Given triangle $ABC$. Point $B_1$ is marked on line $AC$ so that $AB = AB_1$, while $B_1$ and $C$ are on the same side of $A$. Through points $C$, $B_1$ and the foot of the bisector of angle $A$ of triangle $ABC$, a circle $\omega$ is drawn, intersecting for second time the circle circumscribed around triangle $ABC$, at point $Q$. Prove that the tangent drawn to $\omega$ at point $Q$ is parallel to $AC$.

Prove that exists triangle that can be partitions in $2050$ congruent triangles.

Two circles $C_{1}$ and $C_{2}$ intersect in $A$ and $B$. A line passing through $B$ intersects $C_{1}$ in $C$ and $C_{2}$ in $D$. Another line passing through $B$ intersects $C_{1}$ in $E$ and $C_{2}$ in $F$, $(CF)$ intersects $C_{1}$ and $C_{2}$ in $P$ and $Q$ respectively. Make sure that in your diagram, $B, E, C, A, P \in C_{1}$ and $B, D, F, A, Q \in C_{2}$ in this order. Let $M$ and $N$ be the middles of the arcs $BP$ and $BQ$ respectively. Prove that if $CD=EF$, then the points $C,F,M,N$ are cocylic in this order.

Consider a tetrahedron $OABC$ with $ABC$ equilateral. Let $S$ be the area of the triangle of sides $OA$, $OB$ and $OC$. Show that $V\leqslant \dfrac12 RS$ where $R$ is the circumradius and $V$ is the volume of the tetrahedron. [i]Stere Ianuș[/i]

Let $ABC$ be a nonequilateral triangle. Prove that there exist two points $P$ and $Q$ in the plane of the triangle, one in the interior and one in the exterior of the circumcircle of $ABC$, such that the orthogonal projections of any of these two points on the sides of the triangle are vertices of an equilateral triangle.

Find the area of the figure on the coordinate plane bounded by the straight lines $x = 0$, $x = 2$ and the graphs of the functions $y =\sqrt{x^3+ 1}$ and $y = - \sqrt[3]{x^2+ 2x}$.

The parabola with equation $ y \equal{} ax^2 \plus{} bx \plus{} c$ and vertex $ (h,k)$ is reflected about the line $ y \equal{} k$. This results in the parabola with equation $ y \equal{} dx^2 \plus{} ex \plus{} f$. Which of the following equals $ a \plus{} b \plus{} c \plus{} d \plus{} e \plus{} f$? $ \textbf{(A)} \ 2b \qquad \textbf{(B)} \ 2c \qquad \textbf{(C)} \ 2a \plus{} 2b \qquad \textbf{(D)} \ 2h \qquad \textbf{(E)} \ 2k$

7. In triangle $ABC$, let $N$ and $M$ be the midpoints of $AB$ and $AC$, respectively. Point $P$ is chosen on the arc $BC$ not containing $A$ of the circumcircle of $ABC$ such that $BNMP$ is cyclic. Given $BC=28$, $AC=30$ and $AB = 26$, the value of $AP$ may be expressed as $m/\sqrt{n}$ for positive integers $m$ and $n$, where $n$ is not divisible by the square of any prime. Compute $m+n$. [i]Proposed by winnertakeover[/i]

Suppose $ABCD$ is a parallelogram and $E$ is a point between $B$ and $C$ on the line $BC$. If the triangles $DEC$, $BED$ and $BAD$ are isosceles what are the possible values for the angle $DAB$?

[b]p1.[/b] Find all solutions $a, b, c, d, e, f$ if it is known that they represent distinct digits and satisfy the following: $\begin{tabular}{ccccc} & a & b & c & a \\ + & & d & d & e \\ & & & d & e \\ \hline d & f & f & d & d \\ \end{tabular}$ [b]p2.[/b] Explain whether it possible that the sum of two squares of positive whole numbers has all digits equal to $1$: $$n^2 + m^2 = 111...111$$ [b]p3. [/b]Two players play the following game on an $8 \times 8$ chessboard. The first player can put a rook on an arbitrary square. Then the second player can put another rook on a free square that is not controlled by the first rook. Then the first player can put a new rook on a free square that is not controlled by the rooks on the board. Then the second player can do the same, etc. A player who cannot put a new rook on the board loses the game. Who has a winning strategy? [b]p4.[/b] Show that the difference $9^{2008} - 7^{2008}$ is divisible by $10$. [b]p5.[/b] Is it possible to find distict positive whole numbers $a, b, c, d, e$ such that $$\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}+\frac{1}{e}= 1?$$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

$ AA_{3}$ and $ BB_{3}$ are altitudes of acute-angled $ \triangle ABC$. Points $ A_{1}$ and $ B_{1}$ are second points of intersection lines $ AA_{3}$ and $ BB_{3}$ with circumcircle of $ \triangle ABC$ respectively. $ A_{2}$ and $ B_{2}$ are points on $ BC$ and $ AC$ respectively. $ A_{1}A_{2}\parallel AC$, $ B_{1}B_{2}\parallel BC$. Point $ M$ is midpoint of $ A_{2}B_{2}$. $ \angle BCA \equal{} x$. Find $ \angle A_{3}MB_{3}$.

