Let $ABC$ be a triangle with all angles $\leq 120^{\circ}$. Let $F$ be the Fermat point of triangle $ABC$, that is, the interior point of $ABC$ such that $\angle AFB = \angle BFC = \angle CFA = 120^\circ$. For each one of the three triangles $BFC$, $CFA$ and $AFB$, draw its Euler line - that is, the line connecting its circumcenter and its centroid. Prove that these three Euler lines pass through one common point. [i]Remark.[/i] The Fermat point $F$ is also known as the [b]first Fermat point[/b] or the [b]first Toricelli point[/b] of triangle $ABC$. [i]Floor van Lamoen[/i]

In a half-plane, bounded by a line \(r\), equilateral triangles \(S_1, S_2, \ldots, S_n\) are placed, each with one side parallel to \(r\), and their opposite vertex is the point of the triangle farthest from \(r\). For each triangle \(S_i\), let \(T_i\) be its medial triangle. Let \(S\) be the region covered by triangles \(S_1, S_2, \ldots, S_n\), and let \(T\) be the region covered by triangles \(T_1, T_2, \ldots, T_n\). Prove that \[\text{area}(S) \leq 4 \cdot \text{area}(T).\]

Each of the sides $AB$ and $CD$ of a convex quadrilateral $ABCD$ is divided into three equal parts, $|AE| = |EF| = |F B|$ , $|DP| = |P Q| = |QC|$. The diagonals of $AEPD$ and $FBCQ$ intersect at $M$ and $N$, respectively. Prove that the sum of the areas of $\vartriangle AMD$ and $\vartriangle BNC$ is equal to the sum of the areas of $\vartriangle EPM$ and $\vartriangle FNQ$.

The diagram shows a polygon made by removing six $2\times 2$ squares from the sides of an $8\times 12$ rectangle. Find the perimeter of this polygon. [img]https://cdn.artofproblemsolving.com/attachments/6/3/c23510c821c159d31aff0e6688edebc81e2737.png[/img]

There are 3 lines $r, s$ and $t$ on a plane. The lines $r$ and $s$ intersect perpendicularly at point $A$. the line $t$ intersects the line $r$ at point $B$ and the line $s$ at point $C$. There exist exactly 4 circumferences on the plane that are simultaneously tangent to all those 3 lines. Prove that the radius of one of those circumferences is equal to the sum of the radius of the other three circumferences.

Let $ABCD$ be a rectangle. A line $r$ moves parallel to $AB$ and intersects diagonal $AC$ , forming two triangles opposite the vertex, inside the rectangle. Prove that the sum of the areas of these triangles is minimal when $r$ passes through the midpoint of segment $AD$ .

A $2\sqrt5$ by $4\sqrt5$ rectangle is rotated by an angle $\theta$ about one of its diagonals. If the total volume swept out by the rotating rectangle is $62\pi$, find the measure of $\theta$ in degrees. [i]Proposed by Connor Gordon[/i]

Let triangle $\vartriangle ABC$ be a triangle with $AB = 5$, $BC = 7$, and $CA = 8$, and let $I$ be the incenter of $\vartriangle ABC$. Let circle $C_A$ denote the circle with center $A$ and radius $\overline{AI}$, denote $C_B$ and circle $C_C$ similarly. Besides all intersecting at $I$, the circles $C_A$,$C_B$,$C_C$ also intersect pairwise at $F$, $G$, and $H$. Compute the area of triangle $\vartriangle FGH$.

Let $O$ be the circumcenter of an acute-angled triangle $ABC$ with ${\angle B<\angle C}$. The line $AO$ meets the side $BC$ at $D$. The circumcenters of the triangles $ABD$ and $ACD$ are $E$ and $F$, respectively. Extend the sides $BA$ and $CA$ beyond $A$, and choose on the respective extensions points $G$ and $H$ such that ${AG=AC}$ and ${AH=AB}$. Prove that the quadrilateral $EFGH$ is a rectangle if and only if ${\angle ACB-\angle ABC=60^{\circ }}$. [i]Proposed by Hojoo Lee, Korea[/i]

Find the perimeter of a triangle whose altitudes are $3,4,$ and $6$. $ \textbf{(A)}\ 12\sqrt\frac35 \qquad \textbf{(B)}\ 16\sqrt\frac35 \qquad \textbf{(C)}\ 20\sqrt\frac35 \qquad \textbf{(D)}\ 24\sqrt\frac35 \qquad \textbf{(E)}\ \text{None}$

Let $P$ be a variable point on arc $BC$ of the circumcircle of triangle $ABC$ not containing $A$. Let $I_1$ and $I_2$ be the incenters of the triangles $PAB$ and $PAC$, respectively. Prove that: [b](a)[/b] The circumcircle of $?PI_1I_2$ passes through a fixed point. [b](b)[/b] The circle with diameter $I_1I_2$ passes through a fixed point. [b](c)[/b] The midpoint of $I_1I_2$ lies on a fixed circle.

