In a convex quadrilateral $ ABCD$ the points $ P$ and $ Q$ are chosen on the sides $ BC$ and $ CD$ respectively so that $ \angle{BAP}\equal{}\angle{DAQ}$. Prove that the line, passing through the orthocenters of triangles $ ABP$ and $ ADQ$, is perpendicular to $ AC$ if and only if the triangles $ ABP$ and $ ADQ$ have the same areas.

Let $n$ be an positive integer. Find the smallest integer $k$ with the following property; Given any real numbers $a_1 , \cdots , a_d $ such that $a_1 + a_2 + \cdots + a_d = n$ and $0 \le a_i \le 1$ for $i=1,2,\cdots ,d$, it is possible to partition these numbers into $k$ groups (some of which may be empty) such that the sum of the numbers in each group is at most $1$.

Find $x$ such that \[\frac{\frac{5}{x-50}+ \frac{7}{x+25}}{\frac{2}{x-50}- \frac{3}{x+25}} = 17.\]

Let $k,n$ be positive integers with $n \geq k \geq 3$. We consider $n+1$ points on the real plane with none three of them on the same line. We colour any segment between the points with one of $k$ possibilities. We say that an angle is a "bicolour angle" iff its vertex is one of the $n+1$ points and the two segments that define it are of different colours. Show that there is always a way to colour the segments that makes more than $n \Big\lfloor{\frac{n}{k}}\Big\rfloor^2 \frac{k(k-1)}{2}$ bicolour angles.

Let $ABCD$ be a rhombus and suppose $E$ and $F$ are the midpoints of $\overline{AD}$ and $\overline{EF}$ are the midpoints of $\overline{AD}$ and $\overline{BC},$ respectively. If $G$ is the intersection of $\overline{AC}$ and $\overline{EF},$ find the ratio of the area of $AEG$ to the area of $AGFB.$

A $n$ by $n$ magic square contains numbers from $1$ to $n^2$ such that the sum of every row and every column is the same. What is this sum?

Find all integers $n \geq 3$ such that the following property holds: if we list the divisors of $n!$ in increasing order as $1 = d_1 < d_2 < \dots < d_k = n!$, then we have \[ d_2 - d_1 \leq d_3 - d_2 \leq \dots \leq d_k - d_{k-1}. \] [i]Proposed by Luke Robitaille.[/i]

Consider five-dimensional Cartesian space $R^5 = \{(x_1, x_2, x_3, x_4, x_5) | x_i \in R\}$, and consider the hyperplanes with the following equations: $\bullet$ $x_i = x_j$ for every $1 \le i < j \le 5$, $\bullet$ $x_1 + x_2 + x_3 + x_4 + x_5 = -1$, $\bullet$ $x_1 + x_2 + x_3 + x_4 + x_5 = 0$, $\bullet$ $x_1 + x_2 + x_3 + x_4 + x_5 = 1$. Into how many regions do these hyperplanes divide $R^5$ ?

Let $n \geq 2$ be a fixed integer. Find the least constant $C$ such the inequality \[\sum_{i<j} x_{i}x_{j} \left(x^{2}_{i}+x^{2}_{j} \right) \leq C \left(\sum_{i}x_{i} \right)^4\] holds for any $x_{1}, \ldots ,x_{n} \geq 0$ (the sum on the left consists of $\binom{n}{2}$ summands). For this constant $C$, characterize the instances of equality.

Find all functions $ f : R \to R$ that satisfies $$xf(y) - yf(x)= f\left(\frac{y}{x}\right)$$ for all $x, y \in R$.

Let $AXYBZ$ be a convex pentagon inscribed in a circle with diameter $\overline{AB}$. The tangent to the circle at $Y$ intersects lines $BX$ and $BZ$ at $L$ and $K$, respectively. Suppose that $\overline{AY}$ bisects $\angle LAZ$ and $AY=YZ$. If the minimum possible value of \[ \frac{AK}{AX} + \left( \frac{AL}{AB} \right)^2 \] can be written as $\tfrac{m}{n} + \sqrt{k}$, where $m$, $n$ and $k$ are positive integers with $\gcd(m,n)=1$, compute $m+10n+100k$. [i]Proposed by Evan Chen[/i]

Assume we have $95$ boxes and $19$ balls distributed in these boxes in an arbitrary manner. We take $6$ new balls at a time and place them in $6$ of the boxes, one ball in each of the six. Can we, by repeating this process a suitable number of times, achieve a situation in which each of the $95$ boxes contains an equal number of balls?

Determine all positive integers $n$ such that $5^n - 1$ can be written as a product of an even number of consecutive integers.

A set $\mathcal{S}$ of distinct positive integers has the following property: for every integer $x$ in $\mathcal{S},$ the arithmetic mean of the set of values obtained by deleting $x$ from $\mathcal{S}$ is an integer. Given that 1 belongs to $\mathcal{S}$ and that 2002 is the largest element of $\mathcal{S},$ what is the greatet number of elements that $\mathcal{S}$ can have?

Place eight rooks on a standard $8 \times 8$ chessboard so that no two are in the same row or column. With the standard rules of chess, this means that no two rooks are attacking each other. Now paint $27$ of the remaining squares (not currently occupied by rooks) red. Prove that no matter how the rooks are arranged and which set of $27$ squares are painted, it is always possible to move some or all of the rooks so that: • All the rooks are still on unpainted squares. • The rooks are still not attacking each other (no two are in the same row or same column). • At least one formerly empty square now has a rook on it; that is, the rooks are not on the same $8$ squares as before.

