It is given a tetrahedron $ABCD$ and a plane $\alpha$ intersecting the three edges passing through $D$. Prove that $\alpha$ divides the surface of the tetrahedron into two parts proportional to the volumes of the bodies formed if and only if $\alpha$ is passing through the center of the inscribed tetrahedron sphere.

An infinite increasing arithmetical progression consists of positive integers and contains a perfect cube. Prove that this progression also contains a term which is a perfect cube but not a perfect square.

Let $M$ be a point in the triangle $ABC$ such that \[\text{area}(ABM)=2 \cdot \text{area}(ACM)\] Show that the locus of all such points is a straight line.

Let $ABCD$ be a rectangle of sides $AB = 4$ and $BC = 3$. The perpendicular on the diagonal $BD$ drawn from $A$ cuts $BD$ at point $H$. We call $M$ the midpoint of $BH$ and $N$ the midpoint of $CD$. Calculate the measure of the segment $MN$.

Let $\Gamma$ be a circle and let $d$ be a line such that $\Gamma$ and $d$ have no common points. Further, let $AB$ be a diameter of the circle $\Gamma$; assume that this diameter $AB$ is perpendicular to the line $d$, and the point $B$ is nearer to the line $d$ than the point $A$. Let $C$ be an arbitrary point on the circle $\Gamma$, different from the points $A$ and $B$. Let $D$ be the point of intersection of the lines $AC$ and $d$. One of the two tangents from the point $D$ to the circle $\Gamma$ touches this circle $\Gamma$ at a point $E$; hereby, we assume that the points $B$ and $E$ lie in the same halfplane with respect to the line $AC$. Denote by $F$ the point of intersection of the lines $BE$ and $d$. Let the line $AF$ intersect the circle $\Gamma$ at a point $G$, different from $A$. Prove that the reflection of the point $G$ in the line $AB$ lies on the line $CF$.

