$A$, $B$, $C$, and $D$ are points on a circle, and segments $\overline{AC}$ and $\overline{BD}$ intersect at $P$, such that $AP=8$, $PC=1$, and $BD=6$. Find $BP$, given that $BP<DP$.

The cells of a square $2011 \times 2011$ array are labelled with the integers $1,2,\ldots, 2011^2$, in such a way that every label is used exactly once. We then identify the left-hand and right-hand edges, and then the top and bottom, in the normal way to form a torus (the surface of a doughnut). Determine the largest positive integer $M$ such that, no matter which labelling we choose, there exist two neighbouring cells with the difference of their labels at least $M$. (Cells with coordinates $(x,y)$ and $(x',y')$ are considered to be neighbours if $x=x'$ and $y-y'\equiv\pm1\pmod{2011}$, or if $y=y'$ and $x-x'\equiv\pm1\pmod{2011}$.) [i](Romania) Dan Schwarz[/i]

Which natural numbers can be expressed as the difference of squares of two integers?

Let $ABC$ be an isosceles triangle such that $AB=AC$. Find angles of triangle $ABC$ if $\frac{AB}{BC}=1+2\cos{\frac{2\pi}{7}}$

Quadrilateral $ABCD$ with side lengths $AB=7, BC = 24, CD = 20, DA = 15$ is inscribed in a circle. The area interior to the circle but exterior to the quadrilateral can be written in the form $\frac{a\pi - b}{c}$, where $a, b,$ and $c$ are positive integers such that $a$ and $c$ have no common prime factor. What is $a+b+c$? $\textbf{(A) } 260 \qquad \textbf{(B) } 855 \qquad \textbf{(C) } 1235 \qquad \textbf{(D) } 1565 \qquad \textbf{(E) } 1997$

Evaluate $ \int_0^{\frac{\pi}{2}} e^{xe^x}\{(x\plus{}1)e^x(\cos x\plus{}\sin x)\plus{}\cos x\minus{}\sin x\}dx$.

To open the magic chest, one needs to say a magic code of length $n$ consisting of digits $0, 1, 2, 3, 4, 5, 6, 7, 8, 9.$ Each time Griphook tells the chest a code it thinks up, the chest's talkative guardian responds by saying the number of digits in that code that match the magic code. (For example, if the magic code is $0423$ and Griphook says $3442,$ the chest's talkative guard will say $1$). Prove that there exists a number $k$ such that for any natural number $n \geq k,$ Griphook can find the magic code by checking at most $4n-2023$ times, regardless of what the magic code of the box is.

Some squares of a $1999\times 1999$ board are occupied with pawns. Find the smallest number of pawns for which it is possible that for each empty square, the total number of pawns in the row or column of that square is at least $1999$.

Solve the system $\begin{cases} (x^3 + y^3)(x^2 + y^2) = 2b^5 \\ x + y = b \end{cases}$ in $C$

Prove that there exists a positive integer that can be written, in at least two ways, as a sum of $2014$-th powers of $2015$ distinct positive integers $x_1 <x_2 <\cdots <x_{2015}$.

On a $2012 \times 2012$ board, some cells on the top-right to bottom-left diagonal are marked. None of the marked cells is in a corner. Integers are written in each cell of this board in the following way. All the numbers in the cells along the upper and the left sides of the board are 1's. All the numbers in the marked cells are 0's. Each of the other cells contains a number that is equal to the sum of its upper neighbour and its left neighbour. Prove that the number in the bottom right corner is not divisible by 2011.

Decide whether for every arrangement of the numbers $1,2,3, . . . ,15$ in a sequence one can color these numbers with at most four different colors in such a way that the numbers of each color form a monotone subsequence.

