On the graph of a polynomial with integral coefficients are two points with integral coordinates. Prove that if the distance between these two points is integral, then the segment connecting them is parallel to the $x$-axis. [i](4 points)[/i]

Given a triangle $ABC$ ($AB> AC$) and a circle circumscribed around it. Construct with a compass and a ruler the midpoint of the arc $BC$ (not containing vertex $A$), with no more than two lines (straight or circles).

Let $ p,q,r $ be prime numbers. Solve the equation $ p^{2q}+q^{2p}=r $

Let $c$ be fixed natural number. Sequence $(a_n)$ is defined by: $a_1=1$, $a_{n+1}=d(a_n)+c$ for $n=1,2,...$. where $d(m)$ is number of divisors of $m$. Prove that there exist $k$ natural such that sequence $a_k,a_{k+1},...$ is periodic.

Determine if there is a way to tile a $5 \times 6$ unit square board by dominos such that one can not use a needle to peer through the tiling? Determine if there is a way to tile a $5 \times 6$ unit square board by dominos such that one can use a needle to through the tiling? What if it is a $6 \times 6$ board?

Juli is a mathematician and devised an algorithm to find a husband. The strategy is: • Start interviewing a maximum of $1000$ prospective husbands. Assign a ranking $r$ to each person that is a positive integer. No two prospects will have same the rank $r$. • Reject the first $k$ men and let $H$ be highest rank of these $k$ men. • After rejecting the first $k$ men, select the next prospect with a rank greater than $H$ and then stop the search immediately. If no candidate is selected after $999$ interviews, the $1000th$ person is selected. Juli wants to find the value of $k$ for which she has the highest probability of choosing the highest ranking prospect among all $1000$ candidates without having to interview all $1000$ prospects. [b](a)[/b] (6 points:) What is the probability that the highest ranking prospect among all $1000$ prospects is the $(m + 1)th$ prospect? [b](b)[/b] (6 points:) Assume the highest ranking prospect is the $(m + 1)th$ person to be interviewed. What is the probability that the highest rank candidate among the first $m$ candidates is one of the first $k$ candidates who were rejected? [b](c)[/b] (6 points:) What is the probability that the prospect with the highest rank is the $(m+1)th$ person and that Juli will choose the $(m+1)th$ man using this algorithm? [b](d)[/b] (16 points:) The total probability that Juli will choose the highest ranking prospect among the $1000$ prospects is the sum of the probability for each possible value of $m+1$ with $m+1$ ranging between $k+1$ and $1000$. Find the sum. To simplify your answer use the formula $In N \approx \frac{1}{N-1}+\frac{1}{N-2}+...+\frac{1}{2}+1$ [b](e)[/b] (6 points:) Find that value of $k$ that maximizes the probability of choosing the highest ranking prospect without interviewing all $1000$ candidates. You may need to know that the maximum of the function $x ln \frac{A}{x-1}$ is approximately $\frac{A + 1}{e}$, where $A$ is a constant and $e$ is Euler’s number, $e = 2.718....$

Of the following expressions the one equal to $ \frac{a^{\minus{}1}b^{\minus{}1}}{a^{\minus{}3} \minus{} b^{\minus{}3}}$ is: $ \textbf{(A)}\ \frac{a^2b^2}{b^2 \minus{} a^2}\qquad \textbf{(B)}\ \frac{a^2b^2}{b^3 \minus{} a^3}\qquad \textbf{(C)}\ \frac{ab}{b^3 \minus{} a^3}\qquad \textbf{(D)}\ \frac{a^3 \minus{} b^3}{ab}\qquad \textbf{(E)}\ \frac{a^2b^2}{a \minus{} b}$

Find all functions $f$ and $g$ defined from $\mathbb{R}_{>0}$ to $\mathbb{R}_{>0}$ such that for all $x, y > 0$ the two equations hold $$ (f(x) + y - 1)(g(y) + x - 1) = {(x + y)}^2 $$ $$ (-f(x) + y)(g(y) + x) = (x + y + 1)(y - x - 1) $$ [i]Note: $\mathbb{R}_{>0}$ denotes the set of positive real numbers.[/i]

The digit-sum of $998$ is $9+9+8=26$. How many 3-digit whole numbers, whose digit-sum is $26$, are even? $\text{(A)}\ 1 \qquad \text{(B)}\ 2 \qquad \text{(C)}\ 3 \qquad \text{(D)}\ 4 \qquad \text{(E)}\ 5$

Let $\Omega$ be the incircle of a triangle $ABC$. Suppose that there exists a circle passing through $B$ and $C$ and tangent to $\Omega$ in $A'$. Suppose the similar points $B'$, $C'$ exist. Prove that the lines $AA', BB'$ and $CC'$ are concurrent.

