(a) Find two quadruples of positive integers $(a,b, c,n)$, each with a different value of $n$ greater than $3$, such that $$\frac{a}{b} +\frac{b}{c} +\frac{c}{a} = n$$ (b) Show that if $a,b, c$ are nonzero integers such that $\frac{a}{b} +\frac{b}{c} +\frac{c}{a}$ is an integer, then $abc$ is a perfect cube. (A perfect cube is a number of the form $n^3$, where $n$ is an integer.)

Find the number of trapeziums that it can be formed with the vertices of a regular polygon.

In a circle $\Gamma_{1}$, centered at $O$, $AB$ and $CD$ are two unequal in length chords intersecting at $E$ inside $\Gamma_{1}$. A circle $\Gamma_{2}$, centered at $I$ is tangent to $\Gamma_{1}$ internally at $F$, and also tangent to $AB$ at $G$ and $CD$ at $H$. A line $l$ through $O$ intersects $AB$ and $CD$ at $P$ and $Q$ respectively such that $EP = EQ$. The line $EF$ intersects $l$ at $M$. Prove that the line through $M$ parallel to $AB$ is tangent to $\Gamma_{1}$

The angles at the vertices $A$ and $C$ in the convex quadrilateral $ABCD$ are not acute. Points $K, L, M$ and $N$ are marked on the sides $AB, BC, CD$ and $DA$ respectively. Prove that the perimeter of $KLMN$ is not less than the double length of the diagonal $AC$.

In acute-angled triangle $ABC$ altitudes $BE,CF$ meet at $H$. A perpendicular line is drawn from $H$ to $EF$ and intersects the arc $BC$ of circumcircle of $ABC$ (that doesn’t contain $A$) at $K$. If $AK,BC$ meet at $P$, prove that $PK=PH$.

$(GDR 4)$ Find the maximal number of regions into which a sphere can be partitioned by $n$ circles.

Pete has an instrument which can locate the midpoint of a line segment, and also the point which divides the line segment into two segments whose lengths are in a ratio of $n : (n + 1)$, where $n$ is any positive integer. Pete claims that with this instrument, he can locate the point which divides a line segment into two segments whose lengths are at any given rational ratio. Is Pete right?

Let $ABCD$ be a convex quadrilateral with $\angle B = \angle D = 90^{\circ}$. Let $E$ be the point of intersection of $BC$ with $AD$ and let $M$ be the midpoint of $AE$. On the extension of $CD$, beyond the point $D$, we pick a point $Z$ such that $MZ = \frac{AE}{2}$. Let $U$ and $V$ be the projections of $A$ and $E$ respectively on $BZ$. The circumcircle of the triangle $DUV$ meets again $AE$ at the point $L$. If $I$ is the point of intersection of $BZ$ with $AE$, prove that the lines $BL$ and $CI$ intersect on the line $AZ$.

Two squares and an isosceles triangle are positioned as shown in the figure (the up left vertex of the large square lies on the side of the triangle). Prove that points $A, B$ and $C$ are collinear. [img]https://cdn.artofproblemsolving.com/attachments/d/c/03515e40f74ced1f8243c11b3e610ef92137ac.png[/img]

In triangular pyramid $S-ABC$, any two of $SA,SB,SC$ are perpendicular. $M$ is the centre of gravity of $\triangle ABC$. $D$ is the midpoint of $AB$, line $DP//SC$. Prove: [b](a)[/b] $DP$ and $SM$ intersect. [b](b)[/b] $DP\cap SM=D'$, then $D'$ is the center of circumsphere of $S-ABC$.

Given an acute triangle $ABC$. Point $H$ denotes the foot of the altitude drawn from $A$. Prove that $$AB + AC \ge BC cos \angle BAC + 2AH sin \angle BAC$$

A sphere is inscribed in a truncated right circular cone as shown. The volume of the truncated cone is twice that of the sphere. What is the ratio of the radius of the bottom base of the truncated cone to the radius of the top base of the truncated cone? [asy] real r=(3+sqrt(5))/2; real s=sqrt(r); real Brad=r; real brad=1; real Fht = 2*s; import graph3; import solids; currentprojection=orthographic(1,0,.2); currentlight=(10,10,5); revolution sph=sphere((0,0,Fht/2),Fht/2); //draw(surface(sph),green+white+opacity(0.5)); //triple f(pair t) {return (t.x*cos(t.y),t.x*sin(t.y),t.x^(1/n)*sin(t.y/n));} triple f(pair t) { triple v0 = Brad*(cos(t.x),sin(t.x),0); triple v1 = brad*(cos(t.x),sin(t.x),0)+(0,0,Fht); return (v0 + t.y*(v1-v0)); } triple g(pair t) { return (t.y*cos(t.x),t.y*sin(t.x),0); } surface sback=surface(f,(3pi/4,0),(7pi/4,1),80,2); surface sfront=surface(f,(7pi/4,0),(11pi/4,1),80,2); surface base = surface(g,(0,0),(2pi,Brad),80,2); draw(sback,rgb(0,1,0)); draw(sfront,rgb(.3,1,.3)); draw(base,rgb(.4,1,.4)); draw(surface(sph),rgb(.3,1,.3)); [/asy] $ \textbf {(A) } \dfrac {3}{2} \qquad \textbf {(B) } \dfrac {1+\sqrt{5}}{2} \qquad \textbf {(C) } \sqrt{3} \qquad \textbf {(D) } 2 \qquad \textbf {(E) } \dfrac {3+\sqrt{5}}{2} $

