A polynomial $p$ with integer coefficients is such that $p(-n) < p(n) < n$ for some integer $n$. Prove that $p(-n) < -n$.

Let $m$ be a positive integer. We say that a sequence of positive integers written on a circle is [i] good [/i], if the sum of any $m$ consecutive numbers on this circle is a power of $m$. 1. Let $n \geq 2$ be a positive integer. Prove that for any [i] good [/i] sequence with $mn$ numbers, we can remove $m$ numbers such that the remaining $mn-m$ numbers form a [i] good [/i] sequence. 2. Prove that in any [i] good [/i] sequence with $m^2$ numbers, we can always find a number that was repeated at least $m$ times in the sequence.

If $\log_{b^2}x+\log_{x^2}b=1, b>0, b \neq 1, x \neq 1$, then $x$ equals: $ \textbf{(A)}\ 1/b^2 \qquad\textbf{(B)}\ 1/b \qquad\textbf{(C)}\ b^2 \qquad\textbf{(D)}\ b \qquad\textbf{(E)}\ \sqrt{b} $

Prove: \[|\sin a_1|+|\sin a_2|+|\sin a_3|+\ldots+|\sin a_n|+|\cos(a_1+a_2+a_3+\ldots+a_n)|\geq 1.\]

Let $a.b,c,d$ be positive real numbers such that $cd = 1$. Prove that there is an integer $n$ such that $ab\le n^2\le (a + c)(b + d)$.

Evaluate $ (\tan 10^\circ)(\tan 20^\circ)(\tan 30^\circ)(\tan 40^\circ)(\tan 50^\circ)(\tan 60^\circ)(\tan 70^\circ)(\tan 80^\circ)$.

Let $\omega_1$ and $\omega_2$ be two circles with no common points, such that any of them is not inside the other one. Let $M, N$ lie on $\omega_1, \omega_2$, such that the tangents at $M$ to $\omega_1$ and $N$ to $\omega_2$ meet at $P$, such that $PM=PN$. The circles $\omega_1$, $\omega_2$ meet $MN$ at $A, B$. The lines $PA, PB$ meet $\omega_1, \omega_2$ at $C, D$. Show that $\angle BCN=\angle ADM$.

Let $BB_1$ and $CC_1$ be the altitudes of the acute-angled triangle $ABC$. From the point $B_1$ the perpendiculars $B_1E$ and $B_1F$ are drawn on the sides $AB$ and $BC$ of the triangle, respectively, and from the point $C_1$ the perpendiculars $C_1 K$ and $C_1L$ on the sides $AC$ and $BC$, respectively. It turned out that the lines $EF$ and $KL$ are perpendicular. Find the measure of the angle $A$ of the triangle $ABC$. (Alexander Dunyak)

The quadrilateral $ ABCD$ is cyclic. Points $ E$ and $ F$ are chosen at the diagonals $ AC$ and $ BD$ in such a way that $ AF\bot CD$ and $ DE\bot AB.$ Prove that $ EF \parallel BC.$

If $$\sum_{k=1}^{40} \left( \sqrt{1 + \frac{1}{k^{2}} + \frac{1}{(k + 1)^{2}}}\right) = a + \frac {b}{c}$$ where $a, b, c \in \mathbb{N}, b < c, gcd(b,c) =1 ,$ then what is the value of $a+ b ?$

Within a group of $ 2009$ people, every two people has exactly one common friend. Find the least value of the difference between the person with maximum number of friends and the person with minimum number of friends.

For all positive integers $m$ and $k$ with $m\ge k$, define $a_{m,k}=\binom{m}{k-1}-3^{m-k}$. Determine all sequences of real numbers $\{x_1, x_2, x_3, \ldots\}$, such that each positive integer $n$ satisfies the equation \[a_{n,1}x_1+ a_{n,2}x_2+ \cdots + a_{n,n}x_n = 0\]

$CH$ is an altitude in a right triangle $ABC$ $(\angle C = 90^{\circ})$. Points $P$ and $Q$ lie on $AC$ and $BC$ respectively such that $HP \perp AC$ and $HQ \perp BC$. Let $M$ be an arbitrary point on $PQ$. A line passing through $M$ and perpendicular to $MH$ intersects lines $AC$ and $BC$ at points $R$ and $S$ respectively. Let $M_1$ be another point on $PQ$ distinct from $M$. Points $R_1$ and $S_1$ are determined similarly for $M_1$. Prove that the ratio $\frac{RR_1}{SS_1}$ is constant.

