An equilateral triangle is originally painted black. Each time the triangle is changed, the middle fourth of each black triangle turns white. After five changes, what fractional part of the original area of the black triangle remains black? [asy] unitsize(36); fill((0,0)--(2,0)--(1,sqrt(3))--cycle,gray); draw((0,0)--(2,0)--(1,sqrt(3))--cycle,linewidth(1)); fill((4,0)--(6,0)--(5,sqrt(3))--cycle,gray); fill((5,0)--(9/2,sqrt(3)/2)--(11/2,sqrt(3)/2)--cycle,white); draw((5,sqrt(3))--(4,0)--(5,0)--(9/2,sqrt(3)/2)--(11/2,sqrt(3)/2)--(5,0)--(6,0)--cycle,linewidth(1)); fill((8,0)--(10,0)--(9,sqrt(3))--cycle,gray); fill((9,0)--(17/2,sqrt(3)/2)--(19/2,sqrt(3)/2)--cycle,white); fill((17/2,0)--(33/4,sqrt(3)/4)--(35/4,sqrt(3)/4)--cycle,white); fill((9,sqrt(3)/2)--(35/4,3*sqrt(3)/4)--(37/4,3*sqrt(3)/4)--cycle,white); fill((19/2,0)--(37/4,sqrt(3)/4)--(39/4,sqrt(3)/4)--cycle,white); draw((9,sqrt(3))--(35/4,3*sqrt(3)/4)--(37/4,3*sqrt(3)/4)--(9,sqrt(3)/2)--(35/4,3*sqrt(3)/4)--(33/4,sqrt(3)/4)--(35/4,sqrt(3)/4)--(17/2,0)--(33/4,sqrt(3)/4)--(8,0)--(9,0)--(17/2,sqrt(3)/2)--(19/2,sqrt(3)/2)--(9,0)--(19/2,0)--(37/4,sqrt(3)/4)--(39/4,sqrt(3)/4)--(19/2,0)--(10,0)--cycle,linewidth(1)); label("Change 1",(3,3*sqrt(3)/4),N); label("$\Longrightarrow $",(3,5*sqrt(3)/8),S); label("Change 2",(7,3*sqrt(3)/4),N); label("$\Longrightarrow $",(7,5*sqrt(3)/8),S); [/asy] $\text{(A)}\ \frac{1}{1024} \qquad \text{(B)}\ \frac{15}{64} \qquad \text{(C)}\ \frac{243}{1024} \qquad \text{(D)}\ \frac{1}{4} \qquad \text{(E)}\ \frac{81}{256}$

Let $\Delta ABC$ be an acute triangle with incenter $I$ and orthocenter $H$. $AI$ meets the circumcircle of $\Delta ABC$ again at $M$. Suppose the length $IM$ is exactly the circumradius of $\Delta ABC$. Show that $AH\geq AI$.

For n points \[ P_1;P_2;...;P_n \] in that order on a straight line. We colored each point by 1 in 5 white, red, green, blue, and purple. A coloring is called acceptable if two consecutive points \[ P_i;P_{i+1} (i=1;2;...n-1) \] is the same color or 2 points with at least one of 2 points are colored white. How many ways acceptable color?

Prove that for any prime number $p$ the equation $2^p+3^p=a^n$ has no solution $(a,n)$ in integers greater than $1$.

We call a rectangle of size $2 \times 3$ (or $3 \times 2$) without one cell in corner a $P$-rectangle. We call a rectangle of size $2 \times 3$ (or $3 \times 2$) without two cells in opposite (under center of rectangle) corners a $S$-rectangle. Using some squares of size $2 \times 2$, some $P$-rectangles and some $S$-rectangles, one form one rectangle of size $1993 \times 2000$ (figures don’t overlap each other). Let $s$ denote the sum of numbers of squares and $S$-rectangles used in such tiling. Find the maximal value of $s$.

Suppose you have identical coins distributed in several piles with one or more coins in each pile. An action consists of taking two piles, which have an even total of coins among them, and redistribute their coins in two piles so that they end up with the same number of coins. A distribution is [i]levelable[/i] if it is possible, by means of 0 or more operations, to end up with all the piles having the same number of coins. Determine all positive integers $n$ such that, for all positive integers $k$, any distribution of $nk$ coins in $n$ piles is levelable.

Numbers $1992,1993, ... ,2000$ are written in a $3 \times 3$ table to form a magic square (i.e. the sums of numbers in rows, columns and big diagonals are all equal). Prove that the number in the center is $1996$. Which numbers are placed in the corners?

Let set $S$ be the smallest set of positive integers satisfying the following properties: [list] [*] $2$ is in set $S$. [*] If $n^2$ is in set $S$, then $n$ is also in set $S$. [*] If $n$ is in set $S$, then $(n+5)^2$ is also in set $S$. [/list] Determine which positive integers are not in set $S$.

Let $f(x)$ be a polynomial such that $2f(x) + f(2 - x) = 5 + x$ for any real number x. Find the value of $f(0) + f(2)$. A. $4$ B. $0$ C.$ 2$ D. $3$ E. $1$

Let $x,y,z \in [0,1]$ , and $|y-z|\leq \frac{1}{2},|z-x|\leq \frac{1}{2},|x-y|\leq \frac{1}{2}$ . Find the maximum and minimum value of $W=x+y+z-yz-zx-xy$.