Let $ABC$ be an acute triangle. Let $\omega$ be a circle whose centre $L$ lies on the side $BC$. Suppose that $\omega$ is tangent to $AB$ at $B'$ and $AC$ at $C'$. Suppose also that the circumcentre $O$ of triangle $ABC$ lies on the shorter arc $B'C'$ of $\omega$. Prove that the circumcircle of $ABC$ and $\omega$ meet at two points. [i]Proposed by Härmel Nestra, Estonia[/i]

$1.$A bottle in the shape of a cone lies on its base. Water is poured into the bottle until its level reaches a distance of 8 centimeters from the vertex of the cone (measured vertically). We now turn the bottle upside down without changing the amount of water it contains; This leaves an empty space in the upper part of the cone that is 2 centimeters high. Find the height of the bottle.

Let ${ABC}$ be an acute angled triangle whose shortest side is ${BC}$. Consider a variable point ${P}$ on the side ${BC}$, and let ${D}$ and ${E}$ be points on ${AB}$ and ${AC}$, respectively, such that ${BD=BP}$ and ${CP=CE}$. Prove that, as ${P}$ traces ${BC}$, the circumcircle of the triangle ${ADE}$ passes through a fixed point.

A tetrahedron with four equilateral triangular faces has a sphere inscribed within it and a sphere circumscribed about it. For each of the four faces, there is a sphere tangent externally to the face at its center and to the circumscribed sphere. A point $ P$ is selected at random inside the circumscribed sphere. The probability that $ P$ lies inside one of the five small spheres is closest to $ \textbf{(A)}\ 0\qquad \textbf{(B)}\ 0.1\qquad \textbf{(C)}\ 0.2\qquad \textbf{(D)}\ 0.3\qquad \textbf{(E)}\ 0.4$

$\omega$ is incircle of $\triangle ABC$ touch $AC$ in $S$. Point $Q$ lies on $\omega$ and midpoints of $AQ$ and $QC$ lies on $\omega$ . Prove that $QS$ bisects $\angle AQC$

Inside of convex quadrilateral $ABCD$ point $E$ was chosen such that $\angle DAE = \angle CAB$ and $\angle ADE = \angle CDB$. Prove that if perpendicular from $E$ to $AD$ passes from the intersection of diagonals of $ABCD$, then $\angle AEB = \angle CED$.

Denote by $\Omega$ the circumcircle of the acute triangle $ABC$. Point $D$ is the midpoint of the arc $BC$ of $\Omega$ not containing $A$. Circle $\omega$ centered at $D$ is tangent to the segment $BC$ at point $E$. Tangents to the circle $\omega$ passing through point $A$ intersect line $BC$ at points $K$ and $L$ such that points $B, K, L, C$ lie on the line $BC$ in that order. Circle $\gamma_1$ is tangent to the segments $AL$ and $BL$ and to the circle $\Omega$ at point $M$. Circle $\gamma_2$ is tangent to the segments $AK$ and $CK$ and to the circle $\Omega$ at point $N$. Lines $KN$ and $LM$ intersect at point $P$. Prove that $\sphericalangle KAP = \sphericalangle EAL$.

Let $ABC$ be a triangle and $I$ its incenter. The circle passing through $A, C$ and $I$ intersect the line $BC$ for second time at point $X$. The circle passing through $B, C$ and $I$ intersects the line $AC$ for second time at point $Y$. Show that the segments $AY$ and $BX$ have equal length.

Let $O$ be the circumcenter and $I$ be the incenter of an acute triangle $ABC$ with $m(\widehat{B}) \neq m(\widehat{C})$. Let $D$, $E$, $F$ be the midpoints of the sides $[BC]$, $[CA]$, $[AB]$, respectively. Let $T$ be the foot of perpendicular from $I$ to $[AB]$. Let $P$ be the circumcenter of the triangle $DEF$ and $Q$ be the midpoint of $[OI]$. If $A$, $P$, $Q$ are collinear, prove that \[\dfrac{|AO|}{|OD|}-\dfrac{|BC|}{|AT|}=4.\]

Let $\Gamma_1$ and $\Gamma_2$ be two circles of unequal radii, with centres $O_1$ and $O_2$ respectively, intersecting in two distinct points $A$ and $B$. Assume that the centre of each circle is outside the other circle. The tangent to $\Gamma_1$ at $B$ intersects $\Gamma_2$ again in $C$, different from $B$; the tangent to $\Gamma_2$ at $B$ intersects $\Gamma_1$ again at $D$, different from $B$. The bisectors of $\angle DAB$ and $\angle CAB$ meet $\Gamma_1$ and $\Gamma_2$ again in $X$ and $Y$, respectively. Let $P$ and $Q$ be the circumcentres of triangles $ACD$ and $XAY$, respectively. Prove that $PQ$ is the perpendicular bisector of the line segment $O_1O_2$. [i]Proposed by Prithwijit De[/i]

Let $\vartriangle ABC$ be an equilateral triangle. Points $D,E, F$ are drawn on sides $AB$,$BC$, and $CA$ respectively such that $[ADF] = [BED] + [CEF]$ and $\vartriangle ADF \sim \vartriangle BED \sim \vartriangle CEF$. The ratio $\frac{[ABC]}{[DEF]}$ can be expressed as $\frac{a+b\sqrt{c}}{d}$ , where $a$, $b$, $c$, and $d$ are positive integers such that $a$ and $d$ are relatively prime, and $c$ is not divisible by the square of any prime. Find $a + b + c + d$. (Here $[P]$ denotes the area of polygon $P$.)

