In the diagram, if points $ A$, $ B$ and $ C$ are points of tangency, then $ x$ equals: [asy]unitsize(5cm); defaultpen(linewidth(.8pt)+fontsize(8pt)); dotfactor=3; pair A=(-3*sqrt(3)/32,9/32), B=(3*sqrt(3)/32, 9/32), C=(0,9/16); pair O=(0,3/8); draw((-2/3,9/16)--(2/3,9/16)); draw((-2/3,1/2)--(-sqrt(3)/6,1/2)--(0,0)--(sqrt(3)/6,1/2)--(2/3,1/2)); draw(Circle(O,3/16)); draw((-2/3,0)--(2/3,0)); label("$A$",A,SW); label("$B$",B,SE); label("$C$",C,N); label("$\frac{3}{8}$",O); draw(O+.07*dir(60)--O+3/16*dir(60),EndArrow(3)); draw(O+.07*dir(240)--O+3/16*dir(240),EndArrow(3)); label("$\frac{1}{2}$",(.5,.25)); draw((.5,.33)--(.5,.5),EndArrow(3)); draw((.5,.17)--(.5,0),EndArrow(3)); label("$x$",midpoint((.5,.5)--(.5,9/16))); draw((.5,5/8)--(.5,9/16),EndArrow(3)); label("$60^{\circ}$",(0.01,0.12)); dot(A); dot(B); dot(C);[/asy]$ \textbf{(A)}\ \frac {3}{16}" \qquad \textbf{(B)}\ \frac {1}{8}" \qquad \textbf{(C)}\ \frac {1}{32}" \qquad \textbf{(D)}\ \frac {3}{32}" \qquad \textbf{(E)}\ \frac {1}{16}"$

A ball was floating in a lake when the lake froze. The ball was removed (without breaking the ice), leaving a hole $24$ cm across as the top and $8$ cm deep. What was the radius of the ball (in centimeters)? $ \textbf{(A)}\ 8 \qquad\textbf{(B)}\ 12 \qquad\textbf{(C)}\ 13 \qquad\textbf{(D)}\ 8\sqrt{3} \qquad\textbf{(E)}\ 6\sqrt{6} $

A point $ N $ is marked on the median $ CM $ of the triangle $ ABC $ so that $ MN \cdot MC = AB ^ 2/4 $. Lines $ AN $ and $ BN $ intersect the circumcircle $ \triangle ABC $ for the second time at points $ P $ and $ Q $, respectively. $ R $ is the point of segment $ PQ $, nearest to $ Q $, such that $ \angle NRC = \angle BNC $. $ S $ is the point of the segment $ PQ $ closest to $ P $ such that $ \angle NSC = \angle ANC $. Prove that $ RN = SN $.

The sequence $\{a_n\}$ is defined as follows: $a_1$ is a positive rational number, $a_n= \frac{p_n}{q_n}$, ($n= 1,2,…$) is a positive integer, where $p_n$ and $q_n$ are positive integers that are relatively prime, then $a_{n+1} = \frac{p_n^2+2015}{p_nq_n}$ Is there a$_1>2015$, making the sequence $\{a_n\}$ a bounded sequence? Justify your conclusion.

In each of $n$ houses on a straight street are one or more boys. At what point should all the boys meet so that the sum of the distances that they walk is as small as possible?

Suppose that $4^{x_1}= 5, 5^{x_2}= 6,6^{x_3} = 7,..., 126^{x_{123}} = 127,127^{x_{124}} = 128$. What is the value of the product $X_1X_2... X_{124}$?

Let $ A_1,A_2,\dots,A_n$ be idempotent matrices with real entries. Prove that: \[ \mbox{N}(A_1)\plus{}\mbox{N}(A_2)\plus{}\dots\plus{}\mbox{N}(A_n)\geq \mbox{rank}(I\minus{}A_1A_2\dots A_n)\] $ \mbox{N}(A)$ is $ \mbox{dim}(\mbox{ker(A)})$

Determine the value of $(101 \times 99)$ - $(102 \times 98)$ + $(103 \times 97)$ − $(104 \times 96)$ + ... ... + $(149 \times 51)$ − $(150 \times 50)$.

Let $ n$ be a positive integer, and let $ \left( a_{1},\ a_{2} ,\ ...,\ a_{n}\right)$, $ \left( b_{1},\ b_{2},\ ...,\ b_{n}\right)$ and $ \left( c_{1},\ c_{2},\ ...,\ c_{n}\right)$ be three sequences of integers such that for any two distinct numbers $ i$ and $ j$ from the set $ \left\{ 1,2,...,n\right\}$, none of the seven integers $ a_{i}\minus{}a_{j}$; $ \left( b_{i}\plus{}c_{i}\right) \minus{}\left( b_{j}\plus{}c_{j}\right)$; $ b_{i}\minus{}b_{j}$; $ \left( c_{i}\plus{}a_{i}\right) \minus{}\left( c_{j}\plus{}a_{j}\right)$; $ c_{i}\minus{}c_{j}$; $ \left( a_{i}\plus{}b_{i}\right) \minus{}\left( a_{j}\plus{}b_{j}\right)$; $ \left( a_{i}\plus{}b_{i}\plus{}c_{i}\right) \minus{}\left( a_{j}\plus{}b_{j}\plus{}c_{j}\right)$ is divisible by $ n$. Prove that: [b]a)[/b] The number $ n$ is odd. [b]b)[/b] The number $ n$ is not divisible by $ 3$. [hide="Source of the problem"][i]Source of the problem:[/i] This question is a generalization of one direction of Theorem 2.1 in: Dean Alvis, Michael Kinyon, [i]Birkhoff's Theorem for Panstochastic Matrices[/i], American Mathematical Monthly, 1/2001 (Vol. 108), pp. 28-37. The original Theorem 2.1 is obtained if you require $ b_{i}\equal{}i$ and $ c_{i}\equal{}\minus{}i$ for all $ i$, and add in a converse stating that such sequences $ \left( a_{1},\ a_{2},\ ...,\ a_{n}\right)$, $ \left( b_{1},\ b_{2},\ ...,\ b_{n}\right)$ and $ \left( c_{1} ,\ c_{2},\ ...,\ c_{n}\right)$ indeed exist if $ n$ is odd and not divisible by $ 3$.[/hide]

A convex polygon is such that any segment dividing the polygon into two parts of equal area which has at least one end at a vertex has length $< 1$. Show that the area of the polygon is $< \pi /4$.

