On the sides $ AB $ and $ AC $ of the triangle $ ABC $ consider the points $ D, $ respectively, $ E, $ such that $$ \overrightarrow{DA} +\overrightarrow{DB} +\overrightarrow{EA} +\overrightarrow{EC} =\overrightarrow{O} . $$ If $ T $ is the intersection of $ DC $ and $ BE, $ determine the real number $ \alpha $ so that: $$ \overrightarrow{TB} +\overrightarrow{TC} =\alpha\cdot\overrightarrow{TA} . $$

Consider an acute-angled triangle $ABC$. Let $P$ be the foot of the altitude of triangle $ABC$ issuing from the vertex $A$, and let $O$ be the circumcenter of triangle $ABC$. Assume that $\angle C \geq \angle B+30^{\circ}$. Prove that $\angle A+\angle COP < 90^{\circ}$.

Let $ABC$ be a triangle with $AB=9$, $BC=10$, $CA=11$, and orthocenter $H$. Suppose point $D$ is placed on $\overline{BC}$ such that $AH=HD$. Compute $AD$.

A circle of radius $R$ is inscribed into an acute triangle. Three tangents to the circle split the triangle into three right angle triangles and a hexagon that has perimeter $Q$. Find the sum of diameters of circles inscribed into the three right triangles. (6)

How many distinct triangles can be drawn using three of the dots below as vertices? [asy]dot(origin^^(1,0)^^(2,0)^^(0,1)^^(1,1)^^(2,1));[/asy] $ \textbf{(A)}\ 9\qquad\textbf{(B)}\ 12\qquad\textbf{(C)}\ 18\qquad\textbf{(D)}\ 20\qquad\textbf{(E)}\ 24 $

The diagonals of a quadrangle inscribed in a circle intersect at point $K$. The projections of the point $ K$ onto the subsequent sides of this quadrangle are points $M, N, P, Q$. Prove that these lines $KM$, $KN$, $KP$, $KQ$ are the angle bisectors of the quadrangle $MNPQ$.

Let $ABCD$ be a rectangle of center $O$, such that $\angle DAC = 60^o$. The angle bisector of $\angle DAC$ meets $DC$ at $S$. Lines $OS$ and $AD$ meet at $L$ and lines $BL$ and $AC$ meet at $M$. Prove that lines $SM$ and $CL$ are parallel.

Let $ABC$ be an acute triangle with $AB > AC$. Let $\Gamma $ be its circumcircle, $H$ its orthocenter, and $F$ the foot of the altitude from $A$. Let $M$ be the midpoint of $BC$. Let $Q$ be the point on $\Gamma$ such that $\angle HQA = 90^{\circ}$ and let $K$ be the point on $\Gamma$ such that $\angle HKQ = 90^{\circ}$. Assume that the points $A$, $B$, $C$, $K$ and $Q$ are all different and lie on $\Gamma$ in this order. Prove that the circumcircles of triangles $KQH$ and $FKM$ are tangent to each other. Proposed by Ukraine

Let $ABC$ be a triangle such that $AB=2$, $CA=3$, and $BC=4$. A semicircle with its diameter on $BC$ is tangent to $AB$ and $AC$. Compute the area of the semicircle.

Let $ABCD$ be a cyclic quadrilateral with $\angle BAD < \angle ADC$. Let $M$ be the midpoint of the arc $CD$ not containing $A$. Suppose there is a point $P$ inside $ABCD$ such that $\angle ADB = \angle CPD$ and $\angle ADP = \angle PCB$. Prove that lines $AD, PM$, and $BC$ are concurrent.

As shown in the figure below, the in-circle of $ABC$ is tangent to sides $AB$ and $AC$ at $D$ and $E$ respectively, and $O$ is the circumcenter of $BCI$. Prove that $\angle ODB = \angle OEC$. [asy]import graph; size(5.55cm); pathpen=linewidth(0.7); pointpen=black; pen fp=fontsize(10); pointfontpen=fp; real xmin=-5.76,xmax=4.8,ymin=-3.69,ymax=3.71; pen zzttqq=rgb(0.6,0.2,0), wwwwqq=rgb(0.4,0.4,0), qqwuqq=rgb(0,0.39,0); pair A=(-2,2.5), B=(-3,-1.5), C=(2,-1.5), I=(-1.27,-0.15), D=(-2.58,0.18), O=(-0.5,-2.92); D(A--B--C--cycle,zzttqq); D(arc(D,0.25,-104.04,-56.12)--(-2.58,0.18)--cycle,qqwuqq); D(arc((-0.31,0.81),0.25,-92.92,-45)--(-0.31,0.81)--cycle,qqwuqq); D(A--B,zzttqq); D(B--C,zzttqq); D(C--A,zzttqq); D(CR(I,1.35),linewidth(1.2)+dotted+wwwwqq); D(CR(O,2.87),linetype("2 2")+blue); D(D--O); D((-0.31,0.81)--O); D(A); D(B); D(C); D(I); D(D); D((-0.31,0.81)); D(O); MP( "A", A, dir(110)); MP("B", B, dir(140)); D("C", C, dir(20)); D("D", D, dir(150)); D("E", (-0.31, 0.81), dir(60)); D("O", O, dir(290)); D("I", I, dir(100)); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); [/asy]

