Find all positive integers $n$ such that $21$ divides $2^{2^n}+2^n+1$.

Let $C$ be a point on the extension of the diameter $AB$ of a circle. A line through $C$ is tangent to the circle at point $N$. The bisector of $\angle ACN$ meets the lines $AN$ and $BN$ at $P$ and $Q$ respectively. Prove that $PN = QN$.

For a point $ P $ situated in the plane determined by a triangle $ ABC, $ prove the following inequality: $$ BC\cdot PB\cdot PC+AC\cdot PC\cdot PA +AB\cdot PA\cdot PB\ge AB\cdot BC\cdot CA $$

Given a real number $\lambda > 1$, let $P$ be a point on the arc $BAC$ of the circumcircle of $\bigtriangleup ABC$. Extend $BP$ and $CP$ to $U$ and $V$ respectively such that $BU = \lambda BA$, $CV = \lambda CA$. Then extend $UV$ to $Q$ such that $UQ = \lambda UV$. Find the locus of point $Q$.

Let a.b.c be positive reals such that their sum is 1. Prove that $ \frac{a^{2}b^{2}}{c^{3}(a^{2}\minus{}ab\plus{}b^{2})}\plus{}\frac{b^{2}c^{2}}{a^{3}(b^{2}\minus{}bc\plus{}c^{2})}\plus{}\frac{a^{2}c^{2}}{b^{3}(a^{2}\minus{}ac\plus{}c^{2})}\geq \frac{3}{ab\plus{}bc\plus{}ac}$

In multiplying two positive integers $a$ and $b$, Ron reversed the digits of the two-digit number $a$. His errorneous product was $161$. What is the correct value of the product of $a$ and $b$? $ \textbf{(A)}\ 116 \qquad \textbf{(B)}\ 161 \qquad \textbf{(C)}\ 204 \qquad \textbf{(D)}\ 214 \qquad \textbf{(E)}\ 224 $

A regular tetrahedral massless frame whose side length is physically variable (with the constraint of the tetrahedron being regular) is dipped in a soap solution of surface tension $T$, taken outside and allowed to settle after a little wiggle.\\ The soap film is formed such that there is no volume in space that is enclosed by any of the surfaces soap film and all the soap film surfaces are planar. You may assume the configuration of the soap film without proof.\\ Now 4 point charges of charge $q$ are fixed at the vertices of the tetrahedron.\\ The system now sets into motion with the shape and nature of soap film being unaltered at all times.\\ [list] [*] Find the side length of the tetrahedron for which the system attains mechanical equilibrium. [/*] [*] Find the differential equation(s) governing the side length with respect to time.[/*] [*] If the amplitude of oscillations are very small, find the time period of oscillations.[/*] [/list]

Non-degenerate quadrilateral $ABCD$ with $AB = AD$ and $BC = CD$ has integer side lengths, and $\angle ABC = \angle BCD = \angle CDA$. If $AB = 3$ and $B \ne D$, how many possible lengths are there for $BC$?

Square $S$ is the unit square with vertices at $(0, 0)$, $(0, 1)$, $(1, 0)$ and $(1, 1)$. We choose a random point $(x, y)$ inside $S$ and construct a rectangle with length $x$ and width $y$. What is the average of $\lfloor p \rfloor$ where $p$ is the perimeter of the rectangle? $\lfloor x \rfloor$ is the greatest integer less than or equal to $x$.

Suppose from $(m+2)\times(n+2)$ rectangle we cut $4$, $1\times1$ corners. Now on first and last row first and last columns we write $2(m+n)$ real numbers. Prove we can fill the interior $m\times n$ rectangle with real numbers that every number is average of it's $4$ neighbors.

Let $N$ be the number of ordered triples $(A,B,C)$ of integers satisfying the conditions (a) $0\leq A<B<C\leq99$, (b) there exist integers $a$, $b$, and $c$, and prime $p$ where $0\leq b < a < c < p$, (c) $p$ divides $A-a$, $B-b$, and $C-c$, and (d) each ordered triple $(A,B,C)$ and each ordered triple $(b,a,c)$ form arithmetic sequences. Find $N$.

