Let $a_1, a_2, ....., a_{2003}$ be sequence of reals number. Call $a_k$ $leading$ element, if at least one of expression $a_k; a_k+a_{k+1}; a_k+a_{k+1}+a_{k+2}; ....; a_k+a{k+1}+a_{k+2}+....+a_{2003}$ is positive. Prove, that if exist at least one $leading$ element, then sum of all $leading$'s elements is positive. Official solution [url=http://www.artofproblemsolving.com/Forum/viewtopic.php?f=125&t=365714&p=2011659#p2011659]here[/url]

Real numbers $a_1,a_2,...,a_n$ ($n \ge 3$) satisfy the conditions $a_1 = a_n = 0$ and $$a_{k-1}+a_{k+1} \ge 2a_k$$ for $k = 2$,$3$$,...,$$n -1$. Prove that none of the numbers $a_1$,$...$,$a_n$ is positive.

Find all continuous function $f:\mathbb{R}^{\geq 0}\rightarrow \mathbb{R}^{\geq 0}$ such that : \[f(xf(y))+f(f(y)) = f(x)f(y)+2 \: \: \forall x,y\in \mathbb{R}^{\geq 0}\] [i]Proposed by Mohammad Ahmadi[/i]

[b]p1.[/b] $\vartriangle DEF$ is constructed from equilateral $\vartriangle ABC$ by choosing $D$ on $AB$, $E$ on $BC$ and $F$ on $CA$ so that $\frac{DB}{AB}=\frac{EC}{BC}=\frac{FA}{CA}=a$, where $a$ is a number between $0$ and $1/2$. (a) Show that $\vartriangle DEF$ is also equilateral. (b) Determine the value of $a$ that makes the area of $\vartriangle DEF$ equal to one half the area of $\vartriangle ABC$. [b]p2.[/b] A bowl contains some red balls and some white balls. The following operation is repeated until only one ball remains in the bowl: Two balls are drawn at random from the bowl. If they have different colors, then the red one is discarded and the white one is returned to the bowl. If they have the same color, then both are discarded and a red ball (from an outside supply of red balls) is added to the bowl. (Note that this operation—in either case—reduces the number of balls in the bowl by one.) (a) Show that if the bowl originally contained exactly $1$ red ball and $ 2$ white balls, then the color of the ball remaining at the end (i.e., after two applications of the operation) does not depend on chance, and determine the color of this remaining ball. (b) Suppose the bowl originally contained exactly $1986$ red balls and $1986$ white balls. Show again that the color of the ball remaining at the end does not depend on chance and determine its color. [b]p3.[/b] Let $a, b$, and $c$ be three consecutive positive integers, with $a < b < c.$ (a) Show that $ab$ cannot be the square of an integer. (b) Show that $ac$ cannot be the square of an integer. (c) Show that $abc$ cannot be the square of an integer. [b]p4.[/b] Consider the system of equations $$\sqrt{x}+\sqrt{y}=2$$ $$ x^2+y^2=5$$ (a) Show (algebraically or graphically) that there are two or more solutions in real numbers $x$ and $y$. (b) The graphs of the two given equations intersect in exactly two points. Find the equation of the straight line passing through these two points of intersection. [b]p5.[/b] Let $n$ and $m$ be positive integers. An $n \times m $ rectangle is tiled with unit squares. Let $r(n, m)$ denote the number of rectangles formed by the edges of these unit squares. Thus, for example, $r(2, 1) = 3$. (a) Find $r(2, 3)$. (b) Find $r(n, 1)$. (c) Find, with justification, a formula for $r(n, m)$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $x,y,z,t$ be real numbers with $x,y,z$ not all equal such that \[x+\frac{1}{y}=y+\frac{1}{z}=z+\frac{1}{x}=t.\] Find all possible values of $ t$ such that $xyz+t=0$.

Given a sequence of natural numbers $ (a_i) $, for each natural number $ n $ the sum of the terms of the sequence that are not greater than $ n $ is a number not less than $ n $. Prove that for every natural number $ k $ it is possible to choose from the sequence $ (a_i) $ a finite sequence with the sum of terms equal to $ k $.

