Denote by $E(n)$ the number of $1$'s in the binary representation of a positive integer $n$. Call $n$ [i]interesting[/i] if $E(n)$ divides $n$. Prove that (a) there cannot be five consecutive interesting numbers, and (b) there are infinitely many positive integers $n$ such that $n$, $n+1$ and $n+2$ are each interesting.

How many triangles are there for which the perimeters are equal to $30$ cm and the lengths of sides are integers in centimeters? A. $16$ B. $17$ C. $18$ D. $19$ E. $20$

Three gamblers play against each other for money. They each start by placing a pile of one-krone coins on the table, and from this point on the total number of coins on the table does not change. The ratio between the number of coins they start with is $6 : 5 : 4$. At the end of the game, the ratio of the number of coins they have is $7 : 6 : 5$ in some order. At the end of the game, one of the gamblers has three coins more than at the beginning. How many coins does this gambler have at the end?

Is: \[ 4\sqrt{4-2\sqrt{3}}+\sqrt{97-56\sqrt{3}} \] an integer?

The boiling points of the halogens, $\ce{F2}, \ce{Cl2}, \ce{Br2}$ and $\ce{I2}$ increase in that order. This is best attributed to differences in $ \textbf{(A) }\text{covalent bond strengths}\qquad$ $\textbf{(B) }\text{dipole forces}\qquad$ $\textbf{(C) }\text{London dispersion forces}\qquad$ $\textbf{(D) }\text{colligative forces}\qquad$

A positive integer $n>1$ is called beautiful if $n$ can be written in one and only one way as $n=a_1+a_2+\cdots+a_k=a_1 \cdot a_2 \cdots a_k$ for some positive integers $a_1, a_2, \ldots, a_k$, where $k>1$ and $a_1 \geq a_2 \geq \cdots \geq a_k$. (For example 6 is beautiful since $6=3 \cdot 2 \cdot 1=3+2+1$, and this is unique. But 8 is not beautiful since $8=4+2+1+1=4 \cdot 2 \cdot 1 \cdot 1$ as well as $8=2+2+2+1+1=2 \cdot 2 \cdot 2 \cdot 1 \cdot 1$, so uniqueness is lost.) Find the largest beautiful number less than 100.

Let $a$ and $b$ be two natural numbers such that $a<b$. Prove that in each set of $b$ consecutive positive integers there are two numbers whose product is divisible by $ab$.

A right triangle has integer side lengths, and the sum of its area and the length of one of its legs equals $75$. Find the side lengths of the triangle.

Suppose $A\subset \{(a_1,a_2,\dots,a_n)\mid a_i\in \mathbb{R},i=1,2\dots,n\}$. For any $\alpha=(a_1,a_2,\dots,a_n)\in A$ and $\beta=(b_1,b_2,\dots,b_n)\in A$, we define \[ \gamma(\alpha,\beta)=(|a_1-b_1|,|a_2-b_2|,\dots,|a_n-b_n|), \] \[ D(A)=\{\gamma(\alpha,\beta)\mid\alpha,\beta\in A\}. \] Please show that $|D(A)|\geq |A|$.

Circle $C_1$ and $C_2$ each have radius $1$, and the distance between their centers is $\frac{1}{2}$. Circle $C_3$ is the largest circle internally tangent to both $C_1$ and $C_2$. Circle $C_4$ is internally tangent to both $C_1$ and $C_2$ and externally tangent to $C_3$. What is the radius of $C_4$? [asy] import olympiad; size(10cm); draw(circle((0,0),0.75)); draw(circle((-0.25,0),1)); draw(circle((0.25,0),1)); draw(circle((0,6/7),3/28)); pair A = (0,0), B = (-0.25,0), C = (0.25,0), D = (0,6/7), E = (-0.95710678118, 0.70710678118), F = (0.95710678118, -0.70710678118); dot(B^^C); draw(B--E, dashed); draw(C--F, dashed); draw(B--C); label("$C_4$", D); label("$C_1$", (-1.375, 0)); label("$C_2$", (1.375,0)); label("$\frac{1}{2}$", (0, -.125)); label("$C_3$", (-0.4, -0.4)); label("$1$", (-.85, 0.70)); label("$1$", (.85, -.7)); import olympiad; markscalefactor=0.005; [/asy] $\textbf{(A) } \frac{1}{14} \qquad \textbf{(B) } \frac{1}{12} \qquad \textbf{(C) } \frac{1}{10} \qquad \textbf{(D) } \frac{3}{28} \qquad \textbf{(E) } \frac{1}{9}$

What is the minimum number of small squares that must be colored black so that a line of symmetry lies on the diagonal $ \overline{BD}$ of square $ ABCD$? [asy]defaultpen(linewidth(1)); for ( int x = 0; x &lt; 5; ++x ) { draw((0,x)--(4,x)); draw((x,0)--(x,4)); } fill((1,0)--(2,0)--(2,1)--(1,1)--cycle); fill((0,3)--(1,3)--(1,4)--(0,4)--cycle); fill((2,3)--(4,3)--(4,4)--(2,4)--cycle); fill((3,1)--(4,1)--(4,2)--(3,2)--cycle); label("$A$", (0, 4), NW); label("$B$", (4, 4), NE); label("$C$", (4, 0), SE); label("$D$", (0, 0), SW);[/asy] $ \textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5 $

