Each of the numbers $x_1, x_2, \ldots, x_{101}$ is $\pm 1$. What is the smallest positive value of $\sum_{1\leq i < j \leq 101} x_i x_j$ ?

In a multiple-choice test, there are 4 problems, each having 3 possible answers. In some group of examinees, it turned out that for every 3 of them, there was a question that each of them gave a different answer to. What is the maximal number of examinees in this group?

Consider the maximum value of circular cone inscribed to a sphere, the ratio of it to the volume of the sphere is________.

Let $f:[0,1]\rightarrow\mathbb{R}$ be a continuous function. Define $\tilde{f}:[0,1]\rightarrow\mathbb{R}$ as $$\tilde{f}(x)=\begin{dcases} \frac{1}{x}\cdot\int_0^x f(t)dt &\text{ for } x>0;\\ f(0) &\text{ for }x=0.\end{dcases}$$ Show that: [list=a] [*] $\tilde{f}$ is continuous in $0$ and differentiable on $(0,1]$. [*] the following equality takes place: $$\int_0^1 f^2(x)dx = \left(\int_0^1 f(x)dx\right)^2 + \int_0^1 \left(f(x)-\tilde{f}(x)\right)^2dx.$$

In the expansion of $(ax+b)^{2000},$ where $a$ and $b$ are relatively prime positive integers, the coefficients of $x^{2}$ and $x^{3}$ are equal. Find $a+b.$

Let $BE$ and $CF$ be the bisectors of a triangle $ABC$. Prove that $2EF \leq BF + CE$.

It is given function $f : A \rightarrow \mathbb{R}$, $(A\subseteq \mathbb{R})$ such that $$f(x+y)=f(x)\cdot f(y)-f(xy)+1; (\forall x,y \in A)$$ If $f : A \rightarrow \mathbb{R}$, $(\mathbb{N} \subseteq A\subseteq \mathbb{R})$ is solution of given functional equation, prove that: $$f(n)=\begin{cases} \frac{c^{n+1}-1}{c-1} \text{, } \forall n \in \mathbb{N}, c \neq 1 \\ n+1 \text{, } \forall n \in \mathbb{N}, c = 1 \end{cases}$$ where $c=f(1)-1$ $a)$ Solve given functional equation for $A=\mathbb{N}$ $b)$ With $A=\mathbb{Q}$, find all functions $f$ which are solutions of the given functional equation and also $f(1997) \neq f(1998)$

You want to hang a weight $P$ so that it is $7$ m below a ceiling. To do this, it is suspended by means of a vertical cable attached to the midpoint $M$ of a chain hung by its ends from two points on the ceiling $A$ and $B$ distant from each other $4$ m. The price of the cable $PM$ is $p$ pta/m and that of the chain $AMB$ is $q$ pta/m. It is requested: a) Determine the lengths of the cable and the chain to obtain the lowest price cost of installation. b) Discuss the solution for the different values of the relation $p/q$ of both prices. (It is assumed that the weight is large enough to be considered rectile lines the chain segments $AM$ and $MB$).

For every nonempty subset $R$ of $S = \{1,2,...,10\}$, we define the alternating sum $A(R)$ as follows: If $r_1,r_2,...,r_k$ are the elements of $R$ in the increasing order, then $A(R) = r_k -r_{k-1} +r_{k-2}- ... +(-1)^{k-1}r_1$. (a) Is it possible to partition $S$ into two sets having the same alternating sum? (b) Determine the sum $\sum_{R} A(R)$, where $R$ runs over all nonempty subsets of $S$.

Find all integer triples $(a, b, c)$ satisfying the equation $$ 5 a^2 + 9 b^2 = 13 c^2. $$

The sequence of real numbers $a_0,a_1,a_2,\ldots$ is defined recursively by \[a_0=-1,\qquad\sum_{k=0}^n\dfrac{a_{n-k}}{k+1}=0\quad\text{for}\quad n\geq 1.\]Show that $ a_{n} > 0$ for all $ n\geq 1$. [i]Proposed by Mariusz Skalba, Poland[/i]

Let $A$ be a ring with $2^n+1$ elements, where $n$ is a positive integer and let \[ M = \{ k \in\mathbb{Z} \mid k \geq 2, \ x^k =x , \ \forall \ x\in A \} . \] Prove that the following statements are equivalent: a) $A$ is a field; b) $M$ is not empty and the smallest element in $M$ is $2^n+1$. [i]Marian Andronache[/i]