[u]Round 6[/u] [b]p16.[/b] Let $a_1, a_2, ... , a_{2011}$ be a sequence of numbers such that $a_1 = 2011$ and $a_1+a_2+...+a_n = n^2 \cdot a_n$ for $n = 1, 2, ... 2011$. (That is, $a_1 = 1^2\cdot a_1$, $a_1 + a_2 = 2^2 \cdot a_2$, $...$) Compute $a_{2011}$. [b]p17.[/b] Three rectangles, with dimensions $3 \times 5$, $4 \times 2$, and $6 \times 4$, are each divided into unit squares which are alternately colored black and white like a checkerboard. Each rectangle is cut along one of its diagonals into two triangles. For each triangle, let m be the total black area and n the total white area. Find the maximum value of $|m - n|$ for the $6$ triangles. [b]p18.[/b] In triangle $ABC$, $\angle BAC = 90^o$, and the length of segment $AB$ is $2011$. Let $M$ be the midpoint of $BC$ and $D$ the midpoint of $AM$. Let $E$ be the point on segment $AB$ such that $EM \parallel CD$. What is the length of segment $BE$? [u]Round 7[/u] [b]p19.[/b] How many integers from $1$ to $100$, inclusive, can be expressed as the difference of two perfect squares? (For example, $3 = 2^2 - 1^2$). [b]p20.[/b] In triangle $ABC$, $\angle ABC = 45$ and $\angle ACB = 60^o$. Let $P$ and $Q$ be points on segment $BC$, $F$ a point on segment $AB$, and $E$ a point on segment $AC$ such that $F Q \parallel AC$ and $EP \parallel AB$. Let $D$ be the foot of the altitude from $A$ to $BC$. The lines $AD$, $F Q$, and $P E$ form a triangle. Find the positive difference, in degrees, between the largest and smallest angles of this triangle. [b]p21.[/b] For real number $x$, $\lceil x \rceil$ is equal to the smallest integer larger than or equal to $x$. For example, $\lceil 3 \rceil = 3$ and $\lceil 2.5 \rceil = 3$. Let $f(n)$ be a function such that $f(n) = \left\lceil \frac{n}{2}\right\rceil + f\left( \left\lceil \frac{n}{2}\right\rceil\right)$ for every integer $n$ greater than $1$. If $f(1) = 1$, find the maximum value of $f(k) - k$, where $k$ is a positive integer less than or equal to $2011$. [u]Round 8[/u] The answer to each of the three questions in this round depends on the answer to one of the other questions. There is only one set of correct answers to these problems; however, each question will be scored independently, regardless of whether the answers to the other questions are correct. [b]p22.[/b] Let $W$ be the answer to problem 24 in this guts round. Let $f(a) = \frac{1}{1 -\frac{1}{1- \frac{1}{a}}}$. Determine$|f(2) + ... + f(W)|$. [b]p23.[/b] Let $X$ be the answer to problem $22$ in this guts round. How many odd perfect squares are less than $8X$? [b]p24.[/b] Let $Y$ be the answer to problem $23$ in this guts round. What is the maximum number of points of intersections of two regular $(Y - 5)$-sided polygons, if no side of the first polygon is parallel to any side of the second polygon? [u]Round 9[/u] [b]p25.[/b] Cross country skiers $s_1, s_2, s_3, ..., s_7$ start a race one by one in that order. While each skier skis at a constant pace, the skiers do not all ski at the same rate. In the course of the race, each skier either overtakes another skier or is overtaken by another skier exactly two times. Find all the possible orders in which they can finish. Write each possible finish as an ordered septuplet $(a, b, c, d, e, f, g)$ where $a, b, c, d, e, f, g$ are the numbers $1-7$ in some order. (So a finishes first, b finishes second, etc.) [b]p26.[/b] Archie the Alchemist is making a list of all the elements in the world, and the proportion of earth, air, fire, and water needed to produce each. He writes the proportions in the form E:A:F:W. If each of the letters represents a whole number from $0$ to $4$, inclusive, how many different elements can Archie list? Note that if Archie lists wood as $2:0:1:2$, then $4:0:2:4$ would also produce wood. In addition, $0:0:0:0$ does not produce an element. [b]p27.[/b] Let $ABCD$ be a rectangle with $AB = 10$ and $BC = 12$. Let $M$ be the midpoint of $CD$, and $P$ be the point on $BM$ such that $DP = DA$. Find the area of quadrilateral $ABPD$. [u]Round 10[/u] [b]p28.[/b] David the farmer has an infinitely large grass-covered field which contains a straight wall. He ties his cow to the wall with a rope of integer length. The point where David ties his rope to the wall divides the wall into two parts of length $a$ and $b$, where $a > b$ and both are integers. The rope is shorter than the wall but is longer than $a$. Suppose that the cow can reach grass covering an area of $\frac{165\pi}{2}$. Find the ratio $\frac{a}{b}$ . You may assume that the wall has $0$ width. [b]p29.[/b] Let $S$ be the number of ordered quintuples $(a, b, x, y, n)$ of positive integers such that $$\frac{a}{x}+\frac{b}{y}=\frac{1}{n}$$ $$abn = 2011^{2011}$$ Compute the remainder when $S$ is divided by $2012$. [b]p30.[/b] Let $n$ be a positive integer. An $n \times n$ square grid is formed by $n^2$ unit squares. Each unit square is then colored either red or blue such that each row or column has exactly $10$ blue squares. A move consists of choosing a row or a column, and recolor each unit square in the chosen row or column – if it is red, we recolor it blue, and if it is blue, we recolor it red. Suppose that it is possible to obtain fewer than $10n$ blue squares after a sequence of finite number of moves. Find the maximum possible value of $n$. PS. You should use hide for answers. First rounds have been posted [url=https://artofproblemsolving.com/community/c4h2786905p24497746]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $\Gamma_1,\Gamma_2$ be two circles centred at the points $(a,0),(b,0);0<a<b$ and having radii $a,b$ respectively.Let $\Gamma$ be the circle touching $\Gamma_1$ externally and $\Gamma_2$ internally. Find the locus of the centre of of $\Gamma$

Point $P$ is taken in the interior of the triangle $ABC$, so that \[\angle PAB = \angle PCB = \dfrac {1} {4} (\angle A + \angle C).\] Let $L$ be the foot of the angle bisector of $\angle B$. The line $PL$ meets the circumcircle of $\triangle APC$ at point $Q$. Prove that $QB$ is the angle bisector of $\angle AQC$. [i]Proposed by S. Berlov[/i]

Given a regular polygon with $p$ sides, where $p$ is a prime number. After rotating this polygon about its center by an integer number of degrees it coincides with itself. What is the maximal possible number for $p$?

Find the largest value of the area of the projection of the cylinder onto the plane if its radius is $r$ and its height is $h$ (orthogonal projection).

