(1) Let $x>0,\ y$ be real numbers. For variable $t$, find the difference of Maximum and minimum value of the quadratic function $f(t)=xt^2+yt$ in $0\leq t\leq 1$. (2) Let $S$ be the domain of the points $(x,\ y)$ in the coordinate plane forming the following condition: For $x>0$ and all real numbers $t$ with $0\leq t\leq 1$ , there exists real number $z$ for which $0\leq xt^2+yt+z\leq 1$ . Sketch the outline of $S$. (3) Let $V$ be the domain of the points $(x,\ y,\ z) $ in the coordinate space forming the following condition: For $0\leq x\leq 1$ and for all real numbers $t$ with $0\leq t\leq 1$, $0\leq xt^2+yt+z\leq 1$ holds. Find the volume of $V$. [i]2011 Tokyo University entrance exam/Science, Problem 6[/i]

There are $100$ numbers on circle, and no one number is divided by other. In same time for all numbers we make next operation: If $(a,b)$ are two neighbors ($a$ is left neighbor) , then we write between $a,b$ number $\frac{a}{(a,b)}$ and erase $a,b$ This operation was repeated some times. What maximum number of $1$ we can receive ? Example: If we have circle with $3$ numbers $4,5,6$ then after operation we receive circle with numbers $\frac{4}{(4,5)}=4,\frac{5}{(5,6)}=5, \frac{6}{(6,4)}=3$.

In acute-angled triangle $ABC,$ let $M$ be the midpoint of $BC$ and let $K$ be a point on side $AB.$ We know that $AM$ meet $CK$ at $F.$ Prove that if $AK = KF$ then $AB = CF .$

Determine all functions $ f: \mathbb{R} \mapsto \mathbb{R}$ with $ x,y \in \mathbb{R}$ such that \[ f(x \minus{} f(y)) \equal{} f(x\plus{}y) \plus{} f(y)\]

Let $ p$ be a prime number with $ p>5$. Consider the set $ X \equal{} \left\{p \minus{} n^2 \mid n\in \mathbb{N} ,\ n^2 < p\right\}$. Prove that the set $ X$ has two distinct elements $ x$ and $ y$ such that $ x\neq 1$ and $ x\mid y$. [i]Albania[/i]

Let $$f(x) = a \cos(x + 1) + b \cos(x + 2) + c \cos(x + 3)$$, where $a, b, c$ are real. Given that $f(x)$ has at least two zeros in the interval $(0, \pi)$, find all its real zeros.

A chess-board $6\times 6$ is tiled with the $2\times 1$ dominos. Prove that you can cut the board onto two parts by a straight line that does not cut dominos.

in a triangle $ABC$, $I$ is the incenter. $BI$ and $CI$ cut the circumcircle of $ABC$ at $E$ and $F$ respectively. $M$ is the midpoint of $EF$. $C$ is a circle with diameter $EF$. $IM$ cuts $C$ at two points $L$ and $K$ and the arc $BC$ of circumcircle of $ABC$ (not containing $A$) at $D$. prove that $\frac{DL}{IL}=\frac{DK}{IK}$.(25 points)

Define the function $\xi : \mathbb Z^2 \to \mathbb Z$ by $\xi(n,k) = 1$ when $n \le k$ and $\xi(n,k) = -1$ when $n > k$, and construct the polynomial \[ P(x_1, \dots, x_{1000}) = \prod_{n=1}^{1000} \left( \sum_{k=1}^{1000} \xi(n,k)x_k \right). \] (a) Determine the coefficient of $x_1x_2 \dots x_{1000}$ in $P$. (b) Show that if $x_1, x_2, \dots, x_{1000} \in \left\{ -1,1 \right\}$ then $P(x_1,x_2,\dots,x_{1000}) = 0$. [i]Proposed by Evan Chen[/i]

For a positive integer $M$, if there exist integers $a$, $b$, $c$ and $d$ so that: \[ M \leq a < b \leq c < d \leq M+49, \qquad ad=bc \] then we call $M$ a GOOD number, if not then $M$ is BAD. Please find the greatest GOOD number and the smallest BAD number.

Matilda drew $12$ quadrilaterals. The first quadrilateral is an rectangle of integer sides and $7$ times more width than long. Every time she drew a quadrilateral she joined the midpoints of each pair of consecutive sides with a segment. It´s is known that the last quadrilateral Matilda drew was the first with area less than $1$. What is the maximum area possible for the first quadrilateral? [asy]size(200); pair A, B, C, D, M, N, P, Q; real base = 7; real altura = 1; A = (0, 0); B = (base, 0); C = (base, altura); D = (0, altura); M = (0.5*base, 0*altura); N = (0.5*base, 1*altura); P = (base, 0.5*altura); Q = (0, 0.5*altura); draw(A--B--C--D--cycle); // Rectángulo draw(M--P--N--Q--cycle); // Paralelogramo dot(M); dot(N); dot(P); dot(Q); [/asy] $\textbf{Note:}$ The above figure illustrates the first two quadrilaterals that Matilda drew.

Prove that if $n$ is large enough, in every $n\times n$ square that a natural number is written on each one of its cells, one can find a subsquare from the main square such that the sum of the numbers is this subsquare is divisible by $1391$.

A sequence of integers $ a_1,a_2,a_3,...$ is such that $ a_1\equal{}2, a_2\equal{}3$, and $ a_{n\plus{}1}\equal{}2a_{n\minus{}1}$ or $ 3a_n\minus{}2a_{n\minus{}1}$ for all $ n \ge 2$. Prove that no number between $ 1600$ and $ 2000$ can be an element of the sequence.

