Let $S$ be a circle with centre $O$. A chord $AB$, not a diameter, divides $S$ into two regions $R_1$ and $R_2$ such that $O$ belongs to $R_2$. Let $S_1$ be a circle with centre in $R_1$, touching $AB$ at $X$ and $S$ internally. Let $S_2$ be a circle with centre in $R_2$, touching $AB$ at $Y$, the circle $S$ internally and passing through the centre of $S$. The point $X$ lies on the diameter passing through the centre of $S_2$ and $\angle YXO=30^o$. If the radius of $S_2$ is $100 $ then what is the radius of $S_1$?

Let $ABCD$ be a trapezoid of parallel bases $ BC$ and $AD$. If $\angle CAD = 2\angle CAB, BC = CD$ and $AC = AD$, determine all the possible values of the measure of the angle $\angle CAB$.

Nine points are drawn on the surface of a regular tetrahedron with an edge of $1$ cm. Prove that among these points there are two located at a distance (in space) no greater than $0.5$ cm.

In $ \triangle{ABC}$, we have $ AB\equal{}1$ and $ AC\equal{}2$. Side $ BC$ and the median from $ A$ to $ BC$ have the same length. What is $ BC$? $ \textbf{(A)}\ \frac{1\plus{}\sqrt2}{2} \qquad \textbf{(B)}\ \frac{1\plus{}\sqrt3}{2} \qquad \textbf{(C)}\ \sqrt2 \qquad \textbf{(D)}\ \frac{3}{2} \qquad \textbf{(E)}\ \sqrt3$

The circle inscribed in the triangle $ABC$ touches the side $[AC]$ in the point $K$. Prove that the line connecting the midpoint of the side $[AC]$ with the centre of the circle halves the segment $[BK]$ .

Triangle $ ABC$ and point $ P$ in the same plane are given. Point $ P$ is equidistant from $ A$ and $ B$, angle $ APB$ is twice angle $ ACB$, and $ \overline{AC}$ intersects $ \overline{BP}$ at point $ D$. If $ PB \equal{} 3$ and $ PD \equal{} 2$, then $ AD\cdot CD \equal{}$ $ \textbf{(A)}\ 5\qquad \textbf{(B)}\ 6\qquad \textbf{(C)}\ 7\qquad \textbf{(D)}\ 8\qquad \textbf{(E)}\ 9$ [asy]defaultpen(linewidth(.8pt)); dotfactor=4; pair A = origin; pair B = (2,0); pair C = (3,1); pair P = (1,2.25); pair D = intersectionpoint(P--B,C--A); dot(A);dot(B);dot(C);dot(P);dot(D); label("$A$",A,SW);label("$B$",B,SE);label("$C$",C,N);label("$D$",D,NE + N);label("$P$",P,N); draw(A--B--P--cycle); draw(A--C--B--cycle);[/asy]

Let $ABC$ be a triangle, and let $T$ be a point on the extension of $AB$ beyond $B$, and $U$ a point on the extension of $AC$ beyond $C$, such that $BT = CU$. Moreover, let $R$ and $S$ be points on the extensions of $AB$ and $AC$ beyond $A$ such that $AS = AT$ and $AR = AU$. Prove that $R$, $S$, $T$, $U$ lie on a circle whose centre lies on the circumcircle of $ABC$.

Let $ Q$ be the centre of the inscribed circle of a triangle $ ABC.$ Prove that for any point $ P,$ \[ a(PA)^2 \plus{} b(PB)^2 \plus{} c(PC)^2 \equal{} a(QA)^2 \plus{} b(QB)^2 \plus{} c(QC)^2 \plus{} (a \plus{} b \plus{} c)(QP)^2, \] where $ a \equal{} BC, b \equal{} CA$ and $ c \equal{} AB.$

In a triangle $ABC$, $\angle C=45$. $AD$ is the altitude of the triangle. $X$ is on $AD$ such that $\angle XBC=90-\angle B$ ($X$ is in the triangle). $AD$ and $CX$ cut the circumcircle of $ABC$ in $M$ and $N$ respectively. if tangent to circumcircle of $ABC$ at $M$ cuts $AN$ at $P$, prove that $P$,$B$ and $O$ are collinear.(25 points) the exam time was 4 hours and 30 minutes.

Replacing any of the coefficients of quadratic trinomial $f(x)=ax^2+bx+c$ with an $1$ will result in a quadratic trinomial with at least one real root. Prove that the resulting trinomial attains a negative value at at least one point. EDIT: Oops I failed, added "with a 1." Also, I am sorry for not knowing these are posted already, however, these weren't posted in the contest lab yet, which made me think they weren't translated yet. Note: fresh translation

$D$ and $E$ are points on the sides $BC$ and $AC$ of a triangle $ABC$ such that $AD$ and $BE$ are angle bisectors of the triangle $ABC$. If $DE$ bisects $\angle ADC$, find $\angle A$. (SI Tokarev)

A plane is coloured into black and white squares in a chessboard pattern. Then, all the white squares are coloured red and blue such that any two initially white squares that share a corner are different colours. (One is red and the other is blue.) Let $\ell$ be a line not parallel to the sides of any squares. For every line segment $I$ that is parallel to $\ell$, we can count the difference between the length of its red and its blue areas. Prove that for every such line $\ell$ there exists a number $C$ that exceeds all those differences that we can calculate.

