Let $ H $ denote the orthocenter of an acute triangle $ ABC, $ and $ A_1,A_2,A_3 $ denote the intersections of the altitudes of this triangle with its circumcircle, and $ A',B',C' $ denote the projections of the vertices of this triangle on their opposite sides. [b]a)[/b] Prove that the sides of the triangle $ A'B'C' $ are parallel to the sides of $ A_1B_1C_1. $ [b]b)[/b] Show that $ B_1C_1\cdot\overrightarrow{HA_1} +C_1A_1\cdot\overrightarrow{HB_1} +A_1B_1\cdot\overrightarrow{HC_1} =0. $ [i]Geoghe Duță[/i]

Let $A_1A_2A_3\ldots A_n$ be a regular $n$-gon. Let $B_1$ and $B_{n-1}$ be the midpoints of its sides $A_1A_2$ and $A_{n-1}A_n$. Also, for every $i\in\left\{2,3,4,\ldots ,n-2\right\}$. Let $S$ be the point of intersection of the lines $A_1A_{i+1}$ and $A_nA_i$, and let $B_i$ be the point of intersection of the angle bisector bisector of the angle $\measuredangle A_iSA_{i+1}$ with the segment $A_iA_{i+1}$. Prove that $\sum_{i=1}^{n-1} \measuredangle A_1B_iA_n=180^{\circ}$. [i]Proposed by Dusan Dukic, Serbia and Montenegro[/i]

Given a rectangular parallelepiped $ABCDA_1B_1C_1D_1$, which has $AD= DC = 3\sqrt2$ cm, and $DD_1 = 8$ cm. Through the diagonal $B_1D$ of the parallelepiped $m$ parallel to line $A_1C_1$ is drawn on the plane $\gamma$. a) Draw a section of a parallelepiped with plane $\gamma$. b) Justify what geometric figure is this section, and find its area. (Alexander Shkolny)

[b]p1.[/b] Find $20 \cdot 18 + 20 + 18 + 1$. [b]p2.[/b] Suzie’s Ice Cream has $10$ flavors of ice cream, $5$ types of cones, and $5$ toppings to choose from. An ice cream cone consists of one flavor, one cone, and one topping. How many ways are there for Sebastian to order an ice cream cone from Suzie’s? [b]p3.[/b] Let $a = 7$ and $b = 77$. Find $\frac{(2ab)^2}{(a+b)^2-(a-b)^2}$ . [b]p4.[/b] Sebastian invests $100,000$ dollars. On the first day, the value of his investment falls by $20$ percent. On the second day, it increases by $25$ percent. On the third day, it falls by $25$ percent. On the fourth day, it increases by $60$ percent. How many dollars is his investment worth by the end of the fourth day? [b]p5.[/b] Square $ABCD$ has side length $5$. Points $K,L,M,N$ are on segments $AB$,$BC$,$CD$,$DA$ respectively,such that $MC = CL = 2$ and $NA = AK = 1$. The area of trapezoid $KLMN$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Find $m + n$. [b]p6.[/b] Suppose that $p$ and $q$ are prime numbers. If $p + q = 30$, find the sum of all possible values of $pq$. [b]p7.[/b] Tori receives a $15 - 20 - 25$ right triangle. She cuts the triangle into two pieces along the altitude to the side of length $25$. What is the difference between the areas of the two pieces? [b]p8.[/b] The factorial of a positive integer $n$, denoted $n!$, is the product of all the positive integers less than or equal to $n$. For example, $1! = 1$ and $5! = 120$. Let $m!$ and $n!$ be the smallest and largest factorial ending in exactly $3$ zeroes, respectively. Find $m + n$. [b]p9.[/b] Sam is late to class, which is located at point $B$. He begins his walk at point $A$ and is only allowed to walk on the grid lines. He wants to get to his destination quickly; how many paths are there that minimize his walking distance? [img]https://cdn.artofproblemsolving.com/attachments/a/5/764e64ac315c950367357a1a8658b08abd635b.png[/img] [b]p10.[/b] Mr. Iyer owns a set of $6$ antique marbles, where $1$ is red, $2$ are yellow, and $3$ are blue. Unfortunately, he has randomly lost two of the marbles. His granddaughter starts drawing the remaining $4$ out of a bag without replacement. She draws a yellow marble, then the red marble. Suppose that the probability that the next marble she draws is blue is equal to $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positiveintegers. What is $m + n$? [b]p11.[/b] If $a$ is a positive integer, what is the largest integer that will always be a factor of $(a^3+1)(a^3+2)(a^3+3)$? [b]p12.[/b] What is the largest prime number that is a factor of $160,401$? [b]p13.[/b] For how many integers $m$ does the equation $x^2 + mx + 2018 = 0$ have no real solutions in $x$? [b]p14.[/b] What is the largest palindrome that can be expressed as the product of two two-digit numbers? A palindrome is a positive integer that has the same value when its digits are reversed. An example of a palindrome is $7887887$. [b]p15.[/b] In circle $\omega$ inscribe quadrilateral $ADBC$ such that $AB \perp CD$. Let $E$ be the intersection of diagonals $AB$ and $CD$, and suppose that $EC = 3$, $ED = 4$, and $EB = 2$. If the radius of $\omega$ is $r$, then $r^2 =\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Determine $m + n$. [b]p16.[/b] Suppose that $a, b, c$ are nonzero real numbers such that $2a^2 + 5b^2 + 45c^2 = 4ab + 6bc + 12ca$. Find the value of $\frac{9(a + b + c)^3}{5abc}$ . [b]p17.[/b] Call a positive integer n spicy if there exist n distinct integers $k_1, k_2, ... , k_n$ such that the following two conditions hold: $\bullet$ $|k_1| + |k_2| +... + |k_n| = n2$, $\bullet$ $k_1 + k_2 + ...+ k_n = 0$. Determine the number of spicy integers less than $10^6$. [b]p18.[/b] Consider the system of equations $$|x^2 - y^2 - 4x + 4y| = 4$$ $$|x^2 + y^2 - 4x - 4y| = 4.$$ Find the sum of all $x$ and $y$ that satisfy the system. [b]p19.[/b] Determine the number of $8$ letter sequences, consisting only of the letters $W,Q,N$, in which none of the sequences $WW$, $QQQ$, or $NNNN$ appear. For example, $WQQNNNQQ$ is a valid sequence, while $WWWQNQNQ$ is not. [b]p20.[/b] Triangle $\vartriangle ABC$ has $AB = 7$, $CA = 8$, and $BC = 9$. Let the reflections of $A,B,C$ over the orthocenter H be $A'$,$B'$,$C'$. The area of the intersection of triangles $ABC$ and $A'B'C'$ can be expressed in the form $\frac{a\sqrt{b}}{c}$ , where $b$ is squarefree and $a$ and $c$ are relatively prime. determine $a+b+c$. (The orthocenter of a triangle is the intersection of its three altitudes.) PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

