A square piece of paper, 4 inches on a side, is folded in half vertically. Both layers are then cut in half parallel to the fold. Three new rectangles are formed, a large one and two small ones. What is the ratio of the perimeter of one of the small rectangles to the perimeter of the large rectangle? [asy] draw((0,8)--(0,0)--(4,0)--(4,8)--(0,8)--(3.5,8.5)--(3.5,8)); draw((2,-1)--(2,9),dashed);[/asy] $ \text{(A)}\ \frac{1}{3}\qquad\text{(B)}\ \frac{1}{2}\qquad\text{(C)}\ \frac{3}{4}\qquad\text{(D)}\ \frac{4}{5}\qquad\text{(E)}\ \frac{5}{6} $

Let $ABC$ be an acute triangle with incenter $O$, orthocenter $H$, altitude $AD. AO$ meets $BC$ at $E$. Line through $D$ parallel to $OH$ meet $AB,AC$ at $M,N$, respectively. Let $I$ be the midpoint of $AE$, and $DI$ intersect $AB,AC$ at $P,Q$ respectively. $MQ$ meets $NP$ at $T$. Prove that $D,O, T$ are collinear. Trần Quang Hùng

Let $ 0 < x <y$ be real numbers. Let $ H\equal{}\frac{2xy}{x\plus{}y}$ , $ G\equal{}\sqrt{xy}$ , $ A\equal{}\frac{x\plus{}y}{2}$ , $ Q\equal{}\sqrt{\frac{x^2\plus{}y^2}{2}}$ be the harmonic, geometric, arithmetic and root mean square (quadratic mean) of $ x$ and $ y$. As generally known $ H<G<A<Q$. Arrange the intervals $ [H,G]$ , $ [G,A]$ and $ [A,Q]$ in ascending order by their length.

Let $EFGH$, $EFDC$, and $EHBC$ be three adjacent square faces of a cube, for which $EC=8$, and let $A$ be the eighth vertex of the cube. Let $I$, $J$, and $K$, be the points on $\overline{EF}$, $\overline{EH}$, and $\overline{EC}$, respectively, so that $EI=EJ=EK=2$. A solid $S$ is obtained by drilling a tunnel through the cube. The sides of the tunnel are planes parallel to $\overline{AE}$, and containing the edges, $\overline{IJ}$, $\overline{JK}$, and $\overline{KI}$. The surface area of $S$, including the walls of the tunnel, is $m+n\sqrt{p}$, where $m$, $n$, and $p$ are positive integers and $p$ is not divisible by the square of any prime. Find $m+n+p$.

In a convex quadrilateral $ABCD$, let $M$, $N$, $P$, and $Q$ be the midpoints of $AB$, $BC$, $CD$, and $DA$ respectively. If $MP$ and $NQ$ divide $ABCD$ in four quadrilaterals with the same area, prove that $ABCD$ is a parallelogram.

Show that if an infinite arithmetic progression of positive integers contains a square and a cube, it must contain a sixth power.

Work base 3. (so each digit is 0,1,2) A good number of size $n$ is a number in which there are no consecutive $1$'s and no consecutive $2$'s. How many good 10-digit numbers are there?

A tetrahedron has three edges of length $2$ and three edges of length $4$, and one of its faces is an equilateral triangle. Compute the radius of the sphere that is tangent to every edge of this tetrahedron.

Andrew generates a finite random sequence $\{a_n\}$ of distinct integers according to the following criteria: [list] [*] $a_0 = 1$, $0 < |a_n| < 7$ for all $n$, and $a_i \neq a_j$ for all $i < j$. [*] $a_{n+1}$ is selected uniformly at random from the set $\{a_n - 1, a_n + 1, -a_n\}$, conditioned on the above rule. The sequence terminates if no element of the set satisfies the first condition. [/list] For example, if $(a_0, a_1) = (1, 2)$, then $a_2$ would be chosen from the set $\{-2,3\}$, each with probability $\tfrac12$. Determine the probability that there exists an integer $k$ such that $a_k = 6$.

If $m,n\in\mathbb N_{\ge2}$, find the best constant $k\in\mathbb R$ for which $$\sum_{j=2}^m\sum_{i=2}^n\frac1{i^j}<k$$ [i]Proposed by Dorin Mărghidanu[/i]

Let $f:S\to\mathbb{R}$ be the function from the set of all right triangles into the set of real numbers, defined by $f(\Delta ABC)=\frac{h}{r}$, where $h$ is the height with respect to the hypotenuse and $r$ is the inscribed circle's radius. Find the image, $Im(f)$, of the function.

Let $n$ be a posititve integer. On a $n \times n$ grid there are $n^2$ unit squares and on these we color the sides with blue such that every unit square has exactly one side with blue. [b]a)[/b] Find the maximun number of blue unit sides we can have on the $n \times n$ grid. [b]b)[/b] Find the minimun number of blue unit sides we can have on the $n \times n$ grid.

