Ann and Sue bought identical boxes of stationery. Ann used hers to write 1-sheet letters and Sue used hers to write 3-sheet letters. Ann used all the envelopes and had 50 sheets of paper left, while Sue used all of the sheets of paper and had 50 envelopes left. The number of sheets of paper in each box was \[ \begin{array}{rlrlrlrlrlrl} \hbox {(A)}& 150 \qquad & \hbox {(B)}& 125 \qquad & \hbox {(C)}& 120 \qquad & \hbox {(D)}& 100 \qquad & \hbox {(E)}& 80 & \end{array} \]

Refer to the diagram below. Let $ABCD$ be a cyclic quadrilateral with center $O$. Let the internal angle bisectors of $\angle A$ and $\angle C$ intersect at $I$ and let those of $\angle B$ and $\angle D$ intersect at $J$. Now extend $AB$ and $CD$ to intersect $IJ$ and $P$ and $R$ respectively and let $IJ$ intersect $BC$ and $DA$ at $Q$ and $S$ respectively. Let the midpoints of $PR$ and $QS$ be $M$ and $N$ respectively. Given that $O$ does not lie on the line $IJ$, show that $OM$ and $ON$ are perpendicular.

How many triples $(a,b,c)$ of positive integers strictly less than $51$ are there such that $a+b+c$ is divisible by $a, b$, and $c$?

A positive integer $N$ is called [i]balanced[/i], if $N=1$ or if $N$ can be written as a product of an even number of not necessarily distinct primes. Given positive integers $a$ and $b$, consider the polynomial $P$ defined by $P(x)=(x+a)(x+b)$. (a) Prove that there exist distinct positive integers $a$ and $b$ such that all the number $P(1)$, $P(2)$,$\ldots$, $P(50)$ are balanced. (b) Prove that if $P(n)$ is balanced for all positive integers $n$, then $a=b$. [i]Proposed by Jorge Tipe, Peru[/i]

Consider a non-zero convex angle $\angle POQ$ and its inner point $M$. Moreover, let $m>0$ be given. Construct a trapezoid $ABCD$ satisfying the following conditions: (1) vertices $A, D$ lie on ray $OP$ and vertices $B,C$ lie on ray $OQ$, (2) diagonals $AC$ and $BD$ intersect in $M$, (3) $AB=m$. Prove that your construction is correct and discuss conditions of solvability.

Prove the inequality $$\prod_{k=2}^n\left(1+\frac{k^{p-1}}{1^p+2^p+\ldots+k^p}\right)<e^{\frac{p-1}2}$$ [i]Proposed by Arkady Alt[/i]

$C_0:x^2+y^2=1,C_1:\frac{x^2}{a^2}+\frac{y^2}{b^2}(a>b>0)$. Find all $(a,b)$ such that for any point $P$ on $C_1$, we can find a parallelogram with an apex $P$, and it is externally tangent to $C_0$, inscribed to $C_1$.

Consider a fixed regular $n$-gon of unit side. When a second regular $n$-gon of unit size rolls around the first one, one of its vertices successively pinpoints the vertices of a closed broken line $\kappa$ as in the figure. [asy] int n=9; draw(polygon(n)); for (int i = 0; i<n;++i) { draw(reflect(dir(360*i/n + 90), dir(360*(i+1)/n + 90))*polygon(n), dashed+linewidth(0.4)); draw(reflect(dir(360*i/n + 90),dir(360*(i+1)/n + 90))*(0,1)--reflect(dir(360*(i-1)/n + 90),dir(360*i/n + 90))*(0,1), linewidth(1.2)); } [/asy] Let $A$ be the area of a regular $n$-gon of unit side, and let $B$ be the area of a regular $n$-gon of unit circumradius. Prove that the area enclosed by $\kappa$ equals $6A-2B$.

Five students named Fred, Gail, Henry, Iggy, and Joan are seated around a circular table in that order. To decide who goes first in a game, they play “countdown”. Henry starts by saying ‘34’, with Iggy saying ‘33’. If they continue to count down in their circular order, who will eventually say ‘1’? $\textbf{(A)}\ \text{Fred} \qquad \textbf{(B)}\ \text{Gail} \qquad \textbf{(C)}\ \text{Henry} \qquad \textbf{(D)}\ \text{Iggy} \qquad \textbf{(E)}\ \text{Joan}$

Let $\kappa$ be an infinite cardinality and let A , B be sets of cardinality $\kappa$. Construct a family $\cal F$ of functions $f : A \to B$ with cardinality $2^\kappa$ such that for all functions $f_1,\cdots, f_n \in\cal F$ and for all $b_1 , ..., b_n \in B$, there exist $a\in A$ such that $f_1(a) = b_1,\cdots, f_n(a) = b_n$.

