Find any polynomial with integer coefficients, the smallest value of which on the entire line is equal to : a) $-\sqrt2$ b) $\sqrt2$

Prove that there exists a partition of the set $A = \{1^3, 2^3, \ldots , 2000^3\}$ into $19$ nonempty subsets such that the sum of elements of each subset is divisible by $2001^2$.

Prove that if $a,b,c$ are positive numbers with $abc=1$, then \[\frac{a}{b} +\frac{b}{c} + \frac{c}{a} \ge a + b + c. \]

[b]a)[/b] Determine the center of the ring of square matrices of a certain dimensions with elements in a given field, and prove that it is isomorphic with the given field. [b]b)[/b] Prove that $$ \left(\mathcal{M}_n\left( \mathbb{R} \right) ,+, \cdot\right)\not\cong \left(\mathcal{M}_n\left( \mathbb{C} \right) ,+,\cdot\right) , $$ for any natural number $ n\ge 2. $ [i]Marian Andronache, Ion Sava[/i]

For $n\in\mathbb{N}$, let $a_{n}$ denote the closest integer to $\sqrt{n}$. Evaluate \[\sum_{n=1}^\infty{\frac{1}{a_{n}^{3}}}.\]

(a) Let $a, b, c, d$ be integers such that $ad\ne bc$. Show that is always possible to write the fraction $\frac{1}{(ax+b)(cx+d)}$in the form $\frac{r}{ax+b}+\frac{s}{cx+d}$ (b) Find the sum $$\frac{1}{1 \cdot 4}+\frac{1}{4 \cdot 7}+\frac{1}{7 \cdot 10}+...+\frac{1}{1995 \cdot 1996}$$

There is a population $P$ of $10000$ bacteria, some of which are friends (friendship is mutual), so that each bacterion has at least one friend and if we wish to assign to each bacterion a coloured membrane so that no two friends have the same colour, then there is a way to do it with $2021$ colours, but not with $2020$ or less. Two friends $A$ and $B$ can decide to merge in which case they become a single bacterion whose friends are precisely the union of friends of $A$ and $B$. (Merging is not allowed if $A$ and $B$ are not friends.) It turns out that no matter how we perform one merge or two consecutive merges, in the resulting population it would be possible to assign $2020$ colours or less so that no two friends have the same colour. Is it true that in any such population $P$ every bacterium has at least $2021$ friends?

A cup with a volume of $8$ fluid ounces is filled at the rate of $0.5$ ounces per second. However, a hole at the bottom of the cup also drains it at the rate of $0.3$ ounces per second. Once the cup is full, how many ounces of water will have drained out of the cup?

Consider an orange and black coloring of a $20 \times 14$ square grid. Let $n$ be the number of colorings such that every row and column has an even number of orange squares. Evaluate $\log_2 n$.

How many three-digit numbers $\underline{abc}$ have the property that when it is added to $\underline{cba}$, the number obtained by reversing its digits, the result is a palindrome? (Note that $\underline{cba}$ is not necessarily a three-digit number since before reversing, $c$ may be equal to $0$.)

In a class with $n$ students in the span of $k$ days, each day are chosen three to be tested. Each two students can be taken in such triple only once. Prove that for the greatest $k$ satisfying these conditions, the following inequalities are true: $\frac{n(n-3)}{6}\leq k\leq \frac{n(n-1)}{6}$.

See all the problems from 5-th Kyiv math festival [url=http://www.mathlinks.ro/Forum/viewtopic.php?p=506789#p506789]here[/url] Squirrels $A$ and $B$ have $360$ nuts. $A$ divides these nuts into five non-empty heaps and $B$ chooses three heaps. If the total number of nuts in these heaps is divisible by the total number of nuts in other two heaps then $A$ wins. Otherwise $B$ wins. Which of the squirrels has a winning strategy?

Inside the acute-angled triangle $ABC$, mark the point $O$ so that $\angle AOB=90^o$, a point $M$ on the side $BC$ such that $\angle COM=90^o$, and a point $N$ on the segment $BO$ such that $\angle OMN = 90^o$. Let $P$ be the point of intersection of the lines $AM$ and $CN$, and let $Q$ be a point on the side $AB$ that such $\angle POQ = 90^o$. Prove that the lines $AN, CO$ and $MQ$ intersect at one point.

All pages of a magazine are numbered and printed on both sides. One sheet with two sides is missing. The numbers of the remaining pages sum to $963$. How many pages did the magazine have originally and which pages are missing?

