On the sides $BC$ and $AC$ of the isosceles triangle $ABC$ ($AB = BC$), points $E$ and $D$ are marked, respectively, so that $DE \parallel AB$. On the extendsion of side $CB$ beyond the point $B$, point $K$ was arbitrarily marked. Let $P$ be the intersection point of the lines $AB$ and $KD$. Let $Q$ be the intersection point of the lines $AK$ and $DE$. Prove that $CA$ is the bisector of angle $\angle PCQ$.

How many ways are there to color the vertices of a cube red, blue, or green such that no edge connects two vertices of the same color? Rotations and reflections are considered distinct colorings.

Find all possible values of integer $n > 3$ such that there is a convex $n$-gon in which, each diagonal is the perpendicular bisector of at least one other diagonal. Proposed by Mahdi Etesamifard

[u]Round 5[/u] [i]Each of the three problems in this round depends on the answer to two of the other problems. There is only one set of correct answers to these problems; however, each problem will be scored independently, regardless of whether the answers to the other problems are correct. [/i] [b]p13.[/b] Let $B$ be the answer to problem $14$, and let $C$ be the answer to problem $15$. A quadratic function $f(x)$ has two real roots that sum to $2^{10} + 4$. After translating the graph of $f(x)$ left by $B$ units and down by $C$ units, the new quadratic function also has two real roots. Find the sum of the two real roots of the new quadratic function. [b]p14.[/b] Let $A$ be the answer to problem $13$, and let $C$ be the answer to problem $15$. In the interior of angle $\angle NOM = 45^o$, there is a point $P$ such that $\angle MOP = A^o$ and $OP = C$. Let $X$ and $Y$ be the reflections of $P$ over $MO$ and $NO$, respectively. Find $(XY)^2$. [b]p15.[/b] Let $A$ be the answer to problem $13$, and let $B$ be the answer to problem $14$. Totoro hides a guava at point $X$ in a flat field and a mango at point $Y$ different from $X$ such that the length $XY$ is $B$. He wants to hide a papaya at point $Z$ such that $Y Z$ has length $A$ and the distance $ZX$ is a nonnegative integer. In how many different locations can he hide the papaya? [u]Round 6[/u] [b]p16.[/b] Let $ABCD$ be a trapezoid such that $AB$ is parallel to $CD$, $AB = 4$, $CD = 8$, $BC = 5$, and $AD = 6$. Given that point $E$ is on segment $CD$ and that $AE$ is parallel to $BC$, find the ratio between the area of trapezoid $ABCD$ and the area of triangle $ABE$. [b]p17.[/b] Find the maximum possible value of the greatest common divisor of $\overline{MOO}$ and $\overline{MOOSE}$, given that $S$, $O$, $M$, and $E$ are some nonzero digits. (The digits $S$, $O$, $M$, and $E$ are not necessarily pairwise distinct.) [b]p18.[/b] Suppose that $125$ politicians sit around a conference table. Each politician either always tells the truth or always lies. (Statements of a liar are never completely true, but can be partially true.) Each politician now claims that the two people beside them are both liars. Suppose that the greatest possible number of liars is $M$ and that the least possible number of liars is $N$. Determine the ordered pair $(M,N)$. [u]Round 7[/u] [b]p19.[/b] Define a [i]lucky [/i] number as a number that only contains $4$s and $7$s in its decimal representation. Find the sum of all three-digit lucky numbers. [b]p20.[/b] Let line segment $AB$ have length $25$ and let points $C$ and $D$ lie on the same side of line $AB$ such that $AC = 15$, $AD = 24$, $BC = 20$, and $BD = 7$. Given that rays $AC$ and $BD$ intersect at point $E$, compute $EA + EB$. [b]p21.[/b] A $3\times 3$ grid is filled with positive integers and has the property that each integer divides both the integer directly above it and directly to the right of it. Given that the number in the top-right corner is $30$, how many distinct grids are possible? [u]Round 8[/u] [b]p22.[/b] Define a sequence of positive integers $s_1, s_2, ... , s_{10}$ to be [i]terrible [/i] if the following conditions are satisfied for any pair of positive integers $i$ and $j$ satisfying $1 \le i < j \le 10$: $\bullet$ $s_i > s_j $ $\bullet$ $j - i + 1$ divides the quantity $s_i + s_{i+1} + ... + s_j$ Determine the minimum possible value of $s_1 + s_2 + ...+ s_{10}$ over all terrible sequences. [b]p23.[/b] The four points $(x, y)$ that satisfy $x = y^2 - 37$ and $y = x^2 - 37$ form a convex quadrilateral in the coordinate plane. Given that the diagonals of this quadrilateral intersect at point $P$, find the coordinates of $P$ as an ordered pair. [b]p24.[/b] Consider a non-empty set of segments of length $1$ in the plane which do not intersect except at their endpoints. (In other words, if point $P$ lies on distinct segments $a$ and $b$, then $P$ is an endpoint of both $a$ and $b$.) This set is called $3$-[i]amazing [/i] if each endpoint of a segment is the endpoint of exactly three segments in the set. Find the smallest possible size of a $3$-amazing set of segments. PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h2934024p26255963]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $\Gamma$ be a circle of center $S$ and radius $r$ and let be $A$ a point outside the circle. Let $BC$ be a diameter of $\Gamma$ such that $B$ does not belong to the line $AS$ and consider the point $O$ where the perpendicular bisectors of triangle $ABC$ intersect, that is, the circumcenter of $ABC$. Determine all possible locations of point $O$ when $B$ varies in circle $\Gamma$.