In isosceles trapezoid $ABCD$ with $AB < CD$ and $BC = AD$, the angle bisectors of $\angle A$ and $\angle B$ intersect $CD$ at $E$ and $F$ respectively, and intersect each other outside the trapezoid at $G$. Given that $AD = 8$, $EF = 3$, and $EG = 4$, the area of $ABCD$ can be expressed as $\frac{a\sqrt{b}}{c}$ for positive integers $a, b$, and $c$, with $a$ and $c$ relatively prime and $b$ squarefree. Find $10000a +100b +c$.

Let $ABC$ be an acute, non-isosceles triangle with circumcircle $(O)$. $BE, CF$ are the heights of $\triangle ABC$, and $BE, CF$ intersect at $H$. Let $M$ be the midpoint of $AH$, and $K$ be the point on $EF$ such that $HK \perp EF$. A line not going through $A$ and parallel to $BC$ intersects the minor arc $AB$ and $AC$ of $(O)$ at $P$, $Q$, respectively. Show that the tangent line of $(CQE)$ at $E$, the tangent line of $(BPF)$ at $F$, and $MK$ concur.

Let $ABC$ be a triangle. Points $D$, $E$, and $F$ are respectively on the sides $\overline{BC}$, $\overline{CA}$, and $\overline{AB}$ of $\triangle ABC$. Suppose that \[ \frac{AE}{AC} = \frac{CD}{CB} = \frac{BF}{BA} = x \] for some $x$ with $\frac{1}{2} < x < 1$. Segments $\overline{AD}$, $\overline{BE}$, and $\overline{CF}$ cut the triangle into 7 nonoverlapping regions: 4 triangles and 3 quadrilaterals. The total area of the 4 triangles equals the total area of the 3 quadrilaterals. Compute the value of $x$.

Triangle $ ABC$ has fixed vertices $ B$ and $ C$, so that $ BC \equal{} 2$ and $ A$ is variable. Denote by $ H$ and $ G$ the orthocenter and the centroid, respectively, of triangle $ ABC$. Let $ F\in(HG)$ so that $ \frac {HF}{FG} \equal{} 3$. Find the locus of the point $ A$ so that $ F\in BC$.

Let $ABC$ be an acute, non-isosceles triangle with $AD$,$BE$, $CF$ are altitudes and $d$ is the tangent line of the circumcircle of triangle $ABC$ at $A$. The line through $H$ and parallel to $EF$ cuts $DE$, $DF$ at $Q, P$ respectively. Prove that $d$ is tangent to the ex-circle respect to vertex $D$ of triangle $DPQ$.

Suppose that $ ABCD$ is a convex quadrilateral. Let $ F \equal{} AB\cap CD$, $ E \equal{} AD\cap BC$ and $ T \equal{} AC\cap BD$. Suppose that $ A,B,T,E$ lie on a circle which intersects with $ EF$ at $ P$. Prove that if $ M$ is midpoint of $ AB$, then $ \angle APM \equal{} \angle BPT$.