As shown in the figure, $\odot O_1$ and $\odot O_2$ touches each other externally at a point $T$, quadrilateral $ABCD$ is inscribed in $\odot O_1$, and the lines $DA$, $CB$ are tangent to $\odot O_2$ at points $E$ and $F$ respectively. Line $BN$ bisects $\angle ABF$ and meets segment $EF$ at $N$. Line $FT$ meets the arc $\widehat{AT}$ (not passing through the point $B$) at another point $M$ different from $A$. Prove that $M$ is the circumcenter of $\triangle BCN$.

Let $ABC$ be a triangle with orthocenter $H$ and let $P$ be the second intersection of the circumcircle of triangle $AHC$ with the internal bisector of the angle $\angle BAC$. Let $X$ be the circumcenter of triangle $APB$ and $Y$ the orthocenter of triangle $APC$. Prove that the length of segment $XY$ is equal to the circumradius of triangle $ABC$.

In a triangle $ABC$, let $D$ and $E$ be the midpoints of $AC$ and $BC$ respectively. The distance from the midpoint of $BD$ to the midpoint of $AE$ is $4.5$. What is the length of side $AB$?

$ABCD$ is a convex quadrilateral inscribed in a circle with centre $O$, and with mutually perpendicular diagonals. Prove that the broken line $AOC$ divides the quadrilateral into two parts of equal area. (V Varvarkin)

Given $\triangle ABC$ with circumcircle $\Gamma$ and circumcentre $O$, let $X$ be a point on $\Gamma$. Let $XC_1$, $XB_1$ to be feet of perpendiculars from $X$ to lines $AB$ and $AC$. Define $\omega_C$ as the circle with centre the midpoint of $AB$ and passing through $C_1$ . Define $\omega_B$ similarly. Prove that $\omega_B$ and $\omega_C$ has a common point on $XO$.

Let $ABCD$ be a cyclic quadrilateral with $AB<CD$, and let $P$ be the point of intersection of the lines $AD$ and $BC$.The circumcircle of the triangle $PCD$ intersects the line $AB$ at the points $Q$ and $R$. Let $S$ and $T$ be the points where the tangents from $P$ to the circumcircle of $ABCD$ touch that circle. (a) Prove that $PQ=PR$. (b) Prove that $QRST$ is a cyclic quadrilateral.

Each face of a tetrahedron can be placed in a circle of radius $ 1$. Show that the tetrahedron can be placed in a sphere of radius $ \frac{3}{2\sqrt2}$.

A circumscribed quadrilateral $ABCD$ is given. It is known that $\angle{ACB} = \angle{ACD}$. On the angle bisector of $\angle{C}$, a point $E$ is marked such that $AE \bot BD$. Point $F$ is the foot of the perpendicular line from point $E$ to the side $BC$. Prove that $AB = BF$.

A triangle $ABC$ is given. $N$ and $M$ are the midpoints of $AB$ and $BC$, respectively. The bisector of angle $B$ meets the segment $MN$ at $E$. $H$ is the base of the altitude drawn from $B$ in the triangle $ABC$. The point $T$ on the circumcircle of $ABC$ is such that the circumcircles of $TMN$ and $ABC$ are tangent. Prove that points $T, H, E, B$ are concyclic. [i]Proposed by M. Yumatov[/i]

Given a triangle $ABC$ with the are $1$. Let $A',B'$ and $C' $ are the midpoints of the sides $[BC], [CA]$ and $[AB]$ respectively. What is the minimal possible area of the common part of two triangles $A'B'C'$ and $KLM$, if the points $K,L$ and $M$ are lying on the segments $[AB'], [CA']$ and $[BC']$ respectively?

A semicircle of diameter $ 1$ sits at the top of a semicircle of diameter $ 2$, as shown. The shaded area inside the smaller semicircle and outside the larger semicircle is called a lune. Determine the area of this lune. [asy]unitsize(2.5cm); defaultpen(fontsize(10pt)+linewidth(.8pt)); filldraw(Circle((0,.866),.5),grey,black); label("1",(0,.866),S); filldraw(Circle((0,0),1),white,black); draw((-.5,.866)--(.5,.866),linetype("4 4")); clip((-1,0)--(1,0)--(1,2)--(-1,2)--cycle); draw((-1,0)--(1,0)); label("2",(0,0),S);[/asy]$ \textbf{(A)}\ \frac {1}{6}\pi \minus{} \frac {\sqrt {3}}{4} \qquad \textbf{(B)}\ \frac {\sqrt {3}}{4} \minus{} \frac {1}{12}\pi \qquad \textbf{(C)}\ \frac {\sqrt {3}}{4} \minus{} \frac {1}{24}\pi\qquad\textbf{(D)}\ \frac {\sqrt {3}}{4} \plus{} \frac {1}{24}\pi$ $ \textbf{(E)}\ \frac {\sqrt {3}}{4} \plus{} \frac {1}{12}\pi$

Let $ k$ and $ n$ be integers with $ 0\le k\le n \minus{} 2$. Consider a set $ L$ of $ n$ lines in the plane such that no two of them are parallel and no three have a common point. Denote by $ I$ the set of intersections of lines in $ L$. Let $ O$ be a point in the plane not lying on any line of $ L$. A point $ X\in I$ is colored red if the open line segment $ OX$ intersects at most $ k$ lines in $ L$. Prove that $ I$ contains at least $ \dfrac{1}{2}(k \plus{} 1)(k \plus{} 2)$ red points. [i]Proposed by Gerhard Woeginger, Netherlands[/i]

A circle centred at $(a, b)$ contains the origin $(0,0)$. Denote by $S^+$ the total area of the parts of the circle in the first and third quadrants, and by $S^-$ the total area of the parts of the circle in the second and the fourth quadrants. Compute $S^+ -S^-$. (G Galperin)