Olympiad problems

2009 Balkan MO Shortlist, G6

Tags: trapezoid , reflection , rotation , circumcircle , geometric transformation , trigonometry , geometry

Two circles $O_1$ and $O_2$ intersect each other at $M$ and $N$. The common tangent to two circles nearer to $M$ touch $O_1$ and $O_2$ at $A$ and $B$ respectively. Let $C$ and $D$ be the reflection of $A$ and $B$ respectively with respect to $M$. The circumcircle of the triangle $DCM$ intersect circles $O_1$ and $O_2$ respectively at points $E$ and $F$ (both distinct from $M$). Show that the circumcircles of triangles $MEF$ and $NEF$ have same radius length.

KoMaL A Problems 2022/2023, A. 844

Tags: reflection , geometry , geometric transformation , circumcircle

The inscribed circle of triangle $ABC$ is tangent to sides $BC$, $AC$ and $AB$ at points $D$, $E$ and $F$, respectively. Let $E'$ be the reflection of point $E$ across line $DF$, and $F'$ be the reflection of point $F$ across line $DE$. Let line $EF$ intersect the circumcircle of triangle $AE'F'$ at points $X$ and $Y$. Prove that $DX=DY$. [i]Proposed by Márton Lovas, Budapest[/i]

2020-21 KVS IOQM India, 19

Tags: geometry , semicircle , paper folding , folding

A semicircular paper is folded along a chord such that the folded circular arc is tangent to the diameter of the semicircle. The radius of the semicircle is $4$ units and the point of tangency divides the diameter in the ratio $7 :1$. If the length of the crease (the dotted line segment in the figure) is $\ell$ then determine $ \ell^2$. [img]https://cdn.artofproblemsolving.com/attachments/5/6/63fed83742c8baa92d9e63962a77a57d43556f.png[/img]

2015 JHMT, 8

Tags: geometry

In a triangle $ABC$, let $D$ and $E$ trisect $BC$, so $BD = DE = EC$. Let $F$ be the point on $AB$ such that $\frac{AF}{F B}= 2$, and $G$ on $AC$ such that $\frac{AG}{GC} =\frac12$ . Let $P$ be the intersection of $DG$ and $EF$, and extend $AP$ to intersect $BC$ at a point $X$. Find $\frac{BX}{XC}$

1980 Swedish Mathematical Competition, 3

Tags: geometry , combinatorics , integer , combinatorial geometry

Let $T(n)$ be the number of dissimilar (non-degenerate) triangles with all side lengths integral and $\leq n$. Find $T(n+1)-T(n)$.

1977 AMC 12/AHSME, 23

Tags: geometry , quadratic , 3d geometry , vieta

If the solutions of the equation $x^2+px+q=0$ are the cubes of the solutions of the equation $x^2+mx+n=0$, then $\textbf{(A) }p=m^3+3mn\qquad\textbf{(B) }p=m^3-3mn\qquad$ $\textbf{(C) }p+q=m^3\qquad\textbf{(D) }\left(\frac{m}{n}\right)^2=\frac{p}{q}\qquad \textbf{(E) }\text{none of these}$

Durer Math Competition CD 1st Round - geometry, 2016.C1

Tags: concyclic , geometry

Let $P$ be an arbitrary point of the side line $AB$ of the triangle $ABC$. Mark the perpendicular projection of $P$ on the side lines $AC$ and $BC$ as $A_1$ and $B_1$ respectively. Denote $C_1$ he foot of the alttiude starting from $C$. Prove that the points $A_1$, $B_1$, $C_1$, $C$ and $P$ lie on a circle.

Indonesia Regional MO OSP SMA - geometry, 2005.4

Tags: geometry , integer , perimeter , area

The lengths of the three sides $a, b, c$ with $a \le b \le c$, of a right triangle is an integer. Find all the sequences $(a, b, c)$ so that the values of perimeter and area of the triangle are the same.

2017 Hanoi Open Mathematics Competitions, 13

Tags: distance , geometric inequalities , inequalities , geometry

Let $ABC$ be a triangle. For some $d>0$ let $P$ stand for a point inside the triangle such that $|AB| - |P B| \ge d$, and $|AC | - |P C | \ge d$. Is the following inequality true $|AM | - |P M | \ge d$, for any position of $M \in BC $?

VI Soros Olympiad 1999 - 2000 (Russia), 9.5

Tags: concurrency , trapezoid , concyclic , geometry

A straight line is drawn through an arbitrary internal point $K$ of the trapezoid $ABCD$, intersecting the bases of $BC$ and $AD$ at points $P$ and $Q$, respectively. The circles circumscribed around the triangles $BPK$ and $DQK$ intersect, besides the point $K$, also at the point $L$. Prove that the point $L$ lies on the diagonal $BD$.

2006 Austrian-Polish Competition, 3

Tags: tetrahedron , geometry , 3d geometry

$ABCD$ is a tetrahedron. Let $K$ be the center of the incircle of $CBD$. Let $M$ be the center of the incircle of $ABD$. Let $L$ be the gravycenter of $DAC$. Let $N$ be the gravycenter of $BAC$. Suppose $AK$, $BL$, $CM$, $DN$ have one common point. Is $ABCD$ necessarily regular?

1981 IMO Shortlist, 11

Tags: inequalities , law of sines , quadrilateral , trigonometry , geometry

On a semicircle with unit radius four consecutive chords $AB,BC, CD,DE$ with lengths $a, b, c, d$, respectively, are given. Prove that \[a^2 + b^2 + c^2 + d^2 + abc + bcd < 4.\]

MathLinks Contest 3rd, 3

Tags: geometry , combinatorics

An integer point of the usual Euclidean $3$-dimensional space is a point whose three coordinates are all integers. A set $S$ of integer points is called a [i]covered [/i] set if for all points $A, B$ in $S$ each integer point in the segment $[AB]$ is also in $S$. Determine the maximum number of elements that a covered set can have if it does not contain $2004$ collinear points.