Take a point $O$ inside $\Delta A B C$ such that $\angle B O C=90^{\circ}$, $\angle C A O=\angle A B O$, $\angle B A O=\angle B C O .$ Find the value of $\frac{A C}{O C}$ [list=1] [*] $\sqrt{2}$ [*] $\sqrt{\frac{3}{2}}$ [*] 2 [*] None of these [/list]

In a convex quadrilateral $ABCD$, $P$ and $Q$ are points on sides $BC$ and $DC$ such that $B\hat{A}P = D\hat{A}Q$. If the line that passes through the orthocenters of $\triangle ABP$ and $\triangle ADQ$ is perpendicular to $AC$, prove that the area of these triangles are equals.

In triangle $ABC$, points $D, E$, and $F$ are marked on sides $AC, BC$, and $AB$ respectively, so that $AD = AB$, $EC = DC$, $BF = BE$. After that, they erased everything except points $E, F$ and $D$. Reconstruct the triangle $ABC$ (no study required).

Let $S$ be the incenter of triangle $ABC$. $A_1, B_1, C_1$ are the intersections of $AS, BS, CS$ with the circumcircle of triangle $ABC$ respectively. Prove that $SA_1 + SB_1 + SC_1 \geq SA + SB + SC.$

Rectangle $ABCD$ has $AB=6$ and $BC=3$. Point $M$ is chosen on side $AB$ so that $\angle AMD = \angle CMD$. What is the degree measure of $\angle AMD$? $ \textbf{(A)}\ 15 \qquad \textbf{(B)}\ 30 \qquad \textbf{(C)}\ 45 \qquad \textbf{(D)}\ 60 \qquad \textbf{(E)}\ 75 $

$ABCD$ is inscribed. Bisector of angle between diagonals intersect $AB$ anc $CD$ at $X$ and $Y$. $M,N$ are midpoints of $AD,BC$. $XM=YM$ Prove, that $XN=YN$.

Let $M$ be the intersection of two diagonal, $AC$ and $BD$, of a rhombus $ABCD$, where angle $A<90^\circ$. Construct $O$ on segment $MC$ so that $OB<OC$ and let $t=\frac{MA}{MO}$, provided that $O \neq M$. Construct a circle that has $O$ as centre and goes through $B$ and $D$. Let the intersections between the circle and $AB$ be $B$ and $X$. Let the intersections between the circle and $BC$ be $B$ and $Y$. Let the intersections of $AC$ with $DX$ and $DY$ be $P$ and $Q$, respectively. Express $\frac{OQ}{OP}$ in terms of $t$.

Consider an acute non-isosceles triangle. In a single step it is allowed to cut any one of the available triangles into two triangles along its median. Is it possible that after a finite number of cuttings all triangles will be isosceles? [i]Proposed by E. Bakaev[/i]

Let $ ABC $ be an obtuse triangle with $ AB=AC, M $ the symmetric point of $ A $ with respect to $ C, $ and $ P $ the intersection of the line $ AB $ with the perpendicular bisector of the segment $ \overline{AB} . $ Knowing that $ PM $ is perpendicular to $ BC, $ show that $ APM $ is equilateral.

A convex polygon on the plane is called [i]wide[/i] if the projection of the polygon onto any line in the same plane is a segment with length at least 1. Prove that a circle of radius $\tfrac{1}{3}$ can be placed completely inside any wide polygon.

An acute angle with the vertex $O$ and the rays $Op_1$ and $Op_2$ is given in a plane. Let $k_1$ be a circle with the center on $Op_1$ which is tangent to $Op_2$. Let $k_2$ be the circle that is tangent to both rays $Op_1$ and $Op_2$ and to the circle $k_1$ from outside. Find the locus of tangency points of $k_1$ and $k_2$ when center of $k_1$ moves along the ray $Op_1$.

Given a positive integer $n \ge 2$, let $\mathcal{P}$ and $\mathcal{Q}$ be two sets, each consisting of $n$ points in three-dimensional space. Suppose that these $2n$ points are distinct. Show that it is possible to label the points of $\mathcal{P}$ as $P_1,P_2,\dots,P_n$ and the points of $\mathcal{Q}$ as $Q_1,Q_2,\dots,Q_n$ such that for any indices $i$ and $j$, the balls of diameters $P_iQ_i$ and $P_jQ_j$ have at least one common point.

Point $X$ is chosen on side $AD$ of square $ABCD$. The inscribed circle of triangle $ABX$ touches $AX$, $BX$, and $AB$ at points $N$, $K$, and $F$, respectively. Prove that the ray $NK$ passes through the center $O$ of the square $ABCD$. (Dmytro Shvetsov)