How many strings of length $5$ formed from the digits $0$,$1$,$2$,$3$,$4$ are there such that for each $j\in\{1,2,3,4\}$, at least $j$ of the digits are less than $j$? (For example, $02214$ satisfies the condition because it contains at least $1$ digit less than $1$, at least $2$ digits less than $2$, at least $3$ digits less than $3$, and at least $4$ digits less than $4$. The string $23404$ does not satisfy the condition because it does not contain at least $2$ digits less than $2$.) $\textbf{(A) }500\qquad\textbf{(B) }625\qquad\textbf{(C) }1089\qquad\textbf{(D) }1199\qquad\textbf{(E) }1296$

Call a positive integer $n$ "Fantastic" if none of its digits are zero and it is possible to remove one of its digits and reach to an integer which is a divisor of $n$ . ( for example , 25 is fantastic , as if we remove digit 2 , resulting number would be 5 which is divisor of 25 ) Prove that the number of Fantastic numbers is finite.

Let $\alpha\in\mathbb{Q}^+$. Determine all functions $f:\mathbb{Q}^+\to\mathbb{Q}^+$ that for all $x,y\in\mathbb{Q}^+$ satisfy the equation \[ f\left(\frac{x}{y}+y\right) ~=~ \frac{f(x)}{f(y)}+f(y)+\alpha x.\] Here $\mathbb{Q}^+$ denote the set of positive rational numbers. (Proposed by Walther Janous)

Let $a_1,a_2,\dots,a_n$ be positive real numbers whose product is $1$. Show that the sum \[\textstyle\frac{a_1}{1+a_1}+\frac{a_2}{(1+a_1)(1+a_2)}+\frac{a_3}{(1+a_1)(1+a_2)(1+a_3)}+\cdots+\frac{a_n}{(1+a_1)(1+a_2)\cdots(1+a_n)}\] is greater than or equal to $\frac{2^n-1}{2^n}$.

Let $a = \frac{m^{m+1} + n^{n+1}}{m^m + n^n}$, where $m$ and $n$ are positive integers. Prove that $a^m + a^n \geq m^m + n^n$.

Let $ABCDEF GH$ be a unit cube such that $ABCD$ is one face of the cube and $\overline{AE}$, $\overline{BF}$, $\overline{CG}$, and $\overline{DH}$ are all edges of the cube. Points $I, J, K$, and $L$ are the respective midpoints of $\overline{AF}$, $\overline{BG}$, $\overline{CH}$, and $\overline{DE}$. The inscribed circle of $IJKL$ is the largest cross-section of some sphere. Compute the volume of this sphere.

We are given an angle $0^o < \phi \le 180^o$ and a circular disc. An ant begins its journey from an interior point of the disc, travelling in a straight line in a certain direction. When it reaches the edge of the disc, it does the following: it turns clockwise by the angle $\phi $, and if its new direction does not point towards the interior of the disc, it turns by the angle $\phi $ again, and repeats this until it faces the interior. Then it continues its journey in this new direction and turns as before every time when it reaches the edge. For what values of $\phi $ is it true that for any starting point and initial direction the ant eventually returns to its starting position?

In triangle $ABC$, let $A'$, $B'$ and $C'$ be points on the sides $BC$, $CA$ and $AB$, respectively, such that$$\frac{BA'}{A'C}=\frac{CB'}{B'A}=\frac{AC'}{C'B}.$$ The line parallel to $B'C'$ passing through $A'$ intersects line $AC$ at $P$ and line $AB$ at $Q$. Prove that$$\frac{PQ}{B'C'} \geqslant 2.$$

The incircle of the triangle $ABC$ touches the side $AC$ at the point $D$. Another circle passes through $D$ and touches the rays $BC$ and $BA$, the latter at the point $A$. Determine the ratio $\frac{AD}{DC}$.

$n\ge 2 $ is an integer.Prove that the number of natural numbers $m$ so that $0 \le m \le n^2-1,x^n+y^n \equiv m (mod n^2)$ has no solutions is at least $\binom{n}{2}$

Given the equations $x^2+kx+6=0$ and $x^2-kx+6=0$. If, when the roots of the equation are suitably listed, each root of the second equation is $5$ more than the corresponding root of the first equation, then $k$ equals: $\textbf{(A)}\ 5 \qquad \textbf{(B)}\ -5 \qquad \textbf{(C)}\ 7 \qquad \textbf{(D)}\ -7 \qquad \textbf{(E)}\ \text{none of these}$

Find all positive integers $x,y$ such that $2^x+19^y$ is a perfect cube.

In a planet $Papella$ year has $400$ days with days coundting from $1-400$ . A holiday would be that day which is divisible by $6$ . The new gonverment decide to reform a new calendar and split in $10$ months with $40$ day each month , and they decide that day of month which is divisible by $6$ to be holiday . Show that after reform the number of holidays after one year decreased less then $ 10 $ percent .