$ABC$ is right triangle with right angle near vertex $B, M$ is the midpoint of $AC$. The square $BKLM$ is built on $BM$, such that segments $ML$ and $BC$ intersect. Segment $AL$ intersects $BC$ in point $E$. Prove that lines $AB,CL$ and$ KE$ intersect in one point.

Let $ABC$ be an isosceles triangle with vertex $A$ and $AB> BC$. Let $k$ be the circle with center $A$ passsing through $B$ and $C$. Let $H$ be the second intersection of $k$ with the altitude of the triangle $ABC$ through $B$. Further let $G$ be the second intersection of $k$ with the median through $B$ in triangle $ABC$. Let $X$ be the intersection of the lines $AC$ and $GH$. Show that $C$ is the midpoint of $AX$.

Let $ABCD$ be a parallelogram and let $P{}$ be a point inside it such that $\angle PDA= \angle PBA$. Let $\omega_1$ be the excircle of $PAB$ opposite to the vertex $A{}$. Let $\omega_2$ be the incircle of the triangle $PCD$. Prove that one of the common tangents of $\omega_1$ and $\omega_2$ is parallel to $AD$. [i]Ivan Frolov[/i]

Let $ABC$ be a triangle with circumcircle $\Gamma$ and incenter $I$ and let $M$ be the midpoint of $\overline{BC}$. The points $D$, $E$, $F$ are selected on sides $\overline{BC}$, $\overline{CA}$, $\overline{AB}$ such that $\overline{ID} \perp \overline{BC}$, $\overline{IE}\perp \overline{AI}$, and $\overline{IF}\perp \overline{AI}$. Suppose that the circumcircle of $\triangle AEF$ intersects $\Gamma$ at a point $X$ other than $A$. Prove that lines $XD$ and $AM$ meet on $\Gamma$. [i]Proposed by Evan Chen, Taiwan[/i]

A cylinder is inscribed within a sphere of radius 10 such that its volume is [i]almost-half[/i] that of the sphere. If [i]almost-half[/i] is defined such that the cylinder has volume $\frac12+\frac{1}{250}$ times the sphere’s volume, find the sum of all possible heights for the cylinder.

Consider the solid with 4 triangles and 4 regular hexagons as faces, where each triangle borders 3 hexagons, and all the sides are of length 1. Compute the [i]square[/i] of the volume of the solid. Express your result in reduced fraction and concatenate the numerator with the denominator (e.g., if you think that the square is $\frac{1734}{274}$, then you would submit 1734274).

[b]p1.[/b] Ana, Bob, Cho, Dan, and Eve want to use a microwave. In order to be fair, they choose a random order to heat their food in (all orders have equal probability). Ana's food needs $5$ minutes to cook, Bob's food needs $7$ minutes, Cho's needs $1$ minute, Dan's needs $12$ minutes, and Eve's needs $5$ minutes. What is the expected number of minutes Bob has to wait for his food to be done? [b]p2.[/b] $ABC$ is an equilateral triangle. $H$ lies in the interior of $ABC$, and points $X$, $Y$, $Z$ lie on sides $AB, BC, CA$, respectively, such that $HX\perp AB$, $HY \perp BC$, $HZ\perp CA$. Furthermore, $HX =2$, $HY = 3$, $HZ = 4$. Find the area of triangle $ABC$. [b]p3.[/b] Amy, Ben, and Chime play a dice game. They each take turns rolling a die such that the $first$ person to roll one of his favorite numbers wins. Amy's favorite number is $1$, Ben's favorite numbers are $2$ and $3$, and Chime's are $4$, $5$, and $6$. Amy rolls first, Ben rolls second, and Chime rolls third. If no one has won after Chime's turn, they repeat the sequence until someone has won. What's the probability that Chime wins the game? [b]p4.[/b] A point $P$ is chosen randomly in the interior of a square $ABCD$. What is the probability that the angle $\angle APB$ is obtuse? [b]p5.[/b] Let $ABCD$ be the quadrilateral with vertices $A = (3, 9)$, $B = (1, 1)$, $C = (5, 3)$, and $D = (a, b)$, all of which lie in the first quadrant. Let $M$ be the midpoint of $AB$, $N$ the midpoint of $BC$, $O$ the midpoint of $CD$, and $P$ the midpoint of $AD$. If $MNOP$ is a square, find $(a, b)$. [b]p6.[/b] Let $M$ be the number of positive perfect cubes that divide $60^{60}$. What is the prime factorization of $M$? [b]p7.[/b] Given that $x$, $y$, and $z$ are complex numbers with $|x|=|y| =|z|= 1$, $x + y + z = 1$ and $xyz = 1$, find $|(x + 2)(y + 2)(z + 2)|$. [b]p8.[/b] If $f(x)$ is a polynomial of degree $2008$ such that $f(m) = \frac{1}{m}$ for $m = 1, 2, ..., 2009$, find $f(2010)$. [b]p9.[/b] A drunkard is randomly walking through a city when he stumbles upon a $2 \times 2$ sliding tile puzzle. The puzzle consists of a $2 \times 2$ grid filled with a blank square, as well as $3$ square tiles, labeled $1$, $2$, and $3$. During each turn you may fill the empty square by sliding one of the adjacent tiles into it. The following image shows the puzzle's correct state, as well as two possible moves you can make: [img]https://cdn.artofproblemsolving.com/attachments/c/6/7ddd9305885523deeee2a530dc90505875d1cc.png[/img] Assuming that the puzzle is initially in an incorrect (but solvable) state, and that the drunkard will make completely random moves to try and solve it, how many moves is he expected to make before he restores the puzzle to its correct state? [b]p10.[/b] How many polynomials $p(x)$ exist such that the coeffients of $p(x)$ are a rearrangement of $\{0, 1, 2, .., deg \, p(x)\}$ and all of the roots of $p(x)$ are rational? (Note that the leading coefficient of $p(x)$ must be nonzero.) PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ABCD$ be a quadrilateral with $AD = 20$ and $BC = 13$. The area of $\triangle ABC$ is $338$ and the area of $\triangle DBC$ is $212$. Compute the smallest possible perimeter of $ABCD$. [i]Proposed by Evan Chen[/i]

