The centre of the circle passing through the midpoints of the sides of am isosceles triangle $ABC$ lies on the circumcircle of triangle $ABC$. If the larger angle of triangle $ABC$ is $\alpha^{\circ}$ and the smaller one $\beta^{\circ}$ then what is the value of $\alpha-\beta$?

Let $O$ be the circumcenter of triangle $ABC$, and $\omega$ be the circumcircle of triangle $BOC$. Line $AO$ intersects with circle $\omega$ again at the point $G$. Let $M$ be the midpoint of side $BC$, and the perpendicular bisector of $BC$ meets circle $\omega$ at the points $O$ and $N$. Prove that the midpoint of the segment $AN$ lies on the radical axis of the circumcircle of triangle $OMG$, and the circle whose diameter is $AO$.

In the figure, points $A$, $B$, $C$ and $D$ are on the same line and are the centers of four tangent circles at the same point. Segment $AB$ measures $8$ and segment $CD$ measures $4$. The circumferences woth centers $A$ and $C$ are of equal size. We know that the sum of the areas of the two medium circles is equivalent to the sum of the areas of the small and large circles. What is the length of segment $AD$? [img]https://cdn.artofproblemsolving.com/attachments/d/4/378243b9f4203e103af266e551eadccfc96adf.png[/img]

Logan is constructing a scaled model of his town. The city's water tower stands $ 40$ meters high, and the top portion is a sphere that holds $ 100,000$ liters of water. Logan's miniature water tower holds $ 0.1$ liters. How tall, in meters, should Logan make his tower? $ \textbf{(A)}\ 0.04\qquad \textbf{(B)}\ \frac{0.4}{\pi}\qquad \textbf{(C)}\ 0.4\qquad \textbf{(D)}\ \frac{4}{\pi}\qquad \textbf{(E)}\ 4$

The closed curve in the figure is made up of $9$ congruent circular arcs each of length $\frac{2\pi}{3}$, where each of the centers of the corresponding circles is among the vertices of a regular hexagon of side $2$. What is the area enclosed by the curve? [asy] size(170); defaultpen(fontsize(6pt)); dotfactor=4; label("$\circ$",(0,1)); label("$\circ$",(0.865,0.5)); label("$\circ$",(-0.865,0.5)); label("$\circ$",(0.865,-0.5)); label("$\circ$",(-0.865,-0.5)); label("$\circ$",(0,-1)); dot((0,1.5)); dot((-0.4325,0.75)); dot((0.4325,0.75)); dot((-0.4325,-0.75)); dot((0.4325,-0.75)); dot((-0.865,0)); dot((0.865,0)); dot((-1.2975,-0.75)); dot((1.2975,-0.75)); draw(Arc((0,1),0.5,210,-30)); draw(Arc((0.865,0.5),0.5,150,270)); draw(Arc((0.865,-0.5),0.5,90,-150)); draw(Arc((0.865,-0.5),0.5,90,-150)); draw(Arc((0,-1),0.5,30,150)); draw(Arc((-0.865,-0.5),0.5,330,90)); draw(Arc((-0.865,0.5),0.5,-90,30)); [/asy] $ \textbf{(A)}\ 2\pi+6\qquad\textbf{(B)}\ 2\pi+4\sqrt3 \qquad\textbf{(C)}\ 3\pi+4 \qquad\textbf{(D)}\ 2\pi+3\sqrt3+2 \qquad\textbf{(E)}\ \pi+6\sqrt3 $

$AB, AC$ are two chords of the circle centered at $O$. The diameter, which is perpendicular to $BC$, intersects $AB, AC$ at $F, G$ respectively ($F$ is in the circle). The tangent from $G$ tangents the circle at $T$. Prove that $F$ is the projection of $T$ on $OG. $

