Show that if $ \left| ax^2+bx+c\right|\le 1, $ for all $ x\in [-1,1], $ then $ |a|+|b|+|c|\le 4. $

An urn contains balls of $k$ different colors; there are $n_i$ balls of $i-th$ color. Balls are selected at random from the urn, one by one, without replacement, until among the selected balls $m$ balls of the same color appear. Find the greatest number of selections.

The board depicts the triangle $ABC$, the altitude $AH$ and the angle bisector $AL$ which intersectthe inscribed circle in the triangle at the points $M$ and $N, P$ and $Q$, respectively. After that, the figure was erased, leaving only the points $H, M$ and $Q$. Restore the triangle $ABC$. (Bogdan Rublev)

In an $n\times n$ array all the fields are colored with a different color. In one move one can choose a row, move all the fields one place to the right, and move the last field (from the right) to the leftmost field of the row; or one can choose a column, move all the fields one place downwards, and move the field at the bottom of the column to the top field of the same column. For what values of $n$ is it possible to reach any arrangement of the $n^2$ fields using these kinds of steps? [i]Proposed by Ádám Schweitzer[/i]

Let $a_n=\int_0^{\frac{\pi}{2}} \sin 2t\ (1-\sin t)^{\frac{n-1}{2}}dt\ (n=1,2,\cdots)$ Evaluate \[\sum_{n=1}^{\infty} (n+1)(a_n-a_{n+1})\]

Find all $f:\mathbb{Z}\to\mathbb{Z}$ such that \[f\left(\left\lfloor\frac{f(x)+f(y)}{2}\right\rfloor\right)+f(x)=f(f(y))+\left\lfloor\frac{f(x)+f(y)}{2}\right\rfloor\] holds for all $x,y\in\mathbb{Z}$. [i]Proposed by usjl[/i]

Quadrilateral $ABCD$ is inscribed in a circle such that the midpoints of its sides also lie on a (different) circle. Let $M$ and $N$ be the midpoints of $\overline{AB}$ and $\overline{CD}$ respectively, and let $P$ be the foot of the perpendicular from the intersection of $\overline{AC}$ and $\overline{BD}$ onto $\overline{BC}$. If the side lengths of $ABCD$ are $1$, $3$, $\sqrt 2$, and $2\sqrt 2$ in some order, compute the greatest possible area of the circumcircle of triangle $MNP$. [i]Proposed by Connor Gordon[/i]

Points $A(11,9)$ and $B(2,-3)$ are vertices of $\triangle ABC$ with $AB=AC$. The altitude from $A$ meets the opposite side at $D(-1, 3)$. What are the coordinates of point $C$? $\textbf{(A) } (-8, 9)\qquad \textbf{(B) } (-4, 8)\qquad \textbf{(C) } (-4,9)\qquad \textbf{(D) } (-2, 3)\qquad \textbf{(E) } (-1, 0)$

$ABCDE$ is a convex pentagon such that $DC = DE$ and $\angle C = \angle E = 90^{\cdot}$. $F$ is a point on the side $AB$ such that $\frac{AF}{BF}= \frac{AE}{BC}$. Show that $\angle FCE = \angle ADE$ and $\angle FEC = \angle BDC$.

One of the factors of $ x^4\plus{}2x^2\plus{}9$ is: $ \textbf{(A)}\ x^2\plus{}3 \qquad \textbf{(B)}\ x\plus{}1 \qquad \textbf{(C)}\ x^2\minus{}3 \qquad \textbf{(D)}\ x^2\minus{}2x\minus{}3 \qquad \textbf{(E)}\ \text{none of these}$

How many subsets of $\{2,3,4,5,6,7,8,9\}$ contain at least one prime number? $\textbf{(A)} \text{ 128} \qquad \textbf{(B)} \text{ 192} \qquad \textbf{(C)} \text{ 224} \qquad \textbf{(D)} \text{ 240} \qquad \textbf{(E)} \text{ 256}$

Let $S$ be the set of circles in the coordinate plane that are tangent to each of the three circles with equations $x^{2}+y^{2}=4$, $x^{2}+y^{2}=64$, and $(x-5)^{2}+y^{2}=3$. What is the sum of the areas of all circles in $S$? $\textbf{(A)}~48\pi\qquad\textbf{(B)}~68\pi\qquad\textbf{(C)}~96\pi\qquad\textbf{(D)}~102\pi\qquad\textbf{(E)}~136\pi\qquad$

