Olympiad problems

2020 JHMT, 5

Tags: geometry

Let $A$ and $B$ be fixed points in the Euclidean plane with $AB = 6$. Let $R$ be the region of points in the plane such that, for each $P \in R$, there exists a point $C$ such that $AC = 3$ and $P$ does not lie outside $\vartriangle ABC$. Compute the greatest integer less than or equal to the area of $R$.

2019 Switzerland Team Selection Test, 1

Tags: geometry , orthocenter , similarity

Let $ABC$ be a triangle and $D, E, F$ be the foots of altitudes drawn from $A,B,C$ respectively. Let $H$ be the orthocenter of $ABC$. Lines $EF$ and $AD$ intersect at $G$. Let $K$ the point on circumcircle of $ABC$ such that $AK$ is a diameter of this circle. $AK$ cuts $BC$ in $M$. Prove that $GM$ and $HK$ are parallel.

1999 Mongolian Mathematical Olympiad, Problem 6

Tags: vector , geometry , analytic geometry

Find the minimum possible length of the sum of $1999$ unit vectors in the coordinate plane whose both coordinates are nonnegative.

1973 Yugoslav Team Selection Test, Problem 2

Tags: geometry , construction

A circle $k$ is drawn using a given disc (e.g. a coin). A point $A$ is chosen on $k$. Using just the given disc, determine the point $B$ on $k$ so that $AB$ is a diameter of $k$. (You are allowed to choose an arbitrary point in one of the drawn circles, and using the given disc it is possible to construct either of the two circles that passes through the points at a distance that is smaller than the radius of the circle.)

2015 Middle European Mathematical Olympiad, 6

Tags: geometry , incenter

Let $I$ be the incentre of triangle $ABC$ with $AB>AC$ and let the line $AI$ intersect the side $BC$ at $D$. Suppose that point $P$ lies on the segment $BC$ and satisfies $PI=PD$. Further, let $J$ be the point obtained by reflecting $I$ over the perpendicular bisector of $BC$, and let $Q$ be the other intersection of the circumcircles of the triangles $ABC$ and $APD$. Prove that $\angle BAQ=\angle CAJ$.

2009 Princeton University Math Competition, 2

Tags: geometry , inradius

A triangle has sides of lengths 5, 6, 7. What is 60 times the square of the radius of the inscribed circle?

1940 Putnam, B4

Tags: geometry , 3d geometry , sphere

Prove that the locus of the point of intersection of three mutually perpendicular planes tangent to the surface $$ax^2 + by^2 +cz^2 =1\;\;\; (\text{where}\;\;abc \ne 0)$$ is the sphere $$x^2 +y^2 +z^2 =\frac{1}{a}+\frac{1}{b}+\frac{1}{c}.$$

2018 IMO Shortlist, C1

Tags: combinatorics , number theory , algebra , geometry , projective geometry

Let $n\geqslant 3$ be an integer. Prove that there exists a set $S$ of $2n$ positive integers satisfying the following property: For every $m=2,3,...,n$ the set $S$ can be partitioned into two subsets with equal sums of elements, with one of subsets of cardinality $m$.

2012 Online Math Open Problems, 28

Tags: geometry , 3d geometry , octahedron

A fly is being chased by three spiders on the edges of a regular octahedron. The fly has a speed of $50$ meters per second, while each of the spiders has a speed of $r$ meters per second. The spiders choose their starting positions, and choose the fly's starting position, with the requirement that the fly must begin at a vertex. Each bug knows the position of each other bug at all times, and the goal of the spiders is for at least one of them to catch the fly. What is the maximum $c$ so that for any $r<c,$ the fly can always avoid being caught? [i]Author: Anderson Wang[/i]

2007 Bulgaria National Olympiad, 1

Tags: geometry , trigonometry , geometric transformation , reflection , rhombus , inradius , circumcircle

The quadrilateral $ABCD$, where $\angle BAD+\angle ADC>\pi$, is inscribed a circle with centre $I$. A line through $I$ intersects $AB$ and $CD$ in points $X$ and $Y$ respectively such that $IX=IY$. Prove that $AX\cdot DY=BX\cdot CY$.

2019 District Olympiad, 2

Tags: vector , geometry

Let $H$ be the orthocenter of the acute triangle $ABC.$ In the plane of the triangle $ABC$ we consider a point $X$ such that the triangle $XAH$ is right and isosceles, having the hypotenuse $AH,$ and $B$ and $X$ are on each part of the line $AH.$ Prove that $\overrightarrow{XA}+\overrightarrow{XC}+\overrightarrow{XH}=\overrightarrow{XB}$ if and only if $ \angle BAC=45^{\circ}.$

2008 Hong Kong TST, 3

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

Let $ ABCDE$ be an arbitrary convex pentagon. Suppose that $ BD\cap CE \equal{} A'$, $ CE \cap DA \equal{} B'$, $ DA\cap EB \equal{} C'$, $ EB\cap AC \equal{} D'$ and $ AC \cap BD \equal{} E'$. Suppose also that $ (ABD')\cap (AC'E) \equal{} A''$, $ (BCE')\cap (BD'A) \equal{} B''$, $ (CDA')\cap (CE'B) \equal{} C''$, $ (DEB')\cap DA'C \equal{} D''$ and $ (EAC')\cap (EB'D) \equal{} E''$. Prove that $ AA''$, $ BB''$, $ CC''$, $ DD''$ and $ EE''$ are concurrent.