Let $ \Gamma_1$ and $\Gamma_2$ be two different circles with the radius of same length and centers at points $O_1$ and $O_2$, respectively. Circles $\Gamma_1$ and $\Gamma_2$ are tangent at point $P$. The line $\ell$ passing through $O_1$ is tangent to $\Gamma_2$ at point $A$. The line $\ell$ intersects $\Gamma_1$ at point $X$ with $X$ between $A$ and $O_1$. Let $M$ be the midpoint of $AX$ and $Y$ the intersection of $PM$ and $\Gamma_2$ with $Y\ne P$. Prove that $XY$ is parallel to $O_1O_2$.

Prove that given $n$ points in the plane, not all aligned, there exists a line that passes through exactly two of them. [hide=original wording]Demostrar que dados n puntos en el plano, no todos alineados, existe una recta que pasa por exactamente dos de ellos.[/hide]

Positive integers $x>1$ and $y$ satisfy an equation $2x^2-1=y^{15}$. Prove that 5 divides $x$.

Ten students take a test consisting of $4$ different papers in Algebra, Geometry, Number Theory and Combinatorics. First, the proctor distributes randomly the Algebra paper to each student. Then the remaining papers are distributed one at a time in the following order: Geometry, Number Theory, Combinatorics in such a way that no student receives a paper before he finishes the previous one. In how many ways can the proctor distribute the test papers given that a student may for example nish the Number Theory paper before another student receives the Geometry paper, and that he receives the Combinatorics paper after that the same other student receives the Combinatorics papers.

Find all four-digit numbers which are equal to the cube of the sum of their digits.

On the three distinct lines $a, b$, and $c$ three points $A, B$, and $C$ are given, respectively. Construct three collinear points $X, Y,Z$ on lines $a, b, c$, respectively, such that $\frac{BY}{AX} = 2$ and $ \frac{CZ}{AX} = 3$.

Solve the equation $p+\sqrt{q^{2}+r}=\sqrt{s^{2}+t}$ in prime numbers.

Given a convex quadrangle $ABCD$ with $|AD| = |BD| = |CD|$ and $\angle ADB = \angle DCA$, $\angle CBD = \angle BAC$, find the sizes of the angles of the quadrangle.

Alice and Bob play a game together as a team on a $100 \times 100$ board with all unit squares initially white. Alice sets up the game by coloring exactly $k$ of the unit squares red at the beginning. After that, a legal move for Bob is to choose a row or column with at least $10$ red squares and color all of the remaining squares in it red. What is the smallest $k$ such that Alice can set up a game in such a way that Bob can color the entire board red after finitely many moves? Proposed by [i]Nikola Velov, Macedonia[/i]

a) The digits of a natural number were rearranged. Prove that the sum of given and obtained numbers can't equal $999...9$ ($1967$ of nines). b) The digits of a natural number were rearranged. Prove that if the sum of the given and obtained numbers equals $1010$, than the given number was divisible by $10$.

Let $r_1=2$ and $r_n = \prod^{n-1}_{k=1} r_i + 1$, $n \geq 2.$ Prove that among all sets of positive integers such that $\sum^{n}_{k=1} \frac{1}{a_i} < 1,$ the partial sequences $r_1,r_2, ... , r_n$ are the one that gets nearer to 1.

$(MON 3)$ Let $A_k (1 \le k \le h)$ be $n-$element sets such that each two of them have a nonempty intersection. Let $A$ be the union of all the sets $A_k,$ and let $B$ be a subset of $A$ such that for each $k (1\le k \le h)$ the intersection of $A_k$ and $B$ consists of exactly two different elements $a_k$ and $b_k$. Find all subsets $X$ of the set $A$ with $r$ elements satisfying the condition that for at least one index $k,$ both elements $a_k$ and $b_k$ belong to $X$.

The figure below is cut along the lines into polygons (which need not be convex). No polygon contains a $2 \times 2$ square. What is the smallest possible number of polygons? [missing figure]

Let $P(x)=a x^{75}+b$ be a polynomial where \(a\) and \(b\) are coprime integers in the set \(\{1,2,\dots,151\}\), and suppose it satisfies the following condition: there exists at most one prime \(p\) such that for every positive integer \(k\), \(p\mid P(k)\). Prove that for every prime \(q \neq p\) there exists a positive integer \(k\) for which $q^2 \mid P(k).$

Let $S$ be a subset of $\{1,2,3,...,24\}$ with $n(S)=10$. Show that $S$ has two $2$-element subsets $\{x,y\}$ and $\{u,v\}$ such that $x+y=u+v$