Let $ABCD$ be a convex quadrilateral with $\angle ABC>90$, $CDA>90$ and $\angle DAB=\angle BCD$. Denote by $E$ and $F$ the reflections of $A$ in lines $BC$ and $CD$, respectively. Suppose that the segments $AE$ and $AF$ meet the line $BD$ at $K$ and $L$, respectively. Prove that the circumcircles of triangles $BEK$ and $DFL$ are tangent to each other. $\emph{Slovakia}$

In parallelogram $ABCD$ with acute angle $A$ a point $N$ is chosen on the segment $AD$, and a point $M$ on the segment $CN$ so that $AB = BM = CM$. Point $K$ is the reflection of $N$ in line $MD$. The line $MK$ meets the segment $AD$ at point $L$. Let $P$ be the common point of the circumcircles of $AMD$ and $CNK$ such that $A$ and $P$ share the same side of the line $MK$. Prove that $\angle CPM = \angle DPL$.

Let $ \Gamma(I,r)$ and $ \Gamma(O,R)$ denote the incircle and circumcircle, respectively, of a triangle $ ABC$. Consider all the triangels $ A_iB_iC_i$ which are simultaneously inscribed in $ \Gamma(O,R)$ and circumscribed to $ \Gamma(I,r)$. Prove that the centroids of these triangles are concyclic.

Prove that this triangle cut out of paper can be folded so that the surface of a regular unit tetradragon (i.e., a triangular pyramid, all edges of which are equal to $1$) is obtained if: a) this triangle is isosceles, the lateral sides are equal to $2$ , the angle between them is $120^o$, b) two sides of this triangle are equal to $2$ and $2\sqrt3$, the angle between them is $150^o$.

A ray originating at point $P$ intersects a circle with center $O$ at points $A$ and $B$, with $PB > PA$. Segment $\overline{OP}$ intersects the circle at point $C$. Given that $PA = 31$, $PC = 17$, and $\angle PBO = 60^o$, find the radius of the circle.

Mari and Juri ordered a round pizza. Juri cut the pizza into four pieces by two straight cuts, none of which passed through the centre point of the pizza. Mari can choose two pieces not aside of these four, and Juri gets the rest two pieces. Prove that if Mari chooses the piece that covers the centre point of the pizza, she will get more pizza than Juri.

In a quadrilateral $ABCD$, the diagonals are perpendicular and intersect at the point $S$. Let $K, L, M$, and $N$ be points symmetric to $S$ with respect to the lines $AB, BC, CD$, and $DA$, respectively, $BN$ intersects the circumcircle of the triangle $SKN$ at point $E$, and $BM$ intersects circumscribed the circle of the triangle $SLM$ at the point $F$. Prove that the quadrilateral $EFLK$ is cyclic .