Prime $p=n^2 +1$ is given. Find the sets of solutions to the below equation: \[x^2 - (n^2 +1)y^2 = n^2.\] (25 points)

Given several points, not all lying on one straight line. Some number is assigned to every point. It is known, that if a straight line contains two or more points, than the sum of the assigned to those points equals zero. Prove that all the numbers equal to zero.

Let $n,k$ be integers such that $n\geq k\geq3$. Consider $n+1$ points in a plane (there is no three collinear points) and $k$ different colors, then, we color all the segments that connect every two points. We say that an angle is good if its vertex is one of the initial set, and its two sides aren't the same color. Show that there exist a coloration such that the \\ total number of good angles is greater than $n \binom{k}{2} \lfloor(\frac{n}{k})\rfloor^2$

How many ways are there to paint an $m \times n$ board, so that each square is painted blue, white, brown or gold, and in each $2 \times 2$ square there is one square of each color?

It’s allowed to replace any of three coefficients of quadratic trinomial by its discriminant. Is it true that from any quadratic trinomial that does not have real roots, we can perform such operation several times to get a quadratic trinomial that have real roots?

Let $\omega$ be the circumcircle of triangle $ABC$. A point $B_1$ is chosen on the prolongation of side $AB$ beyond point B so that $AB_1 = AC$. The angle bisector of $\angle BAC$ meets $\omega$ again at point $W$. Prove that the orthocenter of triangle $AWB_1$ lies on $\omega$ . (A.Tumanyan)

Let $a,b,c\in\{0,1,2,\cdots,9\}$.The quadratic equation $ax^2+bx+c=0$ has a rational root. Prove that the three-digit number $abc$ is not a prime number.

Suppose that $ a$ and $ b$ are are nonzero real numbers, and that the equation $ x^2\plus{}ax\plus{}b\equal{}0$ has solutions $ a$ and $ b$. Then the pair $ (a,b)$ is $ \textbf{(A)}\ (\minus{}2,1) \qquad \textbf{(B)}\ (\minus{}1,2) \qquad \textbf{(C)}\ (1,\minus{}2) \qquad \textbf{(D)}\ (2,\minus{}1) \qquad \textbf{(E)}\ (4,4)$

Given the equilateral triangle $ABC$, find the locus of all points $P$ such that the segments of the lines $AP$ and $BP$ lying inside the triangle are equal.

Find all positive integers $k$ and $\ell$ such that $k^2 -\ell^2 = 1005$.

Four $L$s are equivalent to three $M$s. Nine $M$s are equivalent to fourteen $T$ s. Seven $T$ s are equivalent to two $W$ s. If Kevin has thirty-six $L$s, how many $W$ s would that be equivalent to?

What is the largest factor of $130000$ that does not contain the digit $0$ or $5$?

If $a$ and $b$ are digits, how many are there $4$ digit numbers $\overline{3ab4}$ divisible with $9$ . Which numbers are they ($4$ digit numbers)?

There are $2016$ red and $2016$ blue cards each having a number written on it. For some $64$ distinct positive real numbers, it is known that the set of numbers on cards of a particular color happens to be the set of their pairwise sums and the other happens to be the set of their pairwise products. Can we necessarily determine which color corresponds to sum and which to product? [i](B. Frenkin)[/i] (Translated from [url=http://sasja.shap.homedns.org/Turniry/TG/index.html]here.[/url])

There are $n$ rooms in a sauna, each has unlimited capacity. No room may be attended by a female and a male simultaneously. Moreover, males want to share a room only with males that they don't know and females want to share a room only with females that they know. Find the biggest number $k$ such that any $k$ couples can visit the sauna at the same time, given that two males know each other if and only if their wives know each other.

Let $ABCD$ be a cyclic quadrilateral. Let $P$ be the intersection of $BC$ and $AD$. Line $AC$ intersects the circumcircle of triangle $BDP$ in points $S$ and $T$, with $S$ between $A$ and $C$. Line $BD$ intersects the circumcircle of triangle $ACP$ in points $U$ and $V$, with $U$ between $B$ and $D$. Prove that $PS$ = $PT$ = $PU$ = $PV$.

\[\int_1^2 \frac{\sqrt{1-\frac 1x}}{x^2-1}\mathrm dx\] [i]Proposed by Connor Gordon[/i]

Let $\alpha,\beta,\gamma$ be the three real roots of the polynomial $x^3-x^2-2x+1=0$. Find all possible values of $\tfrac{\alpha}{\beta}+\tfrac{\beta}{\gamma}+\tfrac{\gamma}{\alpha}$.

Let $n$ be a positive integer. There are given $n$ lines such that no two are parallel and no three meet at a single point. a) Prove that there exists a line such that the number of intersection points of these $n$ lines on both of its sides is at least $$\left \lfloor \frac{(n-1)(n-2)}{10} \right \rfloor.$$ Notice that the points on the line are not counted. b) Find all $n$ for which there exists a configurations where the equality is achieved.

A deck consists of 13 types of cards: \( A > K > Q > J > 10 > 9 > \dots > 3 > 2 \), each card appearing 4 times. In total, there are 52 cards. Marius and Alexandru each receive half of the standard deck of cards, placing them face down. On each turn, the players simultaneously draw the top card from their hands, and the player with the most valuable card takes both cards and places them under all of his other cards, with Marius deciding the order in which the two cards are placed. In case of a tie, each player places their own card under the rest of their cards. The game ends when one of the players runs out of cards. Is it possible that, although Alexandru initially has all four \( A \) cards, the game lasts forever?

In the triangle $ABC$, angle bisector $CD$ intersects the circumcircle of $ABC$ at the point $K$. (a) Prove the equalities: $$\frac1{ID}-\frac1{IK}=\frac1{CI},\enspace\frac{CI}{ID}-\frac{ID}{DK}=1$$where $I$ is the center of the inscribed circle of triangle $ABC$. (b) On the segment $CK$ some point $P$ is chosen whose projections on $AC,BC,AB$ respectively are $P_1,P_2,P_3$. The lines $PP_3$ and $P_1P_2$ intersect at a point $M$. Find the locus of $M$ when $P$ moves around segment $CK$.

Let $ABCD$ be a rectangle with $AB>BC$. Let points $E,F$ be on side $CD$ such that $CE=ED$ and $BC=CF$. Show that if $AC$ is prependicular to $BE$ then $AB=BF$.

Consider a particle at rest which may decay into two (daughter) particles or into three (daughter) particles. Which of the following is true in the two-body case but false in the three-body case? (There are no external forces.) (a) The velocity vectors of the daughter particles must lie in a single plane. (b) Given the total kinetic energy of the system and the mass of each daughter particle, it is possible to determine the speed of each daughter particle. (c) Given the speed(s) of all but one daughter particle, it is possible to determine the speed of the remaining particle. (d) The total momentum of the daughter particles is zero. (e) None of the above.

The set $\{1, 2, \cdots, 49\}$ is divided into three subsets. Prove that at least one of these subsets contains three different numbers $a, b, c$ such that $a + b = c$.

Find all functions $f:\mathbb{R}^+\to\mathbb{R}^+$ such that whenever $a>b>c>d>0$ and $ad=bc$, \[f(a+d)+f(b-c)=f(a-d)+f(b+c).\] [i]Calvin Deng.[/i]