The polynomial $P$ has integer coefficients and $P(x)=5$ for five different integers $x$. Show that there is no integer $x$ such that $-6\le P(x)\le 4$ or $6\le P(x)\le 16$.

Find all real $a,b,c,d,e,f$ that satisfy the system $4a = (b + c + d + e)^4$ $4b = (c + d + e + f)^4$ $4c = (d + e + f + a)^4$ $4d = (e + f + a + b)^4$ $4e = (f + a + b + c)^4$ $4f = (a + b + c + d)^4$

Let $f(x) = x^2 + ax + b cos x$. Find all values of parameter$ a$ and $b$, for which the equations $f(x) = 0$ and $f(f(x)) = 0 $have the same non-empty sets of real roots.

$S$ ist the set of all cubic polynomials $f$ with $|f(\pm 1)| \leq 1$ and $|f(\pm \frac{1}{2})| \leq 1$. Find $\sup_{f \in S} \max_{-1 \leq x \leq 1} |f''(x)|$ and all members of $f$ which give equality.

Prove that $\sin^318^{\circ}+\sin^218^{\circ}=\frac18$.

Given the sequence $(t_n)$ defined as $t_0 = 0$, $t_1 = 6$, $t_{n + 2} = 14t_{n + 1} - t_n$. Prove that for every number $n \ge 1$, $t_n$ is the area of a triangle whose lengths are all numbers integers. Dang Hung Thang, University of Natural Sciences, Hanoi National University.

Let $Q(x) = x^2 + 2x + 3$, and suppose that $P(x)$ is a polynomial such that $$P(Q(x)) = x^6 + 6x^5 + 18x^4 + 32x^3 + 35x^2 + 22x + 8.$$ Compute $P(2)$.

In a sequence $P_n$ of quadratic trinomials each trinomial, starting with the third, is the sum of the two preceding trinomials. The first two trinomials do not have common roots. Is it possible that $P_n$ has an integral root for each $n$?

Prove that for every positive integer $n$ there exists a unique ordered pair $(a,b)$ of positive integers such that \[ n = \frac{1}{2}(a + b - 1)(a + b - 2) + a . \]

For every n = 2; 3; : : : , we put $$A_n = \left(1 - \frac{1}{1+2}\right) X \left(1 - \frac{1}{1+2+3}\right)X \left(1 - \frac{1}{1+2+3+...+n}\right) $$ Determine all positive integer $ n (n \geq 2)$ such that $\frac{1}{A_n}$ is an integer.

Find maximum of the expression $(a -b^2)(b - a^2)$, where $0 \le a,b \le 1$.

Each of given $100$ numbers was increased by $1$. Then each number was increased by $1$ once more. Given that the first time the sum of the squares of the numbers was not changed find how this sum was changed the second time.

Find all pairs of integers $(m,n)$ such that $m^6 = n^{n+1} + n -1$.

[b]p1.[/b] I open my $2010$-page dictionary, whose pages are numbered $ 1$ to $2010$ starting on page $ 1$ on the right side of the spine when opened, and ending with page $2010$ on the left. If I open to a random page, what is the probability that the two page numbers showing sum to a multiple of $6$? [b]p2.[/b] Let $A$ be the number of positive integer factors of $128$. Let $B$ be the sum of the distinct prime factors of $135$. Let $C$ be the units’ digit of $381$. Let $D$ be the number of zeroes at the end of $2^5\cdot 3^4 \cdot 5^3 \cdot 7^2\cdot 11^1$. Let $E$ be the largest prime factor of $999$. Compute $\sqrt[3]{\sqrt{A + B} +\sqrt[3]{D^C+E}}$. [b]p3. [/b] The root mean square of a set of real numbers is defined to be the square root of the average of the squares of the numbers in the set. Determine the root mean square of $17$ and $7$. [b]p4.[/b] A regular hexagon $ABCDEF$ has area $1$. The sides$ AB$, $CD$, and $EF$ are extended to form a larger polygon with $ABCDEF$ in the interior. Find the area of this larger polygon. [b]p5.[/b] For real numbers $x$, let $\lfloor x \rfloor$ denote the greatest integer less than or equal to $x$. For example, $\lfloor 3\rfloor = 3$ and $\lfloor 5.2 \rfloor = 5$. Evaluate $\lfloor -2.5 \rfloor + \lfloor \sqrt 2 \rfloor + \lfloor -\sqrt 2 \rfloor + \lfloor 2.5 \rfloor$. [b]p6.[/b] The mean of five positive integers is $7$, the median is $8$, and the unique mode is $9$. How many possible sets of integers could this describe? [b]p7.[/b] How many three digit numbers x are there such that $x + 1$ is divisible by $11$? [b]p8.[/b] Rectangle $ABCD$ is such that $AD = 10$ and $AB > 10$. Semicircles are drawn with diameters $AD$ and $BC$ such that the semicircles lie completely inside rectangle $ABCD$. If the area of the region inside $ABCD$ but outside both semicircles is $100$, determine the shortest possible distance between a point $X$ on semicircle $AD$ and $Y$ on semicircle $BC$. [b]p9.[/b] $ 8$ distinct points are in the plane such that five of them lie on a line $\ell$, and the other three points lie off the line, in a way such that if some three of the eight points lie on a line, they lie on $\ell$. How many triangles can be formed using some three of the $ 8$ points? [b]p10.[/b] Carl has $10$ Art of Problem Solving books, all exactly the same size, but only $9$ spaces in his bookshelf. At the beginning, there are $9$ books in his bookshelf, ordered in the following way. $A - B - C - D - E - F - G - H - I$ He is holding the tenth book, $J$, in his hand. He takes the books out one-by-one, replacing each with the book currently in his hand. For example, he could take out $A$, put $J$ in its place, then take out $D$, put $A$ in its place, etc. He never takes the same book out twice, and stops once he has taken out the tenth book, which is $G$. At the end, he is holding G in his hand, and his bookshelf looks like this. $C - I - H - J - F - B - E - D - A$ Give the order (start to finish) in which Carl took out the books, expressed as a $9$-letter string (word). PS. You had better use hide for answers.

