Prove that for every positive integer $n$ $$ \frac{2}{3} n \sqrt{n} < \sqrt{1} + \sqrt{2} +\ldots +\sqrt{n} < \frac{4n+3}{6} \sqrt{n}.$$

[b]p1.[/b] On the island of Nevermind some people are liars; they always lie. The remaining habitants of the island are truthlovers; they tell only the truth. Three habitants of the island, $A, B$, and $C$ met this morning. $A$ said: “All of us are liars”. $B$ said: “Only one of us is a truthlover”. Who of them is a liar and who of them is a truthlover? [b]p2.[/b] Pinocchio has $9$ pieces of paper. He is allowed to take a piece of paper and cut it in $5$ pieces or $7$ pieces which increases the number of his pieces. Then he can take again one of his pieces of paper and cut it in $5$ pieces or $7$ pieces. He can do this again and again as many times as he wishes. Can he get $2004$ pieces of paper? [b]p3.[/b] In Dragonland there are coins of $1$ cent, $2$ cents, $10$ cents, $20$ cents, and $50$ cents. What is the largest amount of money one can have in coins, yet still not be able to make exactly $1$ dollar? [b]p4.[/b] Find all solutions $a, b, c, d, e$ if it is known that they represent distinct digits and satisfy the following: $\begin{tabular}{ccccc} & a & b & c & d \\ + & a & c & a & c \\ \hline c & d & e & b & c \\ \end{tabular}$ [b]p5.[/b] Two players play the following game. On the lowest left square of an $8\times 8$ chessboard there is a rook. The first player is allowed to move the rook up or to the right by an arbitrary number of squares. The second player is also allowed to move the rook up or to the right by an arbitrary number of squares. Then the first player is allowed to do this again, and so on. The one who moves the rook to the upper right square wins. Who has a winning strategy? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all pairs of positive integers $(a, b)$ such that $f(x)=x$ is the only function $f:\mathbb{R}\to \mathbb{R}$ that satisfies $$f^a(x)f^b(y)+f^b(x)f^a(y)=2xy$$ for all $x, y\in \mathbb{R}$.

A majority of the 30 students in Ms. Deameanor's class bought pencils at the school bookstore. Each of these students bought the same number of pencils, and this number was greater than 1. The cost of a pencil in cents was greater than the number of pencils each student bought, and the total cost of all the pencils was $\$17.71$. What was the cost of a pencil in cents? $\textbf{(A)}\,7 \qquad\textbf{(B)}\,11 \qquad\textbf{(C)}\,17 \qquad\textbf{(D)}\,23 \qquad\textbf{(E)}\,77$

Compute the number of ways to shade in some subset of the $16$ cells in a $4 \times 4$ grid such that each of the $25$ vertices of the grid is a corner of at least one shaded cell.

Given a right angled triangle $ABC$ with $\angle BAC = 90^{\circ}$. The points $D,E,F$ lie on sides $BC,CA,AB$ respectively so that $AD$ is perpendicular to $BC$ and $EF$ is parallel to $BC$. A point $G$ lies on side $AC$ such that $AG=CE$. Prove that $\angle GDF = 90^{\circ}$.

There are ${n}$ tokens in a pack. Some of them (at least one, but not all) are white and the rest are black. All tokens are extracted randomly from the pack, one by one, without putting them back. Let ${X_i}$ be the ratio of white tokens in the pack before the ${i^{\text{th}}}$ extraction and let \[ \displaystyle T =\max \{ |X_i-X_j| : 1 \leq i \leq j \leq n\}.\] Prove that ${\Bbb{E}(T) \leq H(\Bbb{E}(X_1))},$ where ${H(x)=-x\ln x -(1-x)\ln(1-x)}.$ [i]Proposed by Tamás Móri[/i]

A set of positive integers $X$ is called [i]connected[/i] if $|X|\ge 2$ and there exist two distinct elements $m$ and $n$ of $X$ such that $m$ is a divisor of $n$. Determine the number of connected subsets of the set $\{1,2,\ldots,10\}$.

Initially five variables are defined: $a_1=1, a_2=0, a_3=0, a_4=0, a_5=0.$ On a turn, Evan can choose an integer $2 \le i \le 5.$ Then, the integer $a_{i-1}$ will be added to $a_i$. For example, if Evan initially chooses $i = 2,$ then now $a_1=1, a_2=0+1=1, a_3=0, a_4=0, a_5=0.$ Find the minimum number of turns Evan needs to make $a_5$ exceed $1,000,000.$

All sets of $49$ distinct positive integers less than or equal to $100$ are considered. Leandro assigned each of these sets a positive integer less than or equal to $100$. Prove that there is a set $L$ of $50$ distinct positive integers less than or equal to $100$, such that for each number $x$ of $L$ the number that Leandro assigned to the set of $49$ numbers $L-\{ x\}$ is different from $x$. Clarification: $L-\{x\}$ denotes the set that results from removing the number $x$ from $L$.

