Found problems: 111
Brazil L2 Finals (OBM) - geometry, 2010.5
The diagonals of an cyclic quadrilateral $ABCD$ intersect at $O$. The circumcircles of triangle $AOB$ and $COD$ intersect lines $BC$ and $AD$, for the second time, at points $M, N, P$and $Q$. Prove that the $MNPQ$ quadrilateral is inscribed in a circle of center $O$.
Mathley 2014-15, 5
A quadrilateral $ABCD$ is inscribed in a circle $(O)$. Another circle $(I)$ is tangent to the diagonals $AC, BD$ at $M, N$ respectively. Suppose that $MN$ meets $AB,CD$ at $P, Q$ respectively. The circumcircle of triangle $IMN$ meets the circumcircles of $IAB, ICD$ at $K, L$ respectively, which are distinct from $I$. Prove that the lines $PK, QL$, and $OI$ are concurrent.
Tran Minh Ngoc, a student of Ho Chi Minh City College, Ho Chi Minh
2014 Saudi Arabia GMO TST, 3
Let $ABCDE$ be a cyclic pentagon such that the diagonals $AC$ and $AD$ intersect $BE$ at $P$ and $Q$, respectively, with $BP \cdot QE = PQ^2$. Prove that $BC \cdot DE = CD \cdot PQ$.
2019 Regional Competition For Advanced Students, 2
The convex pentagon $ABCDE$ is cyclic and $AB = BD$. Let point $P$ be the intersection of the diagonals $AC$ and $BE$. Let the straight lines $BC$ and $DE$ intersect at point $Q$. Prove that the straight line $PQ$ is parallel to the diagonal $AD$.
2006 Estonia Team Selection Test, 2
The center of the circumcircle of the acute triangle $ABC$ is $O$. The line $AO$ intersects $BC$ at $D$. On the sides $AB$ and $AC$ of the triangle, choose points $E$ and $F$, respectively, so that the points $A, E, D, F$ lie on the same circle. Let $E'$ and $F'$ projections of points $E$ and $F$ on side $BC$ respectively. Prove that length of the segment $E'F'$ does not depend on the position of points $E$ and $F$.
2003 Portugal MO, 4
In a village there are only $10$ houses, arranged in a circle of a radius $r$ meters. Each has is the same distance from each of the two closest houses. Every year on Sunday of Pascoa, the village priest makes the Easter visit, leaving the parish house (point $A$) and following the path described in Figure 1. This year the priest decided to take the path represented in the Figure 2. Prove that this year the priest will walk another $10r$ meters.
[img]https://cdn.artofproblemsolving.com/attachments/a/9/a6315f4a63f28741ca6fbc75c19a421eb1da06.png[/img]
1981 All Soviet Union Mathematical Olympiad, 305
Given points $A,B,M,N$ on the circumference. Two chords $[MA_1]$ and $[MA_2]$ are orthogonal to lines $(NA)$ and $(NB)$ respectively. Prove that $(AA_1)$ and $(BB_1)$ lines are parallel.
2010 Oral Moscow Geometry Olympiad, 1
Convex $n$-gon $P$, where $n> 3$, is cut into equal triangles by diagonals that do not intersect inside it. What are the possible values of $n$ if the $n$-gon is cyclic?
2014 Saudi Arabia GMO TST, 1
Let $ABC$ be a triangle with $\angle A < \angle B \le \angle C$, $M$ and $N$ the midpoints of sides $CA$ and $AB$, respectively, and $P$ and $Q$ the projections of $B$ and $C$ on the medians $CN$ and $BM$, respectively. Prove that the quadrilateral $MNPQ$ is cyclic.
2020 Novosibirsk Oral Olympiad in Geometry, 7
The quadrilateral $ABCD$ is known to be inscribed in a circle, and that there is a circle with center on side $AD$ tangent to the other three sides. Prove that $AD = AB + CD$.
1999 German National Olympiad, 3
A mathematician investigates methods of finding area of a convex quadrilateral obtains the following formula for the area $A$ of a quadrilateral with consecutive sides $a,b,c,d$:
$A =\frac{a+c}{2}\frac{b+d}{2}$ (1) and $A = \sqrt{(p-a)(p-b)(p-c)(p-d)}$ (2) where $p = (a+b+c+d)/2$.
However, these formulas are not valid for all convex quadrilaterals. Prove that (1) holds if and only if the quadrilateral is a rectangle, while (2) holds if and only if the quadrilateral is cyclic.
Croatia MO (HMO) - geometry, 2019.7
On the side $AB$ of the cyclic quadrilateral $ABCD$ there is a point $X$ such that diagonal $AC$ bisects the segment $DX$, and the diagonal $BD$ bisects the segment $CX$. What is the smallest possible ratio $|AB | : |CD|$ in such a quadrilateral ?