Let $\Gamma$ be a circle, $P$ be a point outside it, and $A$ and $B$ the intersection points between $\Gamma$ and the tangents from $P$ to $\Gamma$. Let $K$ be a point on the line $AB$, distinct from $A$ and $B$ and let $T$ be the second intersection point of $\Gamma$ and the circumcircle of the triangle $PBK$.Also, let $P'$ be the reflection of $P$ in point $A$. Show that $\angle PBT=\angle P'KA$

Given $k$ parallel lines $l_1, \ldots, l_k$ and $n_i$ points on the line $l_i, i = 1, 2, \ldots, k$, find the maximum possible number of triangles with vertices at these points.

Determine all functions $ f : \mathbb{R} \to \mathbb{R} $ continuous at zero and such that for every real number $ x $ the equality holds $$ 2f(2x) = f(x) + x.$$

Let $\triangle{ABC}$ be a scalene triangle with incentre $I$ and circumcircle $\omega$. Lines $AI, BI, CI$ intersect $\omega$ for the second time at points $D, E, F$, respectively. The parallel lines from $I$ to the sides $BC, AC, AB$ intersect $EF, DF, DE$ at points $K, L, M$, respectively. Prove that the points $K, L, M$ are collinear. [i](Cyprus)[/i]

Prove that $\frac{a}{(a + 1)(b + 1)} +\frac{ b}{(b + 1)(c + 1)} + \frac{c}{(c + 1)(a + 1)} \ge \frac34$ where $a, b$ and $c$ are positive real numbers satisfying $abc = 1$.

For a positive integer $n,$ the factorial notation $n!$ represents the product of the integers from $n$ to $1.$ (For example, $6! = 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1.$) What value of $N$ satisfies the following equation? $$5! \cdot 9! = 12 \cdot N!$$ $\textbf{(A) }10 \qquad \textbf{(B) }11 \qquad \textbf{(C) }12 \qquad \textbf{(D) }13 \qquad \textbf{(E) }14$

Let $m$ and $n$ be positive integers and $S$ be a subset with $(2^m-1)n+1$ elements of the set $\{1,2,3,\ldots, 2^mn\}$. Prove that $S$ contains $m+1$ distinct numbers $a_0,a_1,\ldots, a_m$ such that $a_{k-1} \mid a_{k}$ for all $k=1,2,\ldots, m$.

An isosceles trapezoid $ABCD$ with bases $AB$ and $CD$ has $AB=13$, $CD=17$, and height $3$. Let $E$ be the intersection of $AC$ and $BD$. Circles $\Omega$ and $\omega$ are circumscribed about triangles $ABE$ and $CDE$. Compute the sum of the radii of $\Omega$ and $\omega$.

Let $p(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$ be a monic polynomial of degree $n>2$, with real coefficients and all its roots real and different from zero. Prove that for all $k=0,1,2,\cdots,n-2$, at least one of the coefficients $a_k,a_{k+1}$ is different from zero.

Let $ x$ be an integer and $ y,z,w$ be odd positive integers. Prove that $ 17$ divides $ x^{y^{z^w}}\minus{}x^{y^z}$.

The side $AB$ of $\Delta ABC$ touches the corresponding excircle at point $T$. Let $J$ be the center of the excircle inscribed into $\angle A$, and $M$ be the midpoint of $AJ$. Prove that $MT = MC$.

Let $ABCD$ be a rectangle whose vertices are labeled in counterclockwise order with $AB=32$ and $AD=60.$ Rectangle $A'B'C'D'$ is constructed by rotating $ABCD$ counterclockwise about $A$ by $60^\circ.$ Given that lines $BB'$ and $DD'$ intersect at point $X,$ compute $CX.$

A $3 \times 3$ square is filled with numbers: $a, b, c, d, e, f, g, h, i$ in the following way: [img]https://cdn.artofproblemsolving.com/attachments/8/9/737c41e9d0dbfdc81be1b986b8e680290db55e.png[/img] Given that the square is magic (sums of the numbers in each row, column and each of two diagonals are the same), show that a) $2(a + c + g + i) = b + d + f + h + 4e$. (3) b) $2(a^3 + c^3 + g^3 + i^3) = b^3 + d^3 + f^3 + h^3 + 4e^3$. (3)

Let $f(x)$ be a continuous function, where $f(0)\neq0$. Find $\lim_{x \to 0} \frac{2\int_{0}^{x}(x-t)f(t)dt}{x\int_{0}^{x}f(x-t)dt}$.

1. The transformation $ n \to 2n \minus{} 1$ or $ n \to 3n \minus{} 1$, where $ n$ is a positive integer, is called the 'change' of $ n$. Numbers $ a$ and $ b$ are called 'similar', if there exists such positive integer, that can be got by finite number of 'changes' from both $ a$ and $ b$. Find all positive integers 'similar' to $ 2005$ and less than $ 2005$.