Let $n$ be a positive integer. We define $f(n)$ as the number of finite sequences $(a_1, a_2, \ldots , a_k)$ of positive integers such that $a_1 < a_2 < a_3 < \cdots < a_k$ and $$a_1+a_2^2+a_3^3+\cdots + a_k^k \leq n.$$ Determine the positive constants $\alpha$ and $C$ such that $$\lim\limits_{n\rightarrow \infty} \frac{f(n)}{n^\alpha}=C.$$

Let $n$ be a positive integer and let $V$ be a $(2n-1)$-dimensional vector space over the two-element field. Prove that for arbitrary vectors $v_1,\dots,v_{4n-1} \in V,$ there exists a sequence $1\leq i_1<\dots<i_{2n}\leq 4n-1$ of indices such that $v_{i_1}+\dots+v_{i_{2n}}=0.$

Given a parallelogram with area $1$ and we will construct lines where this lines connect a vertex with a midpoint of the side no adjacent to this vertex; with the $8$ lines formed we have a octagon inside of the parallelogram. Determine the area of this octagon

In a wrestling tournament, there are $100$ participants, all of different strengths. The stronger wrestler always wins over the weaker opponent. Each wrestler fights twice and those who win both of their fights are given awards. What is the least possible number of awardees?

Compute the number of ordered pairs $(m,n)$ of positive integers such that $(2^m-1)(2^n-1)\mid2^{10!}-1.$ [i]Proposed by Luke Robitaille[/i]

Let $ a_{0} \equal{} 1994$ and $ a_{n \plus{} 1} \equal{} \frac {a_{n}^{2}}{a_{n} \plus{} 1}$ for each nonnegative integer $ n$. Prove that $ 1994 \minus{} n$ is the greatest integer less than or equal to $ a_{n}$, $ 0 \leq n \leq 998$

As shown below, convex pentagon $ ABCDE$ has sides $ AB \equal{} 3$, $ BC \equal{} 4$, $ CD \equal{} 6$, $ DE \equal{} 3$, and $ EA \equal{} 7$. The pentagon is originally positioned in the plane with vertex $ A$ at the origin and vertex $ B$ on the positive $ x$-axis. The pentagon is then rolled clockwise to the right along the $ x$-axis. Which side will touch the point $ x \equal{} 2009$ on the $ x$-axis? [asy]size(250); defaultpen(linewidth(.8pt)+fontsize(8pt)); dotfactor=4; pair A=(0,0), Ep=7*dir(105), B=3*dir(0); pair D=Ep+B; pair C=intersectionpoints(Circle(D,6),Circle(B,4))[1]; pair[] ds={A,B,C,D,Ep}; dot(ds); draw(B--C--D--Ep--A); draw((6,6)..(8,4)..(8,3),EndArrow(3)); xaxis("$x$",-8,14,EndArrow(3)); label("$E$",Ep,NW); label("$D$",D,NE); label("$C$",C,E); label("$B$",B+(.2,.1),ENE); label("$A$",A+(-.1,.1),WNW); label("$(0,0)$",A,S); label("$3$",midpoint(A--B),N); label("$4$",midpoint(B--C),NW); label("$6$",midpoint(C--D),NE); label("$3$",midpoint(D--Ep),S); label("$7$",midpoint(Ep--A),W);[/asy]$ \textbf{(A)}\ \overline{AB} \qquad \textbf{(B)}\ \overline{BC} \qquad \textbf{(C)}\ \overline{CD} \qquad \textbf{(D)}\ \overline{DE} \qquad \textbf{(E)}\ \overline{EA}$

Positive integers $a$ and $b$ satisfy the equality $a+d(a)=b^2+2$ where $d(n)$ denotes the number of divisors of $n$. Prove that $a+b$ is even.

Find the side sizes of an isosceles trapezoid that has longest side $13$ cm, perimeter $28$ cm and area $27$ cm$^2$. Is there such a trapezoid, if we we ask for area $27.001$ cm$^2$ ?

Prove that the polynomial $P_n(x)=1+x+\frac{x^2}{2!}+\cdots +\frac{x^n}{n!}$ has no real zeros if $n$ is even and has exatly one real zero if $n$ is odd

In a non-isosceles triangle $ABC$ the bisectors of angles $A$ and $B$ are inversely proportional to the respective sidelengths. Find angle $C$.

A Boy Scouts camp holds a campfire. The camp has scarfs of m colors with n scarves of each color, and gives each of its $mn$ scouts a scarf, where $m, n \ge 2$ are integers. The camp then divides its scouts into troops by the color of their scarfs. At the beginning of the campfire, the scouts are seated in a circle so that scouts in the same troop are seated next to each other. The camp organizer then proceeds to select, round by round, representatives to perform a show, with the following conditions: there must be at least two representatives in each round, they must come from the same troop, and any specific set of representatives can only perform once. (For example, if $\{A, B\}$ has performed, then $\{A, B\}$ cannot perform again, but $\{A, B, C\}$ can still perform.) This process is repeated until all valid sets of representatives have performed. At this point, the organizers order each scout to hand their scarfs to the scout to the left, and re-group the scouts into troops, again according to their scarf color, and the process above is resumed, until the set of valid sets of representatives is exhausted again. (The sets of representatives after re-grouping must also be distinct from the sets before re-grouping.) When that happens, the organizers order another re-group, and resumes the process, and this repeats until there can be no further performances. Find, in simple form, the total number of performances that will be performed.