In a triangle $ABC$ points $A_1,B_1,C_1$ are the midpoints of $BC,CA,AB$ respectively,and $A_2,B_2,C_2$ are the midpoints of the altitudes from $A,B,C$ respectively. Show that the lines $A_1A_2,B_1B_2,C_1,C_2$ are concurrent.

Let $ABC$ be an isosceles triangle ($AB=AC$). An arbitrary point $M$ is chosen on the extension of the $BC$ beyond point $B$. Prove that the sum of the radius of the circle inscribed in the triangle $AM​​B$ and the radius of the circle tangent to the side $AC$ and the extensions of the sides $AM, CM$ of the triangle $AMC$ does not depend on the choice of point $M$.

[b]p1[/b]. Points $P_1,P_2,P_3,... ,P_n$ lie on a plane such that $P_aP_b = 1$,$P_cP_d = 2$, and $P_eP_f = 2018$ for not necessarily distinct indices $a,b,c,d,e, f \in \{1, 2,... ,n\}$. Find the minimum possible value of $n$. [b]p2.[/b] Find the coefficient of the $x^2y^4$ term in the expansion of $(3x +2y)^6$. [b]p3.[/b] Find the number of positive integers $n < 1000$ such that $n$ is a multiple of $27$ and the digit sum of $n$ is a multiple of $11$. [b]p4.[/b] How many times do the minute hand and hour hand of a $ 12$-hour analog clock overlap in a $366$-day leap year? [b]p5.[/b] Find the number of ordered triples of integers $(a,b,c)$ such that $(a +b)(b +c)(c + a) = 2018$. [b]p6.[/b] Let $S$ denote the set of the first $2018$ positive integers. Call the score of a subset the sum of its maximal element and its minimal element. Find the sum of score $(x)$ over all subsets $s \in S$ [b]p7.[/b] How many ordered pairs of integers $(a,b)$ exist such that $1 \le a,b \le 20$ and $a^a$ divides $b^b$? [b]p8.[/b] Let $f$ be a function such that for every non-negative integer $p$, $f (p)$ equals the number of ordered pairs of positive integers $(a,n)$ such that $a^n = a^p \cdot n$. Find $\sum^{2018}_{p=0}f (p)$. [b]p9.[/b] A point $P$ is randomly chosen inside a regular octagon $A_1A_2A_3A_4A_5A_6A_7A_8$. What is the probability that the projections of $P$ onto the lines $\overleftrightarrow{A_i A_{i+1}}$ for $i = 1,2,... ,8$ lie on the segments $\overline{A_iA_{i+1}}$ for $i = 1,2,... ,8$ (where indices are taken $mod \,\, 8$)? [b]p10. [/b]A person keeps flipping an unfair coin until it flips $3$ tails in a row. The probability of it landing on heads is $\frac23$ and the probability it lands on tails is $\frac13$ . What is the expected value of the number of the times the coin flips? PS. You had better use hide for answers.

