Let $a_1, \ldots, a_n$ be $n$ positive numbers and $0 < q < 1.$ Determine $n$ positive numbers $b_1, \ldots, b_n$ so that: [i]a.)[/i] $ a_{k} < b_{k}$ for all $k = 1, \ldots, n,$ [i]b.)[/i] $q < \frac{b_{k+1}}{b_{k}} < \frac{1}{q}$ for all $k = 1, \ldots, n-1,$ [i]c.)[/i] $\sum \limits^n_{k=1} b_k < \frac{1+q}{1-q} \cdot \sum \limits^n_{k=1} a_k.$

Given a triangle $ABC$ and a plane $\pi$ having no common points with the triangle, find a point $M$ such that the triangle determined by the points of intersection of the lines $MA,MB,MC$ with $\pi$ is congruent to the triangle $ABC$.

Let $p_1, p_2, \ldots, p_{25}$ are primes which don’t exceed 2004. Find the largest integer $T$ such that every positive integer $\leq T$ can be expressed as sums of distinct divisors of $(p_1\cdot p_2 \cdot \ldots \cdot p_{25})^{2004}.$

Let $a$ and $b$ be positive integers satisfying $3a < b$ and $a^2 + ab + b^2 = (b + 3)^2 + 27.$ Find the minimum possible value of $a + b.$

A grid is called [i]groovy[/i] if each cell of the grid is labeled with the smallest positive integer that does not appear below it in the same column or to the left of it in the same row. Compute the sum of the entries of a groovy $14 \times 14$ grid whose bottom left entry is $1.$

[b](a)[/b] Prove that $\frac{1}{n+1} \cdot \binom{2n}{n}$ is an integer for $n \geq 0.$ [b](b)[/b] Given a positive integer $k$, determine the smallest integer $C_k$ with the property that $\frac{C_k}{n+k+1} \cdot \binom{2n}{n}$ is an integer for all $n \geq k.$

Let $a, b$ be two lines intersecting each other at $O$. Point $M$ is not on either $a$ or $b$. A variable circle $(C)$ passes through $O,M$ intersecting $a, b$ at $A,B$ respectively, distinct from $O$. Prove that the midpoint of $AB$ is on a fixed line. Hạ Vũ Anh

Prove that $\tau ((n+1)!) \leq 2 \tau (n!)$ for all positive integers $n$.

An irreducible polynomial is a not-constant polynomial that cannot be factored into product of two non-constant polynomials. Consider the following statements :- [b]Statement 1 :[/b] $p(x)$ be any monic irreducible polynomial with integer coefficients and degree $\geq 4$. Then $p(n)$ is a prime for at least one natural number $n$ [b]Statement 2 :[/b] $n^2+1$ is prime for infinitely many values of natural number $n$ Show that if [b]Statement 1[/b] is true then [b]Statement 2[/b] is also true

Let $n$ be a positive integer and $S = \{ m \mid 2^n \le m < 2^{n+1} \}$. We call a pair of non-negative integers $(a, b)$ [i]fancy[/i] if $a + b$ is in $S$ and is a palindrome in binary. Find the number of [i]fancy[/i] pairs $(a, b)$.

Seven positive integers are written on a piece of paper. No matter which five numbers one chooses, each of the remaining two numbers divides the sum of the five chosen numbers. How many distinct numbers can there be among the seven?

Given positive real numbers $x_1, x_2, \ldots , x_n$ find the polygon $A_0A_1\ldots A_n$ with $A_iA_{i+1} = x_{i+1}$ and which has greatest area.

Let $n$ be an odd integer greater than $11$; $k\in \mathbb{N}$, $k \geq 6$, $n=2k-1$. We define \[d(x,y) = \left | \{ i\in \{1,2,\dots, n \} \bigm | x_i \neq y_i \} \right |\] for $T=\{ (x_1, x_2, \dots, x_n) \bigm | x_i \in \{0,1\}, i=1,2,\dots, n \}$ and $x=(x_1,x_2,\dots, x_n), y=(y_1, y_2, \dots, y_n) \in T$. Show that $n=23$ if $T$ has a subset $S$ satisfying [list=i] [*]$|S|=2^k$ [*]For each $x \in T$, there exists exacly one $y\in S$ such that $d(x,y)\leq 3$[/list]

Given triangle $ ABC$. Points $ D,E,F$ outside triangle $ ABC$ are chosen such that triangles $ ABD$, $ BCE$, and $ CAF$ are equilateral triangles. Prove that cicumcircles of these three triangles are concurrent.