A numismatist Fred has some coins. A diameter of any coin is no more than $10$ cm. All the coins are contained in a one-layer box of dimensions $30$ cm by $70$ cm. He is presented with a new coin. Its diameter is $25$ cm. Prove that it is possible to put all the coins in a one-layer box of dimensions $55$ cm by $55$ cm. (Fedja Nazarov, St Petersburg)

Let $ABC$ be any triangle and $D, E, F$ the midpoints of $BC, CA, AB$. The medians $AD, BE$ and $CF$ intersect at point $S$. At least two of the quadrilaterals $AF SE, BDSF, CESD$ are cyclic. Show that the triangle $ABC$ is equilateral.

Let \( BE \) and \( CF \) be the medians of \( \triangle ABC \), and \( G \) be their intersection point. On segments \( GF \) and \( GE \), points \( K \) and \( L \), respectively, are chosen such that \( BK = CL = AG \). Prove that \[ \angle BKF + \angle CLE = \angle BGC. \] [i]Proposed by Vadym Solomka[/i]

Prove that the inequality $ r^2+r_a^2+r_b^2+ r_c^2 \ge 2S$ holds for an arbitrary triangle, where $r$ is the radius of the circle inscribed in the triangle, $r_a$, $r_b$, $r_c$ are the radii of its three excribed circles, $S$ is the area of the triangle.

We removed the middle square of $2 \times 2$ from the $8 \times 8$ board. a) How many checkers can be placed on the remaining $60$ boxes so that there are no two not jeopardize? b) How many at least checkers can be placed on the board so that they are at risk all $60$ squares? (A lady is threatening the box she stands on, as well as any box she can get to in one move without going over any of the four removed boxes.)

Given a convex quadrilateral $ABCD$. Let $O$ be the intersection point of diagonals $AC$ and $BD$ and let $I , K , H$ be feet of perpendiculars from $B , O , C$ to $AD$, respectively. Prove that $AD \times BI \times CH \le AC \times BD \times OK$.

Let $ABCD$ be a trapezoid ($AB<CD,AB\parallel CD$) and $P\equiv AD\cap BC$. Suppose that $Q$ be a point inside $ABCD$ such that $\angle QAB=\angle QDC=90-\angle BQC$. Prove that $\angle PQA=2\angle QCD$.

Suppose that $A$ and $B$ are sets of real numbers such that $$A\subset B+\alpha \mathbb{Z}\quad \text{and}\quad B\subset A+\alpha\mathbb{Z}\quad \text{for all}\quad \alpha>0$$ (where $X+\alpha\mathbb=\{x+\alpha n|x\in\mathbb{X}, n\in\mathbb{Z}\}$ (a) Does it follow that $A=B$ (b) The same question, with the assumption that $B$ is bounded

Let $a,b, c$ be $3$ distinct numbers from $\{1, 2,3, 4, 5, 6\}$ Show that $7$ divides $abc + (7 - a)(7 - b)(7 - c)$

In the diagram, congruent rectangles $ABCD$ and $DEFG$ have a common vertex $D$. Sides $BC$ and $EF$ meet at $H$. Given that $DA = DE = 8$, $AB = EF = 12$, and $BH = 7$. Find the area of $ABHED$. [img]https://cdn.artofproblemsolving.com/attachments/f/b/7225fa89097e7b20ea246b3aa920d2464080a5.png[/img]

For a given cyclic quadrilateral $ABCD$, let $I$ be a variable point on the diagonal $AC$ such that $I$ and $A$ are on the same side of the diagonal $BD$. Assume $E,F$ lie on the diagonal $BD$ such that $IE\parallel AB$ and $IF\parallel AD$. Show that $\angle BIE =\angle DCF $

Let $n\geq 2$ be an integer and let $f$ be a $4n$-variable polynomial with real coefficients. Assume that, for any $2n$ points $(x_1,y_1),\dots,(x_{2n},y_{2n})$ in the Cartesian plane, $f(x_1,y_1,\dots,x_{2n},y_{2n})=0$ if and only if the points form the vertices of a regular $2n$-gon in some order, or are all equal. Determine the smallest possible degree of $f$. (Note, for example, that the degree of the polynomial $$g(x,y)=4x^3y^4+yx+x-2$$ is $7$ because $7=3+4$.) [i]Ankan Bhattacharya[/i]

Evaluate $ \int_{0}^{\pi}\sqrt{(\cos x\plus{}\cos 2x\plus{}\cos 3x)^{2}\plus{}(\sin x\plus{}\sin 2x\plus{}\sin 3x)^{2}}\ dx$.

Let $n\ge2$ be an integer. Prove that if $k^2+k+n$ is prime for all integers $k$ such that $0\le k\le\sqrt{n\over3}$, then $k^2+k+n$ is prime for all integers $k$ such that $0\le k\le n-2$.[i](IMO Problem 6)[/i] [b][i]Original Formulation[/i][/b] Let $f(x) = x^2 + x + p$, $p \in \mathbb N.$ Prove that if the numbers $f(0), f(1), \cdots , f(\sqrt{p\over 3} )$ are primes, then all the numbers $f(0), f(1), \cdots , f(p - 2)$ are primes. [i]Proposed by Soviet Union. [/i]

In an acute-angled triangle $ABC$, let $D$ be a point on the segment $BC$. Let $R$ and $S$ be the feet of the perpendiculars from $D$ to $AC$ and $AB$, respectively. The line $DR$ intersects the circumcircle of $BDS$ at $X$, with $X \neq D$. Similarly, the line $DS$ intersects the circumcircle of $CDR$ at $Y$, with $Y \neq D$. Prove that if $XY$ is parallel to $RS$, then $D$ is the midpoint of $BC$.