Vitya cut the chessboard along the borders of the cells into pieces of the same perimeter. It turned out that not all of the received parts are equal. What is the largest possible number of parts that Vitya could get?

Let $ABC$ be a triangle with $AB = AC$. Suppose that the bisector of $\angle ABC$ meets the side $AC$ at point $D$ such that $BC = BD+AD$. Find the measure of $\angle BAC$.

In a triangle, the edges are extended past both vertices by the length of the edge opposite to the respective vertex. Show that the area of the resulting hexagon is at least $13$ times the area of the original triangle.

A circle touches the sides $AB,BC, CD,DA$ of a square at points $K,L,M,N$ respectively, and $BU, KV$ are parallel lines such that $U$ is on $DM$ and $V$ on $DN$. Prove that $UV$ touches the circle.

(i) Find the angles of $\triangle ABC$ if the length of the altitude through $B$ is equal to the length of the median through $C$ and the length of the altitude through $C$ is equal to the length of the median through $B$. (ii) Find all possible values of $\angle ABC$ of $\triangle ABC$ if the length of the altitude through $A$ is equal to the length of the median through $C$ and the length of the altitude through $C$ is equal to the length of the median through $B$.

Let $P$ be a point in the interior of the equilateral triangle $\triangle ABC$ such that $\sphericalangle{APC}=120^\circ$. Let $M$ be the intersection of $CP$ with $AB$, and $N$ the intersection of $AP$ and $BC$. Find the locus of the circumcentre of the triangle $MBN$ as $P$ varies.

A triangle with side lengths in the ratio $ 3: 4: 5$ is inscribed in a circle of radius $ 3$. What is the area of the triangle? $ \textbf{(A)}\ 8.64 \qquad \textbf{(B)}\ 12 \qquad \textbf{(C)}\ 5\pi \qquad \textbf{(D)}\ 17.28 \qquad \textbf{(E)}\ 18$

The sides of the triangle $ABC$ are extended in both directions and on these extensions $6$ equal segments $AA_1 , AA_2, BB_1,BB_2, CC_1, CC_2$ are drawn (fig.). It turned out that all $6$ points $A_1,A_2,B_1,B_2,C_1, C_2$ lie on the same circle, is $\vartriangle ABC$ necessarily equilateral? (Bogdan Rublev) [img]https://cdn.artofproblemsolving.com/attachments/0/3/a499f6e6d978ce63d2ab40460dc73b62882863.png[/img]

Given some square carpets with the total area $4$. Prove that they can fully cover the unit square.

Let $\mathcal{V}_c$ denote the infinite parallel ruler with the parallel edges being at distance $c$ from each other. The following construction steps are allowed using ruler $\mathcal V_c$: [list] [*] the line through two given points; [*] line $\ell'$ parallel to a given line $\ell $at distance $c$ (there are two such lines, both of which can be constructed using this step); [*] for given points $A$ and $B$ with $|AB|\ge c$ two parallel lines at distance $c$ such that one of them passes through $A$, and the other one passes through $B$ (if $|AB|>c$, there exists two such pairs of parallel lines, and both can be constructed using this step). [/list] On the perimeter of a circular piece of paper three points are given that form a scalene triangle. Let $n$ be a given positive integer. Prove that based on the three points and $n$ there exists $C>0$ such that for any $0<c\le C$ it is possible to construct $n$ points using only $\mathcal V_c$ on one of the excircles of the triangle. [i]We are not allowed to draw anything outside our circular paper. We can construct on the boundary of the paper; it is allowed to take the intersection point of a line with the boundary of the paper.[/i] [i]Proposed by Áron Bán-Szabó[/i]

In the figure below, parallelograms $ABCD$ and $BFEC$ have areas $1234$ cm$^2$ and $2804$ cm$^2$, respectively. Points $M$ and $N$ are chosen on sides $AD$ and $FE$, respectively, so that segment $MN$ passes through $B$. Find the area of $\vartriangle MNC$. [img]https://cdn.artofproblemsolving.com/attachments/b/6/8b57b632191bdb3a27ab7c59e2376dab23950b.png[/img]

Let $M$ be a point in the interior of triangle $ABC$. Let $A'$ lie on $BC$ with $MA'$ perpendicular to $BC$. Define $B'$ on $CA$ and $C'$ on $AB$ similarly. Define \[ p(M) = \frac{MA' \cdot MB' \cdot MC'}{MA \cdot MB \cdot MC}. \] Determine, with proof, the location of $M$ such that $p(M)$ is maximal. Let $\mu(ABC)$ denote this maximum value. For which triangles $ABC$ is the value of $\mu(ABC)$ maximal?