Find all pair of positive integers $(m,n)$ such that $$mn(m^2+6mn+n^2)$$is a perfect square. [i]Proposed by Li4 and Untro368[/i]

[b]Problem 1.[/b]Find all $ (x,y)$ such that: \[ \{\begin{matrix} \displaystyle\dfrac {1}{\sqrt {1 + 2x^2}} + \dfrac {1}{\sqrt {1 + 2y^2}} & = & \displaystyle\dfrac {2}{\sqrt {1 + 2xy}} \\ \sqrt {x(1 - 2x)} + \sqrt {y(1 - 2y)} & = & \displaystyle\dfrac {2}{9} \end{matrix}\; \]

How many quadruples of positive integers $(a,b,m,n)$ are there such that all of the following statements hold? 1. $a,b<5000$ 2. $m,n<22$ 3. $gcd(m,n)=1$ 4. $(a^2+b^2)^m=(ab)^n$

Let $X$ be a variable point on the side $BC$ of a triangle $ABC$. Let $B'$ and $C'$ be points on the rays $[XB$ and $[XC$, respectively, satisfying $B'X=BC=C'X$. The line passing through $X$ and parallel to $AB'$ cuts the line $AC$ at $Y$ and the line passing through $X$ and parallel to $AC'$ cuts the line $AB$ at $Z$. Prove that all lines $YZ$ pass through a fixed point as $X$ varies on the line segment $BC$.

There are two parallel lines $p_1$ and $p_2$. Points $A$ and $B$ lie on $p_1$, and $C$ on $p_2$. We will move the segment $BC$ parallel to itself and consider all the triangles $AB'C '$ thus obtained. Find the locus of the points in these triangles: a) points of intersection of heights, b) the intersection points of the medians, c) the centers of the circumscribed circles.

Let $\Omega$ be the inscribed circle of a triangle $\vartriangle ABC$. Let $D, E$ and $F$ be the tangency points of $\Omega$ and the sides $BC, CA$ and $AB$, respectively, and let $AD, BE$ and $CF$ intersect $\Omega$ at $K, L$ and $M$, respectively, such that $D, E, F, K, L$ and $M$ are all distinct. The tangent line of $\Omega$ at $K$ intersects $EF$ at $X$, the tangent line of $\Omega$ at $L$ intersects $DE$ at $Y$ , and the tangent line of $\Omega$ at M intersects $DF$ at $Z$. Prove that $X,Y$ and $Z$ are collinear.

The rhombus $ABCD$ is given. Let $E$ be one of the points of intersection of the circles $\Gamma_B$ and $\Gamma_C$, where $\Gamma_B$ is the circle centered at $B$ and passing through $C$, and $\Gamma_C$ is the circle centered at $C$ and passing through $B$. The line $ED$ intersects $\Gamma_B$ at point $F$. Find the value of angle $\angle AFB$. [i](S. Mazanik)[/i]

Let $a$, $b$, and $n$ be positive integers with $\gcd (a, b)=1$. Without using Dirichlet's theorem, show that there are infinitely many $k \in \mathbb{N}$ such that $\gcd(ak+b, n)=1$.

Let $x, y$ be nonzero real numbers that satisfy the following equation: $$x^3 + y^3 + 3x^2y^2 = x^3y^3.$$ Determine all the values that the expression $\frac{1}{x}+ \frac{1}{y}$ can take.

Let $ n > 4$ be a non-negative integer. Given is the in a circle inscribed convex $ n$-gon $ A_0A_1A_2\dots A_{n \minus{} 1}A_n$ $ (A_n \equal{} A_0)$ where the side $ A_{i \minus{} 1}A_i \equal{} i$ (for $ 1 \le i \le n$). Moreover, let $ \phi_i$ be the angle between the line $ A_iA_{i \plus{} 1}$ and the tangent to the circle in the point $ A_i$ (where the angle $ \phi_i$ is less than or equal $ 90^o$, i.e. $ \phi_i$ is always the smaller angle of the two angles between the two lines). Determine the sum $ \Phi \equal{} \sum_{i \equal{} 0}^{n \minus{} 1} \phi_i$ of these $ n$ angles.

is there any polynomial of $deg=2007$ with integer coefficients,such that for any integer $n$,$f(n),f(f(n)),f(f(f(n))),...$ is coprime to each other?

Three groups $A, B$ and $C$ of mathematicians from different countries have invited to a ceremony. We have formed meetings such that three mathematicians participate in every meeting and there is exactly one mathematician from each group in every meeting. Also every two mathematicians have participated in exactly one meeting with each other. [b](a)[/b] Prove that if this is possible, then number of mathematicians of the groups is equal. [b](b)[/b] Prove that if there exist $3$ mathematicians in each group, then that work is possible. [b](c)[/b] Prove that if number mathematicians of the groups be equal, then that work is possible.

There are 12 gold stars arranged in a circle on a blue background. Giorgi wants to label each star with one of the letters $G$, $E$, or $O$, such that no two consecutive stars have the same letter. Determine the number of distinct ways Giorgi can label the stars. [img]https://i.imgur.com/qIxdJ8j.jpeg[/img] [i]Proposed by Giorgi Arabidze, Georgia [/i]

Four orange lights are located at the points $(2,0)$, $(4,0)$, $(6,0)$ and $(8,0)$ in the $xy$-plane. Four yellow lights are located at the points $(1,0)$, $(3,0)$, $(5,0)$, $(7,0)$. Sparky chooses one or more of the lights to turn on. In how many ways can he do this such that the collection of illuminated lights is symmetric around some line parallel to the $y$-axis? [i]Proposed by Evan Chen[/i]

Let $P$ be the product of any three consecutive positive odd integers. The largest integer dividing all such $P$ is: $\textbf{(A)}\ 15 \qquad \textbf{(B)}\ 6 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ 1 $