There are two positive numbers that may be inserted between $3$ and $9$ such that the first three are in geometric progression while the last three are in arithmetic progression. The sum of those two positive numbers is $\textbf{(A) }13\textstyle\frac{1}{2}\qquad\textbf{(B) }11\frac{1}{4}\qquad\textbf{(C) }10\frac{1}{2}\qquad\textbf{(D) }10\qquad \textbf{(E) }9\frac{1}{2}$

For every positive integer $ n$, we define $ n?$ as $ 1?\equal{}1$ and $ n?\equal{}\frac{n}{(n\minus{}1)?}$ for $ n \ge 2$. Prove that $ \sqrt{1992}<1992?<\frac{4}{3} \sqrt{1992}.$

Circles $A$, $B$, and $C$ are externally tangent circles. Line $PQ$ is drawn such that $PQ$ is tangent to $A$ at $P$, tangent to $B$ at $Q$, and does not intersect with $C$. Circle $D$ is drawn such that it passes through the centers of $A$, $B$, and $C$. Let $R$ be the point on $D$ furthest from $PQ$. If $A$, $B$, and $C$ have radii $3$, $2$, and $1$, respectively, the area of triangle $PQR$ can be expressed in the form of $a+b\sqrt{c}$, where $a$, $b$, and $c$ are integers with $c$ not divisible by any prime square. What is $a + b + c$?

Let $P, Q$ be points taken on the side $BC$ of a triangle $ABC$, in the order $B, P, Q, C$. Let the circumcircles of $\vartriangle PAB$, $\vartriangle QAC$ intersect at $M$ ($\ne A$) and those of $\vartriangle PAC, \vartriangle QAB$ at N. Prove that $A, M, N$ are collinear if and only if $P$ and $Q$ are symmetric in the midpoint $A' $ of $BC$.

Let $ABC$ be a triangle with $\angle A = 90^o, \angle B = 60^o$ and $BC = 1$ cm. Draw outside of $\vartriangle ABC$ three equilateral triangles $ABD,ACE$ and $BCF$. Determine the area of $\vartriangle DEF$.

Kilua and Ndoti play the following game in a square $ABCD$: Kilua chooses one of the sides of the square and draws a point $X$ at this side. Ndoti chooses one of the other three sides and draws a point Y. Kilua chooses another side that hasn't been chosen and draws a point Z. Finally, Ndoti chooses the last side that hasn't been chosen yet and draws a point W. Each one of the players can draw his point at a vertex of $ABCD$, but they have to choose the side of the square that is going to be used to do that. For example, if Kilua chooses $AB$, he can draws $X$ at the point $B$ and it doesn't impede Ndoti of choosing $BC$. A vertex cannot de chosen twice. Kilua wins if the area of the convex quadrilateral formed by $X$, $Y$, $Z$, and $W$ is greater or equal than a half of the area of $ABCD$. Otherwise, Ndoti wins. Which player has a winning strategy? How can he play?

For a positive integer $n$, define $n?=1^n\cdot2^{n-1}\cdot3^{n-2}\cdots\left(n-1\right)^2\cdot n^1$. Find the positive integer $k$ for which $7?9?=5?k?$. [i]Proposed by Tristan Shin[/i]

With expressions containing the symbol $*$, the following transformations can be performed: 1) rewrite the expression in the form $x * (y * z) as ((1 * x) * y) * z$; 2) rewrite the expression in the form $x * 1$ as $x$. Conversions can only be performed with an integer expression, but not with its parts. For example, $(1 *1) * (1 *1)$ can be rewritten according to the first rule as $((1 * (1 * 1)) * 1) * 1$ (taking $x = 1 * 1$, $y = 1$ and $z = 1$), but not as $1 * (1 * 1)$ or $(1* 1) * 1$ (in the last two cases, the second rule would be applied separately to the left or right side $1 * 1$). Find all positive integers $n$ for which the expression $\underbrace{1 * (1 * (1 * (...* (1 * 1)...))}_{n units}$ it is possible to lead to a form in which there is not a single asterisk. Note. The expressions $(x * y) * $z and $x * (y * z)$ are considered different, also, in the general case, the expressions $x * y$ and $y * x$ are different.

Given a positive integer $\displaystyle n = \prod_{i=1}^s p_i^{\alpha_i}$, we write $\Omega(n)$ for the total number $\displaystyle \sum_{i=1}^s \alpha_i$ of prime factors of $n$, counted with multiplicity. Let $\lambda(n) = (-1)^{\Omega(n)}$ (so, for example, $\lambda(12)=\lambda(2^2\cdot3^1)=(-1)^{2+1}=-1$). Prove the following two claims: i) There are infinitely many positive integers $n$ such that $\lambda(n) = \lambda(n+1) = +1$; ii) There are infinitely many positive integers $n$ such that $\lambda(n) = \lambda(n+1) = -1$. [i](Romania) Dan Schwarz[/i]

The sequence $a_1,a_2,a_3\ldots $, consisting of natural numbers, is defined by the rule: \[a_{n+1}=a_{n}+2t(n)\] for every natural number $n$, where $t(n)$ is the number of the different divisors of $n$ (including $1$ and $n$). Is it possible that two consecutive members of the sequence are squares of natural numbers?

