation $x^6+x^5+4=y^2. $ [i]Ioan Cucurezeanu[/i]

[u]Round 1[/u] [b]p1.[/b] Nineteen witches, all of different heights, stand in a circle around a campfire. Each witch says whether she is taller than both of her neighbors, shorter than both, or in-between. Exactly three said “I am taller.” How many said “I am in-between”? [b]p2.[/b] Alex is writing a sequence of $A$’s and $B$’s on a chalkboard. Any $20$ consecutive letters must have an equal number of $A$’s and $B$’s, but any 22 consecutive letters must have a different number of $A$’s and $B$’s. What is the length of the longest sequence Alex can write?. [b]p3.[/b] A police officer patrols a town whose map is shown. The officer must walk down every street segment at least once and return to the starting point, only changing direction at intersections and corners. It takes the officer one minute to walk each segment. What is the fastest the officer can complete a patrol? [img]https://cdn.artofproblemsolving.com/attachments/a/3/78814b37318adb116466ede7066b0d99d6c64d.png[/img] [b]p4.[/b] A zebra is a new chess piece that jumps in the shape of an “L” to a location three squares away in one direction and two squares away in a perpendicular direction. The picture shows all the moves a zebra can make from its given position. Is it possible for a zebra to make a sequence of $64$ moves on an $8\times 8$ chessboard so that it visits each square exactly once and returns to its starting position? [img]https://cdn.artofproblemsolving.com/attachments/2/d/01a8af0214a2400b279816fc5f6c039320e816.png[/img] [b]p5.[/b] Ann places the integers $1, 2,..., 100$ in a $10 \times 10$ grid, however she wants. In each round, Bob picks a row or column, and Ann sorts it from lowest to highest (left-to-right for rows; top-to-bottom for columns). However, Bob never sees the grid and gets no information from Ann. After eleven rounds, Bob must name a single cell that is guaranteed to contain a number that is at least $30$ and no more than $71$. Can he find a strategy to do this, no matter how Ann originally arranged the numbers? [u]Round 2[/u] [b]p6.[/b] Evelyn and Odette are playing a game with a deck of $101$ cards numbered $1$ through $101$. At the start of the game the deck is split, with Evelyn taking all the even cards and Odette taking all the odd cards. Each shuffles her cards. On every move, each player takes the top card from her deck and places it on a table. The player whose number is higher takes both cards from the table and adds them to the bottom of her deck, first the opponent’s card, then her own. The first player to run out of cards loses. Card $101$ was played against card $2$ on the $10$th move. Prove that this game will never end. [img]https://cdn.artofproblemsolving.com/attachments/8/1/aa16fe1fb4a30d5b9e89ac53bdae0d1bdf20b0.png[/img] [b]p7.[/b] The Vogon spaceship Tempest is descending on planet Earth. It will land on five adjacent buildings within a $10 \times 10$ grid, crushing any teacups on roofs of buildings within a $5 \times 1$ length of blocks (vertically or horizontally). As Commander of the Space Force, you can place any number of teacups on rooftops in advance. When the ship lands, you will hear how many teacups the spaceship breaks, but not where they were. (In the figure, you would hear $4$ cups break.) What is the smallest number of teacups you need to place to ensure you can identify at least one building the spaceship landed on? [img]https://cdn.artofproblemsolving.com/attachments/8/7/2a48592b371bba282303e60b4ff38f42de3551.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $n$ be a positive integer greater than $2$. Find all functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ satisfying: $(f(x+y))^{n} = f(x^{n})+f(y^{n}),$ for all integers $x,y$

Let $n\geqslant 1$ be an integer, and let $x_0,x_1,\ldots,x_{n+1}$ be $n+2$ non-negative real numbers that satisfy $x_ix_{i+1}-x_{i-1}^2\geqslant 1$ for all $i=1,2,\ldots,n.$ Show that \[x_0+x_1+\cdots+x_n+x_{n+1}>\bigg(\frac{2n}{3}\bigg)^{3/2}.\][i]Pakawut Jiradilok and Wijit Yangjit, Thailand[/i]

Let $x_1, ..., x_n$ $(n \geq 2)$ be real numbers from the interval $[1,2]$. Prove that $$|x_1-x_2|+...+|x_n-x_1| + \frac{1}{3} (|x_1-x_3|+...+|x_n-x_2|) \leq \frac{2}{3} (x_1+...+x_n)$$ and determine all cases of equality.

Prove that the circle with radius $2$ can be completely covered with $7$ unit circles

Let $ABC $ be a triangle and $D $ be the feet of $A $-altitude.\\ $E,F $ are defined on segments $AD,BC $,respectively such that $\frac {AE}{DE}=\frac{BF}{CF} $.\\ Assume that $G $ lies on $AF $ such that $BG\perp AF $.Prove that $EF $ is tangent to the circumcircle of $CFG $. [i]Proposed by Mehdi Etesami Fard[/i]

Let $f(n)=\displaystyle\sum_{k=1}^n \dfrac{1}{k}$. Then there exists constants $\gamma$, $c$, and $d$ such that \[f(n)=\ln(x)+\gamma+\dfrac{c}{n}+\dfrac{d}{n^2}+O\left(\dfrac{1}{n^3}\right),\] where the $O\left(\dfrac{1}{n^3}\right)$ means terms of order $\dfrac{1}{n^3}$ or lower. Compute the ordered pair $(c,d)$.

