[b]p1.[/b] Compute the sum of sharp angles at all five nodes of the star below. ( [url=http://www.math.msu.edu/~mshapiro/NewOlympiad/Olymp2009/10_12_2009.pdf]figure missing[/url] ) [b]p2.[/b] Arrange the integers from $1$ to $15$ in a row so that the sum of any two consecutive numbers is a perfect square. In how many ways this can be done? [b]p3.[/b] Prove that if $p$ and $q$ are prime numbers which are greater than $3$ then $p^2 -q^2$ is divisible by $ 24$. [b]p4.[/b] A city in a country is called Large Northern if comparing to any other city of the country it is either larger or farther to the North (or both). Similarly, a city is called Small Southern. We know that in the country all cities are Large Northern city. Show that all the cities in this country are simultaneously Small Southern. [b]p5.[/b] You have four tall and thin glasses of cylindrical form. Place on the flat table these four glasses in such a way that all distances between any pair of centers of the glasses' bottoms are equal. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $A$ be a real $4\times 2$ matrix and $B$ be a real $2\times 4$ matrix such that \[ AB = \left(% \begin{array}{cccc} 1 & 0 & -1 & 0 \\ 0 & 1 & 0 & -1 \\ -1 & 0 & 1 & 0 \\ 0 & -1 & 0 & 1 \\ \end{array}% \right). \] Find $BA$.

[u]Round 5[/u] [b]p13.[/b] Let $\{a\} _{n\ge 1}$ be an arithmetic sequence, with $a_ 1 = 0$, such that for some positive integers $k$ and $x$ we have $a_{k+1} = {k \choose x}$, $a_{2k+1} ={k \choose {x+1}}$ , and $a_{3k+1} ={k \choose {x+2}}$. Let $\{b\}_{n\ge 1}$ be an arithmetic sequence of integers with $b_1 = 0$. Given that there is some integer $m$ such that $b_m ={k \choose x}$, what is the number of possible values of $b_2$? [b]p14.[/b] Let $A = arcsin \left( \frac14 \right)$ and $B = arcsin \left( \frac17 \right)$. Find $\sin(A + B) \sin(A - B)$. [b]p15.[/b] Let $\{f_i\}^{9}_{i=1}$ be a sequence of continuous functions such that $f_i : R \to Z$ is continuous (i.e. each $f_i$ maps from the real numbers to the integers). Also, for all $i$, $f_i(i) = 3^i$. Compute $\sum^{9}_{k=1} f_k \circ f_{k-1} \circ ... \circ f_1(3^{-k})$. [u]Round 6[/u] [b]p16.[/b] If $x$ and $y$ are integers for which $\frac{10x^3 + 10x^2y + xy^3 + y^4}{203}= 1134341$ and $x - y = 1$, then compute $x + y$. [b]p17.[/b] Let $T_n$ be the number of ways that n letters from the set $\{a, b, c, d\}$ can be arranged in a line (some letters may be repeated, and not every letter must be used) so that the letter a occurs an odd number of times. Compute the sum $T_5 + T_6$. [b]p18.[/b] McDonald plays a game with a standard deck of $52$ cards and a collection of chips numbered $1$ to $52$. He picks $1$ card from a fully shuffled deck and $1$ chip from a bucket, and his score is the product of the numbers on card and on the chip. In order to win, McDonald must obtain a score that is a positive multiple of $6$. If he wins, the game ends; if he loses, he eats a burger, replaces the card and chip, shuffles the deck, mixes the chips, and replays his turn. The probability that he wins on his third turn can be written in the form $\frac{x^2 \cdot y}{z^3}$ such that $x, y$, and $z$ are relatively prime positive integers. What is $x + y + z$? (NOTE: Use Ace as $1$, Jack as $11$, Queen as $12$, and King as $13$) [u]Round 7[/u] [b]p19.[/b] Let $f_n(x) = ln(1 + x^{2^n}+ x^{2^{n+1}}+ x^{3\cdot 2^n})$. Compute $\sum^{\infty}_{k=0} f_{2k} \left( \frac12 \right)$. [b]p20.[/b] $ABCD$ is a quadrilateral with $AB = 183$, $BC = 300$, $CD = 55$, $DA = 244$, and $BD = 305$. Find $AC$. [b]p21.[/b] Define $\overline{xyz(t + 1)} = 1000x + 100y + 10z + t + 1$ as the decimal representation of a four digit integer. You are given that $3^x5^y7^z2^t = \overline{xyz(t + 1)}$ where $x, y, z$, and t are non-negative integers such that $t$ is odd and $0 \le x, y, z,(t + 1) \le 9$. Compute$3^x5^y7^z$ PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c4h2782822p24445934]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Assume the quartic $x^4-ax^3+bx^2-ax+d=0$ has four real roots $\frac{1}{2}\leq x_1,x_2,x_3,x_4\leq 2.$ Find the maximum possible value of $\frac{(x_1+x_2)(x_1+x_3)x_4}{(x_4+x_2)(x_4+x_3)x_1}.$