Three real numbers $ a,b,c$ satisfy the inequality $ |ax^2\plus{}bx\plus{}c| \le 1$ for all $ x \in [\minus{}1,1]$. Prove that $ |cx^2\plus{}bx\plus{}a| \le 2$ for all $ x \in [\minus{}1,1]$.

Find all the pairs of integers $(m,n)$ satisfying the equality $3(m^2+n^2)-7(m+n)=-4$

Let $ f$ be a function defined on the set of non-negative integers and taking values in the same set. Given that (a) $ \displaystyle x \minus{} f(x) \equal{} 19\left[\frac{x}{19}\right] \minus{} 90\left[\frac{f(x)}{90}\right]$ for all non-negative integers $ x$; (b) $ 1900 < f(1990) < 2000$, find the possible values that $ f(1990)$ can take. (Notation : here $ [z]$ refers to largest integer that is $ \leq z$, e.g. $ [3.1415] \equal{} 3$).

Find all functions $f\colon \mathbb{R} \rightarrow \mathbb{R}$ such that for all $x,y \in \mathbb{R}$, \[f(x+yf(x))+y = xy + f(x+y).\] [i]Proposed by Giannis Galamatis, Greece[/i]

Consider an octagon with equal angles and with rational sides. Prove that it has a center of symmetry.

A sequence of integers $x_1, x_2, ...$ is [i]double-dipped[/i] if $x_{n+2} = ax_{n+1} + bx_n$ for all $n \ge 1$ and some fixed integers $a, b$. Ri begins to form a sequence by randomly picking three integers from the set $\{1, 2, ..., 12\}$, with replacement. It is known that if Ri adds a term by picking anotherelement at random from $\{1, 2, ..., 12\}$, there is at least a $\frac13$ chance that his resulting four-term sequence forms the beginning of a double-dipped sequence. Given this, how many distinct three-term sequences could Ri have picked to begin with?

Prove that these polynomials are irreducible in $\mathbb{Q}[x]:$ a) $\frac{{{x^p}}}{{p!}} + \frac{{{x^{p - 1}}}}{{(p - 1)!}} + ... + \frac{{{x^2}}}{2} + x + 1,$ with $p$ is a prime number. b) $x^{2^n}+1,$ with $n$ is a positive integer.

Find all $n \in N$ for which the value of the expression $x^n+y^n+z^n$ is constant for all $x,y,z \in R$ such that $x+y+z=0$ and $xyz=1$. D. Bazylev

Find all functions $f:\mathbb{N}\rightarrow \mathbb{N}$ such that for all $m,n\in\mathbb{N}$, \[(2^m+1)f(n)f(2^mn)=2^mf(n)^2+f(2^mn)^2+(2^m-1)^2n. \]

Real numbers $a,b,c$ with $0\le a,b,c\le 1$ satisfy the condition $$a+b+c=1+\sqrt{2(1-a)(1-b)(1-c)}.$$ Prove that $$\sqrt{1-a^2}+\sqrt{1-b^2}+\sqrt{1-c^2}\le \frac{3\sqrt 3}{2}.$$ [i] (Nora Gavrea)[/i]

Let $f$ be any function that maps the set of real numbers into the set of real numbers. Prove that there exist real numbers $x$ and $y$ such that \[f\left(x-f(y)\right)>yf(x)+x\] [i]Proposed by Igor Voronovich, Belarus[/i]