Let $x_1,x_2,...,x_n$ be real numbers in an interval $[m,M]$ such that $\sum_{i=1}^n x_i = 0$. Show that $\sum_{i=1}^n x_i ^2 \le -nmM$

Let $a_1, a_2,...,a_n$ be positive real numbers, prove that $$\sum {\frac{a_{i+1}}{a_i}} \ge \sum{\sqrt{\frac{a_{i+1}^2+1}{a_i^2+1}}}$$ $a_{n+1}=a_1$

Prove that for all integers $n\ge3$ there exists a set $A_n=\{a_1,a_2,\ldots,a_n\}$ of $n$ distinct natural numbers such that, for each $i=1,2,\ldots,n$, $$\prod_{\small{\begin{matrix}1\le k\le n\\k\ne i\end{matrix}}}a_k\equiv1\pmod{a_i}.$$

Let $p_0(x),p_1(x),p_2(x),\ldots$ be polynomials such that $p_0(x)=x$ and for all positive integers $n$, $\dfrac{d}{dx}p_n(x)=p_{n-1}(x)$. Define the function $p(x):[0,\infty)\to\mathbb{R}$ by $p(x)=p_n(x)$ for all $x\in [n,n+1)$. Given that $p(x)$ is continuous on $[0,\infty)$, compute \[\sum_{n=0}^\infty p_n(2009).\]

Let $\alpha, \beta$ be real numbers such that $\sin\alpha\sin\beta=\frac{1}{3}$. Prove that the set of possible values of $\cos \alpha \cos \beta$ is the interval $\left[-\frac{2}{3}, \frac{2}{3}\right]$.

Let $P$ be an odd-degree integer-coefficient polynomial. Suppose that $xP(x)=yP(y)$ for infinitely many pairs $x,y$ of integers with $x\ne y$. Prove that the equation $P(x)=0$ has an integer root. [i]Victor Wang[/i]

Find all functions $f: \mathbb{Q} \mapsto \mathbb{C}$ satisfying (i) For any $x_1, x_2, \ldots, x_{1988} \in \mathbb{Q}$, $f(x_{1} + x_{2} + \ldots + x_{1988}) = f(x_1)f(x_2) \ldots f(x_{1988})$. (ii) $\overline{f(1988)}f(x) = f(1988)\overline{f(x)}$ for all $x \in \mathbb{Q}$.

It is known that a sequence of positive real numbers \(\left(x_n\right)\) satisfies the relation: \[ x_{n+1} = x_n + \sqrt{x_n + \frac{1}{4}} + \sqrt{x_{n+1} + \frac{1}{4}}, \quad n \geq 1 \] Prove that the following inequality holds: \[ \frac{1}{x_2} + \frac{1}{x_3} + \cdots + \frac{1}{x_{2025}} < \frac{1}{\sqrt{x_1}} \] [i]Proposed by Oleksii Masalitin[/i]

Find all functions $f : R \to R$ satisfying the equality $f (2^x + 2y) =2^y f ( f (x)) f (y) $for every $x, y \in R$.

Let ${x}$ be a real number such that ${x^3 + 2x^2 + 10x = 20.}$ Demonstrate that both ${x}$ and ${x^2}$ are irrational.

Let $m$ and $n$ be two natural numbers and let $d = gcd (m, n)$ their greatest common divisor. Let $a_1, a_2,...$ and $b_1, b_2, ...$ be two sequences of integers which are periodic with periods $m$ and $n$ respectively (this means that $a_{i + m} = a_i$ and $b_{i + n} = b_i$ for all natural numbers $i \ge 1$, note that there could be smaller periods). Prove that if the two sequences on the first $m + n - d$ terms match (i.e. $a_i = b_i$ for all $i \in \{1, 2, ..., m + n - d\}$), then they are the same (so $a_i = b_i$ for all natural $i \ge 1$).

The positive numbers $a, b, c$ and $d$ satisfy $a\le b\le c\le d$ and $a + b + c + d \le 1$ . Prove that $a^2 + 3b^2 + 5c^2 + 7 d^2 \ge 1$ .

Prove that if $u, v$ are any nonnegative real numbers, and $a,b$ positive real numbers such that $a + b = 1$, then $$u^a v^b \le au + bv.$$

For every positive integer $n$, determine the greatest possible value of the quotient $$\frac{1-x^{n}-(1-x)^{n}}{x(1-x)^n+(1-x)x^n}$$ where $0 < x < 1$.

Determine if there exists a (three-variable) polynomial $P(x,y,z)$ with integer coefficients satisfying the following property: a positive integer $n$ is [i]not[/i] a perfect square if and only if there is a triple $(x,y,z)$ of positive integers such that $P(x,y,z) = n$.

