Let $ABC$ be a triangle. Its excircles touch sides $BC, CA, AB$ at $D, E, F$, respectively. Prove that the perimeter of triangle $ABC$ is at most twice that of triangle $DEF$.

[b]Problem 1.[/b] Points $D$ and $E$ are chosen on the sides $AB$ and $AC$, respectively, of a triangle $\triangle ABC$ such that $DE\parallel BC$. The circumcircle $k$ of triangle $\triangle ADE$ intersects the lines $BE$ and $CD$ at the points $M$ and $N$ (different from $E$ and $D$). The lines $AM$ and $AN$ intersect the side $BC$ at points $P$ and $Q$ such that $BC=2\cdot PQ$ and the point $P$ lies between $B$ and $Q$. Prove that the circle $k$ passes through the point of intersection of the side $BC$ and the angle bisector of $\angle BAC$. [i]Nikolai Nikolov[/i]

As shown in figure , $A,B$ are two fixed points of circle $\odot O$, $C$ is the midpoint of the major arc $AB$, $D$ is any point of the minor arc $AB$. Tangent at $D$ intersects tangents at $A,B$ at points $E,F$ respectively. Segments $CE$ and $CF$ intersect chord $AB$ at points $G$ and $H$ respectively. Prove that the length of line segment $GH$ has a fixed value. [img]https://cdn.artofproblemsolving.com/attachments/9/2/85227f169193f61e313293e9128f6ece2ff1f7.png[/img]

What is the perimeter of a rectangle of area $32$ inscribed in a circle of radius $4$?

Given is a convex cyclic pentagon $ABCDE$ and a point $P$ inside it, such that $AB=AE=AP$ and $BC=CE$. The lines $AD$ and $BE$ intersect in $Q$. Points $R$ and $S$ are on segments $CP$ and $BP$ such that $DR=QR$ and $SR||BC$. Show that the circumcircles of $BEP$ and $PQS$ are tangent to each other.

Let $P$ be a point inside a triangle $ABC$ such that \[\angle P AB = \angle P BC = \angle P CA\] Suppose $AP, BP, CP$ meet the circumcircles of triangles $P BC, P CA, P AB$ at $X, Y, Z$ respectively $(\neq P)$ . Prove that \[[XBC] + [Y CA] + [ZAB] \ge 3[ABC]\]

A plane passing through the center of a cube intersects the cube in a cyclic hexagon. Show that this hexagon is regular.

Given is a triangle with side lengths $a,b$ and $c$, incenter $I$ and centroid $S$. Prove: If $a+b=3c$, then $S \neq I$ and line $SI$ is perpendicular to one of the sides of the triangle.

Let $A_1A_2...A_{2018}$ be regular $2018$-gon. Radius of it's circumcircle is $R$. Prove that: $$A_1A_{1008}-A_1A_{1006}+A_1A_{1004}-A_1A_{1002} + ... + A_1A_4 -A_1A_2=R$$

Let $ABC$ be an acute-angled triangle with $\angle BAC = 60^{\circ}$ and with incenter $I$ and circumcenter $O$. Let $H$ be the point diametrically opposite(antipode) to $O$ in the circumcircle of $\triangle BOC$. Prove that $IH=BI+IC$.

Let triangle $ABC$ satisfy $2BC = AB+AC$ and have incenter $I$ and circumcircle $\omega$. Let $D$ be the intersection of $AI$ and $\omega$ (with $A, D$ distinct). Prove that $I$ is the midpoint of $AD$.

On the coordinate plane is given a polygon $\mathcal P$ with area greater than $1$. Prove that there exist two different points $(x_1,y_1)$ and $(x_2,y_2)$ inside the polygon $\mathcal P$ such that $x_1-x_2$ and $y_1-y_2$ are both integers.

Let $ABC$ be a triangle with $AB \ne AC$ and incenter $I$. The incircle touches $BC$ at $D$. Let $M$ be the midpoint of $BC$ . Show that the line $IM$ bisects segment $AD$ .

Point $E$ is the midpoint of the base $AD$ of the trapezoid $ABCD$. Segments $BD$ and $CE$ intersect at point $F$. It is known that $AF$ is perpendicular to $BD$. Prove that $BC = FC$.