(1) Evaluate the following definite integrals. (a) $\int_0^{\frac{\pi}{2}} \cos ^ 2 x\sin x\ dx$ (b) $\int_0^{\frac{\pi}{2}} (\pi - 2x)\cos x\ dx$ (c) $\int_0^{\frac{\pi}{2}} x\cos ^ 3 x\ dx$ (2) Let $a$ be a positive constant. Find the area of the cross section cut by the plane $z=\sin \theta \ \left(0\leq \theta \leq \frac{\pi}{2}\right)$ of the solid such that \[x^2+y^2+z^2\leq a^2,\ \ x^2+y^2\leq ax,\ \ z\geq 0\] , then find the volume of the solid. [i]1984 Yamanashi Medical University entrance exam[/i] Please slove the problem without multi integral or arcsine function for Japanese high school students aged 17-18 those who don't study them. Thanks in advance. kunny

Given a triangle $ABC$ and a point $O$ on a plane. Let $\Gamma$ be the circumcircle of $ABC$. Suppose that $CO$ intersects with $AB$ at $D$, and $BO$ and $CA$ intersect at $E$. Moreover, suppose that $AO$ intersects with $\Gamma$ at $A,F$. Let $I$ be the other intersection of $\Gamma$ and the circumcircle of $ADE$, and $Y$ be the other intersection of $BE$ and the circumcircle of $CEI$, and $Z$ be the other intersection of $CD$ and the circumcircle of $BDI$. Let $T$ be the intersection of the two tangents of $\Gamma$ at $B,C$, respectively. Lastly, suppose that $TF$ intersects with $\Gamma$ again at $U$, and the reflection of $U$ w.r.t. $BC$ is $G$. Show that $F,I,G,O,Y,Z$ are concyclic.

The perimeter of a rectangle is $100$ and its diagonal has length $x$. What is the area of this rectangle? $\textbf{(A) }625-x^2\qquad\textbf{(B) }625-\dfrac{x^2}2\qquad\textbf{(C) }1250-x^2\qquad\textbf{(D) }1250-\dfrac{x^2}2\qquad\textbf{(E) }2500-\dfrac{x^2}2$

In the plane, consider a circle with center $S$ and radius $1.$ Let $ABC$ be an arbitrary triangle having this circle as its incircle, and assume that $SA\leq SB\leq SC.$ Find the locus of [b]a.)[/b] all vertices $A$ of such triangles; [b]b.)[/b] all vertices $B$ of such triangles; [b]c.)[/b] all vertices $C$ of such triangles.

The numbers $a$, $b$, $c$, $d$ satisfy $0<a \leq b \leq d \leq c$ and ${a+c=b+d}$. Prove that for every internal point $P$ of a segment with length $a$ this segment is a side of a circumscribed quadrilateral with consecutive sides $a$, $b$, $c$, $d$, such that its incircle contains~$P$.

Find the geometric locus of the center of a rectangle whose vertices lie on the perimeter of a given triangle.

Let $ P$ be a point on the diagonal $ BD$ of a rectangle $ ABCD$, $ F$ be the projection of $ P$ on $ BC$, and $ H \not\equal{} B$ be the point on $ BC$ such that $ BF\equal{}FH$. If lines $ PC$ and $ AH$ intersect at $ Q$, prove that the areas of triangles $ APQ$ and $ CHQ$ are equal.

$AD$ and $BE$ are altitudes of an acute triangle $ABC$. From $D$, perpendiculars are dropped to $AB$ at $G$ and $AC$ at $K$. From $E$, perpendiculars are dropped to $AB$ at $F$ and $BC$ at $H$. Prove that $FG$ is parallel to $HK$ and $FK = GH$.

In the non-isosceles triangle $ABC$,$D$ is the midpoint of side $BC$,$E$ is the midpoint of side $CA$,$F$ is the midpoint of side $AB$.The line(different from line $BC$) that is tangent to the inscribed circle of triangle $ABC$ and passing through point $D$ intersect line $EF$ at $X$.Define $Y,Z$ similarly.Prove that $X,Y,Z$ are collinear.