1999 Singapore MO Open, 4
Let $ABCD$ be a quadrilateral with each interior angle less than $180^o$. Show that if $A, B, C, D$ do not lie on a circle, then $AB \cdot CD + AD\cdot BC > AC \cdot BD$
1967 IMO Longlists, 20
In the space $n \geq 3$ points are given. Every pair of points determines some distance. Suppose all distances are different. Connect every point with the nearest point. Prove that it is impossible to obtain (closed) polygonal line in such a way.
2006 Tournament of Towns, 4
Given triangle $ABC, BC$ is extended beyond $B$ to the point $D$ such that $BD = BA$. The bisectors of the exterior angles at vertices $B$ and $C$ intersect at the point $M$. Prove that quadrilateral $ADMC$ is cyclic. (4)
1955 Moscow Mathematical Olympiad, 296
There are four points $A, B, C, D$ on a circle. Circles are drawn through each pair of neighboring points. Denote the intersection points of neighboring circles by $A_1, B_1, C_1, D_1$. (Some of these points may coincide with previously given ones.) Prove that points $A_1, B_1, C_1, D_1$ lie on one circle.
1989 Bundeswettbewerb Mathematik, 3
Over each side of a cyclic quadrilateral erect a rectangle whose height is equal to the length of the opposite side. Prove that the centers of these rectangles form another rectangle.
2013 Sharygin Geometry Olympiad, 1
All angles of a cyclic pentagon $ABCDE$ are obtuse. The sidelines $AB$ and $CD$ meet at point $E_1$, the sidelines $BC$ and $DE$ meet at point $A_1$. The tangent at $B$ to the circumcircle of the triangle $BE_1C$ meets the circumcircle $\omega$ of the pentagon for the second time at point $B_1$. The tangent at $D$ to the circumcircle of the triangle $DA_1C$ meets $\omega$ for the second time at point $D_1$. Prove that $B_1D_1 // AE$
1998 German National Olympiad, 6b
Prove that the following statement holds for all odd integers $n \ge 3$:
If a quadrilateral $ABCD$ can be partitioned by lines into $n$ cyclic quadrilaterals, then $ABCD$ is itself cyclic.
2017 Thailand Mathematical Olympiad, 2
A cyclic quadrilateral $ABCD$ has circumcenter $O$, its diagonals $AC$ and $BD$ intersect at $G$. Let $P, Q, R, S$ be the circumcenters of $\vartriangle AGB, \vartriangle BGC, \vartriangle CGD, \vartriangle DGA$ respectively. Lines $P R$ and $QS$ intersect at $M$. Show that $M$ is the midpoint of $OG$.
Kharkiv City MO Seniors - geometry, 2013.10.4
The pentagon $ABCDE$ is inscribed in the circle $\omega$. Let $T$ be the intersection point of the diagonals $BE$ and $AD$. A line is drawn through the point $T$ parallel to $CD$, which intersects $AB$ and $CE$ at points $X$ and $Y$, respectively. Prove that the circumscribed circle of the triangle $AXY$ is tangent to $\omega$.
1972 All Soviet Union Mathematical Olympiad, 167
The $7$-gon $A_1A_2A_3A_4A_5A_6A_7$ is inscribed in a circle. Prove that if the centre of the circle is inside the $7$-gon , than $$\angle A_1+ \angle A_2 + \angle A_3 < 450^o$$
1975 Chisinau City MO, 89
A closed line on a plane is such that any quadrangle inscribed in it has the sum of opposite angles equal to $180^o$. Prove that this line is a circle.
1967 Czech and Slovak Olympiad III A, 3
Consider a table of cyclic permutations ($n\ge2$)
\[
\begin{matrix}
1, & 2, & \ldots, & n-1, & n \\
2, & 3, & \ldots, & n, & 1, \\
\vdots & \vdots & \ddots & \vdots & \vdots \\
n, & 1, & \ldots, & n-2, & n-1.
\end{matrix}
\]
Then multiply each number of the first row by that number of the $k$-th row that is in the same column. Sum all these products and denote $s_k$ the result (e.g. $s_2=1\cdot2+2\cdot3+\cdots+(n-1)\cdot n+n\cdot1$).
a) Find a recursive relation for $s_k$ in terms of $s_{k-1}$ and determine the explicit formula for $s_k$.
b) Determine both an index $k$ and the value of $s_k$ such that the sum $s_k$ is minimal.
2017 Singapore Senior Math Olympiad, 2
In the cyclic quadrilateral $ABCD$, the sides $AB, DC$ meet at $Q$, the sides $AD,BC$ meet at $P, M$ is the midpoint of $BD$, If $\angle APQ=90^o$, prove that $PM$ is perpendicular to $AB$.