Restore the acute triangle $ABC$ given the vertex $A$, the foot of the altitude drawn from the vertex $B$ and the center of the circle circumscribed around triangle $BHC$ (point $H$ is the orthocenter of triangle $ABC$).

In the quadrilateral $ABCD$ point $E$ - the midpoint of the side $AB$, point $F$ - the midpoint of the side $BC$, point $G$ - the midpoint $AD$ . It turned out that the segment $GE$ is perpendicular to $AB$, and the segment $GF$ is perpendicular to the segment $BC$. Find the value of the angle $GCD$, if it is known that $\angle ADC = 70 {} ^ \circ$.

Two triangles $ABC$ and $BDE$ have vertexes $C$ and $E$ on the same side of the line $AB{}$ and $AB=a<BD$. Denote $\{P\}=CE\cap AB$ and $\gamma=m(\angle CPA)$. Let $r_1$ be the radius of the inscribed cricle of triangle $PAC$ and $r_2$ the radius of the excircle of triangle $PDE$, tangent to the side $DE$. Find $r_1+r_2$.

Let $ABC$ be an equilateral triangle, and let $D$ and $F$ be points on sides $BC$ and $AB$, respectively, with $FA=5$ and $CD=2$. Point $E$ lies on side $CA$ such that $\angle DEF = 60^\circ$. The area of triangle $DEF$ is $14\sqrt{3}$. The two possible values of the length of side $AB$ are $p \pm q\sqrt{r}$, where $p$ and $q$ are rational, and $r$ is an integer not divisible by the square of a prime. Find $r$.

Given triangle $ABC$ with $\angle A = a$. Let $AL$ be the bisector of the triangle $ABC$. Let the incircle of $\vartriangle ABC$ touch the sides $AB$ and $BC$ at points $P$ and $Q$ respectively. Let $X$ be the intersection point of the lines $AQ$ and $LP$. Prove that the lines $BX$ and $AL$ are perpendicular. (V. Karamzin)

Let $ABC$ be an acute-angled triangle with circumcircle $\omega$ and circumcentre $O$. Points $D\neq B$ and $E\neq C$ lie on $\omega$ such that $BD\perp AC$ and $CE\perp AB$. Let $CO$ meet $AB$ at $X$, and $BO$ meet $AC$ at $Y$. Prove that the circumcircles of triangles $BXD$ and $CYE$ have an intersection lie on line $AO$. [i]Ivan Chan Kai Chin, Malaysia[/i]

Two circles are tangent externally at one point. Common external tangents are drawn to them and the tangent points are connected. Prove that the sum of the lengths of the opposite sides of the quadrilateral obtained are equal.

On a straight line $4$ points are successively set , $A, P, Q,W $, which are the points of intersection of the bisector $AL $ of the triangle $ABC$ with the circumscribed and inscribed circle. Knowing only these points, construct a triangle $ABC $.

Given an acute triangle $T$ with sides $a,b,c$, find the tetrahedra with base $T$ whose all faces are acute triangles of the same area.

What is the area enclosed by the geoboard quadrilateral below? [asy] int i,j; for(i=0; i<11; i=i+1) { for(j=0; j<11; j=j+1) { dot((i,j)); } } draw((0,5)--(4,0)--(10,10)--(3,4)--cycle, linewidth(0.7)); [/asy] $\textbf{(A)} 15\qquad \textbf{(B)} 18\tfrac12\qquad \textbf{(C)} 22\tfrac12\qquad \textbf{(D)} 27\qquad \textbf{(E)} 41\qquad$