Prove that the only function $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying the following is $f(x)=x$. (i) $\forall x \not= 0$, $f(x) = x^2f(\frac{1}{x})$. (ii) $\forall x, y$, $f(x+y) = f(x)+f(y)$. (iii) $f(1)=1$.

For all $k=1,2,\ldots,2008$,$a_k>0$.Prove that iff $\sum_{k=1}^{2008}a_k>1$,there exists a function $f:N\rightarrow R$ satisfying (1)$0=f(0)<f(1)<f(2)<\ldots$; (2)$f(n)$ has a finite limit when $n$ approaches infinity; (3)$f(n)-f({n-1})=\sum_{k=1}^{2008}a_kf({n+k})-\sum_{k=0}^{2007}a_{k+1}f({n+k})$,for all $n=1,2,3,\ldots$.

Suppose $a, b, c$ are real numbers such that $abc \ne 0$. Determine $x, y, z$ in terms of $a, b, c$ such that $bz + cy = a, cx + az = b, ay + bx = c$. Prove also that $\frac{1 - x^2}{a^2} = \frac{1 - y^2}{b^2} = \frac{1 - z^2}{c^2}$.

Find all function $f:\mathbb{R}\rightarrow \mathbb{R}$ such that $f(x+y)-2f(x-y)+f(x)-2f(y)=y-2,\forall x,y\in \mathbb{R}$.

Find all polynomials $f(x)$ with real coefficients having the property $f(g(x))=g(f(x))$ for every polynomial $g(x)$ with real coefficients.

Let $P(x)$ be a monic polynomial of degree $n$ with nonnegative coefficients and the free term equal to $1$. Prove that if all the roots of $P(x)$ are real, then $P(x) \geq (x+1)^{n}$ holds for every $x \geq 0$.

Suppose that $ a_1,\cdots , a_{25}$ are non-negative integers, and $ k$ is the smallest of them. Prove that $$\big[\sqrt{a_1}\big]+\big[\sqrt{a_2}\big]+\cdots+\big[\sqrt{a_{25}}\big ]\geq\big[\sqrt{a_1+a_2+\cdots+a_{25}+200k}\big].$$ (As usual, $[x]$ denotes the integer part of the number $x$ , that is, the largest integer not exceeding $x$.)

Bradley is driving at a constant speed. When he passes his school, he notices that in $20$ minutes he will be exactly $\frac14$ of the way to his destination, and in $45$ minutes he will be exactly $\frac13$ of the way to his destination. Find the number of minutes it takes Bradley to reach his destination from the point where he passes his school.

Let $x,y,z,t$ be positive numbers.Prove that $\frac{xyzt}{(x+y)(z+t)}\leq\frac{(x+z)^2(y+t)^2}{4(x+y+z+t)^2}.$

A function $f$ has its domain equal to the set of integers $0$, $1$, $\ldots$, $11$, and $f(n)\geq 0$ for all such $n$, and $f$ satisfies [list] [*]$f(0)=0$ [*]$f(6)=1$ [*]If $x\geq 0$, $y\geq 0$, and $x+y\leq 11$, then $f(x+y)=\tfrac{f(x)+f(y)}{1-f(x)f(y)}$.[/list] Find $f(2)^2+f(10)^2$.

