Let $B = (-1, 0)$ and $C = (1, 0)$ be fixed points on the coordinate plane. A nonempty, bounded subset $S$ of the plane is said to be [i]nice[/i] if $\text{(i)}$ there is a point $T$ in $S$ such that for every point $Q$ in $S$, the segment $TQ$ lies entirely in $S$; and $\text{(ii)}$ for any triangle $P_1P_2P_3$, there exists a unique point $A$ in $S$ and a permutation $\sigma$ of the indices $\{1, 2, 3\}$ for which triangles $ABC$ and $P_{\sigma(1)}P_{\sigma(2)}P_{\sigma(3)}$ are similar. Prove that there exist two distinct nice subsets $S$ and $S'$ of the set $\{(x, y) : x \geq 0, y \geq 0\}$ such that if $A \in S$ and $A' \in S'$ are the unique choices of points in $\text{(ii)}$, then the product $BA \cdot BA'$ is a constant independent of the triangle $P_1P_2P_3$.

Let $B_n$ be the set of all sequences of length $n$, consisting of zeros and ones. For every two sequences $a,b \in B_n$ (not necessarily different) we define strings $\varepsilon_0\varepsilon_1\varepsilon_2 \dots \varepsilon_n$ and $\delta_0\delta_1\delta_2 \dots \delta_n$ such that $\varepsilon_0=\delta_0=0$ and $$ \varepsilon_{i+1}=(\delta_i-a_{i+1})(\delta_i-b_{i+1}), \quad \delta_{i+1}=\delta_i+(-1)^{\delta_i}\varepsilon_{i+1} \quad (0 \leq i \leq n-1). $$. Let $w(a,b)=\varepsilon_0+\varepsilon_1+\varepsilon_2+\dots +\varepsilon_n$ . Find $f(n)=\sum\limits_{a,b \in {B_n}} {w(a,b)} $. .

Let $M, N, P, Q$ be points on sides $AB, BC, CD, DA$ of a convex quadrilateral $ABCD$ such that $AQ = DP = CN = BM$. Prove that if $MNPQ$ is a square, then $ABCD$ is also a square.

Let $n \ge 2$ be an integer. Show that there exist $n+1$ numbers $x_1, x_2, \ldots, x_{n+1} \in \mathbb{Q} \setminus \mathbb{Z}$, so that $\{ x_1^3 \} + \{ x_2^3 \} + \cdots + \{ x_n^3 \}=\{ x_{n+1}^3 \}$, where $\{ x \}$ is the fractionary part of $x$.

Given quadrilateral $ABCD$ and the directions of its sides. Inscribe a rectangle in $ABCD$.

Billy the kid likes to play on escalators! Moving at a constant speed, he manages to climb up one escalator in $24$ seconds and climb back down the same escalator in $40$ seconds. If at any given moment the escalator contains $48$ steps, how many steps can Billy climb in one second?

Minivan chooses a prime number. Then every second, he adds either the digit $1$ or the digit $3$ to the right end of his number (after the unit digit), such that the new number is also a prime. Can he continue indefinitely? [i](Proposed by Wong Jer Ren)[/i]

Convex pentagon $ABCDE$ has $\overline{BC}=17$, $\overline{AB}=2\overline{CD}$, and $\angle E=90^\circ$. Additionally, $\overline{BD}-\overline{CD}=\overline{AC}$, and $\overline{BD}+\overline{CD}=25$. Let $M$ and $N$ be the midpoints of $BC$ and $AD$ respectively. Ray $EA$ is extended out to point $P$, and a line parallel to $AD$ is drawn through $P$, intersecting line $EM$ at $Q$. Let $G$ be the midpoint of $AQ$. Given that $N$ and $G$ lie on $EM$ and $PM$ respectively, and the perimeter of $\triangle QBC$ is $42$, find the length of $\overline{EM}$. [i]Proposed by Adam Bertelli[/i]

Let $\mathbb Q_{>0}$ be the set of all positive rational numbers. Let $f:\mathbb Q_{>0}\to\mathbb R$ be a function satisfying the following three conditions: (i) for all $x,y\in\mathbb Q_{>0}$, we have $f(x)f(y)\geq f(xy)$; (ii) for all $x,y\in\mathbb Q_{>0}$, we have $f(x+y)\geq f(x)+f(y)$; (iii) there exists a rational number $a>1$ such that $f(a)=a$. Prove that $f(x)=x$ for all $x\in\mathbb Q_{>0}$. [i]Proposed by Bulgaria[/i]

A mathematical frog jumps along the number line. The frog starts at $1$, and jumps according to the following rule: if the frog is at integer $n$, then it can jump either to $n+1$ or to $n + 2^{m_n+1}$ where $2^{m_n}$ is the largest power of $2$ that is a factor of $n.$ Show that if $k \geq 2$ is a positive integer and $i$ is a nonnegative integer, then the minimum number of jumps needed to reach $2^ik$ is greater than the minimum number of jumps needed to reach $2^i.$

Melinda has three empty boxes and $12$ textbooks, three of which are mathematics textbooks. One box will hold any three of her textbooks, one will hold any four of her textbooks, and one will hold any five of her textbooks. If Melinda packs her textbooks into these boxes in random order, the probability that all three mathematics textbooks end up in the same box can be written as $\frac{m}{n}$, where $m$ and $n$ Are relatively prime positive integers. Find $m+n$.

