Find the largest positive integer $N$ such that the equation $99x + 100y + 101z = N$ has an unique solution in the positive integers $x, y, z$. [i](7 points)[/i]

Let $n\geqslant 3$ be an integer. Prove that there exists a set $S$ of $2n$ positive integers satisfying the following property: For every $m=2,3,...,n$ the set $S$ can be partitioned into two subsets with equal sums of elements, with one of subsets of cardinality $m$.

Prove that a triangle inscribed in a parallelogram has at most half the area of the parallelogram.

A website offers for $1000$ colones, the possibility of playing $4$ shifts a certain game of randomly, in each turn the user will have the same probability $p$ of winning the game and obtaining $1000$ colones (per shift). But to calculate $p$ he asks you to roll $3$ dice and add the results, with what p will be the probability of obtaining this sum. Olcoman visits the website, and upon rolling the dice, he realizes that the probability of losing his money is from $\left( \frac{103}{108}\right)^4$. a) Determine the probability $p$ that Olcoman wins a game and the possible outcomes with the dice, to get to this one. b) Which sums (with the dice) give the maximum probability of having a profit of exactly $1000$ colones? Calculate this probability and the value of $p$ for this case.

Triangle $ABC$ has $AB=27$, $AC=26$, and $BC=25$. Let $I$ denote the intersection of the internal angle bisectors of $\triangle ABC$. What is $BI$? $ \textbf{(A)}\ 15\qquad\textbf{(B)}\ 5+\sqrt{26}+3\sqrt{3}\qquad\textbf{(C)}\ 3\sqrt{26}\qquad\textbf{(D)}\ \frac{2}{3}\sqrt{546}\qquad\textbf{(E)}\ 9\sqrt{3} $

Consider a graph with $n$ vertices and $\frac{7n}{4}$ edges. (a) Prove that there are two cycles of equal length. (25 points) (b) Can you give a smaller function than $\frac{7n}{4}$ that still fits in part (a)? Prove your claim. We say function $a(n)$ is smaller than $b(n)$ if there exists an $N$ such that for each $n>N$ ,$a(n)<b(n)$ (At most 5 points) [i]Proposed by Afrooz Jabal'ameli[/i]

On the plane, a non-isosceles triangle is given, a circle circumscribed around it and the center of its inscribed circle are marked. Using only a ruler without tick marks and drawing no more than seven lines, construct the diameter of the circumcircle.

A Haiku is a Japanese poem of seventeen syllables, in three lines of five, seven, and five. In how many ways Can you add three integers Summing seventeen? Order matters here. For example, eight, three, six Is not eight, six, three. All nonnegative, Do not need to be distinct. What is your answer? [i]Proposed by Derek Gao[/i]

How many pairs of postive integers $(a,b)$ with $a\geq b$ are there such that $a^2+b^2$ divides both $a^3+b$ and $a+b^3$? $ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ \text{Infinitely many} $

Suppose that $a, b, c$, and p are positive integers such that $p$ is a prime number and $$a^2 + b^2 + c^2 = ab + bc + ca + 2021p$$. Compute the least possible value of $\max \,(a, b, c)$.

Restore the acute triangle $ABC$ given the vertex $A$, the foot of the altitude drawn from the vertex $B$ and the center of the circle circumscribed around triangle $BHC$ (point $H$ is the orthocenter of triangle $ABC$).

$(SWE 1)$ Six points $P_1, . . . , P_6$ are given in $3-$dimensional space such that no four of them lie in the same plane. Each of the line segments $P_jP_k$ is colored black or white. Prove that there exists one triangle $P_jP_kP_l$ whose edges are of the same color.

Prove that the area enclosed by three semicircles, tangent at their ends, is equal to the area of the circle whose diameter is $CD$, perpendicular to the diameter $AB$.

$21$ numbers are written in a row. $u,v,w$ are three consecutive numbers so $v=\frac{2uw}{u+w}$ . The first number is $\frac{1}{100}$ , the last one is $\frac{1}{101}$ . Find the $15$th number.

In the $(x,y)$-plane sketch the curve given parametrically by $x=2t-4t^3$, $y=t^2-3t^4$.

In the sequence $1, 0, 1, 0, 1, 0, 3, 5, \cdots$, each member after the sixth one is equal to the last digit of the sum of the six members just preceeding it. Prove that in this sequence one cannot find the following group of six consecutive members: \[0, 1, 0, 1, 0, 1\]

Given a triangle $ABC$ in which $|AB|>|AC|$. Let $P$ be the midpoint of the side $BC$, and $S$ the point in which the angle bisector of $\angle BAC$ intersects that side. The parallel with the line $AS$ through the point $P$ intersects lines $AB$ and $AC$ at points $X$ and $Y$ respectively . Let $Z$ be the point be such that $Y$ is the midpoint of the length $XZ$ and let the lines $BY$ and $CZ$ intersect at point $D$. Prove that the angle bisector of $\angle BDC$ is parallel to the lines $AS$.

In triangle $ABC$, $\angle BAC= 60^o$. Point $P$ is taken inside the triangle so that angles $\angle APB=\angle BPC= \angle CP A=120^o$. It is known that $AP = a$. Find the area of triangle $BPC$.

How many ways are there to write $91$ as the sum of at least $2$ consecutive positive integers?

Let $ ABC$ be an acute angled triangle with $ \angle B > \angle C$. Let $ D$ be the foot of the altitude from $ A$ on $ BC$, and let $ E$ be the foot of the perpendicular from $ D$ on $ AC$. Let $ F$ be a point on the segment $ (DE)$. Show that the lines $ AF$ and $ BF$ are perpendicular if and only if $ EF\cdot DC \equal{} BD \cdot DE$.

How many ordered triples of integers $(x, y, z)$ are there such that \[ x^2 + y^2 + z^2 = 34 \, ? \]

Suppose that $ \sin a \plus{} \sin b \equal{} \sqrt {\frac {5}{3}}$ and $ \cos a \plus{} \cos b \equal{} 1.$ What is $ \cos(a \minus{} b)?$ $ \textbf{(A)}\ \sqrt {\frac {5}{3}} \minus{} 1 \qquad \textbf{(B)}\ \frac {1}{3}\qquad \textbf{(C)}\ \frac {1}{2}\qquad \textbf{(D)}\ \frac {2}{3}\qquad \textbf{(E)}\ 1$

The numbers $1, 2,\ldots, 2006$ are written around the circumference of a circle. A [i]move[/i] consists of exchanging two adjacent numbers. After a sequence of such moves, each number ends up $13$ positions to the right of its initial position. lf the numbers $1, 2,\ldots, 2006$ are partitioned into $1003$ distinct pairs, then show that in at least one of the moves, the two numbers of one of the pairs were exchanged.

Let $A, B, C, D, E$ points in circle of radius r, in that order, such that $AC = BD = CE = r$. The points $H_1, H_2, H_3$ are the orthocenters of the triangles $ACD$, $BCD$ and $BCE$, respectively. Prove that $H_1H_2H_3$ is a right triangle .

For positive numbers $a_1,a_2,\ldots,a_n$ ($n\ge2$) denote $s=a_1+\ldots+a_n$. Prove that $$\frac{a_1}{s-a_1}+\ldots+\frac{a_n}{s-a_n}\ge\frac n{n-1}.$$