Define a function $f$ on positive real numbers to satisfy \[f(1)=1 , f(x+1)=xf(x) \textrm{ and } f(x)=10^{g(x)},\] where $g(x) $ is a function defined on real numbers and for all real numbers $y,z$ and $0\leq t \leq 1$, it satisfies \[g(ty+(1-t)z) \leq tg(y)+(1-t)g(z).\] (1) Prove: for any integer $n$ and $0 \leq t \leq 1$, we have \[t[g(n)-g(n-1)] \leq g(n+t)-g(n) \leq t[g(n+1)-g(n)].\] (2) Prove that \[\frac{4}{3} \leq f(\frac{1}{2}) \leq \frac{4}{3} \sqrt{2}.\]

Let $a, b, c$ be real numbers such that $a+b+c=abc$. Prove that $\frac{1}{a^2+1}+\frac{1}{b^2+1}+\frac{1}{c^2+1}\ge \frac{3}{4}$.

Solve the system $\begin{cases} x^2 + y^2 - 2z^2 = 2a^2 \\ x + y + 2z = 4(a^2 + 1) \\ z^2 - xy = a^2 \end{cases}$

Prove that the number \[ 3 \underbrace{99\ldots9}_{2025} \underbrace{60\ldots01}_{2025} \] is a square of a positive integer.

The zeroes of a fourth degree polynomial $f(x)$ form an arithmetic progression. Prove that the three zeroes of the polynomial $f'(x)$ also form an arithmetic progression.

Given a positive real number $a_1$, we recursively define $a_{n+1} = 1+a_1 a_2 \cdots \cdot a_n.$ Furthermore, let $$b_n = \frac{1}{a_1 } + \frac{1}{a_2 } +\cdots + \frac{1}{a_n }.$$ Prove that $b_n < \frac{2}{a_1}$ for all positive integers $n$ and that this is the smallest possible bound.

Find all real $x$ in the interval $[0, 2\pi)$ such that \[27 \cdot 3^{3 \sin x} = 9^{\cos^2 x}.\]

[u]Round 1[/u] [b]p1.[/b] A box contains $1$ ball labelledW, $1$ ball labelled $E$, $1$ ball labelled $L$, $1$ ball labelled $C$, $1$ ball labelled $O$, $8$ balls labelled $M$, and $1$ last ball labelled $E$. One ball is randomly drawn from the box. The probability that the ball is labelled $E$ is $\frac{1}{a}$ . Find $a$. [b]p2.[/b] Let $$G +E +N = 7$$ $$G +E +O = 15$$ $$N +T = 22.$$ Find the value of $T +O$. [b]p3.[/b] The area of $\vartriangle LMT$ is $22$. Given that $MT = 4$ and that there is a right angle at $M$, find the length of $LM$. [u]Round 2[/u] [b]p4.[/b] Kevin chooses a positive $2$-digit integer, then adds $6$ times its unit digit and subtracts $3$ times its tens digit from itself. Find the greatest common factor of all possible resulting numbers. [b]p5.[/b] Find the maximum possible number of times circle $D$ can intersect pentagon $GRASS'$ over all possible choices of points $G$, $R$, $A$, $S$, and $S'$. [b]p6.[/b] Find the sum of the digits of the integer solution to $(\log_2 x) \cdot (\log_4 \sqrt{x}) = 36$. [u]Round 3[/u] [b]p7.[/b] Given that $x$ and $y$ are positive real numbers such that $x^2 + y = 20$, the maximum possible value of $x + y$ can be written as $\frac{a}{b}$ where $a$ and $b$ are relatively prime positive integers. Find $a +b$. [b]p8.[/b] In $\vartriangle DRK$, $DR = 13$, $DK = 14$, and $RK = 15$. Let $E$ be the point such that $ED = ER = EK$. Find the value of $\lfloor DE +RE +KE \rfloor$. [b]p9.[/b] Subaru the frog lives on lily pad $1$. There is a line of lily pads, numbered $2$, $3$, $4$, $5$, $6$, and $7$. Every minute, Subaru jumps from his current lily pad to a lily pad whose number is either $1$ or $2$ greater, chosen at random from valid possibilities. There are alligators on lily pads $2$ and $5$. If Subaru lands on an alligator, he dies and time rewinds back to when he was on lily pad number $1$. Find how many times Subaru is expected to die before he reaches pad $7$. [u]Round 4[/u] [b]p10.[/b] Find the sum of the following series: $$\sum^{\infty}_{i=1} = \frac{\sum^i_{j=1} j}{2^i}=\frac{1}{2^1}+\frac{1+2}{2^2}+\frac{1+2+3}{2^3}+\frac{1+2+3+4}{2^4}+... $$ [b]p11.[/b] Let $\phi (x)$ be the number of positive integers less than or equal to $x$ that are relatively prime to $x$. Find the sum of all $x$ such that $\phi (\phi(x)) = x -3$. Note that $1$ is relatively prime to every positive integer. [b]p12.[/b] On a piece of paper, Kevin draws a circle. Then, he draws two perpendicular lines. Finally, he draws two perpendicular rays originating from the same point (an $L$ shape). What is the maximum number of sections into which the lines and rays can split the circle? [u]Round 5 [/u] [b]p13.[/b] In quadrilateral $ABCD$, $\angle A = 90^o$, $\angle C = 60^o$, $\angle ABD = 25^o$, and $\angle BDC = 5^o$. Given that $AB = 4\sqrt3$, the area of quadrilateral $ABCD$ can be written as $a\sqrt{b}$. Find $10a +b$. [b]p14.[/b] The value of $$\sum^6_{n=2} \left( \frac{n^4 +1}{n^4 -1}\right) -2 \sum^6_{n=2}\left(\frac{n^3 -n^2+n}{n^4 -1}\right)$$ can be written as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $100m+n$. [b]p15.[/b] Positive real numbers $x$ and $y$ satisfy the following $2$ equations. $$x^{1+x^{1+x^{1+...}}}= 8$$ $$\sqrt[24]{y +\sqrt[24]{y + \sqrt[24]{y +...}}} = x$$ Find the value of $\lfloor y \rfloor$. PS. You should use hide for answers. Rounds 6-9 have been posted [url=https://artofproblemsolving.com/community/c3h3167130p28823260]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $S$ be a finite set of rational numbers. For each positive integer $k$, let $b_k=0$ if we can select $k$ (not necessarily distinct) numbers in $S$ whose sum is $0$, and $b_k=1$ otherwise. Prove that the binary number $0.b_1b_2b_3…$ is a rational number. Would this statement remain true if we allowed $S$ to be infinite?