$A$ and $B$ are distinct points on a circle $T$. $C$ is a point distinct from $B$ such that $|AB|=|AC|$, and such that $BC$ is tangent to $T$ at $B$. Suppose that the bisector of $\angle ABC$ meets $AC$ at a point $D$ inside $T$. Show that $\angle ABC>72^\circ$.

The vertex $E$ of a square $EFGH$ is at the center of square $ABCD$. The length of a side of $ABCD$ is $1$ and the length of a side of $EFGH$ is $2$. Side $EF$ intersects $CD$ at $I$ and $EH$ intersects $AD$ at $J$. If angle $EID=60^\circ$, the area of quadrilateral $EIDJ$ is $\textbf{(A) }\dfrac14\qquad\textbf{(B) }\dfrac{\sqrt3}6\qquad\textbf{(C) }\dfrac13\qquad\textbf{(D) }\dfrac{\sqrt2}4\qquad\textbf{(E) }\dfrac{\sqrt3}2$

Let $ABCD$ be a trapezium inscribed in a circle $k$ with diameter $AB$. A circle with center $B$ and radius $BE$,where $E$ is the intersection point of the diagonals $AC$ and $BD$ meets $k$ at points $K$ and $L$. If the line ,perpendicular to $BD$ at $E$,intersects $CD$ at $M$,prove that $KM\perp DL$.

$ABC$ is an isosceles triangle, with $AB=AC$. $D$ is a moving point such that $AD\parallel BC$, $BD>CD$. Moving point $E$ is on the arc of $BC$ in circumcircle of $ABC$ not containing $A$, such that $EB<EC$. Ray $BC$ contains point $F$ with $\angle ADE=\angle DFE$. If ray $FD$ intersects ray $BA$ at $X$, and intersects ray $CA$ at $Y$, prove that $\angle XEY$ is a fixed angle.

a) Find the locus of centroids for triangles whose vertices lie on the sides of a given triangle (each side contains a single vertex). b) Find the locus of centroids for tetrahedrons whose vertices lie on the faces of a given tetrahedron (each face contains a single vertex).

Segments $AC$ and $BD$ intersect at point $P$ such that $PA = PD$ and $PB = PC$. Let $E$ be the foot of the perpendicular from $P$ to the line $CD$. Prove that the line $PE$ and the perpendicular bisectors of the segments $PA$ and $PB$ are concurrent.

$2021$ points on the plane in the convex position, no three collinear and no four concyclic, are given. Prove that there exist two of them such that every circle passing through these two points contains at least $673$ of the other points in its interior. (A finite set of points on the plane are in convex position if the points are the vertices of a convex polygon.)

In the acute triangle $ABC$, with $AB \ne BC$, let $T$ denote the midpoint of the side $[AC], A_1$ and $C_1$ denote the feet of the altitudes drawn from $A$ and $C$, respectively. Let $Z$ be the intersection point of the tangents in $A$ and $C $ to the circumcircle of triangle $ABC, X$ be the intersection point of lines $ZA$ and $A_1C_1$ and $Y$ be the intersection point of lines $ZC$ and $A_1C_1$. a) Prove that $T$ is the incircle of triangle $XYZ$. b) The circumcircles of triangles $ABC$ and $A_1BC_1$ meet again at $D$. Prove that the orthocenter $H$ of triangle $ABC$ is on the line $TD$. c) Prove that the point $D$ lies on the circumcircle of triangle $XYZ$.

Points$ D, E, F$ are chosen on the sides $AB$, $BC$, $AC$ of a triangle $ABC$, so that $DE = BE$ and $FE = CE$. Prove that the centre of the circle circumscribed around triangle $ADF$ lies on the bisectrix of angle $DEF$.