Consider the polynomial $P(n) = n^3 -n^2 -5n+ 2$. Determine all integers $n$ for which $P(n)^2$ is a square of a prime. [hide="Remark."]I'm not sure if the statement of this problem is correct, because if $P(n)^2$ be a square of a prime, then $P(n)$ should be that prime, and I don't think the problem means that.[/hide]

Let $a_1$, $a_2$ ,..., $a_{99}$ be a sequence of digits from the set ${0,...,9}$ such that if for some $n$ ∈ $N$, $a_n = 1$, then $a_{n+1} \ne 2$, and if $a_n = 3$ then $a_{n+1} \ne 4$. Prove that there exist indices $k,l$ ∈ ${1,...,98}$ such that $a_k = a_l$ and $a_{k+1} = a_{l+1}$.

Let $k$ be a natural number such that $k\ge 7$. How many $(x,y)$ such that $0\le x,y<2^k$ satisfy the equation $73^{73^x}\equiv 9^{9^y} \pmod {2^k}$? [i]Proposed by Mahyar Sefidgaran[/i]

We define the following operation which will be applied to a row of bars being situated side-by-side on positions $ 1,2,...,N$. Each bar situated at an odd numbered position is left as is, while each bar at an even numbered position is replaced by two bars. After that, all bars will be put side-by-side in such a way that all bars form a new row and are situated on positions $ 1,...,M.$ From an initial number $ a_0>0$ of bars there originates a sequence $ (a_n)_{n \ge 0},$ where $ a_n$ is the number of bars after having applied the operation $ n$ times. $ (a)$ Prove that for no $ n>0$ can we have $ a_n\equal{}1997.$ $ (b)$ Determine all natural numbers that can only occur as $ a_0$ or $ a_1$.

Define mapping $F : \mathbb{R}^4\rightarrow \mathbb{R}^4$ as $F(x,\ y,\ z,\ w)=(xy,\ y,\ z,\ w)$ and let mapping $f : S^3\rightarrow \mathbb{R}^4$ be restriction of $F$ to 3 dimensional ball $S^3=\{(x,\ y,\ z,\ w)\in{\mathbb{R}^4} | x^2+y^2+z^2+w^2=1\}$. Find the rank of $df_p$, or the differentiation of $f$ at every point $p$ in $S^3$.

Let $ABCD$ be a square. Let $N,P$ be two points on sides $AB, AD$, respectively such that $NP=NC$, and let $Q$ be a point on $AN$ such that $\angle QPN = \angle NCB$. Prove that \[ \angle BCQ = \dfrac{1}{2} \angle AQP .\]

Find the geodesic curvature of the line $y=1$ in the upper half-plane with the Lobachevskii—Poincare metric \[ds^2=(dx^2+dy^2)/y^2\]

Find all pairs $ (x, y)$ of integers such that $ x^3 \minus{} y^3 \equal{} 2xy \plus{} 8$.

Let $n$ be a natural number. Solve the equation $$||\cdots|||x-1|-2|-3|-\ldots-(n-1)|-n|=0.$$

Let $a$ and $b$ be integers such that $a - b = a^2c - b^2d$ for some consecutive integers $c$ and $d$. Prove that $|a - b|$ is a perfect square.

A circle is drawn on a plane, the center of which is not indicated. On this circle, point $A$ is marked and a second circle with center at $A$ is constructed. The second circle has a radius greater than the radius of the first and intersects the first at two points. Construct the center of the first circle using only a compass, drawing no more than five more circles.

A tree grows in the following manner. On the first day, one branch grows out of the ground. On the second day, a leaf grows on the branch and the branch tip splits up into two new branches. On each subsequent day, a new leaf grows on every existing branch, and each branch tip splits up into two new branches. How many leaves does the tree have at the end of the tenth day?

There are $12$ candies on the table, four of which are rare candies. Chad has a friend who can tell rare candies apart from regular candies, but Chad can’t. Chad’s friend is allowed to take four candies from the table, but may not take any rare candies. Can his friend always take four candies in such a way that Chad will then be able to identify the four rare candies? If so, describe a strategy. If not, prove that it cannot be done. Note that Chad does not know anything about how the candies were selected (e.g. the order in which they were selected). However, Chad and his friend may communicate beforehand.