Let $ABCD$ be a convex quadrilateral, such that $AB = CD$, $\angle BCD = 2 \angle BAD$, $\angle ABC = 2 \angle ADC$ and $\angle BAD \neq \angle ADC$. Determine the measure of the angle between the diagonals $AC$ and $BD$.

For every rational number $m>0$ we consider the function $f_m:\mathbb{R}\rightarrow\mathbb{R},f_m(x)=\frac{1}{m}x+m$. Denote by $G_m$ the graph of the function $f_m$. Let $p,q,r$ be positive rational numbers. a) Show that if $p$ and $q$ are distinct then $G_p\cap G_q$ is non-empty. b) Show that if $G_p\cap G_q$ is a point with integer coordinates, then $p$ and $q$ are integer numbers. c) Show that if $p,q,r$ are consecutive natural numbers, then the area of the triangle determined by intersections of $G_p,G_q$ and $G_r$ is equal to $1$.

Given a rectangle $ABCD$ with $AB = a$ and $BC = b$. Point $O$ is the intersection of the two diagonals. Extend the side of the $BA$ so that $AE = AO$, also extend the diagonal of $BD$ so that $BZ = BO.$ Assume that triangle $EZC$ is equilateral. Prove that (i) $b = a\sqrt3$ (ii) $EO$ is perpendicular to $ZD$

Bisector of angle $BAC$ of triangle $ABC$ intersects circumcircle of this triangle in point $D \neq A$. Points $K$ and $L$ are orthogonal projections on line $AD$ of points $B$ and $C$, respectively. Prove that $AD \ge BK + CL$.

Sphere $S$ with radius $100$ has diameter $\overline{AB}$ and center $C$. Four small spheres all with radius $17$ have centers that lie in a plane perpendicular to $\overline{AB}$ such that each of the four spheres is internally tangent to $S$ and externally tangent to two of the other small spheres. Find the radius of the smallest sphere that is both externally tangent to two of the four spheres with radius $17$ and internally tangent to $S$ at a point in the plane perpendicular to $\overline{AB}$ at $C$.

The points $K$ and $M$ are the centres of the $AB$ and $CD$ sides of the convex quadrangle $ABCD$. The points $L$ and $M$ belong to two other sides and $KLMN$ is a rectangle. Prove that $KLMN$ area is a half of $ABCD$ area.

You have three colors $\{\text{red}, \text{blue}, \text{green}\}$ with which you can color the faces of a regular octahedron (8 triangle sided polyhedron, which is two square based pyramids stuck together at their base), but you must do so in a way that avoids coloring adjacent pieces with the same color. How many different coloring schemes are possible? (Two coloring schemes are considered equivalent if one can be rotated to fit the other.)

$(BEL 3)$ Construct the circle that is tangent to three given circles.

Dinesh has several squares and regular pentagons, all with side length $ 1$. He wants to arrange the shapes alternately to form a closed loop (see diagram). How many pentagons would Dinesh need to do so? [img]https://cdn.artofproblemsolving.com/attachments/8/9/6345d7150298fe26cfcfba554656804ed25a6d.jpg[/img]

A convex quadrilateral $ABCD$ has perpendicular diagonals. The perpendicular bisectors of the sides $AB$ and $CD$ meet at a unique point $P$ inside $ABCD$. Prove that the quadrilateral $ABCD$ is cyclic if and only if triangles $ABP$ and $CDP$ have equal areas.

On the sides $AB$ and $AD$ of the square $ABCD$, the points $N$ and $P$ are selected respectively such that $NC=NP$. The point $Q$ is chosen on the segment $AN$ so that $\angle QPN = \angle NCB$. Prove that $2\angle BCQ = \angle AQP$.

Triangle $ABC$ of area $1$ is given. Point $A'$ lies on the extension of side $BC$ beyond point $C$ with $BC = CA'$. Point $B'$ lies on extension of side $CA$ beyond $A$ and $CA = AB'$. $C'$ lies on extension of $AB$ beyond $B$ with $AB = BC'$. Find the area of triangle $A'B'C'$.

Consider all pairs of distinct points on the Cartesian plane $(A, B)$ with integer coordinates. Among these pairs of points, find all for which there exist two distinct points $(X, Y)$ with integer coordinates, such that the quadrilateral $AXBY$ is convex and inscribed. [i]Proposed by Anton Trygub[/i]

Let be given a triangle $ ABC$ and its internal angle bisector $ BD$ $ (D\in BC)$. The line $ BD$ intersects the circumcircle $ \Omega$ of triangle $ ABC$ at $ B$ and $ E$. Circle $ \omega$ with diameter $ DE$ cuts $ \Omega$ again at $ F$. Prove that $ BF$ is the symmedian line of triangle $ ABC$.

In triangle $ABC$, $\angle ABC = 54^\circ$ and $\angle ACB = 42^\circ$. Point $D$ is the foot of the altitude from vertex $A$ to $BC$, and $I$ is the incenter of $\triangle ABC$. Point $K$ lies on line $AD$, such that $D$ is between $A$ and $K$ and $AK$ is equal to the diameter of the circumcircle of $\triangle ABC$. Find the measure of $\angle KID$.