Which positive integers $a$ have the property that \(n!-a\) is a perfect square for infinitely many positive integers \(n\)?

Rumburak kidnapped $31$ members of party $A$ , $28$ members of party $B$, $23$ members of party $C$, $19$ members of Party $D$ and each of them in a separate cell. After work out occasionally they could walk in the yard and talk. Once three people started to talk to each other members of three different parties, Rumburak re-registered them to the fourth party as a punishment.(They never talked to each other more than three kidnapped.) a) Could it be that after some time all were abducted by members of one party? Which? b) Determine all four positive integers of which the sum is $101$ and which as the numbers of kidnapped members of the four parties allow the Rumburaks all of them became members of one party over time.

How many permutations $(s_1, s_2,...,s_n) $of $(1,2 ,...,n)$ are there satisfying the condition $s_i > s_j$ for all $i \ge j + 3$ when $n = 5$ and when $n = 7$?

Let $F(n)$ be the set of polynomials $P(x) = a_0+a_1x+\cdots+a_nx^n$, with $a_0, a_1, . . . , a_n \in \mathbb R$ and $0 \leq a_0 = a_n \leq a_1 = a_{n-1 } \leq \cdots \leq a_{[n/2] }= a_{[(n+1)/2]}.$ Prove that if $f \in F(m)$ and $g \in F(n)$, then $fg \in F(m + n).$

Consider $9$ points in space, no four of which are coplanar. Each pair of points is joined by an edge (that is, a line segment) and each edge is either colored blue or red or left uncolored. Find the smallest value of $\,n\,$ such that whenever exactly $\,n\,$ edges are colored, the set of colored edges necessarily contains a triangle all of whose edges have the same color.

A map $ F : P(X) \rightarrow P(X)$, where $ P(X)$ denotes the set of all subsets of $ X$, is called a $ \textit{closure operation}$ on $ X$ if for arbitrary $ A,B \subset X$, the following conditions hold: (i) $ A \subset F(A);$ (ii) $ A \subset B \Rightarrow F(A) \subset F(B);$ (iii) $ F(F(A))\equal{}F(A)$. The cardinal number $ \min \{ |A| : \;A \subset X\ ,\;F(A)\equal{}X\ \}$ is called the $ \textit{density}$ of $ F$ and is denoted by $ d(F)$. A set $ H \subset X$ is called $ \textit{discrete}$ with respect to $ F$ if $ u \not \in F(H\minus{}\{ u \})$ holds for all $ u \in H$. Prove that if the density of the closure operation $ F$ is a singular cardinal number, then for any nonnegative integer $ n$, there exists a set of size $ n$ that is discrete with respect to $ F$. Show that the statement is not true when the existence of an infinite discrete subset is required, even if $ F$ is the closure operation of a topological space satisfying the $ T_1$ separation axiom. [i]A. Hajnal[/i]

Let $S(n)$ be the sum of digits of $n$. Determine all the pairs $(a, b)$ of positive integers, such that the expression $S(an + b) - S(n)$ has a finite number of values, where $n$ is varying in the positive integers.

A regular $2004$-sided polygon is given, with all of its diagonals drawn. After some sides and diagonals are removed, every vertex has at most five segments coming out of it. Prove that one can color the vertices with two colors such that at least $\frac{3}{5}$ of the remaining segments have ends with different colors.

Find the smallest $n$ such that every subset of $\{1, 2, 3, . . . , 2004 \}$ with $n$ elements contains at least two elements that are relatively prime.

The quadrilateral $ABCD$ has the following equality $\angle ABC=\angle BCD=150^{\circ}$. Moreover, $AB=18$ and $BC=24$, the equilateral triangles $\triangle APB,\triangle BQC,\triangle CRD$ are drawn outside the quadrilateral. If $P(X)$ is the perimeter of the polygon $X$, then the following equality is true $P(APQRD)=P(ABCD)+32$. Determine the length of the side $CD$.

Let $a$ and $b$ be positive reals such that $$a=1+\frac{a}{b}$$$$b=3+\frac{4+a}{b-2}$$ What is $a$? $\text{(A) }\sqrt{2}\qquad\text{(B) }2+\sqrt{2}\qquad\text{(C) }2+\sqrt{2}+\sqrt[3]{2}\qquad\text{(D) }\sqrt{2}+\sqrt[3]{2}\qquad\text{(E) }\sqrt[3]{2}$

To every vertex of a regular pentagon a real number is assigned. We may perform the following operation repeatedly: we choose two adjacent vertices of the pentagon and replace each of the two numbers assigned to these vertices by their arithmetic mean. Is it always possible to obtain the position in which all five numbers are zeroes, given that in the initial position the sum of all five numbers is equal to zero?

Suppose that $ ABC$ is an acute triangle such that $ \tan A$, $ \tan B$, $ \tan C$ are the three roots of the equation $ x^3 \plus{} px^2 \plus{} qx \plus{} p \equal{} 0$, where $ q\neq 1$. Show that $ p \le \minus{} 3\sqrt 3$ and $ q > 1$.