For all integers $n\geq 2$, let $f(n)$ denote the largest positive integer $m$ such that $\sqrt[m]{n}$ is an integer. Evaluate \[f(2)+f(3)+\cdots+f(100).\]

For positive integers $a$ and $b$, an $(a,b)$-shuffle of a deck of $a+b$ cards is any shuffle that preserves the relative order of the top $a$ cards and the relative order of the bottom $b$ cards. Let $n$, $k$, $a_1$, $a_2$, $\dots$, $a_k$, $b_1$, $b_2$, $\dots$, $b_k$ be fixed positive integers such that $a_i+b_i=n$ for all $1\leq i\leq k$. Big Bird has a deck of $n$ cards and will perform an $(a_i,b_i)$-shuffle for each $1\leq i\leq k$, in ascending order of $i$. Suppose that Big Bird can reverse the order of the deck. Prove that Big Bird can also achieve any of the $n!$ permutations of the cards. [i]Linus Tang[/i]

There are two airlines A and B and finitely many airports. For each pair of airports, there is exactly one airline among A and B whose flights operates in both directions. Each airline plans to develop world travel packages which pass each airport exactly once using only its flights. Let $a$ and $b$ be the number of possible packages which belongs to A and B respectively. Prove that $a-b$ is a multiple of $4$. The official statement of the problem has been changed. The above is the form which appeared during the contest. Now the condition 'the number of airports is no less than 4'is attached. Cite the following link. [url]https://artofproblemsolving.com/community/c6h2923697p26140823[/url]

Let $w$ be the circumcircle of non-isosceles acute triangle $ABC$. Tangent lines to $w$ in $A$ and $B$ intersect at point $S$. Let M be the midpoint of $AB$, and $H$ be the orthocenter of triangle $ABC$. The line $HA$ intersects lines $CM$ and $CS$ at points $M_a$ and $S_a$, respectively. The points $M_b$ and $S_b$ are defined analogously. Prove that $M_aS_b$ and $M_bS_a$ are the altitudes of triangle $M_aM_bH$.

The diagonals of trapezoid $ABCD$ with bases $AB$ and $CD$ meet at $P$. Prove the inequality $S_{PAB} + S_{PCD} > S_{PBC} + S_{PDA}$, where $S_{XYZ}$ denotes the area of triangle $XYZ$.

In a triangle $ABC$, let $D$, $E$ and $F$ be the touchpoints of the inscribed circle and the sides $AB$, $BC$ and $CA$. Show that the triangles $DEF$ and $ABC$ are similar if and only if $ABC$ is equilateral.

If $M=\{(x,y)||\tan\pi x|+\sin^2\pi x=0\},N=\{(x,y)|x^2+y^2\leq2\}$, then $|M\cap N|$ is equal to $\text{(A)}4\qquad\text{(B)}5\qquad\text{(C)}8\qquad\text{(D)}9$

In triangle $ABC$ the angular bisector from $A$ intersects the side $BC$ at the point $D$, and the angular bisector from $B$ intersects the side $AC$ at the point $E$. Furthermore $|AE| + |BD| = |AB|$. Prove that $\angle C = 60^o$ [img]https://1.bp.blogspot.com/-8ARqn8mLn24/XzP3P5319TI/AAAAAAAAMUQ/t71-imNuS18CSxTTLzYXpd806BlG5hXxACLcBGAsYHQ/s0/2018%2BMohr%2Bp5.png[/img]

Suppose that the differentiable functions $a, b, f, g:\mathbb{R} \rightarrow \mathbb{R} $ satisfy \[ f(x)\geq 0, f'(x) \geq 0,g(x)\geq 0, g'(x) \geq 0 \text{ for all } x \in \mathbb{R}, \] \[\lim_{x\rightarrow \infty} a(x)=A\geq 0,\lim_{x\rightarrow \infty} b(x)=B\geq 0, \lim_{x\rightarrow \infty} f(x)=\lim_{x\rightarrow \infty} g(x)=\infty,\] and \[\frac{f'(x)}{g'(x)}+a(x)\frac{f(x)}{g(x)}=b(x).\] Prove that $\lim_{x\rightarrow\infty}\frac{f(x)}{g(x)}=\frac{B}{A+1}$.

The incircle of a triangle $ABC$ has centre $I$ and touches sides $AB, BC, CA$ in points $C_1, A_1, B_1$ respectively. Denote by $L$ the foot of a bissector of angle $B$, and by $K$ the point of intersecting of lines $B_1I$ and $A_1C_1$. Prove that $KL\parallel BB_1$. [b]proposed by L. Emelyanov, S. Berlov[/b]

Solve: $a, b, m, n\in \mathbb{N}$ $a^2+b^2=m^2-n^2, ab=2mn$

If \[2011^{2011^{2012}} = x^x\] for some positive integer $x$, how many positive integer factors does $x$ have? [i]Author: Alex Zhu[/i]

A frog is at the center of a circular shaped island with radius $r$. The frog jumps $1/2$ meters at first. After the first jump, it turns right or left at exactly $90^\circ$, and it always jumps one half of its previous jump. After a finite number of jumps, what is the least $r$ that yields the frog can never fall into the water? $ \textbf{(A)}\ \frac{\sqrt 5}{3} \qquad\textbf{(B)}\ \frac{\sqrt {13}}{5} \qquad\textbf{(C)}\ \frac{\sqrt {19}}{6} \qquad\textbf{(D)}\ \frac{1}{\sqrt 2} \qquad\textbf{(E)}\ \frac34 $

In the triangle $ABC$, the altitude $BH$ and the angle bisector $BL$ are drawn, the inscribed circle $w$ touches the side of the $AC$ at the point $K$. It is known that $\angle BKA = 45^o$. Prove that the circle with diameter $HL$ touches the circle $w$. (Anton Trygub)

Every positive integer can be represented as a sum of one or more consecutive positive integers. For each $n$ , find the number of such represententation of $n$.