For any positive integer $k$, denote the sum of digits of $k$ in its decimal representation by $S(k)$. Find all polynomials $P(x)$ with integer coefficients such that for any positive integer $n \geq 2016$, the integer $P(n)$ is positive and $$S(P(n)) = P(S(n)).$$ [i]Proposed by Warut Suksompong, Thailand[/i]

Prove that, for all $n\in\mathbb{N}$, on $ [n-4\sqrt{n}, n+4\sqrt{n}]$ exists natural number $k=x^3+y^3$ where $x$, $y$ are nonnegative integers.

Let $ABC$ be a triangle and its incircle meets $BC, AC, AB$ at $D, E$ and $F$ respectively. Let point $ P $ on the incircle and inside $ \triangle AEF $. Let $ X=PB \cap DF , Y=PC \cap DE, Q=EX \cap FY $. Prove that the points $ A$ and $Q$ lies on $DP$ simultaneously or located opposite sides from $DP$.

A positive integer is called [i]piola [/i] if the $9$ is the remainder obtained by dividing it by $2, 3, 4, 5, 6, 7, 8, 9$ and $10$ and it's digits are all different and nonzero. How many [i]piolas[/i] are there between $ 1$ and $100000$?

Let $H(D)$ denote the space of functions holomorphic on the disc $D=\{ z\colon |z|<1 \}$, endowed with the topology of uniform convergence on each compact subset of $D$. If $f(z)=\sum_{n=0}^{\infty} a_nz^n$, then we shall denote $S_n(f,z)=\sum_{k=0}^n a_kz^k$. A function $f\in H(D)$ is called [i]universal[/i] if, for every continuous function $g\colon\partial D\rightarrow \mathbb{C}$ and for every $\varepsilon >0$, there are partial sums $S_{n(j)}(f,z)$ approximating $g$ uniformly on the arc $\{ e^{it} \colon 0\le t\le 2\pi - \varepsilon\}$. Prove that the set of universal functions contains a dense $G_{\delta}$ subset of $H(D)$.

The bisector $AD$ is drawn in the triangle $ABC$. Circle $k$ passes through the vertex $A$ and touches the side $BC$ at point $D$. Prove that the circle circumscribed around $ABC$ touches the circle $k$ at point $A$.

Suppose that \[x_1+1=x_2+2=x_3+3=\cdots=x_{2008}+2008=x_1+x_2+x_3+\cdots+x_{2008}+2009.\] Find the value of $\left\lfloor|S|\right\rfloor$, where $S=\displaystyle\sum_{n=1}^{2008}x_n$.

Four identical squares and one rectangle are placed together to form one large square as shown. The length of the rectangle is how many times as large as its width? [asy]unitsize(8mm); defaultpen(linewidth(.8pt)); draw(scale(4)*unitsquare); draw((0,3)--(4,3)); draw((1,3)--(1,4)); draw((2,3)--(2,4)); draw((3,3)--(3,4));[/asy]$ \textbf{(A)}\ \frac {5}{4} \qquad \textbf{(B)}\ \frac {4}{3} \qquad \textbf{(C)}\ \frac {3}{2} \qquad \textbf{(D)}\ 2 \qquad \textbf{(E)}\ 3$

Find the largest prime factor of $45^5-1.$

Luis' friends played a prank on him in his geometry homework. They erased the entire triangle but left traces equivalent to two sides measuring $a$ and $b$, with $b > a$, and the height $h$ falling on the side measuring $b$, with $h < a$. Help Luis reconstruct the original triangle using only a straightedge and compass. Since Luis' method does not involve measurements, prove that his method results in a triangle longer than its given sides and height.

Let $$P=\prod_{i=0}^{2016} (i^3-i-1)^2.$$ The remainder when $P$ is divided by the prime $2017$ is not zero. Compute this remainder.

Let $C$ be a three-dimensional cube with edge length 1. There are 8 equilateral triangles whose vertices are vertices of $C$. The 8 planes that contain these 8 equilateral triangles divide $C$ into several nonoverlapping regions. Find the volume of the region that contains the center of $C$.

Let $n > 1$ and $x_i \in \mathbb{R}$ for $i = 1,\cdots, n$. Set \[S_k = x_1^k+ x^k_2+\cdots+ x^k_n\] for $k \ge 1$. If $S_1 = S_2 =\cdots= S_{n+1}$, show that $x_i \in \{0, 1\}$ for every $i = 1, 2,\cdots, n.$

Given a segment $AB$ in the plane. Let $C$ be another point in the same plane,$H,I,G$ denote the orthocenter,incenter and centroid of triangle $ABC$. Find the locus of $M$ for which $A,B,H,I$ are concyclic.

There are $m$ real numbers $x_i \geq 0$ ($i=1,2,...,m$), $n \geq 2$, $\sum_{i=1}^{m} x_i=S$. Prove that\\ \[ \sum_{i=1}^{m} \sqrt[n]{\frac{x_i}{S-x_i}} \geq 2, \] The equation holds if and only if there are exactly two of $x_i$ are equal(not equal to $0$), and the rest are equal to $0$.