We are given an $18\times 18$ table, all of whose cells may be black or white. Initially all the cells are coloured white. We may perform the following operation: choose one column or one row and change the colour of all cells in this column or row. Is it possible by repeating the operation to obtain a table with exactly $16$ black cells?

For a sequence $a_1<a_2<\cdots<a_n$ of integers, a pair $(a_i,a_j)$ with $1\leq i<j\leq n$ is called [i]interesting[/i] if there exists a pair $(a_k,a_l)$ of integers with $1\leq k<l\leq n$ such that $$\frac{a_l-a_k}{a_j-a_i}=2.$$ For each $n\geq 3$, find the largest possible number of interesting pairs in a sequence of length $n$.

Given points $A, M, M_1$ and rational number $\lambda \neq -1$. Construct the triangle $ABC$, such that $M$ lies on $BC$ and $M_1$ lies on $B_1C_1$ ($B_1, C_1$ are the projections of $B, C$ on $AC, AB$ respectively), and $\frac{BM}{MC}=\frac{B_1M_1}{M_1C_1}=\lambda.$

Arithmetical progression $a_1, a_2, a_3, a_4,...$ contains $a_1^2 , a_2^2$ and $a_3^2$ at some positions. Prove that all terms of this progression are integers.

Two teams are in a best-two-out-of-three playoff: the teams will play at most $3$ games, and the winner of the playoff is the first team to win $2$ games. The first game is played on Team A's home field, and the remaining games are played on Team B's home field. Team A has a $\frac{2}{3}$ chance of winning at home, and its probability of winning when playing away from home is $p$. Outcomes of the games are independent. The probability that Team A wins the playoff is $\frac{1}{2}$. Then $p$ can be written in the form $\frac{1}{2}(m - \sqrt{n})$, where $m$ and $n$ are positive integers. What is $m + n$? $\textbf{(A) } 10 \qquad \textbf{(B) } 11 \qquad \textbf{(C) } 12 \qquad \textbf{(D) } 13 \qquad \textbf{(E) } 14$

Let $ x, y, z $ be positive real numbers. Prove that \[ \frac{2x^2 + xy}{(y+ \sqrt{zx} + z )^2} + \frac{2y^2 + yz}{(z+ \sqrt{xy} + x )^2} + \frac{2z^2 + zx}{(x+ \sqrt{yz} +y )^2} \ge 1 \]

Find all positive integers $n$ for which there exist two distinct numbers of $n$ digits, $\overline{a_1a_2\ldots a_n}$ and $\overline{b_1b_2\ldots b_n}$, such that the number of $2n$ digits $\overline{a_1a_2\ldots a_nb_1b_2\ldots b_n}$ is divisible by $\overline{b_1b_2\ldots b_na_1a_2\ldots a_n}$.

Given a positive integer $ n $, let $ A, B $ be two co-prime positive integers such that $$ \frac{B}{A} = \left(\frac{n\left(n+1\right)}{2}\right)!\cdot\prod\limits_{k=1}^{n}{\frac{k!}{\left(2k\right)!}} $$ Prove that $ A $ is a power of $ 2 $.

The lengths of the sides of a triangle in inches are three consecutive integers. The length of the shorter side is $30\%$ of the perimeter. What is the length of the longest side? $ \textbf{(A)}\ 7 \qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 9\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 11 $

Given a finite set of points in the plane, each with integer coordinates, is it always possible to color the points red or white so that for any straight line $L$ parallel to one of the coordinate axes the difference (in absolute value) between the numbers of white and red points on $L$ is not greater than $1$?

Let $a$ and $b$ be positive real numbers such that $ab(a-b)=1$. Which of the followings can $a^2+b^2$ take? $ \textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 2\sqrt 2 \qquad\textbf{(D)}\ \sqrt {11} \qquad\textbf{(E)}\ \text{None of the preceding} $

In triangle $ABC$, $|AB| <|BC|$ holds. Point $I$ is the center of the circle inscribed in that triangle. Let $M$ be the midpoint of the side $AC$, and $N$ be the midpoint of the arc $AC$ of the circumcircle of that triangle containing point $B$. Prove that $\angle IMA = \angle INB$.

A right cone in $xyz$-space has its apex at $(0,0,0)$, and the endpoints of a diameter on its base are $(12,13,-9)$ and $(12,-5,15)$. The volume of the cone can be expressed as $a\pi$. What is $a$?

Let $a$ and $b$ be positive real numbers such that $a+b = 1$. Prove that $$\frac12 \le \frac{a^3+b^3}{a^2+b^2} \le 1$$

Let $x$ be the first term in the sequence $31, 331, 3331, . . .$ which is divisible by $17$. How many digits long is$ x$?

Let $N$ be a positive integer; a divisor of $N$ is called [i]common[/i] if it's great than $1$ and different of $N$. A positive integer is called [i]special[/i] if it has, at least, two common divisors and it is multiple of all possible differences between any two of their common divisors. Find all special integers.

Let $ABC$ be an acute-angled triangle. The circle $\Gamma$ with $BC$ as diameter intersects $AB$ and $AC$ again at $P$ and $Q$, respectively. Determine $\angle BAC$ given that the orthocenter of triangle $APQ$ lies on $\Gamma$.

Let $ABC$ be a triangle whose angles measure $A$, $B$, $C$, respectively. Suppose $\tan A$, $\tan B$, $\tan C$ form a geometric sequence in that order. If $1\le \tan A+\tan B+\tan C\le 2015$, find the number of possible integer values for $\tan B$. (The values of $\tan A$ and $\tan C$ need not be integers.) [i] Proposed by Justin Stevens [/i]

Find all integers $n$ satisfying the following property. There exist nonempty finite integer sets $A$ and $B$ such that for any integer $m$, exactly one of these three statements below is true: (a) There is $a \in A$ such that $m \equiv a \pmod n$, (b) There is $b \in B$ such that $m \equiv b \pmod n$, and (c) There are $a \in A$ and $b \in B$ such that $m \equiv a + b \pmod n$.