$100$ red points divide a blue circle into $100$ arcs such that their lengths are all positive integers from $1$ to $100$ in an arbitrary order. Prove that there exist two perpendicular chords with red endpoints.

Find all real numbers $x$ that can be written as $$x= \frac{a_0}{a_1a_2..a_n}+\frac{a_1}{a_2a_3..a_n}+\frac{a_2}{a_3a_4..a_n}+...+\frac{a_{n-2}}{a_{n-1}a_n}+\frac{a_{n-1}}{a_n}$$ where $n, a_1,a_2,...,a_n$ are positive integers and $1 = a_0 \le a_1 <... < a_n$

Find all functions $f:\mathbb Z\rightarrow \mathbb Z$ such that, for all integers $a,b,c$ that satisfy $a+b+c=0$, the following equality holds: \[f(a)^2+f(b)^2+f(c)^2=2f(a)f(b)+2f(b)f(c)+2f(c)f(a).\] (Here $\mathbb{Z}$ denotes the set of integers.) [i]Proposed by Liam Baker, South Africa[/i]

$C$ is a cube side $1$. The $12$ lines containing the sides of the cube meet at plane $p$ in $12$ points. What can you say about the $12$ points?

Fix a positive integer $n$. Let $a_1, a_2, \ldots$ be a sequence of positive integers such that for all $1 \leq j \leq n$, $a_j=j$, and for all $j>n$, $a_j$ is the largest value of $\min(a_i,a_{j-i})$ among $i=1,2, \ldots j-1$. For example, if $n=3$, we have $a_1=1$, $a_2=2$, $a_3=3$, and $a_4=2$ since $\min(a_1,a_3)=1$, $\min(a_2,a_2)=2$, and $\min(a_3,a_1)=1$. We will determine the values of $a_k$ for sufficiently large $k$. (a) Show that $a_i \in \{1,2,3, \ldots n\}$ for all $i$. (b) Show that if $a_x \geq n-1$ and $a_y \geq n-1$, $a_{x+y} \geq n-1$. (c) Show that for some positive integer $N$, $a_k \in \{n-1,n\}$ for all $k \geq N$. (d) Show that $a_k = n$ if and only if $n \mid k$.

Suppose $L$ is the circle inscribed in the square $T_1$, and $T_2$ is the square inscribed in $L$, so that vertices of $T_1$ lie on the straight lines containing the sides of $T_2$. Find the angles of the convex octagon whose vertices are at the tangency points of $L$ with the sides of $T_1$ and at the vertices of $T_2$. (S Markelov)

Does there exist any continuous function $ f$ such that $ f(f(x))=-x^{2019}\ \forall\ x \in \mathbb{R}$

As shown in figure , the inscribed circle of $ABC$ is intersects $BC$, $CA$, $AB$ at points $D$, $E$, $F$, repectively, and $P$ is a point inside the inscribed circle. The line segments $PA$, $PB$ and $PC$ intersect respectively the inscribed circle at points $X$, $Y$ and $Z$. Prove that the three lines $XD$, $YE$ and $ZF$ have a common point. [img]https://cdn.artofproblemsolving.com/attachments/e/9/bbfb0394b9db7aa5fb1e9a869134f0bca372c1.png[/img]

There are $16$ cities in a certain kingdom. The king wants to have a system of roads constructed so that one can go along those roads from any city to any other one without going through more than one intermediate city and so that no more than $5$ roads go out of any city. (a) Prove that this is possible. (b) Prove that if we replace the number $5$ by the number $4$ in the statement of the problem the king’s desire will become unrealizable. (D. Fomin, Leningrad)

Find all functions $f: \mathbb{R}\rightarrow \mathbb{R}$ such that \[ f(xy-1) + f(x)f(y) = 2xy-1 \] for all x and y

A six-digit number is [i]balanced [/i] when all digits are different from zero and the sum of the first three digits is equal to the sum of the last three digits. Prove that the sum of all six-digit balanced numbers is divisible by $13$.

Let be the matrix $ A=\begin{pmatrix} 1& 2& -1\\ 2&2 &0\\1& 4& -3 \end{pmatrix} . $ [b]a)[/b] Show that the equation $ AX=\begin{pmatrix} 2\\ 1\\5 \end{pmatrix} $ has infinite solutions in $ \mathcal{M}_1^3\left( \mathbb{C} \right) . $ [b]b)[/b] Find the rank of the adugate of $ A. $