Let $n \ge 2$ and $ a_1, a_2, \ldots , a_n $, nonzero real numbers not necessarily distinct. We define matrix $A = (a_{ij})_{1 \le i,j \le n} \in M_n( \mathbb{R} )$ , $a_{i,j} = max \{ a_i, a_j \}$, $\forall i,j \in \{ 1,2 , \ldots , n \} $. Show that $\mathbf{rank}(A) $= $\mathbf{card} $ $\{ a_k | k = 1,2, \ldots n \} $

Find all values of $a$ such that absolute value of one of the roots of the equation $x^2 + (a - 2)x - 2a^2 + 5a - 3 = 0$ is twice of absolute value of the other root.

Each of the lateral sides of the trapezoid, whose bases are equal to $ a$ and $b$, serves as a side of a regular triangle. One triangle is located entirely outside the trapezoid, and the other has common points with it. Find the distance between the centers of these triangles.

Let $N$ be the smallest natural number that, when written to its left, forms an integer with twice as many digits that is a perfect square. Find the remainder when $N$ is divided by $1000$.

Beto plays the following game with his computer: initially the computer randomly picks $30$ integers from $1$ to $2015$, and Beto writes them on a chalkboard (there may be repeated numbers). On each turn, Beto chooses a positive integer $k$ and some if the numbers written on the chalkboard, and subtracts $k$ from each of the chosen numbers, with the condition that the resulting numbers remain non-negative. The objective of the game is to reduce all $30$ numbers to $0$, in which case the game ends. Find the minimal number $n$ such that, regardless of which numbers the computer chooses, Beto can end the game in at most $n$ turns.

To [i]dissect [/i] a polygon means to divide it into several regions by cutting along finitely many line segments. For example, the diagram below shows a dissection of a hexagon into two triangles and two quadrilaterals: [img]https://cdn.artofproblemsolving.com/attachments/0/a/378e477bcbcec26fc90412c3eada855ae52b45.png[/img] An [i]integer-ratio[/i] right triangle is a right triangle whose side lengths are in an integer ratio. For example, a triangle with sides $3,4,5$ is an[i] integer-ratio[/i] right triangle, and so is a triangle with sides $\frac52 \sqrt3 ,6\sqrt3, \frac{13}{2} \sqrt3$. On the other hand, the right triangle with sides$ \sqrt2 ,\sqrt5, \sqrt7$ is not an [i]integer-ratio[/i] right triangle. Determine, with proof, all integers $n$ for which it is possible to completely [i]dissect [/i] a regular $n$-sided polygon into [i]integer-ratio[/i] right triangles.

In the isosceles trapezoid with the area of $28$, a circle of radius $2$ is inscribed. Find the length of the side of the trapezoid.

(a). Given any natural number \(N\), prove that there exists a strictly increasing sequence of \(N\) positive integers in harmonic progression. (b). Prove that there cannot exist a strictly increasing infinite sequence of positive integers which is in harmonic progression.

There are a number of cards on a table. A number is written on each card. The "pick and replace" operation involves the following: two random cards are taken from the table and replaced by one new card. If the numbers $a$ and $b$ appear on the two packed cards, the number $a + b + ab$ is set on the new card. If we start with ten cards with the numbers $1, 2, 3, 4, 5, 6, 7, 8, 9$ and $10$ respectively, what value(s) can the number have that "grab and replace" nine times is on the only card still on the table? Prove your answer

Let $M$ be a matrix with $r$ rows and $c$ columns. Each entry of $M$ is a nonnegative integer. Let $a$ be the average of all $rc$ entries of $M$. If $r > {(10 a + 10)}^c$, prove that $M$ has two identical rows.

Let $a, b, c$ be sides of a triangle whose perimeter does not exceed $2 \cdot \pi.$, Prove that $\sin a, \sin b, \sin c$ are sides of a triangle.

Assume that it is possible to color more than half of the surfaces of a given polyhedron so that no two colored surfaces have a common edge. (a) Describe one polyhedron with the above property. (b) Prove that one cannot inscribe a sphere touching all the surfaces of a polyhedron with the above property.

If $0<a,b<1$ and $p,q\geq 0 ,\ p+q=1$ are real numbers , then prove that: \[a^pb^q+(1-a)^p(1-b)^q\le 1\]

Twenty-one rectangles of size $3\times 1$ are placed on an $8\times 8$ chessboard, leaving only one free unit square. What position can the free square lie at?