[b]p1.[/b] The entrance fee the county fair is $64$ cents. Unfortunately, you only have nickels and quarters so you cannot give them exact change. Furthermore, the attendent insists that he is only allowed to change in increments of six cents. What is the least number of coins you will have to pay? [b]p2.[/b] At the county fair, there is a carnival game set up with a mouse and six cups layed out in a circle. The mouse starts at position $A$ and every ten seconds the mouse has equal probability of jumping one cup clockwise or counter-clockwise. After a minute if the mouse has returned to position $A$, you win a giant chunk of cheese. What is the probability of winning the cheese? [b]p3.[/b] A clown stops you and poses a riddle. How many ways can you distribute $21$ identical balls into $3$ different boxes, with at least $4$ balls in the first box and at least $1$ ball in the second box? [b]p4.[/b] Watch out for the pig. How many sets $S$ of positive integers are there such that the product of all the elements of the set is $125970$? [b]p5.[/b] A good word is a word consisting of two letters $A$, $B$ such that there is never a letter $B$ between any two $A$'s. Find the number of good words with length $8$. [b]p6.[/b] Evaluate $\sqrt{2 -\sqrt{2 +\sqrt{2-...}}}$ without looking. [b]p7.[/b] There is nothing wrong with being odd. Of the first $2006$ Fibonacci numbers ($F_1 = 1$, $F_2 = 1$), how many of them are even? [b]p8.[/b] Let $f$ be a function satisfying $f (x) + 2f (27- x) = x$. Find $f (11)$. [b]p9.[/b] Let $A$, $B$, $C$ denote digits in decimal representation. Given that $A$ is prime and $A -B = 4$, nd $(A,B,C)$ such that $AAABBBC$ is a prime. [b]p10.[/b] Given $\frac{x^2+y^2}{x^2-y^2} + \frac{x^2-y^2}{x^2+y^2} = k$ , find $\frac{x^8+y^8}{x^8-y^8}$ in term of $k$. [b]p11.[/b] Let $a_i \in \{-1, 0, 1\}$ for each $i = 1, 2, 3, ..., 2007$. Find the least possible value for $\sum^{2006}_{i=1}\sum^{2007}_{j=i+1} a_ia_j$. [b]p12.[/b] Find all integer solutions $x$ to $x^2 + 615 = 2^n$ for any integer $n \ge 1$. [b]p13.[/b] Suppose a parabola $y = x^2 - ax - 1$ intersects the coordinate axes at three points $A$, $B$, and $C$. The circumcircle of the triangle $ABC$ intersects the $y$ - axis again at point $D = (0, t)$. Find the value of $t$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

We write the decimal expressions of $\sqrt{2}$ and $\sqrt{3}$ as \[\sqrt{2}=1.a_1a_2a_3\dots\quad\quad\sqrt{3}=1.b_1b_2b_3\dots\] where each $a_i$ or $b_i$ is a digit between 0 and 9. Prove that there exist at least 1000 values of $i$ between $1$ and $10^{1000}$ such that $a_i\neq b_i$.

In a room there are several children and a pile of 1000 sweets. The children come to the pile one after another in some order. Upon reaching the pile each of them divides the current number of sweets in the pile by the number of children in the room, rounds the result if it is not integer, takes the resulting number of sweets from the pile and leaves the room. All the boys round upwards and all the girls round downwards. The process continues until everyone leaves the room. Prove that the total number of sweets received by the boys does not depend on the order in which the children reach the pile. [i]Maxim Didin[/i]

Let acute scalene triangle $ABC$ have orthocenter $H$ and altitude $AD$ with $D$ on side $BC$. Let $M$ be the midpoint of side $BC$, and let $D'$ be the reflection of $D$ over $M$. Let $P$ be a point on line $D'H$ such that lines $AP$ and $BC$ are parallel, and let the circumcircles of $\triangle AHP$ and $\triangle BHC$ meet again at $G \neq H$. Prove that $\angle MHG = 90^\circ$. [i]Proposed by Daniel Hu.[/i]

Three faces of a rectangular prism have diagonal lengths of $7$, $8$, and $9$ inches. How many cubic inches are in the volume of the rectangular prism? $\text{(A) }48\sqrt{11}\qquad\text{(B) }160\qquad\text{(C) }14\sqrt{95}\qquad\text{(D) }35\sqrt{15}\qquad\text{(E) }504$

Eleven distinct points $ P_1,P_2,...,P_{11}$ are given on a line so that $ P_i P_j \le 1$ for every $ i,j$. Prove that the sum of all distances $ P_i P_j, 1 \le i <j \le 11$, is smaller than $ 30$.

What is the largest positive integer $n$ less than $10, 000$ such that in base $4$, $n$ and $3n$ have the same number of digits; in base $8$, $n$ and $7n$ have the same number of digits; and in base $16$, $n$ and $15n$ have the same number of digits? Express your answer in base $10$.

Find all integers $n$ with $n \geq 4$ for which there exists a sequence of distinct real numbers $x_1, \ldots, x_n$ such that each of the sets $$\{x_1, x_2, x_3\}, \{x_2, x_3, x_4\},\ldots,\{x_{n-2}, x_{n-1}, x_n\}, \{x_{n-1}, x_n, x_1\},\text{ and } \{x_n, x_1, x_2\}$$ forms a 3-term arithmetic progression when arranged in increasing order.

Let $ABCD$ be a cyclic quadrilateral, and angle bisectors of $\angle BAD$ and $\angle BCD$ meet at point $I$. Show that if $\angle BIC = \angle IDC$, then $I$ is the incenter of triangle $ABD$.

Show that for any two distinct odd primes $p, q$, there exists a positive integer $n$ such that $$\{d(n), d(n + 2) \} = \{p, q\}$$ where $d(n)$ is the smallest prime factor of $n$. [i]Proposed By - ltf0501[/i]