On the sides $AB$ and $AC$ of the triangle $ABC$, the points $P$ and $Q$ are chosen, respectively, so that $PQ\parallel BC$. Segments $BQ$ and $CP$ intersect at point $O$. Point $A'$ is symmetric to point $A$ relative to line $BC$. The segment $A'O$ intersects the circumcircle $w$ of the triangle $APQ$ at the point $S$. Prove that circumcircle of $BSC$ is tangent to the circle $w$.

Let $P_1$, $P_2$, $P_3$ be pairwise distinct parabolas in the plane. Find the maximum possible number of intersections between two or more of the $P_i$. In other words, find the maximum number of points that can lie on two or more of the parabolas $P_1$, $P_2$, $P_3$ .

For which of the following $ n$, $ n\times n$ chessboard cannot be covered using at most one unit square piece and many L-shaped pieces (an L-shaped piece is a 2x2 piece with one square removed)? $\textbf{(A)}\ 96 \qquad\textbf{(B)}\ 97 \qquad\textbf{(C)}\ 98 \qquad\textbf{(D)}\ 99 \qquad\textbf{(E)}\ 100$

In acute triangles $ABC$, $AD, BE ,CF$ are altitudes, with $D, E, F$ on the sides $BC, CA, AB$, respectively. Prove that $$DE + DF \le BC$$

[hide=D stands for Dedekind, Z stands for Zermelo]they had two problem sets under those two names[/hide] [b]D1.[/b] A Giant Hopper is $200$ meters away from you. It can hop $50$ meters. How many hops would it take for it to reach you? [b]D2.[/b] A rope of length $6$ is used to form the edges of an equilateral triangle (a triangle with equal side lengths). What is the length of one of these edges? [b]D3 / Z1.[/b] Point $E$ is on side $AB$ of rectangle $ABCD$. Find the area of triangle $ECD$ divided by the area of rectangle $ABCD$. [b]D4 / Z2.[/b] Garb and Grunt have two rectangular pastures of area $30$. Garb notices that his has a side length of $3$, while Grunt’s has a side length of $5$. What’s the positive difference between the perimeters of their pastures? [b]D5.[/b] Let points $A$ and $B$ be on a circle with radius $6$ and center $O$. If $\angle AOB = 90^o$, find the area of triangle $AOB$. [b]D6 / Z3.[/b] A scalene triangle (the $3$ side lengths are all different) has integer angle measures (in degrees). What is the largest possible difference between two angles in the triangle? [b]D7.[/b] Square $ABCD$ has side length $6$. If triangle $ABE$ has area $9$, find the sum of all possible values of the distance from $E$ to line $CD$. [b]D8 / Z4.[/b] Let point $E$ be on side $\overline{AB}$ of square $ABCD$ with side length $2$. Given $DE = BC+BE$, find $BE$. [b]Z5.[/b] The two diagonals of rectangle $ABCD$ meet at point $E$. If $\angle AEB = 2\angle BEC$, and $BC = 1$, find the area of rectangle $ABCD$. [b]Z6.[/b] In $\vartriangle ABC$, let $D$ be the foot of the altitude from $A$ to $BC$. Additionally, let $X$ be the intersection of the angle bisector of $\angle ACB$ and $AD$. If $BD = AC = 2AX = 6$, find the area of $ABC$. [b]Z7.[/b] Let $\vartriangle ABC$ have $\angle ABC = 40^o$. Let $D$ and $E$ be on $\overline{AB}$ and $\overline{AC}$ respectively such that DE is parallel to $\overline{BC}$, and the circle passing through points $D$, $E$, and $C$ is tangent to $\overline{AB}$. If the center of the circle is $O$, find $\angle DOE$. [b]Z8.[/b] Consider $\vartriangle ABC$ with $AB = 3$, $BC = 4$, and $AC = 5$. Let $D$ be a point of $AC$ other than $A$ for which $BD = 3$, and $E$ be a point on $BC$ such that $\angle BDE = 90^o$. Find $EC$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].