Two circles in a plane intersect. $A$ is one of the points of intersection. Starting simultaneously from $A$ two points move with constant speed, each travelling along its own circle in the same sense. The two points return to $A$ simultaneously after one revolution. Prove that there is a fixed point $P$ in the plane such that the two points are always equidistant from $P.$

Find $\lim_{a\rightarrow +0} \int_a^1 \frac{x\ln x}{(1+x)^3}dx.$ [i]2010 Nara Medical University entrance exam[/i]

A cow lives on a cubic planet of side length $12$. It is tied on a leash $12$ units long that is staked at the center of one of the faces of the cube. The total surface area that the cow can graze is $A \pi+B( \sqrt3 -1)$. Find $A + B$.

We have a chessboard and we call a $1\times1$ square a room. A robot is standing on one arbitrary vertex of the rooms. The robot starts to move and in every one movement, he moves one side of a room. This robot has $2$ memories $A,B$. At first, the values of $A,B$ are $0$. In each movement, if he goes up, $1$ unit is added to $A$, and if he goes down, $1$ unit is waned from $A$, and if he goes right, the value of $A$ is added to $B$, and if he goes left, the value of $A$ is waned from $B$. Suppose that the robot has traversed a traverse (!) which hasn’t intersected itself and finally, he has come back to its initial vertex. If $v(B)$ is the value of $B$ in the last of the traverse, prove that in this traverse, the interior surface of the shape that the robot has moved on its circumference is equal to $|v(B)|$.

Prove that, for any integers $a,b,c$, there exists a positive integer $n$ such that $\sqrt{n^3+an^2+bn+c}$ is not an integer.

All vertices of a convex polyhedron with 100 edges are cut off by some planes. The planes do not intersect either inside or on the surface of the polyhedron. For this new polyhedron find a) the number of vertices; [i](1 point)[/i] b) the number of edges. [i](2 points)[/i]

Let $ABCDEF$ be a convex hexagon with the following properties. (a) $\overline{AC}$ and $\overline{AE}$ trisect $\angle BAF$. (b) $\overline{BE} \parallel \overline{CD}$ and $\overline{CF} \parallel \overline{DE}$. (c) $AB = 2AC = 4AE = 8AF$. Suppose that quadrilaterals $ACDE$ and $ADEF$ have area $2014$ and $1400$, respectively. Find the area of quadrilateral $ABCD$.

Points $ A_{1}$, $ B_{1}$, $ C_{1}$ are chosen on the sides $ BC$, $ CA$, $ AB$ of a triangle $ ABC$ respectively. The circumcircles of triangles $ AB_{1}C_{1}$, $ BC_{1}A_{1}$, $ CA_{1}B_{1}$ intersect the circumcircle of triangle $ ABC$ again at points $ A_{2}$, $ B_{2}$, $ C_{2}$ respectively ($ A_{2}\neq A, B_{2}\neq B, C_{2}\neq C$). Points $ A_{3}$, $ B_{3}$, $ C_{3}$ are symmetric to $ A_{1}$, $ B_{1}$, $ C_{1}$ with respect to the midpoints of the sides $ BC$, $ CA$, $ AB$ respectively. Prove that the triangles $ A_{2}B_{2}C_{2}$ and $ A_{3}B_{3}C_{3}$ are similar.

Let $A,B,C,D,E$ be five different points on the circumference of a circle in that (cyclic) order. Let $F$ be the intersection of chords $BD$ and $CE$. Show that if $AB=AE=AF$ then lines $AF$ and $CD$ are perpendicular.

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~$P$.

In a triangle $ABC$, let $A_0$ be the point where the interior angle bisector of angle $A$ meets with side $BC$. Similarly define $B_0$ and $C_0$. Prove that $\angle B_0A_0C_0 = 90^o$ if and only if $\angle BAC = 120^o$.

Two circles $\omega_1$ and $\omega_2$, which have centers $O_1$ and $O_2$, respectively, intersect at $A$ and $B$. A line $\ell$ that passes through $B$ cuts to $\omega_1$ again at $C$ and cuts to $\omega_2$ again in $D$, so that points $C, B, D$ appear in that order. The tangents of $\omega_1$ and $\omega_2$ in $C$ and $D$, respectively, intersect in $E$. Line $AE$ intersects again to the circumscribed circumference of the triangle $AO_1O_2$ in $F$. Try that the length of the $EF$ segment is constant, that is, it does not depend on the choice of $\ell$.

Two circles $ \gamma_{1}$ and $ \gamma_{2}$ meet at points $ X$ and $ Y$. Consider the parallel through $ Y$ to the nearest common tangent of the circles. This parallel meets again $ \gamma_{1}$ and $ \gamma_{2}$ at $ A$, and $ B$ respectively. Let $ O$ be the center of the circle tangent to $ \gamma_{1},\gamma_{2}$ and the circle $ AXB$, situated outside $ \gamma_{1}$ and $ \gamma_{2}$ and inside the circle $ AXB.$ Prove that $ XO$ is the bisector line of the angle $ \angle{AXB}.$ [b]Radu Gologan[/b]

Let $ABC$ be an acute triangle with orthocenter $H$ and altitudes $AA_1$, $BB_1$, $CC_1$. The lines $AB$ and $A_1B_1$ intersect at $C_2$ and $\ell_C$ is the line through the midpoint of $CH$, perpendicular to $CC_2$. The lines $\ell_A$ and $\ell_B$ are defined analogously. Prove that the lines $\ell_A$, $\ell_B$ and $\ell_C$ are concurrent.