Let $a, b$ be positive real numbers satisfying $a + b = 1$. Show that if $x_1,x_2,...,x_5$ are positive real numbers such that $x_1x_2...x_5 = 1$, then $(ax_1+b)(ax_2+b)...(ax_5+b)>1$

Let $p$ and $q$ two positive integers. Determine the greatest value of $n$ for which there exists sets $A_1,\ A_2,\ldots,\ A_n$ and $B_1,\ B_2,\ldots,\ B_n$ such that: [LIST] [*] The sets $A_1,\ A_2,\ldots,\ A_n$ have $p$ elements each one. [/*] [*] The sets $B_1,\ B_2,\ldots,\ B_n$ have $q$ elements each one. [/*] [*] For all $1\leq i,\ j \leq n$, sets $A_i$ and $B_j$ are disjoint if and only if $i=j$. [/LIST]

The positive integers $ \alpha, \beta, \gamma$ are the roots of a polynomial $ f(x)$ with degree $ 4$ and the coefficient of the first term is $ 1$. If there exists an integer such that $ f(\minus{}1)\equal{}f^2(s)$. Prove that $ \alpha\beta$ is not a perfect square.

Find all functions $f: \mathbb{R}\backslash\{0,1\} \rightarrow \mathbb{R}$ such that \[ f(x)+f\left(\frac{1}{1-x}\right) = 1+\frac{1}{x(1-x)}. \]

Suppose that $f:\{1, 2,\ldots ,1600\}\rightarrow\{1, 2,\ldots ,1600\}$ satisfies $f(1)=1$ and \[f^{2005}(x)=x\quad\text{for}\ x=1,2,\ldots ,1600. \] $(a)$ Prove that $f$ has a fixed point different from $1$. $(b)$ Find all $n>1600$ such that any $f:\{1,\ldots ,n\}\rightarrow\{1,\ldots ,n\}$ satisfying the above condition has at least two fixed points.

Show that for each positive integer $n,$ all the roots of the polynomial \[\sum_{k=0}^n 2^{k(n-k)}x^k\] are real numbers.

How many solutions does the equation: $$[\frac{x}{20}]=[\frac{x}{17}]$$ have over the set of positve integers? $[a]$ denotes the largest integer that is less than or equal to $a$. [i]Proposed by Karl Czakler[/i]

Find all polynomials $P(x)$ with real coefficients that satisfy \[P(x\sqrt{2})=P(x+\sqrt{1-x^2})\]for all real $x$ with $|x|\le 1$.

Find all values ​​of $ r$ such that the inequality $$r (ab + bc + ca) + (3- r) \left( \frac{1}{a}+\frac{1}{b}+\frac{1}{c} \right) \ge 9$$ is true for $a,b,c$ arbitrary positive reals

How many positive integers $n$ are there such that $n+2$ divides $(n+18)^2$?

Prove the identity \[\sum\limits_{k=0}^n\binom{n}{k}\left(\tan\frac{x}{2}\right)^{2k}\left(1+\frac{2^k}{\left(1-\tan^2\frac{x}{2}\right)^k}\right)=\sec^{2n}\frac{x}{2}+\sec^n x\] for any natural number $n$ and any angle $x.$

Let $a$, $b$ and $c$ be rational numbers for which $a+bc$, $b+ac$ and $a+b$ are all non-zero and for which we have \[\frac{1}{a+bc}+\frac{1}{b+ac}=\frac{1}{a+b}.\] Prove that $\sqrt{(c-3)(c+1)}$ is rational.

Find all real pairs $(a,b)$ that solve the system of equation \begin{align*} a^2+b^2 &= 25, \\ 3(a+b)-ab &= 15. \end{align*} [i](German MO 2016 - Problem 1)[/i]

Find all polynomials $P(x)$ with real coefficients such that \[(x-2010)P(x+67)=xP(x) \] for every integer $x$.