Let $ABCD$ be the cyclic quadrilateral. Suppose that there exists some line $l$ parallel to $BD$ which is tangent to the inscribed circles of triangles $ABC, CDA$. Show that $l$ passes through the incenter of $BCD$ or through the incenter of $DAB$. [i](Proposed by Fedir Yudin)[/i]

[i]20 problems for 25 minutes.[/i] [b]p1.[/b] Compute the value of $2 + 20 + 201 + 2016$. [b]p2.[/b] Gleb is making a doll, whose prototype is a cube with side length $5$ centimeters. If the density of the toy is $4$ grams per cubic centimeter, compute its mass in grams. [b]p3.[/b] Find the sum of $20\%$ of $16$ and $16\%$ of $20$. [b]p4.[/b] How many times does Akmal need to roll a standard six-sided die in order to guarantee that two of the rolled values sum to an even number? [b]p5.[/b] During a period of one month, there are ten days without rain and twenty days without snow. What is the positive difference between the number of rainy days and the number of snowy days? [b]p6.[/b] Joanna has a fully charged phone. After using it for $30$ minutes, she notices that $20$ percent of the battery has been consumed. Assuming a constant battery consumption rate, for how many additional minutes can she use the phone until $20$ percent of the battery remains? [b]p7.[/b] In a square $ABCD$, points $P$, $Q$, $R$, and $S$ are chosen on sides $AB$, $BC$, $CD$, and $DA$ respectively, such that $AP = 2PB$, $BQ = 2QC$, $CR = 2RD$, and $DS = 2SA$. What fraction of square $ABCD$ is contained within square $PQRS$? [b]p8.[/b] The sum of the reciprocals of two not necessarily distinct positive integers is $1$. Compute the sum of these two positive integers. [b]p9.[/b] In a room of government officials, two-thirds of the men are standing and $8$ women are standing. There are twice as many standing men as standing women and twice as many women in total as men in total. Find the total number of government ocials in the room. [b]p10.[/b] A string of lowercase English letters is called pseudo-Japanese if it begins with a consonant and alternates between consonants and vowels. (Here the letter "y" is considered neither a consonant nor vowel.) How many $4$-letter pseudo-Japanese strings are there? [b]p11.[/b] In a wooden box, there are $2$ identical black balls, $2$ identical grey balls, and $1$ white ball. Yuka randomly draws two balls in succession without replacement. What is the probability that the first ball is strictly darker than the second one? [b]p12.[/b] Compute the real number $x$ for which $(x + 1)^2 + (x + 2)^2 + (x + 3)^2 = (x + 4)^2 + (x + 5)^2 + (x + 6)^2$. [b]p13.[/b] Let $ABC$ be an isosceles right triangle with $\angle C = 90^o$ and $AB = 2$. Let $D$, $E$, and $F$ be points outside $ABC$ in the same plane such that the triangles $DBC$, $AEC$, and $ABF$ are isosceles right triangles with hypotenuses $BC$, $AC$, and $AB$, respectively. Find the area of triangle $DEF$. [b]p14.[/b] Salma is thinking of a six-digit positive integer $n$ divisible by $90$. If the sum of the digits of n is divisible by $5$, find $n$. [b]p15.[/b] Kiady ate a total of $100$ bananas over five days. On the ($i + 1$)-th day ($1 \le i \le 4$), he ate i more bananas than he did on the $i$-th day. How many bananas did he eat on the fifth day? [b]p16.[/b] In a unit equilateral triangle $ABC$; points $D$,$E$, and $F$ are chosen on sides $BC$, $CA$, and $AB$, respectively. If lines $DE$, $EF$, and $FD$ are perpendicular to $CA$, $AB$ and $BC$, respectively, compute the area of triangle $DEF$. [b]p17.[/b] Carlos rolls three standard six-sided dice. What is the probability that the product of the three numbers on the top faces has units digit 5? [b]p18.[/b] Find the positive integer $n$ for which $n^{n^n}= 3^{3^{82}}$. [b]p19.[/b] John folds a rope in half five times then cuts the folded rope with four knife cuts, leaving five stacks of rope segments. How many pieces of rope does he now have? [b]p20.[/b] An integer $n > 1$ is conglomerate if all positive integers less than n and relatively prime to $n$ are not composite. For example, $3$ is conglomerate since $1$ and $2$ are not composite. Find the sum of all conglomerate integers less than or equal to $200$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ABCD$ be a trapezoid with base sides $AB$ and $CD$ and let $AB=a$, $BC=b$, $CD=c$, $DA=d$, $AC=m$ and $BD=n$. We know that $m^2+n^2=(a+c)^2$ $a)$ Prove that lines $AC$ and $BD$ are perpendicular $b)$ Prove that $ac<bd$