[b]p1.[/b] James has $8$ Instagram accounts, $3$ Facebook accounts, $4$ QQ accounts, and $3$ YouTube accounts. If each Instagram account has $19$ pictures, each Facebook account has $5$ pictures and $9$ videos, each QQ account has a total of $17$ pictures, and each YouTube account has $13$ videos and no pictures, how many pictures in total does James have in all these accounts? [b]p2.[/b] If Poonam can trade $7$ shanks for $4$ shinks, and she can trade $10$ shinks for $17$ shenks. How many shenks can Poonam get if she traded all of her $105$ shanks? [b]p3.[/b] Jerry has a bag with $3$ red marbles, $5$ blue marbles and $2$ white marbles. If Jerry randomly picks two marbles from the bag without replacement, the probability that he gets two different colors can be expressed as a fraction $\frac{m}{n}$ in lowest terms. What is $m + n$? [b]p4.[/b] Bob's favorite number is between $1200$ and $4000$, divisible by $5$, has the same units and hundreds digits, and the same tens and thousands digits. If his favorite number is even and not divisible by $3$, what is his favorite number? [b]p5.[/b] Consider a unit cube $ABCDEFGH$. Let $O$ be the center of the face $EFGH$. The length of $BO$ can be expressed in the form $\frac{\sqrt{a}}{b}$, where $a$ and $b$ are simplified to lowest terms. What is $a + b$? [b]p6.[/b] Mr. Eddie Wang is a crazy rich boss who owns a giant company in Singapore. Even though Mr. Wang appears friendly, he finds great joy in firing his employees. His immediately fires them when they say "hello" and/or "goodbye" to him. It is well known that $1/2$ of the total people say "hello" and/or "goodbye" to him everyday. If Mr. Wang had $2050$ employees at the end of yesterday, and he hires $2$ new employees at the beginning of each day, in how many days will Mr. Wang first only have $6$ employees left? [b]p7.[/b] In $\vartriangle ABC$, $AB = 5$, $AC = 6$. Let $D,E,F$ be the midpoints of $\overline{BC}$, $\overline{AC}$, $\overline{AB}$, respectively. Let $X$ be the foot of the altitude from $D$ to $\overline{EF}$. Let $\overline{AX}$ intersect $\overline{BC}$ at $Y$ . Given $DY = 1$, the length of $BC$ is $\frac{p}{q}$ for relatively prime positive integers $p, q$: Find $p + q$. [b]p8.[/b] Given $\frac{1}{2006} = \frac{1}{a} + \frac{1}{b}$ where $a$ is a $4$ digit positive integer and $b$ is a $6$ digit positive integer, find the smallest possible value of $b$. [b]p9.[/b] Pocky the postman has unlimited stamps worth $5$, $6$ and $7$ cents. However, his post office has two very odd requirements: On each envelope, an odd number of $7$ cent stamps must be used, and the total number of stamps used must also be odd. What is the largest amount of postage money Pocky cannot make with his stamps, in cents? [b]p10.[/b] Let $ABCDEF$ be a regular hexagon with side length $2$. Let $G$ be the midpoint of side $DE$. Now let $O$ be the intersection of $BG$ and $CF$. The radius of the circle inscribed in triangle $BOC$ can be expressed in the form $\frac{a\sqrt{b}-\sqrt{c}}{d} $ where $a$, $b$, $c$, $d$ are simplified to lowest terms. What is $a + b + c + d$? [b]p11.[/b] Estimation (Tiebreaker): What is the total number of characters in all of the participants' email addresses in the Accuracy Round? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Given a finite number of points in the plane as well as many different rays starting at the origin. It is always possible to pair the points with the rays so that they parallell displaced rays starting in respective points do not intersect?

A convex hexagon $A_1B_1A_2B_2A_3B_3$ it is inscribed in a circumference $\Omega$ with radius $R$. The diagonals $A_1B_2$, $A_2B_3$, $A_3B_1$ are concurrent in $X$. For each $i=1,2,3$ let $\omega_i$ tangent to the segments $XA_i$ and $XB_i$ and tangent to the arc $A_iB_i$ of $\Omega$ that does not contain the other vertices of the hexagon; let $r_i$ the radius of $\omega_i$. $(a)$ Prove that $R\geq r_1+r_2+r_3$ $(b)$ If $R= r_1+r_2+r_3$, prove that the six points of tangency of the circumferences $\omega_i$ with the diagonals $A_1B_2$, $A_2B_3$, $A_3B_1$ are concyclic

Two circles with radii 1 meet in points $X, Y$, and the distance between these points also is equal to $1$. Point $C$ lies on the first circle, and lines $CA, CB$ are tangents to the second one. These tangents meet the first circle for the second time in points $B', A'$. Lines $AA'$ and $BB'$ meet in point $Z$. Find angle $XZY$.