[u]Round 1[/u] [b]p1.[/b] A box contains $1$ ball labelledW, $1$ ball labelled $E$, $1$ ball labelled $L$, $1$ ball labelled $C$, $1$ ball labelled $O$, $8$ balls labelled $M$, and $1$ last ball labelled $E$. One ball is randomly drawn from the box. The probability that the ball is labelled $E$ is $\frac{1}{a}$ . Find $a$. [b]p2.[/b] Let $$G +E +N = 7$$ $$G +E +O = 15$$ $$N +T = 22.$$ Find the value of $T +O$. [b]p3.[/b] The area of $\vartriangle LMT$ is $22$. Given that $MT = 4$ and that there is a right angle at $M$, find the length of $LM$. [u]Round 2[/u] [b]p4.[/b] Kevin chooses a positive $2$-digit integer, then adds $6$ times its unit digit and subtracts $3$ times its tens digit from itself. Find the greatest common factor of all possible resulting numbers. [b]p5.[/b] Find the maximum possible number of times circle $D$ can intersect pentagon $GRASS'$ over all possible choices of points $G$, $R$, $A$, $S$, and $S'$. [b]p6.[/b] Find the sum of the digits of the integer solution to $(\log_2 x) \cdot (\log_4 \sqrt{x}) = 36$. [u]Round 3[/u] [b]p7.[/b] Given that $x$ and $y$ are positive real numbers such that $x^2 + y = 20$, the maximum possible value of $x + y$ can be written as $\frac{a}{b}$ where $a$ and $b$ are relatively prime positive integers. Find $a +b$. [b]p8.[/b] In $\vartriangle DRK$, $DR = 13$, $DK = 14$, and $RK = 15$. Let $E$ be the point such that $ED = ER = EK$. Find the value of $\lfloor DE +RE +KE \rfloor$. [b]p9.[/b] Subaru the frog lives on lily pad $1$. There is a line of lily pads, numbered $2$, $3$, $4$, $5$, $6$, and $7$. Every minute, Subaru jumps from his current lily pad to a lily pad whose number is either $1$ or $2$ greater, chosen at random from valid possibilities. There are alligators on lily pads $2$ and $5$. If Subaru lands on an alligator, he dies and time rewinds back to when he was on lily pad number $1$. Find how many times Subaru is expected to die before he reaches pad $7$. [u]Round 4[/u] [b]p10.[/b] Find the sum of the following series: $$\sum^{\infty}_{i=1} = \frac{\sum^i_{j=1} j}{2^i}=\frac{1}{2^1}+\frac{1+2}{2^2}+\frac{1+2+3}{2^3}+\frac{1+2+3+4}{2^4}+... $$ [b]p11.[/b] Let $\phi (x)$ be the number of positive integers less than or equal to $x$ that are relatively prime to $x$. Find the sum of all $x$ such that $\phi (\phi(x)) = x -3$. Note that $1$ is relatively prime to every positive integer. [b]p12.[/b] On a piece of paper, Kevin draws a circle. Then, he draws two perpendicular lines. Finally, he draws two perpendicular rays originating from the same point (an $L$ shape). What is the maximum number of sections into which the lines and rays can split the circle? [u]Round 5 [/u] [b]p13.[/b] In quadrilateral $ABCD$, $\angle A = 90^o$, $\angle C = 60^o$, $\angle ABD = 25^o$, and $\angle BDC = 5^o$. Given that $AB = 4\sqrt3$, the area of quadrilateral $ABCD$ can be written as $a\sqrt{b}$. Find $10a +b$. [b]p14.[/b] The value of $$\sum^6_{n=2} \left( \frac{n^4 +1}{n^4 -1}\right) -2 \sum^6_{n=2}\left(\frac{n^3 -n^2+n}{n^4 -1}\right)$$ can be written as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $100m+n$. [b]p15.[/b] Positive real numbers $x$ and $y$ satisfy the following $2$ equations. $$x^{1+x^{1+x^{1+...}}}= 8$$ $$\sqrt[24]{y +\sqrt[24]{y + \sqrt[24]{y +...}}} = x$$ Find the value of $\lfloor y \rfloor$. PS. You should use hide for answers. Rounds 6-9 have been posted [url=https://artofproblemsolving.com/community/c3h3167130p28823260]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].