Two circles meet at points $A$ and $B$. A line through $B$ intersects the first circle again at $K$ and the second circle at $M$. A line parallel to $AM$ is tangent to the first circle at $Q$. The line $AQ$ intersects the second circle again at $R$. $(a)$ Prove that the tangent to the second circle at $R$ is parallel to $AK$. $(b)$ Prove that these two tangents meet on $KM$.

Points $A$ and $B$ are fixed on a circle $c_1$. Circle $c_2$, whose centre lies on $c_1$, touches line $AB$ at $B$. Another line through $A$ intersects $c_2$ at points $D$ and $E$, where $D$ lies between $A$ and $E$. Line $BD$ intersects $c_1$ again at $F$. Prove that line $EB$ is tangent to $c_1$ if and only if $D$ is the midpoint of the segment $BF$.

On $\triangle ABC$ if there existed a point $D$ in $AC$ such that $\angle CBD=\angle ABD+60$ and $\angle BDC=30$ and $AB \cdot BC=BD^2$, then find the angles inside the triangle $\triangle ABC$

(a) Some vertices of a dodecahedron are to be marked so that each face contains a marked vertex. What is the smallest number of marked vertices for which this is possible? (b) Answer the same question, but for an icosahedron. (G. Galperin, Moscow) (Recall that a dodecahedron has $12$ pentagonal faces which meet in threes at each vertex, while an icosahedron has $20$ triangular faces which meet in fives at each vertex.)

Daeun draws a unit circle centered at the origin and inscribes within it a regular hexagon $ABCDEF$. Then Dylan chooses a point $P$ within the circle of radius $2$ centered at the origin. Let $M$ be the maximum possible value of $|PA| \cdot |PB| \cdot |PC| \cdot |PD| \cdot |PE| \cdot |PF|$, and let $N$ be the number of possible points $P$ for which this maximal value is obtained. Find $M + N^2$.

Two decompositions of a square into three rectangles are called substantially different, if reordering the rectangles does not change one into the other. How many substantially different decompositions of a $2010 \times 2010$ square in three rectangles with integer side lengths are there such that the area of one rectangle is equal to the arithmetic mean of the areas of the other rectangles?

Let $\triangle ABC$ have $AB = 1$ cm, $BC = 2$ cm and $AC = \sqrt{3}$ cm. Points $D$, $E$ and $F$ lie on segments $AB$, $AC$ and $BC$ respectively are such that $AE = BD$ and $BF = AD$. The angle bisector of $\angle BAC$ intersects the circumcircle of $\triangle ADE$ for the second time at $M$ and the angle bisector of $\angle ABC$ intersects the circumcircle of $\triangle BDF$ at $N$. Determine the length of $MN$.

Point $P$ lies inside of parallelogram $ABCD$. Perpendicular lines to $PA,PB,PC$ and $PD$ through $A,B,C$ and $D$ construct convex quadrilateral $XYZT$. Prove that $S_{XYZT}\geq 2S_{ABCD}$. [i]Proposed by Siamak Ahmadpour[/i]