[b]p1. [/b] A clock has a long hand for minutes and a short hand for hours. A placement of those hands is [i]natural [/i] if you will see it in a correctly functioning clock. So, having both hands pointing straight up toward $12$ is natural and so is having the long hand pointing toward $6$ and the short hand half-way between $2$ and $3$. A natural placement of the hands is symmetric if you get another natural placement by interchanging the long and short hands. One kind of symmetric natural placement is when the hands are pointed in exactly the same direction. Are there symmetric natural placements of the hands in which the two hands are not pointed in exactly the same direction? If so, describe one such placement. If not, explain why none are possible. [b]p2.[/b] Let $\frac{m}{n}$ be a fraction such that when you write out the decimal expansion of $\frac{m}{n}$ , it eventually ends up with the four digits $2001$ repeated over and over and over. Prove that $101$ divides $n$. [b]p3.[/b] Consider the following two questions: Question $1$: I am thinking of a number between $0$ and $15$. You get to ask me seven yes-or-no questions, and I am allowed to lie at most once in answering your questions. What seven questions can you ask that will always allow you to determine the number? Note: You need to come up with seven questions that are independent of the answers that are received. In other words, you are not allowed to say, "If the answer to question $1$ is yes, then question $2$ is XXX; but if the answer to question $1$ is no, then question $2$ is YYY." Question $2$: Consider the set $S$ of all seven-tuples of zeros and ones. What sixteen elements of $S$ can you choose so that every pair of your chosen seven-tuples differ in at least three coordinates? a. These two questions are closely related. Show that an answer to Question $1$ gives an answer to Question $2$. b. Answer either Question $1$ or Question $2$. [b]p4.[/b] You may wish to use the angle addition formulas for the sine and cosine functions: $\sin (\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta$ $\cos (\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta$ a) Prove the identity $(\sin x)(1 + 2 \cos 2x) = \sin (3x)$. b) For any positive integer $n$, prove the identity $$(sin x)(1 + 2 \cos 2x + 2\cos 4x + ... +2\cos 2nx) = \sin ((2n +1)x)$$ [b]p5.[/b] Define the set $\Omega$ in the $xy$-plane as the union of the regions bounded by the three geometric figures: triangle $A$ with vertices $(0.5, 1.5)$, $(1.5, 0.5)$ and $(0.5,-0.5)$, triangle $B$ with vertices $(-0.5,1.5)$, $(-1.5,-0.5)$ and $(-0.5, 0.5)$, and rectangle $C$ with corners $(0.5, 1.0)$, $(-0.5, 1.0)$, $(-0.5,-1.0)$, and $(0.5,-1.0)$. a. Explain how copies of $\Omega$ can be used to cover the $xy$-plane. The copies are obtained by translating $\Omega$ in the $xy$-plane, and copies can intersect only along their edges. b. We can define a transformation of the plane as follows: map any point $(x, y)$ to $(x + G, x + y + G)$, where $G = 1$ if $y < -2x$, $G = -1$ if $y > -2x$, and $G = 0$ if $y = -2x$. Prove that every point in $\Omega$ is transformed into another point in $\Omega$, and that there are at least two points in $\Omega$ that are transformed into the same point. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].