The altitudes $AA_1$,$BB_1$,$CC_1$ of $\triangle{ABC}$ intersect at $H$.$O$ is the circumcenter of $\triangle{ABC}$.Let $A_2$ be the reflection of $A$ wrt $B_1C_1$.Prove that: a)$O$,$A_2$,$B_1$,$C$ are all on a circle b)$O$,$H$,$A_1$,$A_2$ are all on a circle

The triangular plot of ACD lies between Aspen Road, Brown Road and a railroad. Main Street runs east and west, and the railroad runs north and south. The numbers in the diagram indicate distances in miles. The width of the railroad track can be ignored. How many square miles are in the plot of land ACD? [asy] size(250); defaultpen(linewidth(0.55)); pair A=(-6,0), B=origin, C=(0,6), D=(0,12); pair ac=C+2.828*dir(45), ca=A+2.828*dir(225), ad=D+2.828*dir(A--D), da=A+2.828*dir(D--A), ab=(2.828,0), ba=(-6-2.828, 0); fill(A--C--D--cycle, gray); draw(ba--ab); draw(ac--ca); draw(ad--da); draw((0,-1)--(0,15)); draw((1/3, -1)--(1/3, 15)); int i; for(i=1; i<15; i=i+1) { draw((-1/10, i)--(13/30, i)); } label("$A$", A, SE); label("$B$", B, SE); label("$C$", C, SE); label("$D$", D, SE); label("$3$", (1/3,3), E); label("$3$", (1/3,9), E); label("$3$", (-3,0), S); label("Main", (-3,0), N); label(rotate(45)*"Aspen", A--C, SE); label(rotate(63.43494882)*"Brown", A--D, NW); [/asy] $\textbf{(A)}\ 2\qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 4.5 \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ 9$

We are given convex quadrilateral $ABCD$. The midpoints of $BC$ and $DA$ are $M$ and $N$ respectively. The diagonal $AC$ divides $MN$ in half. Prove that the areas of triangles $ABC$ and $ACD$ are equal .

Let $J$ be the $A$-excenter of an acute triangle $ABC$. Let $X$, $Y$ be two points on the circumcircle of the triangle $ACJ$ such that $\overline{JX} = \overline{JY} < \overline{JC}$. Let $P$ be a point lies on $XY$ such that $PB$ is tangent to the circumcircle of the triangle $ABC$. Let $Q$ be a point lies on the circumcircle of the triangle $BXY$ such that $BQ$ is parallel to $AC$. Prove that $\angle BAP = \angle QAC$. [i]Proposed by Li4.[/i]

Let $ABC$ be a triangle with $\angle C=90^\circ$. A line joining the midpoint of its altitude $CH$ and the vertex $A$ meets $CB$ at point $K$. Let $L$ be the midpoint of $BC$ ,and $T$ be a point of segment $AB$ such that $\angle ATK=\angle LTB$. It is known that $BC=1$. Find the perimeter of triangle $KTL$.

We are given a convex quadrilateral $ABCD$ in the plane. ([i]i[/i]) If there exists a point $P$ in the plane such that the areas of $\triangle ABP, \triangle BCP, \triangle CDP, \triangle DAP$ are equal, what condition must be satisfied by the quadrilateral $ABCD$? ([i]ii[/i]) Find (with proof) the maximum possible number of such point $P$ which satisfies the condition in ([i]i[/i]).

Let $ A$, $ B$, $ C$, $ A^{\prime}$, $ B^{\prime}$, $ C^{\prime}$, $ X$, $ Y$, $ Z$, $ X^{\prime}$, $ Y^{\prime}$, $ Z^{\prime}$ and $ P$ be pairwise distinct points in space such that $ A^{\prime} \in BC;\ B^{\prime}\in CA;\ C^{\prime}\in AB;\ X^{\prime}\in YZ;\ Y^{\prime}\in ZX;\ Z^{\prime}\in XY;$ $ P \in AX;\ P\in BY;\ P\in CZ;\ P\in A^{\prime}X^{\prime};\ P\in B^{\prime}Y^{\prime};\ P\in C^{\prime}Z^{\prime}$. Prove that $ \frac {BA^{\prime}}{A^{\prime}C}\cdot\frac {CB^{\prime}}{B^{\prime}A}\cdot\frac {AC^{\prime}}{C^{\prime}B} \equal{} \frac {YX^{\prime}}{X^{\prime}Z}\cdot\frac {ZY^{\prime}}{Y^{\prime}X}\cdot\frac {XZ^{\prime}}{Z^{\prime}Y}$.

In an obtuse triangle $ABC$ $(AB>AC)$,$O$ is the circumcentre and $D,E,F$ are the midpoints of $BC,CA,AB$ respectively.Median $AD$ intersects $OF$ and $OE$ at $M$ and $N$ respectively.$BM$ meets $CN$ at point $P$.Prove that $OP\perp AP$

In triangle $\vartriangle ABC$, $E$ and $F$ are the feet of the altitudes from $B$ to $\overline{AC}$ and $C$ to $\overline{AB}$, respectively. Line $\overleftrightarrow{BC}$ and the line through $A$ tangent to the circumcircle of $ABC$ intersect at $X$. Let $Y$ be the intersection of line $\overleftrightarrow{EF}$ and the line through $A$ parallel to $\overline{BC}$. If $XB = 4$, $BC = 8$, and $EF = 4\sqrt3$, compute $XY$.

A circle is divided into $2n$ equal arcs by $2n$ points. Find all $n>1$ such that these points can be joined in pairs using $n$ segments, all of different lengths and such that each point is the endpoint of exactly one segment.

The circumference with the radius $100$ cm is drawn on the cross-lined paper with the side of the squares $1$ cm. It neither comes through the vertices of the squares, nor touches the lines. How many squares can it pass through?