In the tetrahedron $ABCD$, the point $E$ is the projection of the point $D$ on the plane $(ABC)$. Prove that the following statements are equivalent: a) $C = E$ or $CE \parallel AB$ b) For each point M belonging to the segment $CD$, the following equation is satisfied $$S^2_{\vartriangle ABM}= \frac{CM^2}{CD^2}\cdot S^2_{\vartriangle ABD}+\left(1- \frac{CM^2}{CD^2} \right)S^2_{\vartriangle ABC}$$ where $S_{\vartriangle XYZ}$ means the area of ​​triangle $XYZ$.

Consider the regular pyramid $VABCD$ with the vertex in $V$ which measures the angle formed by two opposite lateral edges is $45^o$. The points $M,N,P$ are respectively, the projections of the point $A$ on the line $VC$, the symmetric of the point $M$ with respect to the plane $(VBD)$ and the symmetric of the point $N$ with respect to $O$. ($O$ is the center of the base of the pyramid.) a) Show that the polyhedron $MDNBP$ is a regular pyramid. b) Determine the measure of the angle between the line $ND$ and the plane $(ABC) $

On the extensions of sides $CA$ and $AB$ of triangle $ABC$ beyond points $A$ and $B$, respectively, the segments $AE = BC$ and $BF = AC$ are drawn. A circle is tangent to segment $BF$ at point $N$, side $BC$ and the extension of side $AC$ beyond point $C$. Point $M$ is the midpoint of segment $EF$. Prove that the line $MN$ is parallel to the bisector of angle $A$.

In triangle $ABC$ length of altitude $CH$, with $H \in AB$, is equal to half of side $AB$. If $\angle BAC = 45^{\circ}$ find $\angle ABC$

Given a square cardboard of area $\frac{1}{4}$, and a paper triangle of area $\frac{1}{2}$ such that the square of its sidelength is a positive integer. Prove that the triangle can be folded in some ways such that the squace can be placed inside the folded figure so that both of its faces are completely covered with paper. [i]Proposed by N.Beluhov, Bulgaria[/i]

Let $O$ be the circumcenter of an acute triangle $ABC$ and $R$ be its circumcenter. Consider the disks having $OA,OB,OC$ as diameters, and let $\Delta$ be the set of points in the plane belonging to at least two of the disks. Prove that the area of $\Delta$ is greater than $R^2/8$.

Let $ABC$ be a triangle such that $AB \neq AC$. The internal bisector lines of the angles $ABC$ and $ACB$ meet the opposite sides of the triangle at points $B_0$ and $C_0$, respectively, and the circumcircle $ABC$ at points $B_1$ and $C_1$, respectively. Further, let $I$ be the incentre of the triangle $ABC$. Prove that the lines $B_0C_0$ and $B_1C_1$ meet at some point lying on the parallel through $I$ to the line $BC$. [i]Radu Gologan[/i]

The electric train traveled from platform A to platform B in $X$ minutes ($0< X<60$). Find $X$ if it is known that as at the moment departure from A, and at the time of arrival at B, the angle between hourly and the minute hand was equal to $X$ degrees.

Point $O$ is the intersection point of the heights of an acute triangle $\vartriangle ABC$. Prove that the three circles which pass: a) through $O, A, B$, b) through $O, B, C$, and c) through $O, C, A$, are equal

Let $ ABCD$ be a convex quadrilateral and let $ P$ and $ Q$ be points in $ ABCD$ such that $ PQDA$ and $ QPBC$ are cyclic quadrilaterals. Suppose that there exists a point $ E$ on the line segment $ PQ$ such that $ \angle PAE \equal{} \angle QDE$ and $ \angle PBE \equal{} \angle QCE$. Show that the quadrilateral $ ABCD$ is cyclic. [i]Proposed by John Cuya, Peru[/i]