A cart of mass $m$ moving at $12 \text{ m/s}$ to the right collides elastically with a cart of mass $4.0 \text{ kg}$ that is originally at rest. After the collision, the cart of mass $m$ moves to the left with a velocity of $6.0 \text{ m/s}$. Assuming an elastic collision in one dimension only, what is the velocity of the center of mass ($v_{\text{cm}}$) of the two carts before the collision? $\textbf{(A) } v_{\text{cm}} = 2.0 \text{ m/s}\\ \textbf{(B) } v_{\text{cm}}=3.0 \text{ m/s}\\ \textbf{(C) } v_{\text{cm}}=6.0 \text{ m/s}\\ \textbf{(D) } v_{\text{cm}}=9.0 \text{ m/s}\\ \textbf{(E) } v_{\text{cm}}=18.0 \text{ m/s}$

Find all functions $f,g,h: \mathbb{R} \mapsto \mathbb{R}$ such that $f(x) - g(y) = (x-y) \cdot h(x+y)$ for $x,y \in \mathbb{R}.$

Do there exist infinitely many positive integers $a, b$ such that $$(a^2+1)(b^2+1)((a+b)^2+1)$$ is a perfect square? [i]Proposed Ivan Chan Guan Yu[/i]

Let $\mathbb Z$ be the set of integers. We consider functions $f :\mathbb Z\to\mathbb Z$ satisfying \[f\left(f(x+y)+y\right)=f\left(f(x)+y\right)\] for all integers $x$ and $y$. For such a function, we say that an integer $v$ is [i]f-rare[/i] if the set \[X_v=\{x\in\mathbb Z:f(x)=v\}\] is finite and nonempty. (a) Prove that there exists such a function $f$ for which there is an $f$-rare integer. (b) Prove that no such function $f$ can have more than one $f$-rare integer. [i]Netherlands[/i]

Let $\mathbb Q_{>0}$ be the set of all positive rational numbers. Let $f:\mathbb Q_{>0}\to\mathbb R$ be a function satisfying the following three conditions: (i) for all $x,y\in\mathbb Q_{>0}$, we have $f(x)f(y)\geq f(xy)$; (ii) for all $x,y\in\mathbb Q_{>0}$, we have $f(x+y)\geq f(x)+f(y)$; (iii) there exists a rational number $a>1$ such that $f(a)=a$. Prove that $f(x)=x$ for all $x\in\mathbb Q_{>0}$. [i]Proposed by Bulgaria[/i]

Vijay picks two random distinct primes $1\le p, q\le 10^4$. Let $r$ be the probability that $3^{2205403200}\equiv 1\bmod pq$. Estimate $r$ in the form $0.abcdef$, where $a, b, c, d, e, f$ are decimal digits.

Let $a$ and $b$ be real numbers such that$$\frac{a}{a^2-5} = \frac{b}{5-b^2} = \frac{ab}{a^2b^2-5}$$where $a+b \neq 0$. $a^4 + b^4 =$ ?

There are three non-negative numbers $ a, b, c $ such that the sum of each two is not less than the remaining one. Prove that $$ \sqrt{a+b-c} + \sqrt{a-b+c} + \sqrt{-a+b+c} \leq \sqrt{a} + \sqrt{b} + \sqrt{c}.$$

The continuous function $f(x)$ satisfies $c^2f(x + y) = f(x)f(y)$ for all real numbers $x$ and $y,$ where $c > 0$ is a constant. If $f(1) = c$, find $f(x)$ (with proof).

Let $\mathbb{N}$ be the set of all natural numbers. Let $f:\mathbb{N} \to \mathbb{N}$ be a bijective function. Show that there exists three numbers $a$, $b$, $c$ in arithmatic progression such that $f(a)<f(b)<f(c)$

Let $a$, $b$, $c$, $d$ be real numbers such that $b-d \ge 5$ and all zeros $x_1, x_2, x_3,$ and $x_4$ of the polynomial $P(x)=x^4+ax^3+bx^2+cx+d$ are real. Find the smallest value the product $(x_1^2+1)(x_2^2+1)(x_3^2+1)(x_4^2+1)$ can take.