Form a pentagon by taking a square of side length $1$ and an equilateral triangle of side length $1$ and placing the triangle so that one of its sides coincides with a side of the square. Then "circumscribe" a circle around the pentagon, passing through three of its vertices, so that the circle passes through exactly one vertex of the equilateral triangle, and exactly two vertices of the square. What is the radius of the circle? $\textbf{(A) }\dfrac23\hspace{14.4em}\textbf{(B) }\dfrac34\hspace{14.4em}\textbf{(C) }1$ $\textbf{(D) }\dfrac54\hspace{14.4em}\textbf{(E) }\dfrac43\hspace{14.4em}\textbf{(F) }\dfrac{\sqrt2}2$ $\textbf{(G) }\dfrac{\sqrt3}2\hspace{13.5em}\textbf{(H) }\sqrt2\hspace{13.8em}\textbf{(I) }\sqrt3$ $\textbf{(J) }\dfrac{1+\sqrt3}2\hspace{12em}\textbf{(K) }\dfrac{2+\sqrt6}2\hspace{11.9em}\textbf{(L) }\dfrac76$ $\textbf{(M) }\dfrac{2+\sqrt6}4\hspace{11.5em}\textbf{(N) }\dfrac45\hspace{14.4em}\textbf{(O) }2007$

For all positive integers $m$ and $n$ prove the inequality $$ |n\sqrt{n^2+1}-m|\geqslant \sqrt{2}-1. $$

A rectangle $\mathcal R$ is divided into a set $\mathcal S$ of finitely many smaller rectangles with sides parallel to the sides of $\mathcal R$ such that no three rectangles in $\mathcal S$ share a common corner. An ant is initially located at the bottom-left corner of $\mathcal R$. In one operation, we can choose a rectangle $r$ in $\mathcal S$ such that the ant is currently located at one of the corners of $r$, say $c$, and move the ant to one of the two corners of $r$ adjacent to $c$. Suppose that after a finite number of operations, the ant ends up at the top-right corner of $\mathcal R$. Prove that some rectangle $r$ in $\mathcal S$ was chosen in at least two operations.

Let $m,n \geqslant 2$ be integers, let $X$ be a set with $n$ elements, and let $X_1,X_2,\ldots,X_m$ be pairwise distinct non-empty, not necessary disjoint subset of $X$. A function $f \colon X \to \{1,2,\ldots,n+1\}$ is called [i]nice[/i] if there exists an index $k$ such that \[\sum_{x \in X_k} f(x)>\sum_{x \in X_i} f(x) \quad \text{for all } i \ne k.\] Prove that the number of nice functions is at least $n^n$.

Let $x_1,x_2,\cdots,x_n$ be non-negative real numbers such that $x_ix_j\le 4^{-|i-j|}$ $(1\le i,j\le n)$. Prove that\[x_1+x_2+\cdots+x_n\le \frac{5}{3}.\]

Find the sharpest inequalities of the form $a\cdot AB < AG < b\cdot AB$ and $c\cdot AB < BG < d\cdot AB$ for all triangles $ABC$ with centroid $G$ such that $GA > GB > GC$.

A partition of a set \( A \) is a family of non-empty subsets of \( A \), such that any two distinct subsets in the family are disjoint, and the union of all subsets equals \( A \). We say that a partition of a set of integers \( B \) is [i]separated[/i] if each subset in the partition does [b]not[/b] contain consecutive integers. Prove that, for every positive integer \( n \), the number of partitions of the set \( \{1, 2, \dots, n\} \) is equal to the number of separated partitions of the set \( \{1, 2, \dots, n+1\} \). For example, \( \{\{1,3\}, \{2\}\} \) is a separated partition of the set \( \{1,2,3\} \). On the other hand, \( \{\{1,2\}, \{3\}\} \) is a partition of the same set, but it is not separated since \( \{1,2\} \) contains consecutive integers.

The incircle of triangle $ABC$ touches $BC$ at $D$ and $AB$ at $F$, intersects the line $AD$ again at $H$ and the line $CF$ again at $K$. Prove that $\frac{FD\times HK}{FH\times DK}=3$

a) Find : $A=\{(a,b,c) \in \mathbb{R}^{3} | a+b+c=3 , (6a+b^2+c^2)(6b+c^2+a^2)(6c+a^2+b^2) \neq 0\}$ b) Prove that for any $(a,b,c) \in A$ next inequality hold : \begin{align*} \frac{a}{6a+b^2+c^2}+\frac{b}{6b+c^2+a^2}+\frac{c}{6c+a^2+b^2} \le \frac{3}{8} \end{align*}

Show that the expression $$\frac{n^5 -5n^3 + 4n}{n + 2}$$ where n is any integer, it is always divisible by $24$.