[b]p1.[/b] Fun facts! We know that $1008^2-1007^2 = 1008+1007$ and $1009^2-1008^2 = 1009+1008$. Now compute the following: $$1010^2 - 1009^2 - 1.$$ [b]p2.[/b] Let $m$ be the smallest positive multiple of $2018$ such that the fraction $m/2019$ can be simplified. What is the number $m$? [b]p3.[/b] Given that $n$ satisfies the following equation $$n + 3n + 5n + 7n + 9n = 200,$$ find $n$. [b]p4.[/b] Grace and Somya each have a collection of coins worth a dollar. Both Grace and Somya have quarters, dimes, nickels and pennies. Serena then observes that Grace has the least number of coins possible to make one dollar and Somya has the most number of coins possible. If Grace has $G$ coins and Somya has $S$ coins, what is $G + S$? [b]p5.[/b] What is the ones digit of $2018^{2018}$? [b]p6.[/b] Kaitlyn plays a number game. Each time when Kaitlyn has a number, if it is even, she divides it by $2$, and if it is odd, she multiplies it by $5$ and adds $1$. Kaitlyn then takes the resulting number and continues the process until she reaches $1$. For example, if she begins with $3$, she finds the sequence of $6$ numbers to be $$3, 3 \cdot 5 + 1 = 16, 16/2 = 8, 8/2 = 4, 4/2 = 2, 2/2 = 1.$$ If Kaitlyn's starting number is $51$, how many numbers are in her sequence, including the starting number and the number $1$? [b]p7.[/b] Andrew likes both geometry and piano. His piano has $88$ keys, $x$ of which are white and $y$ of which are black. Each white key has area $3$ and each black key has area $11$. If the keys of his piano have combined area $880$, how many black keys does he have? [b]p8.[/b] A six-sided die contains the numbers $1$, $2$, $3$, $4$, $5$, and $6$ on its faces. If numbers on opposite faces of a die always sum to $7$, how many distinct dice are possible? (Two dice are considered the same if one can be rotated to obtain the other.) [b]p9.[/b] In $\vartriangle ABC$, $AB$ is $12$ and $AC$ is $15$. Alex draws the angle bisector of $BAC$, $AD$, such that $D$ is on $BC$. If $CD$ is $10$, then the area of $\vartriangle ABC$ can be expressed in the form $\frac{m \sqrt{n}}{p}$ where $m, p$ are relatively prime and $n$ is not divisible by the square of any prime. Find $m + n + p$. [b]p10.[/b] Find the smallest positive integer that leaves a remainder of $2$ when divided by $5$, a remainder of $3$ when divided by $6$, a remainder of $4$ when divided by $7$, and a remainder of $5$ when divided by $8$. [b]p11.[/b] Chris has a bag with $4$ marbles. Each minute, Chris randomly selects a marble out of the bag and flips a coin. If the coin comes up heads, Chris puts the marble back in the bag, while if the coin comes up tails, Chris sets the marble aside. What is the expected number of seconds it will take Chris to empty the bag? [b]p12.[/b] A real fixed point $x$ of a function $f(x)$ is a real number such that $f(x) = x$. Find the absolute value of the product of the real fixed points of the function $f(x) = x^4 + x - 16$. [b]p13.[/b] A triangle with angles $30^o$, $75^o$, $75^o$ is inscribed in a circle with radius $1$. The area of the triangle can be expressed as $\frac{a+\sqrt{b}}{c}$ where $b$ is not divisible by the square of any prime. Find $a + b + c$. [b]p14.[/b] Dora and Charlotte are playing a game involving flipping coins. On a player's turn, she first chooses a probability of the coin landing heads between $\frac14$ and $\frac34$ , and the coin magically flips heads with that probability. The player then flips this coin until the coin lands heads, at which point her turn ends. The game ends the first time someone flips heads on an odd-numbered flip. The last player to flip the coin wins. If both players are playing optimally and Dora goes first, let the probability that Charlotte win the game be $\frac{a}{b}$ . Find $a \cdot b$. [b]p15.[/b] Jonny is trying to sort a list of numbers in ascending order by swapping pairs of numbers. For example, if he has the list $1$, $4$, $3$, $2$, Jonny would swap $2$ and $4$ to obtain $1$, $2$, $3$, $4$. If Jonny is given a random list of $400$ distinct numbers, let $x$ be the expected minimum number of swaps he needs. Compute $\left \lfloor \frac{x}{20} \right \rfloor$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

In the convex $ABCDEF$ (has all interior angles less than $180^o$) with the perimeter $s$ the triangles $ACE$ and $BDF$ have perimeters $u$ and $v$ respectively. a) Show the inequalities $\frac{1}{2} \le \frac{s}{u+v}\le 1$ b) Check whether $1$ is replaced by a smaller number or $1/2$ by a larger number can the inequality remains valid for all convex hexagons.

$\angle{A}$ is the least angle in $\Delta{ABC}$. Point $D$ is on the arc $BC$ from the circumcircle of $\Delta{ABC}$. The perpendicular bisectors of the segments $AB,AC$ intersect the line $AD$ at $M,N$, respectively. Point $T$ is the meet point of $BM,CN$. Suppose that $R$ is the radius of the circumcircle of $\Delta{ABC}$. Prove that: \[ BT+CT\leq{2R}. \]

Consider a parameterized curve $C: x=e^{-t}\cos t,\ y=e^{-t}\sin t\left (0\leq t\leq \frac{\pi}{2}\right).$ (1) Find the length $L$ of $C$. (2) Find the area $S$ of the region enclosed by the $x,\ y$ axis and $C$. Please solve the problem without using the formula of area for polar coordinate for Japanese High School Students who don't study it in High School. [i]1997 Kyoto University entrance exam/Science[/i]