Initially, there are $3^{80}$ particles at the origin $(0,0)$. At each step the particles are moved to points above the $x$-axis as follows: if there are $n$ particles at any point $(x,y)$, then $\Bigl \lfloor \frac{n}{3} \Bigr\rfloor$ of them are moves to $(x+1,y+1)$, $\Bigl \lfloor \frac{n}{3} \Bigr\rfloor$ are moved to $(x,y+1)$ and the remaining to $(x-1,y+1)$, For example, after the first step, there are $3^{79}$ particles each at $(1,1),(0,1)$ and $(-1,1)$. After the second step, there are $3^{78}$ particles each at $(-2,2)$ and $(2,2)$, $2 \times 3^{78}$ particles each at $(-1,2)$ and $(1,2)$, and $3^{79}$ particles at $(0,2)$. After $80$ steps, the number of particles at $(79,80)$ is:

The sum of several (not necessarily different) real numbers from $[0,1]$ doesn't exceed $S$. Find the maximum value of $S$ such that it is always possible to partition these numbers into two groups with sums $A\le 8$ and $B\le 4$. [i](I. Gorodnin)[/i]

Given the line $y = \dfrac{3}{4}x + 6$ and a line $L$ parallel to the given line and $4$ units from it. A possible equation for $L$ is: $\textbf{(A)}\ y = \dfrac{3}{4}x + 1 \qquad \textbf{(B)}\ y = \dfrac{3}{4}x\qquad \textbf{(C)}\ y = \dfrac{3}{4}x -\dfrac{2}{3} \qquad$ $ \textbf{(D)}\ y = \dfrac{3}{4}x -1 \qquad \textbf{(E)}\ y = \dfrac{3}{4}x + 2$

Two $n$-sided polygons are said to be of the same type if we can label their vertices in clockwise order as $A_1$, $A_2$, ..., $A_n$ and $B_1$, $B_2$, ..., $B_n$ respectively such that each pair of interior angles $A_i$ and $B_i$ are either both reflex angles or both non-reflex angles. How many different types of $11$-sided polygons are there?

A pet shop sells cats and two types of birds: ducks and parrots. In the shop, $\tfrac{1}{12}$ of animals are ducks, and $\tfrac{1}{4}$ of birds are ducks. Given that there are $56$ cats in the pet shop, how many ducks are there in the pet shop?

Scalene triangle $ABC$ has incenter $I$ and circumcircle $\Omega$ with center $O$. $H$ is the orthocenter of triangle $BIC$, and $T$ is a point on $\Omega$ for which $\angle ATI=90^\circ$. Circle $(AIO)$ intersects line $IH$ again at $X$. Show that the lines $AX, HT$ intersect on $\Omega$.

Let $\triangle ABC$ be a triangle with $AB=5, BC=6, CA=7$. Suppose $P$ is a point inside $\triangle ABC$ such that $\triangle BPA\sim \triangle APC$. If $AP$ intersects $BC$ at $X$, find $\frac{BX}{CX}$. [i]Proposed by Nathan Ramesh

A pair of 8-sided dice have sides numbered 1 through 8. Each side has the same probability (chance) of landing face up. The probability that the product of the two numbers that land face-up exceeds 36 is $\textbf{(A)}\ \dfrac{5}{32} \qquad \textbf{(B)}\ \dfrac{11}{64} \qquad \textbf{(C)}\ \dfrac{3}{16} \qquad \textbf{(D)}\ \dfrac{1}{4} \qquad \textbf{(E)}\ \dfrac{1}{2}$

The paper is marked with the finite number of blue and red dots and some these points are connected by lines. Let's name a point $P$ [i]special [/i] if more than half of the points connected with $P$ has a color other than point $P$. Juku selects one special point and reverses its color. Then Juku selects a new special point and changes its color, etc. Prove that by a finite number of integers Juku ends up in a situation where the paper has not made a special point.