Four points $A$, $B$, $C$, $D$ lie on the hyperbola $y=\frac{1}{x}$. In triangle $BCD$ the point $A_1$ is the circumcenter of the triangle, which vertices are the midpoints of sides of $BCD$. In triangles $ACD$, $ABD$ and $ABC$ points $B_1$, $C_1$ and $D_1$ are chosen similarly. It turned out that points $A_1$, $B_1$, $C_1$ and $D_1$ are pairwise different and concyclic. Prove that the center of that circle coincides with the $(0,0)$ point.

[b]p1.[/b] A man is walking due east at $2$ m.p.h. and to him the wind appears to be blowing from the north. On doubling his speed to $4$ m.p.h. and still walking due east, the wind appears to be blowing from the nortl^eas^. What is the speed of the wind (assumed to have a constant velocity)? [b]p2.[/b] Prove that any triangle can be cut into three pieces which can be rearranged to form a rectangle of the same area. [b]p3.[/b] An increasing sequence of integers starting with $1$ has the property that if $n$ is any member of the sequence, then so also are $3n$ and $n + 7$. Also, all the members of the sequence are solely generated from the first nummber $1$; thus the sequence starts with $1,3,8,9,10, ...$ and $2,4,5,6,7...$ are not members of this sequence. Determine all the other positive integers which are not members of the sequence. [b]p4.[/b] Three prime numbers, each greater than $3$, are in arithmetic progression. Show that their common difference is a multiple of $6$. [b]p5.[/b] Prove that if $S$ is a set of at least $7$ distinct points, no four in a plane, the volumes of all the tetrahedra with vertices in $S$ are not all equal. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