Consider two points $Y$ and $X$ in a plane and a variable point $P$ which is not on $XY$. Let the parallel line to $YP$ through $X$ intersect the internal angle bisector of $\angle XYP$ in $A$, and let the parallel line to $XP$ through $Y$ intersect the internal angle bisector of $\angle YXP$ in $B$. Let $AB$ intersect $XP$ and $YP$ in $S$ and $T$ respectively. Show that the product $|XS|*|YT|$ does not depend on the position of $P$.

If two points are picked randomly on the perimeter of a square, what is the probability that the distance between those points is less than the side length of the square?

In the interior of a square $ABCD$ we construct the equilateral triangles $ABK, BCL, CDM, DAN.$ Prove that the midpoints of the four segments $KL, LM, MN, NK$ and the midpoints of the eight segments $AK, BK, BL, CL, CM, DM, DN, AN$ are the 12 vertices of a regular dodecagon.

Let $D$ and $E$ be points on sides $AB$ and $AC$ respectively of a triangle $ABC$ such that $DE$ is parallel to $BC$ and tangent to the incircle of $ABC$. Prove that \[DE\le\frac{1}{8}(AB+BC+CA) \]

Side $AB$ of triangle $ABC$ was divided into $n$ equal parts (dividing points $B_0 = A, B_1, B_2, ..., B_n = B$), and side $AC$ of this triangle was divided into $(n + 1)$ equal parts (dividing points $C_0 = A, C_1, C_2, ..., C_{n+1} = C$). Colored are the triangles $C_iB_iC_{i+1}$ (where $i = 1,2, ..., n$). What part of the area of the triangle is painted over?

An aquarium has a rectangular base that measures $ 100$ cm by $ 40$ cm and has a height of $ 50$ cm. The aquarium is filled with water to a depth of $ 37$ cm. A rock with volume $ 1000 \text{cm}^3$ is then placed in the aquarium and completely submerged. By how many centimeters does the water level rise? $ \textbf{(A)}\ 0.25 \qquad \textbf{(B)}\ 0.5 \qquad \textbf{(C)}\ 1 \qquad \textbf{(D)}\ 1.25 \qquad \textbf{(E)}\ 2.5$

Given a pointed triangle $ABC$ with orthocenter $H$. Let $D$ be the point such that the quadrilateral $AHCD$ is parallelogram. Let $p$ be the perpendicular to the direction $AB$ through the midpoint $A_1$ of the side $BC$. Denote the intersection of the lines $p$ and $AB$ with $E$, and the midpoint of the length $A_1E$ with $F$. The point where the parallel to the line $BD$ through point $A$ intersects $p$ denote by $G$. Prove that the quadrilateral $AFA_1C$ is cyclic if and only if the lines $BF$ passes through the midpoint of the length $CG$.

In quadrilateral $ABCD$, $AB = CD$, $M$ and $K$ are the midpoints of $BC$ and $AD$.Prove that the angle between $MK$ and $AC$ is equal to the half-sum of angles $BAC$ and $DCA$ [i](Proposed by M.Volchkevich)[/i]

$E$ is a point on the diagonal $BD$ of the square $ABCD$. Show that the points $A, E$ and the circumcenters of $ABE$ and $ADE$ form a square.

Let $ABC$ be an acute angled triangle and point $M$ chosen differently from $A,B,C$. Prove that $M$ is the orthocenter of triangle $ABC$ if and only if \[\frac{BC}{MA}\vec{MA}+\frac{CA}{MB}\vec{MB}+\frac{AB}{MC}\vec{MC}= \vec{0}\]

We want to cover totally a square(side is equal to $k$ integer and $k>1$) with this rectangles: $1$ rectangle ($1\times 1$), $2$ rectangles ($2\times 1$), $4$ rectangles ($3\times 1$),...., $2^n$ rectangles ($n + 1 \times 1$), such that the rectangles can't overlap and don't exceed the limits of square. Find all $k$, such that this is possible and for each $k$ found you have to draw a solution