Segment $ BD$ and $ AE$ intersect at $ C$, as shown, $ AB\equal{}BC\equal{}CD\equal{}CE$, and $ \angle A\equal{}\frac52\angle B$. What is the degree measure of $ \angle D$? [asy]unitsize(2cm); defaultpen(linewidth(.8pt)+fontsize(8pt)); dotfactor=4; pair C=(0,0), Ep=dir(35), D=dir(-35), B=dir(145); pair A=intersectionpoints(Circle(B,1),C--(-1*Ep))[0]; pair[] ds={A,B,C,D,Ep}; dot(ds); draw(A--Ep--D--B--cycle); label("$A$",A,SW); label("$B$",B,NW); label("$C$",C,N); label("$E$",Ep,E); label("$D$",D,E);[/asy]$ \textbf{(A)}\ 52.5 \qquad \textbf{(B)}\ 55 \qquad \textbf{(C)}\ 57.5 \qquad \textbf{(D)}\ 60 \qquad \textbf{(E)}\ 62.5$

Let $f$ be a function whose domain is $[1, 20]$ and whose range is a subset of $[-100, 100].$ Suppose $\tfrac{f(x)}{y} - \tfrac{f(y)}{x} \leq (x - y)^2$ for all $x$ and $y$ in $[1, 20].$ Compute the largest value of $f(x) - f(y)$ over all such functions $f$ and all $x$ and $y$ in the domain $[1, 20].$

Does there exist a triangle, whose side is equal to some of its altitudes, another side is equal to some of its bisectors, and the third is equal to some of its medians?

[b]p1.[/b] An $18$oz glass of apple juice is $6\%$ sugar and a $6$oz glass of orange juice is $12\%$ sugar. The two glasses are poured together to create a cocktail. What percent of the cocktail is sugar? [b]p2.[/b] Find the number of positive numbers that can be expressed as the difference of two integers between $-2$ and $2012$ inclusive. [b]p3.[/b] An annulus is defined as the region between two concentric circles. Suppose that the inner circle of an annulus has radius $2$ and the outer circle has radius $5$. Find the probability that a randomly chosen point in the annulus is at most $3$ units from the center. [b]p4.[/b] Ben and Jerry are walking together inside a train tunnel when they hear a train approaching. They decide to run in opposite directions, with Ben heading towards the train and Jerry heading away from the train. As soon as Ben finishes his $1200$ meter dash to the outside, the front of the train enters the tunnel. Coincidentally, Jerry also barely survives, with the front of the train exiting the tunnel as soon as he does. Given that Ben and Jerry both run at $1/9$ of the train’s speed, how long is the tunnel in meters? [b]p5.[/b] Let $ABC$ be an isosceles triangle with $AB = AC = 9$ and $\angle B = \angle C = 75^o$. Let $DEF$ be another triangle congruent to $ABC$. The two triangles are placed together (without overlapping) to form a quadrilateral, which is cut along one of its diagonals into two triangles. Given that the two resulting triangles are incongruent, find the area of the larger one. [b]p6.[/b] There is an infinitely long row of boxes, with a Ditto in one of them. Every minute, each existing Ditto clones itself, and the clone moves to the box to the right of the original box, while the original Ditto does not move. Eventually, one of the boxes contains over $100$ Dittos. How many Dittos are in that box when this first happens? [b]p7.[/b] Evaluate $$26 + 36 + 998 + 26 \cdot 36 + 26 \cdot 998 + 36 \cdot 998 + 26 \cdot 36 \cdot 998.$$ [b]p8. [/b]There are $15$ students in a school. Every two students are either friends or not friends. Among every group of three students, either all three are friends with each other, or exactly one pair of them are friends. Determine the minimum possible number of friendships at the school. [b]p9.[/b] Let $f(x) = \sqrt{2x + 1 + 2\sqrt{x^2 + x}}$. Determine the value of $$\frac{1}{f(1)}+\frac{1}{f(1)}+\frac{1}{f(3)}+...+\frac{1}{f(24)}.$$ [b]p10.[/b] In square $ABCD$, points $E$ and $F$ lie on segments $AD$ and $CD$, respectively. Given that $\angle EBF = 45^o$, $DE = 12$, and $DF = 35$, compute $AB$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Assume that $X$ is a seperable metric space. Prove that if $f: X\longrightarrow\mathbb R$ is a function that $\lim_{x\rightarrow a}f(x)$ exists for each $a\in\mathbb R$. Prove that set of points in which $f$ is not continuous is countable.

Prove that for $n, p$ integers, $n \geq 4$ and $p \geq 4$, the proposition $\mathcal{P}(n, p)$ \[\sum_{i=1}^{n}\frac{1}{{x_{i}}^{p}}\geq \sum_{i=1}^{n}{x_{i}}^{p}\quad \textrm{for}\quad x_{i}\in \mathbb{R}, \quad x_{i}> 0 , \quad i=1,\ldots,n \ ,\quad \sum_{i=1}^{n}x_{i}= n,\] is false. [i]Dan Schwarz[/i] [hide="Remark"]In the competition, the students were informed (fact that doesn't actually relate to the problem's solution) that the propositions $\mathcal{P}(4, 3)$ are $\mathcal{P}(3, 4)$ true.[/hide]