Prove that there are two natural numbers $ p,q, $ satisfying $$ p<q<n\bigg|p+(p+1)+\cdots +(q-1) +q, $$ if and only if $ n $ is not a power of $ 2. $

As shown in the figure, line segment $\overline{AD}$ is trisected by points $B$ and $C$ so that $AB=BC=CD=2.$ Three semicircles of radius $1,$ $\overarc{AEB},\overarc{BFC},$ and $\overarc{CGD},$ have their diameters on $\overline{AD},$ and are tangent to line $EG$ at $E,F,$ and $G,$ respectively. A circle of radius $2$ has its center on $F. $ The area of the region inside the circle but outside the three semicircles, shaded in the figure, can be expressed in the form \[\frac{a}{b}\cdot\pi-\sqrt{c}+d,\] where $a,b,c,$ and $d$ are positive integers and $a$ and $b$ are relatively prime. What is $a+b+c+d$? [asy] size(6cm); filldraw(circle((0,0),2), gray(0.7)); filldraw(arc((0,-1),1,0,180) -- cycle, gray(1.0)); filldraw(arc((-2,-1),1,0,180) -- cycle, gray(1.0)); filldraw(arc((2,-1),1,0,180) -- cycle, gray(1.0)); dot((-3,-1)); label("$A$",(-3,-1),S); dot((-2,0)); label("$E$",(-2,0),NW); dot((-1,-1)); label("$B$",(-1,-1),S); dot((0,0)); label("$F$",(0,0),N); dot((1,-1)); label("$C$",(1,-1), S); dot((2,0)); label("$G$", (2,0),NE); dot((3,-1)); label("$D$", (3,-1), S); [/asy] $\textbf{(A) } 13 \qquad\textbf{(B) } 14 \qquad\textbf{(C) } 15 \qquad\textbf{(D) } 16\qquad\textbf{(E) } 17$

Find the greatest integer less than or equal to $\sum_{k=1}^{2^{1983}} k^{\frac{1}{1983} -1}.$

It is known that each vertex of the convex polyhedron $P$ belongs to three different faces, and each vertex of $P$ can be dyed black and white, so that the two endpoints of each edge of $P$ are different colors. Proof: The interior of each edge of $P$ can be dyed red, yellow, and blue, so that the colors of the three edges connected to each vertex are different, and each face contains two colors of edges. [i]Created by Liang Xiao[/i]

Let $ABC$ be a triangle with sides $AB = 19$, $BC = 21$ and $AC = 20$. Let $\omega$ be the incircle of $ABC$ with center $I$. Extend $BI$ so that it intersects $AC$ at $E$. If $\omega$ is tangent to $AC$ at the point $D$, then find the length of $DE$.

Determine all functions $f:\mathbb{Z}^+\to \mathbb{Z}$ which satisfy the following condition for all pairs $(x,y)$ of [i]relatively prime[/i] positive integers: \[f(x+y) = f(x+1) + f(y+1).\]

For a triple $(m,n,r)$ of integers with $0 \le r \le n \le m-2$, define $p(m,n,r)=\sum^r_{k=0} (-1)^k \dbinom{m+n-2(k+1)}{n} \dbinom{r}{k}$. Prove that $p(m,n,r)$ is positive and that $\sum^n_{r=0} p(m,n,r)=\dbinom{m+n}{n}$.

On the control board of a nuclear station, there are $n$ electric switches ($n > 0$), all in one row. Each switch has two possible positions: up and down. The switches are connected to each other in such a way that, whenever a switch moves down from its upper position, its right neighbour (if it exists) automatically changes position. At the beginning, all switches are down. The operator of the board first changes the position of the leftmost switch once, then the position of the second leftmost switch twice etc., until eventually he changes the position of the rightmost switch n times. How many switches are up after all these operations?

Points $A$, $B$, and $C$ lie on a circle of radius $5$ such that $AB=6$ and $AC=8$. Find the smaller of the two possible values of $BC$.

Let $ D,E,F $ be the feet of the interior bisectors from $ A,B, $ respectively $ C, $ and let $ A',B',C' $ be the symmetric points of $ A,B, $ respectively, $ C, $ to $ D,E, $ respectively $ F, $ such that $ A,B,C $ lie on $ B'C',A'C', $ respectively, $ A'B'. $ Show that the $ ABC $ is equilateral. [i]Marius Beceanu[/i]

Point $P$ is in the interior of $\triangle ABC$. The side lengths of $ABC$ are $AB = 7$, $BC = 8$, $CA = 9$. The three foots of perpendicular lines from $P$ to sides $BC$, $CA$, $AB$ are $D$, $E$, $F$ respectively. Suppose the minimal value of $\frac{BC}{PD} + \frac{CA}{PE} + \frac{AB}{PF}$ can be written as $\frac{a}{b}\sqrt{c}$, where $\gcd(a,b) = 1$ and $c$ is square free, calculate $abc$. [asy] size(120); pathpen = linewidth(0.7); pointfontpen = fontsize(10); // pointpen = black; pair B=(0,0), C=(8,0), A=IP(CR(B,7),CR(C,9)), P = (2,1.6), D=foot(P,B,C), E=foot(P,A,C), F=foot(P,A,B); D(A--B--C--cycle); D(P--D); D(P--E); D(P--F); D(MP("A",A,N)); D(MP("B",B)); D(MP("C",C)); D(MP("D",D)); D(MP("E",E,NE)); D(MP("F",F,NW)); D(MP("P",P,SE)); [/asy]