2019 Yasinsky Geometry Olympiad, p4

Tags: isosceles , geometry , circumcircle , isosceles triangle

Let $ABC$ be a triangle, $O$ is the center of the circle circumscribed around it, $AD$ the diameter of this circle. It is known that the lines $CO$ and $DB$ are parallel. Prove that the triangle $ABC$ is isosceles. (Andrey Mostovy)

2022 Iranian Geometry Olympiad, 5

Tags: geometry , equilateral triangle , square

a) Do there exist four equilateral triangles in the plane such that each two have exactly one vertex in common, and every point in the plane lies on the boundary of at most two of them? b) Do there exist four squares in the plane such that each two have exactly one vertex in common, and every point in the plane lies on the boundary of at most two of them? (Note that in both parts, there is no assumption on the intersection of interior of polygons.) [i]Proposed by Hesam Rajabzadeh[/i]

2012 Saint Petersburg Mathematical Olympiad, 6

Tags: combinatorics , geometry

On the coordinate plane in the first quarter there are $100$ non-intersecting single unit segments parallel to the coordinate axes. These segments aremirrors (on both sides), they reflect the light according to the rule. "The angle of incidence is equal to the angle of reflection." (If you hit the edge of the mirror, the beam of light does not change its direction.) From the point lying in the unit circle with the center at the origin, a ray of light in the direction of the bisector of the first coordinate angle. Prove that, that this initial point can be chosen so that the ray is reflected from the mirrors not more than $150$ times.

2019 ISI Entrance Examination, 5

Tags: convex quadrilateral , convex set , geometry

A subset $\bf{S}$ of the plane is called [i]convex[/i] if given any two points $x$ and $y$ in $\bf{S}$, the line segment joining $x$ and $y$ is contained in $\bf{S}$. A quadrilateral is called [i]convex[/i] if the region enclosed by the edges of the quadrilateral is a convex set. Show that given a convex quadrilateral $Q$ of area $1$, there is a rectangle $R$ of area $2$ such that $Q$ can be drawn inside $R$.

2008 Iran MO (2nd Round), 3

Tags: geometry , circumcircle , trigonometry

In triangle $ABC$, $H$ is the foot of perpendicular from $A$ to $BC$. $O$ is the circumcenter of $\Delta ABC$. $T,T'$ are the feet of perpendiculars from $H$ to $AB,AC$, respectively. We know that $AC=2OT$. Prove that $AB=2OT'$.

1988 Poland - Second Round, 6

Tags: geometry , vector , 3d geometry

Given is a convex polyhedron with $ k $ faces $ S_1, \ldots, S_k $. Let us denote the vector of length 1 perpendicular to the wall $ S_i $ ($ i = 1, \ldots, k $) directed outside the given polyhedron by $ \overrightarrow{n_i} $, and the surface area of this wall by $ P_i $. Prove that $$ \sum_{i=1}^k P_i \cdot \overrightarrow{n_i} = \overrightarrow{0}.$$

2001 Estonia Team Selection Test, 6

Tags: geometry , circumcircle , incircle

Let $C_1$ and $C_2$ be the incircle and the circumcircle of the triangle $ABC$, respectively. Prove that, for any point $A'$ on $C_2$, there exist points $B'$ and $C'$ such that $C_1$ and $C_2$ are the incircle and the circumcircle of triangle $A'B'C'$, respectively.

2007 Tournament Of Towns, 3

Tags: geometry

Give a construction by straight-edge and compass of a point $C$ on a line $\ell$ parallel to a segment $AB$, such that the product $AC \cdot BC$ is minimum.

1999 Dutch Mathematical Olympiad, 2

Tags: modular arithmetic , geometry , rectangle

A $9 \times 9$ square consists of $81$ unit squares. Some of these unit squares are painted black, and the others are painted white, such that each $2 \times 3$ rectangle and each $3 \times 2$ rectangle contain exactly 2 black unit squares and 4 white unit squares. Determine the number of black unit squares.

2019 USA IMO Team Selection Test, 1

Tags: geometry

Let $ABC$ be a triangle and let $M$ and $N$ denote the midpoints of $\overline{AB}$ and $\overline{AC}$, respectively. Let $X$ be a point such that $\overline{AX}$ is tangent to the circumcircle of triangle $ABC$. Denote by $\omega_B$ the circle through $M$ and $B$ tangent to $\overline{MX}$, and by $\omega_C$ the circle through $N$ and $C$ tangent to $\overline{NX}$. Show that $\omega_B$ and $\omega_C$ intersect on line $BC$. [i]Merlijn Staps[/i]

2007 China National Olympiad, 1

Tags: circumcircle , geometry , power of a point , radical axis , incenter , trigonometry

Let $O, I$ be the circumcenter and incenter of triangle $ABC$. The incircle of $\triangle ABC$ touches $BC, CA, AB$ at points $D, E, F$ repsectively. $FD$ meets $CA$ at $P$, $ED$ meets $AB$ at $Q$. $M$ and $N$ are midpoints of $PE$ and $QF$ respectively. Show that $OI \perp MN$.

1994 China Team Selection Test, 3

Tags: geometry , reflection , trapezoid , symmetry , combinatorics , geometric transformation

Find the smallest $n \in \mathbb{N}$ such that if any 5 vertices of a regular $n$-gon are colored red, there exists a line of symmetry $l$ of the $n$-gon such that every red point is reflected across $l$ to a non-red point.