[b]p1.[/b] Prair takes some set $S$ of positive integers, and for each pair of integers she computes the positive difference between them. Listing down all the numbers she computed, she notices that every integer from $1$ to $10$ is on her list! What is the smallest possible value of $|S|$, the number of elements in her set $S$? [b]p2.[/b] Jake has $2021$ balls that he wants to separate into some number of bags, such that if he wants any number of balls, he can just pick up some bags and take all the balls out of them. What is the least number of bags Jake needs? [b]p3.[/b] Claire has stolen Cat’s scooter once again! She is currently at (0; 0) in the coordinate plane, and wants to ride to $(2, 2)$, but she doesn’t know how to get there. So each second, she rides one unit in the positive $x$ or $y$-direction, each with probability $\frac12$ . If the probability that she makes it to $(2, 2)$ during her ride can be expressed as $\frac{a}{b}$ for positive integers $a, b$ with $gcd(a, b) = 1$, then find $a + b$. [b]p4.[/b] Triangle $ABC$ with $AB = BC = 6$ and $\angle ABC = 120^o$ is rotated about $A$, and suppose that the images of points $B$ and $C$ under this rotation are $B'$ and $C'$, respectively. Suppose that $A$, $B'$ and $C$ are collinear in that order. If the area of triangle $B'CC'$ can be expressed as $a - b\sqrt{c}$ for positive integers $a, b, c$ with csquarefree, find $a + b + c$. [b]p5.[/b] Find the sum of all possible values of $a + b + c + d$ if $a, b, c, $d are positive integers satisfying $$ab + cd = 100,$$ $$ac + bd = 500.$$ [b]p6.[/b] Alex lives in Chutes and Ladders land, which is set in the coordinate plane. Each step they take brings them one unit to the right or one unit up. However, there’s a chute-ladder between points $(1, 2)$ and $(2, 0)$ and a chute-ladder between points $(1, 3)$ and $(4, 0)$, whenever Alex visits an endpoint on a chute-ladder, they immediately appear at the other endpoint of that chute-ladder! How many ways are there for Alex to go from $(0, 0)$ to $(4, 4)$? [b]p7.[/b] There are $8$ identical cubes that each belong to $8$ different people. Each person randomly picks a cube. The probability that exactly $3$ people picked their own cube can be written as $\frac{a}{b}$ , where $a$ and $b$ are positive integers with $gcd(a, b) = 1$. Find $a + b$. [b]p8.[/b] Suppose that $p(R) = Rx^2 + 4x$ for all $R$. There exist finitely many integer values of $R$ such that $p(R)$ intersects the graph of $x^3 + 2021x^2 + 2x + 1$ at some point $(j, k)$ for integers $j$ and $k$. Find the sum of all possible values of $R$. [b]p9.[/b] Let $a, b, c$ be the roots of the polynomial $x^3 - 20x^2 + 22$. Find $\frac{bc}{a^2} +\frac{ac}{b^2} +\frac{ab}{c^2}$. [b]p10.[/b] In any finite grid of squares, some shaded and some not, for each unshaded square, record the number of shaded squares horizontally or vertically adjacent to it, this grid’s score is the sum of all numbers recorded this way. Deyuan shades each square in a blank $n \times n$ grid with probability $k$; he notices that the expected value of the score of the resulting grid is equal to $k$, too! Given that $k > 0.9999$, find the minimum possible value of $n$. [b]p11.[/b] Find the sum of all $x$ from $2$ to $1000$ inclusive such that $$\prod^x_{n=2} \log_{n^n}(n + 1)^{n+2}$$ is an integer. [b]p12.[/b] Let triangle $ABC$ with incenter $I$ and circumcircle $\Gamma$ satisfy $AB = 6\sqrt3$, $BC = 14$, and $CA = 22$. Construct points $P$ and $Q$ on rays $BA$ and $CA$ such that $BP = CQ = 14$. Lines $PI$ and $QI$ meet the tangents from $B$ and $C$ to $\Gamma$, respectively, at points $X$ and $Y$ . If $XY$ can be expressed as $a\sqrt{b}-c$ for positive integers $a, b, c$ with $c$ squarefree, find $a + b + c$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ABCDE$ be a convex pentagon such that $AB=BC=CD$, $\angle{EAB}=\angle{BCD}$, and $\angle{EDC}=\angle{CBA}$. Prove that the perpendicular line from $E$ to $BC$ and the line segments $AC$ and $BD$ are concurrent.

In an acute triangle $ ABC$ segments $ BE$ and $ CF$ are altitudes. Two circles passing through the point $ A$ and $ F$ and tangent to the line $ BC$ at the points $ P$ and $ Q$ so that $ B$ lies between $ C$ and $ Q$. Prove that lines $ PE$ and $ QF$ intersect on the circumcircle of triangle $ AEF$. [i]Proposed by Davood Vakili, Iran[/i]

The square $ABCD$ has side length $1$. The point $E$ lies on the side $CD$. The line through $A$ and $E$ intersects the line through $B$ and $C$ at the point $F$. Prove that $$\frac{1}{|AE|^2}+\frac{1}{|AF|^2}= 1.$$ [img]https://cdn.artofproblemsolving.com/attachments/5/8/4e803eb7748f7a72783065717044cfc06f565f.png[/img]

Let a, b, c be the sidelengths and S the area of a triangle ABC. Denote $x=a+\frac{b}{2}$, $y=b+\frac{c}{2}$ and $z=c+\frac{a}{2}$. Prove that there exists a triangle with sidelengths x, y, z, and the area of this triangle is $\geq\frac94 S$.

Let $P_1P_2\ldots P_n$ be an $n$-gon such that for all $i,j \in \{1,2,\ldots,n\}$ where $i \neq j$, there exists $k \neq i,j$ such that $\angle P_iP_kP_j = 60^\circ$. Prove that $n=3$.