[b]p1.[/b] What is the smallest positive integer divisible by $20$, $12$, and $13$? [b]p2.[/b] Two circles of radius $5$ are placed in the plane such that their centers are $7$ units apart. What is the largest possible distance between a point on one circle and a point on the other? [b]p3.[/b] In a magic square, all the numbers in the rows, columns, and diagonals sum to the same value. How many $2\times 2$ magic squares containing the integers $\{1, 2, 3, 4\}$ are there? [b]p4.[/b] Ethan's sock drawer contains two pairs of white socks and one pair of red socks. Ethan picks two socks at random. What is the probability that he picks two white socks? [b]p5.[/b] The sum of the time on a digital clock is the sum of the digits displayed on the screen. For example, the sum of the time at $10:23$ would be $6$. Assuming the clock is a $12$ hour clock, what is the greatest possible positive difference between the sum of the time at some time and the sum of the time one minute later? [b]p6.[/b] Given the expression $1 \div 2 \div 3 \div 4$, what is the largest possible resulting value if one were to place parentheses $()$ somewhere in the expression? [b]p7.[/b] At a convention, there are many astronomers, astrophysicists, and cosmologists. At $first$, all the astronomers and astrophysicists arrive. At this point, $\frac35$ of the people in the room are astronomers. Then, all the cosmologists come, so now, $30\%$ of the people in the room are astrophysicists. What fraction of the scientists are cosmologists? [b]p8.[/b] At $10:00$ AM, a minuteman starts walking down a $1200$-step stationary escalator at $40$ steps per minute. Halfway down, the escalator starts moving up at a constant speed, while the minuteman continues to walk in the same direction and at the same pace that he was going before. At $10:55$ AM, the minuteman arrives back at the top. At what speed is the escalator going up, in steps per minute? [b]p9.[/b] Given that $x_1 = 57$, $x_2 = 68$, and $x_3 = 32$, let $x_n = x_{n-1} -x_{n-2} +x_{n-3}$ for $n \ge 4$. Find $x_{2013}$. [b]p10.[/b] Two squares are put side by side such that one vertex of the larger one coincides with a vertex of the smaller one. The smallest rectangle that contains both squares is drawn. If the area of the rectangle is $60$ and the area of the smaller square is $24$, what is the length of the diagonal of the rectangle? [b]p11.[/b] On a dield trip, $2$ professors, $4$ girls, and $4$ boys are walking to the forest to gather data on butterflies. They must walk in a line with following restrictions: one adult must be the first person in the line and one adult must be the last person in the line, the boys must be in alphabetical order from front to back, and the girls must also be in alphabetical order from front to back. How many such possible lines are there, if each person has a distinct name? [b]p12.[/b] Flatland is the rectangle with vertices $A, B, C$, and $D$, which are located at $(0, 0)$, $(0, 5)$, $(5, 5)$, and $(5, 0)$, respectively. The citizens put an exact map of Flatland on the rectangular region with vertices $(1, 2)$, $(1, 3)$, $(2, 3)$, and $(2, 2)$ in such a way so that the location of $A$ on the map lies on the point $(1, 2)$ of Flatland, the location of $B$ on the map lies on the point $(1, 3)$ of Flatland, the location of C on the map lies on the point $(2, 3)$ of Flatland, and the location of D on the map lies on the point $(2, 2)$ of Flatland. Which point on the coordinate plane is thesame point on the map as where it actually is on Flatland? [b]p13.[/b] $S$ is a collection of integers such that any integer $x$ that is present in $S$ is present exactly $x$ times. Given that all the integers from $1$ through $22$ inclusive are present in $S$ and no others are, what is the average value of the elements in $S$? [b]p14.[/b] In rectangle $PQRS$ with $PQ < QR$, the angle bisector of $\angle SPQ$ intersects $\overline{SQ}$ at point $T$ and $\overline{QR }$ at $U$. If $PT : TU = 3 : 1$, what is the ratio of the area of triangle $PTS$ to the area of rectangle $PQRS$? [b]p15.[/b] For a function $f(x) = Ax^2 + Bx + C$, $f(A) = f(B)$ and $A + 6 = B$. Find all possible values of $B$. [b]p16.[/b] Let $\alpha$ be the sum of the integers relatively prime to $98$ and less than $98$ and $\beta$ be the sum of the integers not relatively prime to $98$ and less than $98$. What is the value of $\frac{\alpha}{\beta}$ ? [b]p17.[/b] What is the value of the series $\frac{1}{3} + \frac{3}{9} + \frac{6}{27} + \frac{10}{81} + \frac{15}{243} + ...$? [b]p18.[/b] A bug starts at $(0, 0)$ and moves along lattice points restricted to $(i, j)$, where $0 \le i, j \le 2$. Given that the bug moves $1$ unit each second, how many different paths can the bug take such that it ends at $(2, 2)$ after $8$ seconds? [b]p19.[/b] Let $f(n)$ be the sum of the digits of $n$. How many different values of $n < 2013$ are there such that $f(f(f(n))) \ne f(f(n))$ and $f(f(f(n))) < 10$? [b]p20.[/b] Let $A$ and $B$ be points such that $\overline{AB} = 14$ and let $\omega_1$ and $\omega_2$ be circles centered at $A$ and $B$ with radii $13$ and $15$, respectively. Let $C$ be a point on $\omega_1$ and $D$ be a point on $\omega_2$ such that $\overline{CD}$ is a common external tangent to $\omega_1$ and $\omega_2$. Let $P$ be the intersection point of the two circles that is closer to $\overline{CD}$. If $M$ is the midpoint of $\overline{CD}$, what is the length of segment $\overline{PM}$? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Isabella's house has $ 3$ bedrooms. Each bedroom is $ 12$ feet long, $ 10$ feet wide, and $ 8$ feet high. Isabella must paint the walls of all the bedrooms. Doorways and windows, which will not be painted, occupy $ 60$ square feet in each bedroom. How many square feet of walls must be painted? $ \textbf{(A)}\ 678 \qquad \textbf{(B)}\ 768 \qquad \textbf{(C)}\ 786 \qquad \textbf{(D)}\ 867 \qquad \textbf{(E)}\ 876$

Two radii OA and OB of a circle c with midpoint O are perpendicular. Another circle touches c in point Q and the radii in points C and D, respectively. Determine $ \angle{AQC}$.

Let $H$ be the orthocenter of an acute-angled triangle A$BC$. The perpendicular bisector to segment $BH$ meets $BA$ and $BC$ at points $A_0, C_0$ respectively. Prove that the perimeter of triangle $A_0OC_0$ ($O$ is the circumcenter of triangle $ABC$) is equal to $AC$.

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.