Let $\triangle ABC$ be an acute-angled triangle with its incenter $I$. Suppose that $N$ is the midpoint of the arc $\overarc{BAC}$ of the circumcircle of triangle $\triangle ABC$, and $P$ is a point such that $ABPC$ is a parallelogram.Let $Q$ be the reflection of $A$ over $N$ and $R$ the projection of $A$ on $\overline{QI}$. Show that the line $\overline{AI}$ is tangent to the circumcircle of triangle $\triangle PQR$ [i]Proposed by Patrik Bak - Slovakia[/i]

In each square of a $5\times 5$ board is written $1$ or $-1$. In each step, the number of each of the $25$ cells is replaced by the result of the multiplication of the numbers of all its neighboring cells. Initially we have the board of the figure. [img]https://cdn.artofproblemsolving.com/attachments/2/d/29b8e5df29526630102ac400a3a2b2f8fee4f7.gif[/img] Show how the board looks after $2004$ steps. Clarification: Two squares are [i]neighbors [/i] if they have a common side.

Let $ABC$ be a triangle with all angles $\leq 120^{\circ}$. Let $F$ be the Fermat point of triangle $ABC$, that is, the interior point of $ABC$ such that $\angle AFB = \angle BFC = \angle CFA = 120^\circ$. For each one of the three triangles $BFC$, $CFA$ and $AFB$, draw its Euler line - that is, the line connecting its circumcenter and its centroid. Prove that these three Euler lines pass through one common point. [i]Remark.[/i] The Fermat point $F$ is also known as the [b]first Fermat point[/b] or the [b]first Toricelli point[/b] of triangle $ABC$. [i]Floor van Lamoen[/i]

Outside the square $ABCD$, the rhombus $BCMN$ is constructed with angle $BCM$ obtuse . Let $P$ be the intersection point of the lines $BM$ and $AN$ . Prove that $DM \perp CP$ and the triangle $DPM$ is right isosceles .

Find all solutions for (x,y) , both integers such that: $xy=3(\sqrt{x^2+y^2}-1)$

Find all triples $(a,b,c)$ satisfying the following conditions: (i) $a,b,c$ are prime numbers, where $a<b<c<100$. (ii) $a+1,b+1,c+1$ form a geometric sequence.

Let $ G(V,E)$ be a connected graph, and let $ d_G(x,y)$ denote the length of the shortest path joining $ x$ and $ y$ in $ G$. Let $ r_G(x)\equal{} \max \{ d_G(x,y) : \; y \in V \ \}$ for $ x \in V$, and let $ r(G)\equal{} \min \{ r_G(x) : \;x \in V\ \}$. Show that if $ r(G) \geq 2$, then $ G$ contains a path of length $ 2r(G)\minus{}2$ as an induced subgraph. [i]V. T. Sos[/i]

The base of the pyramid is a triangle $ABC$, in which $\angle ACB= 30^o$, and the length of the median from the vertex $B$ is twice less than the side $AC$ and is equal to $\alpha$ . All side edges of the pyramid are inclined to the plane of the base at an angle $a$. Determine the cross-sectional area of ​​the pyramid with a plane passing through the vertex $B$ parallel to the edge $AD$ and inclined to the plane of the base at an angle of $\beta$,

[b]Q13.[/b] Determine the greatest value of the sum $M=11xy+3x+2012yz$, where $x,y,z$ are non negative integers satisfying condition $x+y+z=1000.$

In the city of Urbextorto, the sales tax is $25\%$. A certain clothing store in the city is currently giving an $n\%$ discount on all items, and $n$ is special in that, after both the sales tax and discount are applied, a $\$20$ shirt ends up costing $\$20$. Find the value of $n$. $\textbf{(A) }5\qquad\textbf{(B) }10\qquad\textbf{(C) }20\qquad\textbf{(D) }25\qquad\textbf{(E) }50$

$201$ positive integers are written on a line, such that both the first one and the last one are equal to $19999$. Each one of the remaining numbers is less than the average of its neighbouring numbers, and the differences between each one of the remaining numbers and the average of its neighbouring numbers are all equal to a unique integer. Find the second-to-last term on the line.

Let $A,B \in \mathcal{M}_n(\mathbb{C})$ be two matrices such that $A+B=AB+BA$. Prove that: a) if $n$ is odd, then $\det(AB-BA)=0$; b) if $\text{tr}(A)\neq \text{tr}(B)$, then $\det(AB-BA)=0$.

Prove that $\sum \frac{1}{i_1i_2 \ldots i_k} = n$ is taken over all non-empty subsets $\left\{i_1,i_2, \ldots, i_k\right\}$ of $\left\{1,2,\ldots,n\right\}$. (The $k$ is not fixed, so we are summing over all the $2^n-1$ possible nonempty subsets.)

In a circle $C$ with centre $O$ and radius $r$, let $C_1$, $C_2$ be two circles with centres $O_1$, $O_2$ and radii $r_1$, $r_2$ respectively, so that each circle $C_i$ is internally tangent to $C$ at $A_i$ and so that $C_1$, $C_2$ are externally tangent to each other at $A$. Prove that the three lines $OA$, $O_1 A_2$, and $O_2 A_1$ are concurrent.

A square is subdivided into $K^2$ equal smaller squares. We are given a broken line which passes through the centres of all the smaller squares (such a broken line may intersect itself). Find the minimum number of links in this broken line. (A Andjans, Riga)