For each polynomial $p(x)=a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0$ we define it's derivative as this and we show it by $p'(x)$: \[p'(x)=na_nx^{n-1}+(n-1)a_{n-1}x^{n-2}+...+2a_2x+a_1\] a) For each two polynomials $p(x)$ and $q(x)$ prove that:(3 points) \[(p(x)q(x))'=p'(x)q(x)+p(x)q'(x)\] b) Suppose that $p(x)$ is a polynomial with degree $n$ and $x_1,x_2,...,x_n$ are it's zeros. prove that:(3 points) \[\frac{p'(x)}{p(x)}=\sum_{i=1}^{n}\frac{1}{x-x_i}\] c) $p(x)$ is a monic polynomial with degree $n$ and $z_1,z_2,...,z_n$ are it's zeros such that: \[|z_1|=1, \quad \forall i\in\{2,..,n\}:|z_i|\le1\] Prove that $p'(x)$ has at least one zero in the disc with length one with the center $z_1$ in complex plane. (disc with length one with the center $z_1$ in complex plane: $D=\{z \in \mathbb C: |z-z_1|\le1\}$)(20 points)

Call a $ 7$-digit telephone number $ d_1d_2d_3 \minus{} d_4d_5d_6d_7$ [i]memorable[/i] if the prefix sequence $ d_1d_2d_3$ is exactly the same as either of the sequences $ d_4d_5d_6$ or $ d_5d_6d_7$ (possibly both). Assuming that each $ d_i$ can be any of the ten decimal digits $ 0,1,2,\ldots9$, the number of different memorable telephone numbers is $ \textbf{(A)}\ 19,\!810 \qquad \textbf{(B)}\ 19,\!910 \qquad \textbf{(C)}\ 19,\!990 \qquad \textbf{(D)}\ 20,\!000 \qquad \textbf{(E)}\ 20,\!100$

For how many integers $x$ is the point $(x,-x)$ inside or on the circle of radius $10$ centered at $(5,5)$? $\textbf{(A) } 11 \qquad\textbf{(B) } 12 \qquad\textbf{(C) } 13 \qquad\textbf{(D) } 14 \qquad\textbf{(E) } 15 $

Consider the sequence $x_n>0$ defined with the following recurrence relation: \[x_1 = 0\] and for $n>1$ \[(n+1)^2x_{n+1}^2 + (2^n+4)(n+1)x_{n+1}+ 2^{n+1}+2^{2n-2} = 9n^2x_n^2+36nx_n+32.\] Show that if $n$ is a prime number larger or equal to $5$, then $x_n$ is an integer.

In a dance party there are $n$ girls and $n$ boys, and some $m$ songs are played. Each song is danced to by at least one pair of a boy and a girl, who both receive a [i]malai [/i] each. Prove that for all positive integers $k \le n$, it is possible to select $k$ boys and $n - k$ girls so that the $n$ selected people received at least $m$ malai in total.

The numbers $1, 2, 3,\ldots , 64$ are placed on a chessboard, one number in each square. Consider all squares on the chessboard of size $2 \times 2.$ Prove that there are at least three such squares for which the sum of the $4$ numbers contained exceeds $100.$

Let $n$ be a positive integer. Show that there are infinitely many primes $p$ such that the smallest positive primitive root of $p$ is greater than $n$.

$AB$ and $AC$ are tangents to a circle $\omega$ with center $O$ at $B,C$ respectively. Point $P$ is a variable point on minor arc $BC$. The tangent at $P$ to $\omega$ meets $AB,AC$ at $D,E$ respectively. $AO$ meets $BP,CP$ at $U,V$ respectively. The line through $P$ perpendicular to $AB$ intersects $DV$ at $M$, and the line through $P$ perpendicular to $AC$ intersects $EU$ at $N$. Prove that as $P$ varies, $MN$ passes through a fixed point.

What is the smallest positive integer $n$ such that $\sqrt{n}-\sqrt{n-1}<.01$? $\textbf{(A) }2499\qquad\textbf{(B) }2500\qquad\textbf{(C) }2501\qquad\textbf{(D) }10,000\qquad \textbf{(E) }\text{There is no such integer}$

A straight line passing through the point $(0,4)$ is perpendicular to the line $x-3y-7=0$. Its equation is: $\textbf{(A)}\ y+3x-4=0 \qquad \textbf{(B)}\ y+3x+4=0 \qquad \textbf{(C)}\ y-3x-4=0 \qquad\\ \textbf{(D)}\ 3y+x-12=0 \qquad \textbf{(E)}\ 3y-x-12=0 $

Let $f(x) = (x^4 + 2x^3 + 4x^2 + 2x + 1)^5$. Compute the prime $p$ satisfying $f(p) = 418{,}195{,}493$. [i]Proposed by Eugene Chen[/i]

Let point $T$ be on side $AB$ of $\Delta ABC$ be such that $AT-BT=AC-BC$. The perpendicular from point $T$ to $AB$ intersects $AC$ in point $E$ and the angle bisectors of $\angle B$ and $\angle C$ intersect the circumscribed circle $k$ of $ABC$ in points $M$ and $L$. If $P$ is the second intersection point of the line $ME$ with $k$, then prove that $P,T,L$ are collinear.

Two unequal circles intersect at $X$ and $Y$. Their common tangents intersect at $Z$. One of the tangents touches the circles at $P$ and $Q$. Show that $ZX$ is tangent to the circumcircle of $PXQ$.