Calculate $ \begin{pmatrix}1&0&0& \ldots &0\\\binom{1}{0} &\binom{1}{1} &0& \ldots & 0 \\ \ldots & \ldots & \ldots & \ldots & \ldots \\ \binom{n}{0} &\binom{n}{1} & \binom{n}{2} & \ldots & \binom{n}{n}\end{pmatrix}^{-1} . $

For any polynomial $P(x)=a_0+a_1x+\ldots+a_kx^k$ with integer coefficients, the number of odd coefficients is denoted by $o(P)$. For $i-0,1,2,\ldots$ let $Q_i(x)=(1+x)^i$. Prove that if $i_1,i_2,\ldots,i_n$ are integers satisfying $0\le i_1<i_2<\ldots<i_n$, then: \[ o(Q_{i_{1}}+Q_{i_{2}}+\ldots+Q_{i_{n}})\ge o(Q_{i_{1}}). \]

Find all reals $a$ such that the sequence $\{x(n)\}$, $n=0,1,2, \ldots$ that satisfy: $x(0)=1996$ and $x_{n+1} = \frac{a}{1+x(n)^2}$ for any natural number $n$ has a limit as n goes to infinity.

A cyclic quadrilateral $ABCD$ is given. Let $I{}$ and $J{}$ be the centers of circles inscribed in the triangles $ABC$ and $ADC$. It turns out that the points $B, I, J, D$ lie on the same circle. Prove that the quadrilateral $ABCD$ is tangential.

Angle bisectors $AA', BB'$and $CC'$ are drawn in triangle $ABC$ with angle $\angle B= 120^o$. Find $\angle A'B'C'$.

For how many integers is the number $x^4-51x^2+50$ negative? $ \textbf{(A) }8\qquad\textbf{(B) }10\qquad\textbf{(C) }12\qquad\textbf{(D) }14\qquad\textbf{(E) }16\qquad $

Determine the number of sequences of points $(x_1, y_1),(x_2, y_2), \dots ,(x_{4570}, y_{4570})$ on the plane satisfying the following two properties: $\text{(i)}$ $\{x_1,x_2,\dots,x_{4570}\}=\{1,2,\dots,2014\}$ and $\{y_1,y_2,\dots,y_{4570}\}=\{1,2,\dots,2557\}$ $\text{(ii)} $ For each $i = 1, 2,\dots , 4569$, exactly one of $x_i = x_{i+1}$ and $y_i = y_{i+1}$ holds.

Let be four $ n\times n $ real matrices $ A,B,C,D $ having the property that $ C+D\sqrt{-1} $ is the inverse of $ A+B\sqrt{-1} . $ Show that $ \left| \det\left( A+B\sqrt{-1} \right) \right|^2\cdot\left| \det C \right| =\det A. $ [i]Vasile Pop[/i]

Let $f : Z^2 \to C$ be a function such that $f(x+11, y) = f(x, y+11) = f(x, y)$, and $f(x, y)f(z,w) = f(xz - yw,xw + yz)$. How many possible values can $f(1, 1)$ have?

Tom throws a football to Wes, who is a distance $l$ away. Tom can control the time of flight $t$ of the ball by choosing any speed up to $v_{\text{max}}$ and any launch angle between $0^\circ$ and $90^\circ$. Ignore air resistance and assume Tom and Wes are at the same height. Which of the following statements is [b]incorrect[/b]? $ \textbf{(A)}$ If $v_{\text{max}} < \sqrt{gl}$, the ball cannot reach Wes at all. $ \\ $ $ \textbf{(B)}$ Assuming the ball can reach Wes, as $v_{\text{max}}$ increases with $l$ held fixed, the minimum value of $t$ decreases. $ \\ $ $ \textbf{(C)}$ Assuming the ball can reach Wes, as $v_{\text{max}}$ increases with $l$ held fixed, the maximum value of $t$ increases. $ \\ $ $ \textbf{(D)}$ Assuming the ball can reach Wes, as $l$ increases with $v_{\text{max}}$ held fixed, the minimum value of $t$ increases. $ \\ $ $ \textbf{(E)}$ Assuming the ball can reach Wes, as $l$ increases with $v_{\text{max}}$ held fixed, the maximum value of $t$ increases.

Let $H$ and $O$ be the orthocenter and the circumcenter of the triangle $ABC$. Line $OH$ intersects the sides $AB, AC$ at points $X, Y$ correspondingly, so that $H$ belongs to the segment $OX$. It turned out that $XH = HO = OY$. Find $\angle BAC$. [i](Proposed by Oleksii Masalitin)[/i]

In a quadrilateral $ABCD$, we have $\angle DAB = 110^{\circ} , \angle ABC = 50^{\circ}$ and $\angle BCD = 70^{\circ}$ . Let $ M, N$ be the mid-points of $AB$ and $CD$ respectively. Suppose $P$ is a point on the segment $M N$ such that $\frac{AM}{CN} = \frac{MP}{PN}$ and $AP = CP$ . Find $\angle AP C$.

Every integer is to be coloured blue, green, red, or yellow. Can this be done in such a way that if $a, b, c, d$ are not all $0$ and have the same colour, then $3a-2b \neq 2c-3d$? [size=85][color=#0000FF][Mod edit: Question fixed][/color][/size]

Find all natural numbers for which there exist that many distinct natural numbers such that the factorial of one of these is equal to the product of the factorials of the rest of them.