Let $m$ and $n$ be positive integers. Show that $\frac{(m+n)!}{(m+n)^{m+n}} < \frac{m!}{m^m}\cdot\frac{n!}{n^n}$

Prove that for a natural number $ n > 2 $ the number $ n! $ is the sum of its $ n $ various divisors.

Prove that none of the numbers $2^{2^n}+ 1$, $n = 0, 1, 2, \dots$ is a perfect cube.

[b]Problem Section #1 a) A set contains four numbers. The six pairwise sums of distinct elements of the set, in no particular order, are $189, 320, 287, 264, x$, and y. Find the greatest possible value of: $x + y$. [color=red]NOTE: There is a high chance that this problems was copied.[/color]

The point $X$ is marked inside the triangle $ABC$. The circumcircles of the triangles $AXB$ and $AXC$ intersect the side $BC$ again at $D$ and $E$ respectively. The line $DX$ intersects the side $AC$ at $K$, and the line $EX$ intersects the side $AB$ at $L$. Prove that $LK\parallel BC$.

Let $P, A, B, C$ be points on circle $O$ such that $C$ does not lie on arc $\widehat{BAP}$, $\overline{P A} = 21, \overline{P B} = 56, \overline{P C} = 35$ and $m \angle BP C = 60^\circ$. Now choose point $D$ on the circle such that $C$ does not lie on arc $\widehat{BDP}$ and $\overline{BD} = 39$. What is $AD$?

Let $ \{a_n\}_{n\geq 1}$ be a sequence of real numbers such that $ |a_{n\plus{}1}\minus{}a_n|\leq 1$, for all positive integers $ n$. Let $ \{b_n\}_{n\geq 1}$ be the sequence defined by \[ b_n \equal{} \frac { a_1\plus{} a_2 \plus{} \cdots \plus{}a_n} {n}.\] Prove that $ |b_{n\plus{}1}\minus{}b_n | \leq \frac 12$, for all positive integers $ n$.

If 1,2, and 3 are solutions to the equation $ x^4 \plus{} ax^2 \plus{} bx \plus{} c \equal{} 0,$ then $ a\plus{}c$ equals A. -12 B. 24 C. 35 D. -61 E. -63

The $14 \times 14$ chessboard squares are colored in pattern, as shown in the picture. Can you choose seven fields blacks and seven white squares of this chessboard in such a way, that there is exactly one selected field in each row and column? Justify your answer. [img]https://cdn.artofproblemsolving.com/attachments/e/4/e8ba46030cd0f0e0511f1f9e723e5bd29e9975.png[/img]

Let a triangle $ABC$ inscribed in a circle $\Gamma$ with center $O$. Let $I$ the incenter of triangle $ABC$ and $D, E, F$ the contact points of the incircle with sides $BC, AC, AB$ of triangle $ABC$ respectively . Let also $S$ the foot of the perpendicular line from $D$ to the line $EF$.Prove that line $SI$ passes from the antidiametric point $N$ of $A$ in the circle $\Gamma$.( $AN$ is a diametre of the circle $\Gamma$).

Consider the area encircled by $x^2=4y,x^2=-4y,x=4,x=-4$, rotate it around $y$-axis, the volume of the revolved body is $V_1$. Then consider the figure formed by all points $(x,y)$ that $x^2+y^2\leq16,x^2+(y-2)^2\geq4,x^2+(y-2)^2\geq4$, rotate it around $y$-axis, the volume of the revolved body is $V_2$. The relationship between $V_1$ and $V_2$ is $\text{(A)}V_1=\frac{1}{2}V_2\qquad\text{(B)}V_1=\frac{2}{3}V_2\qquad\text{(C)}V_1=V_2\qquad\text{(D)}V_1=2V_2$

On a blackboard, there are $11$ positive integers. Show that one can choose some (maybe all) of these numbers and place "$+$" and "$-$" in between such that the result is divisible by $2011$.

Find the greatest positive integer $m$, such that one of the $4$ letters $C,G,M,O$ can be placed in each cell of a table with $m$ rows and $8$ columns, and has the following property: For any two distinct rows in the table, there exists at most one column, such that the entries of these two rows in such a column are the same letter.

Point $X$ is taken inside a regular $n$-gon of side length $a$. Let $h_1,h_2,...,h_n$ be the distances from $X$ to the lines defined by the sides of the $n$-gon. Prove that $\frac{1}{h_1}+\frac{1}{h_2}+...+\frac{1}{h_n}>\frac{2\pi}{a}$