Let $E$ be a point on side $[AD]$ of rhombus $ABCD$. Lines $AB$ and $CE$ meet at $F$, lines $BE$ and $DF$ meet at $G$. If $m(\widehat{DAB}) = 60^\circ $, what is$m(\widehat{DGB})$? $ \textbf{a)}\ 45^\circ \qquad\textbf{b)}\ 50^\circ \qquad\textbf{c)}\ 60^\circ \qquad\textbf{d)}\ 65^\circ \qquad\textbf{e)}\ 75^\circ $

Find value of $$\frac{1}{1+x+xy}+\frac{1}{1+y+yz}+\frac{1}{1+z+zx}$$ if $x$, $y$ and $z$ are real numbers usch that $xyz=1$

Will and Lucas are playing a game. Will claims that he has a polynomial $f$ with integer coefficients in mind, but Lucas doesn’t believe him. To see if Will is lying, Lucas asks him on minute $i$ for the value of $f(i)$, starting from minute $ 1$. If Will is telling the truth, he will report $f(i)$. Otherwise, he will randomly and uniformly pick a positive integer from the range $[1,(i+1)!]$. Now, Lucas is able to tell whether or not the values that Will has given are possible immediately, and will call out Will if this occurs. If Will is lying, say the probability that Will makes it to round $20$ is $a/b$. If the prime factorization of $b$ is $p_1^{e_1}... p_k^{e_k}$ , determine the sum $\sum_{i=1}^{k} e_i$.

Let $n \ge 4$ be a natural number. Let $A_1A_2
\cdots
A_n$ be a regular polygon and $X = \{ 1,2,3....,n \} $. A subset $\{ i_1, i_2,\cdots, i_k \} $ of $X$, with $k \ge 3$ and $i_1 < i_2 <
\cdots < i_k$, is called a good subset if the angles of the polygon $A_{i_1}A_{i_2}\cdots
A_{i_k}$ , when arranged in the increasing order, are in an arithmetic progression. If $n$ is a prime, show that a proper good subset of $X$ contains exactly four elements.

Let $ABCD$ be an tangential trapezoid, $E$ is a point of its diagonals intersection, $r_1,r_2,r_3,r_4$ -- the radiuses of the circles inscribed in the triangles $ABE$, $BCE$, $CDE$, $DAE$ respectively. Prove that $$1/(r_1)+1/(r_3) = 1/(r_2)+1/(r_4).$$

3 segments $AA_1$, $BB_1$, $CC_1$ in space share a common midpoint $M$. Turns out, the sphere circumscribed about the tetrahedron $MA_1B_1C_1$ is tangent to plane $ABC$ at point $D$. Point $O$ is the circumcenter of triangle $ABC$. Prove that $MO = MD$.

[b]10.[/b] Consider a point performing a random walk on a planar triangular lattice and suppose that it moves away with equal probability from any lattice point along any one of the six lattice lines issuing from it. Prove that if the walk is continued indefinitely, then the point will return to its starting point with probability 1. [b](P. 5)[/b]

Find all functions $f:(1,+\infty) \rightarrow (1,+\infty)$ that satisfy the following condition: for arbitrary $x,y>1$ and $u,v>0$, inequality $f(x^uy^v)\le f(x)^{\dfrac{1}{4u}}f(y)^{\dfrac{1}{4v}}$ holds.

Given 10 light switches, each can be in two states: on and off. For each pair of switches there is a light bulb which is on if and only if when both switches are on (45 bulbs in total). The bulbs and the switches are unmarked so it is unclear which switches correspond to which bulb. In the beginning all switches are off. How many flips are needed to find out regarding all bulbs which switches are connected to it? On each step you can flip precisely one switch

Circles $\Gamma_1$ and $\Gamma_2$ meet at points $X$ and $Y$. A circle $S_1$ touches internally $\Gamma_1$ at $A$ and $\Gamma_2$ externally at $B$. A circle $S_2$ touches $\Gamma_2$ internally at $C$ and $\Gamma_1$ externally at $D$. Prove that the points $A, B, C, D$ are either collinear or concyclic. (A. Voidelevich)

Determine all three-digit numbers $N$ having the property that $N$ is divisible by $11,$ and $\frac{N}{11}$ is equal to the sum of the squares of the digits of $N.$

Simplify \[ \sqrt[3]{x\sqrt[3]{x\sqrt[3]{x\sqrt{x}}}} \]$ \textbf{(A)}\ \sqrt{x} \qquad \textbf{(B)}\ \sqrt[3]{x^2} \qquad \textbf{(C)}\ \sqrt[27]{x^2} \qquad \textbf{(D)}\ \sqrt[54]{x} \qquad \textbf{(E)}\ \sqrt[81]{x^{80}}$