p1. A set of cards contains $100$ cards, each of which is written with a number from $1$ up to $100$. On each of the two sides of the card the same number is written, side one is red and the other is green. First of all Leny arranges all the cards with red writing face up. Then Leny did the following three steps: I. Turn over all cards whose numbers are divisible by $2$ II. Turn over all the cards whose numbers are divisible by $3$ III. Turning over all the cards whose numbers are divisible by $5$, but didn't turn over all cards whose numbers are divisible by $5$ and $2$. Find the number of Leny cards now numbered in red and face up, p2. Find the area of ​​three intersecting semicircles as shown in the following image. [img]https://cdn.artofproblemsolving.com/attachments/f/b/470c4d2b84435843975a0664fad5fee4a088d5.png[/img] p3. It is known that $x+\frac{1}{x}=7$ . Determine the value of $A$ so that $\frac{Ax}{x^4+x^2+1}=\frac56$. p4. There are $13$ different gifts that will all be distributed to Ami, Ima, Mai,and Mia. If Ami gets at least $4$ gifts, Ima and Mai respectively got at least $3$ gifts, and Mia got at least $2$ gifts, how many possible gift arrangements are there? p5. A natural number is called a [i]quaprimal [/i] number if it satisfies all four following conditions: i. Does not contain zeros. ii. The digits compiling the number are different. iii. The first number and the last number are prime numbers or squares of an integer. iv. Each pair of consecutive numbers forms a prime number or square of an integer. For example, we check the number $971643$. (i) $971643$ does not contain zeros. (ii) The digits who compile $971643$ are different. (iii) One first number and one last number of $971643$, namely $9$ and $3$ is a prime number or a square of an integer. (iv) Each pair of consecutive numbers, namely $97, 71, 16, 64$, and $43$ form prime number or square of an integer. So $971643$ is a quadratic number. Find the largest $6$-digit quaprimal number. Find the smallest $6$-digit quaprimal number. Which digit is never contained in any arbitrary quaprimal number? Explain.

Let $n\geq 3$ be a fixed integer. Each side and each diagonal of a regular $n$-gon is labelled with a number from the set $\left\{1;\;2;\;...;\;r\right\}$ in a way such that the following two conditions are fulfilled: [b]1.[/b] Each number from the set $\left\{1;\;2;\;...;\;r\right\}$ occurs at least once as a label. [b]2.[/b] In each triangle formed by three vertices of the $n$-gon, two of the sides are labelled with the same number, and this number is greater than the label of the third side. [b](a)[/b] Find the maximal $r$ for which such a labelling is possible. [b](b)[/b] [i]Harder version (IMO Shortlist 2005):[/i] For this maximal value of $r$, how many such labellings are there? [hide="Easier version (5th German TST 2006) - contains answer to the harder version"] [i]Easier version (5th German TST 2006):[/i] Show that, for this maximal value of $r$, there are exactly $\frac{n!\left(n-1\right)!}{2^{n-1}}$ possible labellings.[/hide] [i]Proposed by Federico Ardila, Colombia[/i]

Which of the following polygons has the largest area? [asy] size(330); int i,j,k; for(i=0;i<5; i=i+1) { for(j=0;j<5;j=j+1) { for(k=0;k<5;k=k+1) { dot((6i+j, k)); }}} draw((0,0)--(4,0)--(3,1)--(3,3)--(2,3)--(2,1)--(1,1)--cycle); draw(shift(6,0)*((0,0)--(4,0)--(4,1)--(3,1)--(3,2)--(2,1)--(1,1)--(0,2)--cycle)); draw(shift(12,0)*((0,1)--(1,0)--(3,2)--(3,3)--(1,1)--(1,3)--(0,4)--cycle)); draw(shift(18,0)*((0,1)--(2,1)--(3,0)--(3,3)--(2,2)--(1,3)--(1,2)--(0,2)--cycle)); draw(shift(24,0)*((1,0)--(2,1)--(2,3)--(3,2)--(3,4)--(0,4)--(1,3)--cycle)); label("$A$", (0*6+2, 0), S); label("$B$", (1*6+2, 0), S); label("$C$", (2*6+2, 0), S); label("$D$", (3*6+2, 0), S); label("$E$", (4*6+2, 0), S); [/asy] $ \textbf{(A)}\text{A}\qquad\textbf{(B)}\ \text{B}\qquad\textbf{(C)}\ \text{C}\qquad\textbf{(D)}\ \text{D}\qquad\textbf{(E)}\ \text{E} $

On a particular January day, the high temperature in Lincoln, Nebraska, was 16 degrees higher than the low temperature, and the average of the high and low temperatures was $3^{\circ}$. In degrees, what was the low temperature in Lincoln that day? $\textbf{(A) }-13\qquad\textbf{(B) }-8\qquad\textbf{(C) }-5\qquad\textbf{(D) }3\qquad\textbf{(E) }11$