Vasya chose a certain number $x$ and calculated the following: $a_1=1+x^2+x^3, a_2=1+x^3+x^4, a_3=1+x^4+x^5, ..., a_n=1+x^{n+1}+x^{n+2} ,...$ It turned out that $a_2^2 = a_1a_3$. Prove that for all $n\ge 3$, the equality $a_n^2 = a_{n-1}a_{n+1}$ holds.

[u]Set 4[/u] [b]G10.[/b] Let $ABCD$ be a square with side length $1$. It is folded along a line $\ell$ that divides the square into two pieces with equal area. The minimum possible area of the resulting shape is $A$. Find the integer closest to $100A$. [b]G11.[/b] The $10$-digit number $\underline{1A2B3C5D6E}$ is a multiple of $99$. Find $A + B + C + D + E$. [b]G12.[/b] Let $A, B, C, D$ be four points satisfying $AB = 10$ and $AC = BC = AD = BD = CD = 6$. If $V$ is the volume of tetrahedron $ABCD$, then find $V^2$. [u]Set 5[/u] [b]G13.[/b] Nate the giant is running a $5000$ meter long race. His first step is $4$ meters, his next step is $6$ meters, and in general, each step is $2$ meters longer than the previous one. Given that his $n$th step will get him across the finish line, find $n$. [b]G14.[/b] In square $ABCD$ with side length $2$, there exists a point $E$ such that $DA = DE$. Let line $BE$ intersect side $AD$ at $F$ such that $BE = EF$. The area of $ABE$ can be expressed in the form $a -\sqrt{b}$ where $a$ is a positive integer and $b$ is a square-free integer. Find $a + b$. [b]G15.[/b] Patrick the Beetle is located at $1$ on the number line. He then makes an infinite sequence of moves where each move is either moving $1$, $2$, or $3$ units to the right. The probability that he does reach $6$ at some point in his sequence of moves is $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$. [u]Set 6[/u] [b]G16.[/b] Find the smallest positive integer $c$ greater than $1$ for which there do not exist integers $0 \le x, y \le9$ that satisfy $2x + 3y = c$. [b]G17.[/b] Jaeyong is on the point $(0, 0)$ on the coordinate plane. If Jaeyong is on point $(x, y)$, he can either walk to $(x + 2, y)$, $(x + 1, y + 1)$, or $(x, y + 2)$. Call a walk to $(x + 1, y + 1)$ an Brilliant walk. If Jaeyong cannot have two Brilliant walks in a row, how many ways can he walk to the point $(10, 10)$? [b]G18.[/b] Deja vu? Let $ABCD$ be a square with side length $1$. It is folded along a line $\ell$ that divides the square into two pieces with equal area. The maximum possible area of the resulting shape is $B$. Find the integer closest to $100B$. PS. You should use hide for answers. Sets 1-3 have been posted [url=https://artofproblemsolving.com/community/c3h3131303p28367061]here [/url] and 7-9 [url=https://artofproblemsolving.com/community/c3h3131308p28367095]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

For positive real numbers $b_1,b_2,\ldots,b_n$ define $$a_1=\frac{b_1}{b_1+b_2+\ldots+b_n}\enspace\text{ and }\enspace a_k=\frac{b_1+\ldots+b_k}{b_1+\ldots+b_{k-1}}\text{ for }k>1.$$Prove that $a_1+a_2+\ldots+a_n\le\frac1{a_1}+\frac1{a_2}+\ldots+\frac1{a_n}$

Given positive numbers $a_1$ and $b_1$, consider the sequences defined by \[a_{n+1}=a_n+\frac{1}{b_n},\quad b_{n+1}=b_n+\frac{1}{a_n}\quad (n \ge 1)\] Prove that $a_{25}+b_{25} \geq 10\sqrt{2}$.

Let $\alpha$ be the unique real root of the polynomial $x^3-2x^2+x-1$. It is known that $1<\alpha<2$. We define the sequence of polynomials $\left\{{p_n(x)}\right\}_{n\ge0}$ by taking $p_0(x)=x$ and setting \begin{align*} p_{n+1}(x)=(p_n(x))^2-\alpha \end{align*} How many distinct real roots does $p_{10}(x)$ have?