2024 Oral Moscow Geometry Olympiad, 1

Tags: geometry , trapezoid

In an isosceles trapezoid, the diagonals are perpendicular. Find the distance from the center of the circle described around the trapezoid to the point of intersection of its diagonals, if the lengths of the bases are equal to $a$ and $b$.

1977 Poland - Second Round, 6

Tags: combinatorics , geometry , combinatorial geometry , 3d geometry

What is the greatest number of parts into which the plane can be cut by the edges of $ n $ squares?

2016 Thailand Mathematical Olympiad, 1

Tags: geometry , perpendicular bisector , ratio , angle bisector

Let $ABC$ be a triangle with $AB \ne AC$. Let the angle bisector of $\angle BAC$ intersects $BC$ at $P$ and intersects the perpendicular bisector of segment $BC$ at $Q$. Prove that $\frac{PQ}{AQ} =\left( \frac{BC}{AB + AC}\right)^2$

2007 Cuba MO, 2

Tags: combinatorics , combinatorial geometry , prism , 3d geometry , geometry

A prism is called [i]binary [/i] if it can be assigned to each of its vertices a number from the set $\{-1, 1\}$, such that the product of the numbers assigned to the vertices of each face is equal to $-1$. a) Prove that the number of vertices of the binary prisms is divisible for $8$. b) Prove that a prism with $2000$ vertices is binary.

2008 Estonia Team Selection Test, 2

Tags: geometry , cyclic , cyclic quadrilateral , angle bisector

Let $ABCD$ be a cyclic quadrangle whose midpoints of diagonals $AC$ and $BD$ are $F$ and $G$, respectively. a) Prove the following implication: if the bisectors of angles at $B$ and $D$ of the quadrangle intersect at diagonal $AC$ then $\frac14 \cdot |AC| \cdot |BD| = | AG| \cdot |BF| \cdot |CG| \cdot |DF|$. b) Does the converse implication also always hold?

2023 Cono Sur Olympiad, 5

Tags: geometry

Let $ABC$ be an acute triangle and $D, E, F$ are the midpoints of $BC, CA, AB$, respectively. The circle with diameter $AD$ intersects the lines $AB$ and $AC$ at points $P$ and $Q$ , respectively. The lines through $P$ and $Q$ parallel to $BC$ intersect $DE$ at point $R$ and $DF$ at point $S$, respectively. The circumcircle of $DPR$ intersects $AB$ at $X$, the circumcircle of $DQS$ intersects $AC$ in $Y$, and these two circles intersect again point $Z$. Prove that $Z$ is the midpoint of $XY$.

1970 All Soviet Union Mathematical Olympiad, 140

Tags: rectangle , geometry

Two equal rectangles are intersecting in $8$ points. Prove that the common part area is greater than the half of the rectangle's area.

1998 Belarus Team Selection Test, 3

Tags: geometry , parallelogram , hexagon , area , equilateral

Let $ABCDEF$ be a convex hexagon such that $BCEF$ is a parallelogram and $ABF$ an equilateral triangle. Given that $BC = 1, AD = 3, CD+DE = 2$, compute the area of $ABCDEF$

2011 Oral Moscow Geometry Olympiad, 2

Tags: 3d geometry , geometry , line , plane

Line $\ell $ intersects the plane $a$. It is known that in this plane there are $2011$ straight lines equidistant from $\ell$ and not intersecting $\ell$. Is it true that $\ell$ is perpendicular to $a$?

2021 Thailand TSTST, 1

Tags: right triangle , isosceles triangle , geometry

Let $ABC$ be an isosceles triangle with $BC=CA$, and let $D$ be a point inside side $AB$ such that $AD< DB$. Let $P$ and $Q$ be two points inside sides $BC$ and $CA$, respectively, such that $\angle DPB = \angle DQA = 90^{\circ}$. Let the perpendicular bisector of $PQ$ meet line segment $CQ$ at $E$, and let the circumcircles of triangles $ABC$ and $CPQ$ meet again at point $F$, different from $C$. Suppose that $P$, $E$, $F$ are collinear. Prove that $\angle ACB = 90^{\circ}$.

May Olympiad L1 - geometry, 2022.5

Tags: geometry , isosceles

Vero had an isosceles triangle made of paper. Using scissors, he divided it into three smaller triangles and painted them blue, red and green. Having done so, he observed that: $\bullet$ with the blue triangle and the red triangle an isosceles triangle can be formed, $\bullet$ with the blue triangle and the green triangle an isosceles triangle can be formed, $\bullet$ with the red triangle and the green triangle an isosceles triangle can be formed. Show what Vero's triangle looked like and how he might have made the cuts to make this situation be possible.

2007 Italy TST, 1

Tags: rectangle , incenter , geometric transformation , homothety , reflection , geometry

Let $ABC$ an acute triangle. (a) Find the locus of points that are centers of rectangles whose vertices lie on the sides of $ABC$; (b) Determine if exist some points that are centers of $3$ distinct rectangles whose vertices lie on the sides of $ABC$.

2017 Kosovo National Mathematical Olympiad, 5

Tags: geometry

Lines determined by sides $AB$ and $CD$ of the convex quadrilateral $ABCD$ intersect at point $P$. Prove that $\alpha +\gamma =\beta +\delta$ if and only if $PA\cdot PB=PC\cdot PD$, where $\alpha ,\beta ,\gamma ,\delta$ are the measures of the internal angles of vertices $A, B, C, D$ respectively.