Find the positive integers $n$ with $n \geq 4$ such that $[\sqrt{n}]+1$ divides $n-1$ and $[\sqrt{n}]-1$ divides $n+1$. [hide="Remark"]This problem can be solved in a similar way with the one given at [url=http://www.mathlinks.ro/Forum/resources.php?c=1&cid=97&year=2006]Cono Sur Olympiad 2006[/url], problem 5.[/hide]

[u]Round 1[/u] [b]p1.[/b] Define the operation $\clubsuit$ so that $a \,\clubsuit \, b = a^b + b^a$. Then, if $2 \,\clubsuit \,b = 32$, what is $b$? [b]p2. [/b] A square is changed into a rectangle by increasing two of its sides by $p\%$ and decreasing the two other sides by $p\%$. The area is then reduced by $1\%$. What is the value of $p$? [b]p3.[/b] What is the sum, in degrees, of the internal angles of a heptagon? [b]p4.[/b] How many integers in between $\sqrt{47}$ and $\sqrt{8283}$ are divisible by $7$? [u]Round 2[/u] [b]p5.[/b] Some mutant green turkeys and pink elephants are grazing in a field. Mutant green turkeys have six legs and three heads. Pink elephants have $4$ legs and $1$ head. There are $100$ legs and $37$ heads in the field. How many animals are grazing? [b]p6.[/b] Let $A = (0, 0)$, $B = (6, 8)$, $C = (20, 8)$, $D = (14, 0)$, $E = (21, -10)$, and $F = (7, -10)$. Find the area of the hexagon $ABCDEF$. [b]p7.[/b] In Moscow, three men, Oleg, Igor, and Dima, are questioned on suspicion of stealing Vladimir Putin’s blankie. It is known that each man either always tells the truth or always lies. They make the following statements: (a) Oleg: I am innocent! (b) Igor: Dima stole the blankie! (c) Dima: I am innocent! (d) Igor: I am guilty! (e) Oleg: Yes, Igor is indeed guilty! If exactly one of Oleg, Igor, and Dima is guilty of the theft, who is the thief?? [b]p8.[/b] How many $11$-letter sequences of $E$’s and $M$’s have at least as many $E$’s as $M$’s? [u]Round 3[/u] [b]p9.[/b] John is entering the following summation $31 + 32 + 33 + 34 + 35 + 36 + 37 + 38 + 39$ in his calculator. However, he accidently leaves out a plus sign and the answer becomes $3582$. What is the number that comes before the missing plus sign? [b]p10.[/b] Two circles of radius $6$ intersect such that they share a common chord of length $6$. The total area covered may be expressed as $a\pi + \sqrt{b}$, where $a$ and $b$ are integers. What is $a + b$? [b]p11.[/b] Alice has a rectangular room with $6$ outlets lined up on one wall and $6$ lamps lined up on the opposite wall. She has $6$ distinct power cords (red, blue, green, purple, black, yellow). If the red and green power cords cannot cross, how many ways can she plug in all six lamps? [b]p12.[/b] Tracy wants to jump through a line of $12$ tiles on the floor by either jumping onto the next block, or jumping onto the block two steps ahead. An example of a path through the $12$ tiles may be: $1$ step, $2$ steps, $2$ steps, $2$ steps, $1$ step, $2$ steps, $2$ steps. In how many ways can Tracy jump through these $12$ tiles? PS. You should use hide for answers. Last rounds have been posted [url=https://artofproblemsolving.com/community/c4h2784268p24464984]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $\triangle ABC$ be a nondegenerate isosceles triangle with $AB=AC$, and let $D, E, F$ be the midpoints of $BC, CA, AB$ respectively. $BE$ intersects the circumcircle of $\triangle ABC$ again at $G$, and $H$ is the midpoint of minor arc $BC$. $CF\cap DG=I, BI\cap AC=J$. Prove that $\angle BJH=\angle ADG$ if and only if $\angle BID=\angle GBC$. [i]Proposed by David Stoner[/i]

Let $P$, $A$, $B$ and $C$ be points on a line $r$, in that order, so that $AB = BC$. Let $H$ be a point that does not belong to this line and let $S$ be the other intersection of the circles $(HPB)$ and $(HAC)$. Let $I$ be the second intersection of the circle with diameter $HB$ and $(HAC)$. Show that the points $P$, $H$, $I$ lie on the same line if and only if $HS$ is perpendicular to $r$.