Let $ M=\{1,2,...,2013\} $ and let $ \Gamma $ be a circle. For every nonempty subset $ B $ of the set $ M $, denote by $ S(B) $ sum of elements of the set $ B $, and define $ S(\varnothing)=0 $ ( $ \varnothing $ is the empty set ). Is it possible to join every subset $ B $ of $ M $ with some point $ A $ on the circle $ \Gamma $ so that following conditions are fulfilled: $ 1 $. Different subsets are joined with different points; $ 2 $. All joined points are vertices of a regular polygon; $ 3 $. If $ A_1,A_2,...,A_k $ are some of the joined points, $ k>2 $ , such that $ A_1A_2...A_k $ is a regular $ k-gon $, then $ 2014 $ divides $ S(B_1)+S(B_2)+...+S(B_k) $ ?

Find all functions $f : \mathbb{R} \to \mathbb{R}$ such that for all real $x,y$: $$ f(x^2) + xf(y) = f(x) f(x + f(y)) \, . $$

Suppose that $A$, $B$, $C$, and $D$ are distinct points, no three of which lie on a line, in the Euclidean plane. Show that if the squares of the lengths of the line segments $AB$, $AC$, $AD$, $BC$, $BD$, and $CD$ are rational numbers, then the quotient \[\frac{\mathrm{area}(\triangle ABC)}{\mathrm{area}(\triangle ABD)}\] is a rational number.

In isosceles triangle, the condition $AB=AC>BC$ is satisfied. Point $D$ is taken on the circumcircle of $ABC$ such that $\angle CAD=90^{\circ}$.A line parallel to $AC$ which passes from $D$ intersects $AB$ and $BC$ respectively at $E$ and $F$.Show that circumcircle of $ADE$ passes from circumcenter of $DFC$.

Show that for any integer $n$ the number \[a_n=\frac{\bigl(2+\sqrt3\bigr)^n-\bigl(2-\sqrt3\bigr)^n}{2\sqrt3}\] is also integer. Determine all integers $n$ such that $a_n$ is divisible by 3.

Let $ABCD$ a convex quadrilateral such that $AD$ and $BC$ are not parallel. Let $M$ and $N$ the midpoints of $AD$ and $BC$ respectively. The segment $MN$ intersects $AC$ and $BD$ in $K$ and $L$ respectively, Show that at least one point of the intersections of the circumcircles of $AKM$ and $BNL$ is in the line $AB$.

Let $n$ be an positive integer. Find the smallest integer $k$ with the following property; Given any real numbers $a_1 , \cdots , a_d $ such that $a_1 + a_2 + \cdots + a_d = n$ and $0 \le a_i \le 1$ for $i=1,2,\cdots ,d$, it is possible to partition these numbers into $k$ groups (some of which may be empty) such that the sum of the numbers in each group is at most $1$.

Let $A_1A_2A_3 \cdots A_{2013}$ be a cyclic $2013$-gon. Prove that for every point $P$ not the circumcenter of the $2013$-gon, there exists a point $Q\neq P$ such that $\frac{A_iP}{A_iQ}$ is constant for $i \in \{1, 2, 3, \cdots, 2013\}$. [i]Proposed by Robin Park[/i]

Chord $EF$ is the perpendicular bisector of chord $BC$, intersecting it in $M$. Between $B$ and $M$ point $U$ is taken, and $EU$ extended meets the circle in $A$. Then, for any selection of $U$, as described, triangle $EUM$ is similar to triangle: [asy] pair B = (-0.866, -0.5); pair C = (0.866, -0.5); pair E = (0, -1); pair F = (0, 1); pair M = midpoint(B--C); pair A = (-0.99, -0.141); pair U = intersectionpoints(A--E, B--C)[0]; draw(B--C); draw(F--E--A); draw(unitcircle); label("$B$", B, SW); label("$C$", C, SE); label("$A$", A, W); label("$E$", E, S); label("$U$", U, NE); label("$M$", M, NE); label("$F$", F, N); //Credit to MSTang for the asymptote [/asy] $\textbf{(A)}\ EFA \qquad \textbf{(B)}\ EFC \qquad \textbf{(C)}\ ABM \qquad \textbf{(D)}\ ABU \qquad \textbf{(E)}\ FMC$

The incircle $\odot (I)$ of $\triangle ABC$ touch $AC$ and $AB$ at $E$ and $F$ respectively. Let $H$ be the foot of the altitude from $A$, if $R \equiv IC \cap AH, \ \ Q \equiv BI \cap AH$ prove that the midpoint of $AH$ lies on the radical axis between $\odot (REC)$ and $\odot (QFB)$ I hope that this is not repost :)

Compute \[\sum_{k=1}^{1007}\left(\cos\left(\dfrac{\pi k}{1007}\right)\right)^{2014}.\]