Determine the number of functions $f: \mathbb{N} \to \mathbb{N}$ and $g: \mathbb{N} \to \mathbb{N}$ such that for all $n \in \mathbb{N}$: \[ f(g(n)) = n + 2015, \] \[ g(f(n)) = n^2 + 2015. \]

Every point in the plane was colored in red or blue. Prove that one the two following statements is true: $\bullet$ there exist two red points at distance $1$ from each other; $\bullet$ there exist four blue points $B_1$, $B_2$, $B_3$, $B_4$ such that the points $B_i$ and $B_j$ are at distance $|i - j|$ from each other, for all integers $i $ and $j$ such as $1 \le i \le 4$ and $1 \le j \le 4$.

How many integers $n \geq 2$ are there such that whenever $z_1, z_2, ..., z_n$ are complex numbers such that $$|z_1| = |z_2| = ... = |z_n| = 1 \text{ and } z_1 + z_2 + ... + z_n = 0,$$ then the numbers $z_1, z_2, ..., z_n$ are equally spaced on the unit circle in the complex plane? $\textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ 5$

A sequence of $A$s and $B$s is called [i]antipalindromic [/i] if writing it backwards, then turning all the $A$s into $B$s and vice versa, produces the original sequence. For example $ABBAAB$ is antipalindromic. For any sequence of $A$s and $B$s we define the cost of the sequence to be the product of the positions of the $A$s. For example, the string $ABBAAB$ has cost $1\cdot 4 \cdot 5 = 20$. Find the sum of the costs of all antipalindromic sequences of length $2020$.

The fraction \[\dfrac1{99^2}=0.\overline{b_{n-1}b_{n-2}\ldots b_2b_1b_0},\] where $n$ is the length of the period of the repeating decimal expansion. What is the sum $b_0+b_1+\cdots+b_{n-1}$? $\textbf{(A) }874\qquad \textbf{(B) }883\qquad \textbf{(C) }887\qquad \textbf{(D) }891\qquad \textbf{(E) }892\qquad$

For a composite number $n$, let $d_n$ denote its largest proper divisor. Show that there are infinitely many $n$ for which $d_n +d_{n+1}$ is a perfect square.

The circle $k'$ rolls along the inside of circle $k$, the radius of $k$ is twice the radius of $k'$. Describe the path of a point on $k$..

For every natural number $n$ we define \[f(n)=\left\lfloor n+\sqrt{n}+\frac{1}{2}\right\rfloor\] Show that for every integer $k \geq 1$ the equation \[f(f(n))-f(n)=k\] has exactly $2k-1$ solutions.

On June 1, a group of students is standing in rows, with 15 students in each row. On June 2, the same group is standing with all of the students in one long row. On June 3, the same group is standing with just one student in each row. On June 4, the same group is standing with 6 students in each row. This process continues through June 12 with a different number of students per row each day. However, on June 13, they cannot find a new way of organizing the students. What is the smallest possible number of students in the group? $ \textbf{(A) } 21 \qquad \textbf{(B) } 30 \qquad \textbf{(C) } 60 \qquad \textbf{(D) } 90 \qquad \textbf{(E) } 1080 $

Nine people sit in three rows of three chairs each. The probability that two of them, Celery and Drum, sit next to each other in the same row is $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Find $100m+n$. [i]Proposed by Michael Tang[/i]

Let $ABCD$ be a convex quadrilateral. Lines $AB$ and $CD$ meet at $P$, and lines $AD$ and $BC$ meet at $Q$. Let $O$ be a point in the interior of $ABCD$ such that $\angle BOP = \angle DOQ$. Prove that $\angle AOB +\angle COD = 180$.

Let $p$ be an odd prime congruent to 2 modulo 3. Prove that at most $p-1$ members of the set $\{m^2 - n^3 - 1 \mid 0 < m,\ n < p\}$ are divisible by $p$.

Determine all integers $ n\geq 2$ having the following property: for any integers $a_1,a_2,\ldots, a_n$ whose sum is not divisible by $n$, there exists an index $1 \leq i \leq n$ such that none of the numbers $$a_i,a_i+a_{i+1},\ldots,a_i+a_{i+1}+\ldots+a_{i+n-1}$$ is divisible by $n$. Here, we let $a_i=a_{i-n}$ when $i >n$. [i]Proposed by Warut Suksompong, Thailand[/i]

Let $x_1$, $x_2$, $\cdots$, $x_n$ be positive real numbers, and let \[ S = x_1 + x_2 + \cdots + x_n. \] Prove that \[ (1 + x_1)(1 + x_2) \cdots (1 + x_n) \leq 1 + S + \frac{S^2}{2!} + \frac{S^3}{3!} + \cdots + \frac{S^n}{n!} \]

Given a sequence $\{ a_n\}_{n\in \mathbb{Z}^+}$ defined by $a_1=1$ and $a_{2k}=a_{2k-1}+a_k,a_{2k+1}=a_{2k}$ for all positive integer $k$. Prove that, for any positive integer $n$, $a_{2^n}>2^{\frac{n^2}{4}}$.