Olympiad problems

2006 Sharygin Geometry Olympiad, 24

Tags: geometry , Locus , 3D geometry , projections , Plane , circle , sphere

a) Two perpendicular rays are drawn through a fixed point $P$ inside a given circle, intersecting the circle at points $A$ and $B$. Find the geometric locus of the projections of $P$ on the lines $AB$. b) Three pairwise perpendicular rays passing through the fixed point $P$ inside a given sphere intersect the sphere at points $A, B, C$. Find the geometrical locus of the projections $P$ on the $ABC$ plane

1939 Moscow Mathematical Olympiad, 053

Tags: geometry , 3D geometry , sphere , Plane , combinatorics , combinatorial geometry

What is the greatest number of parts that $5$ spheres can divide the space into?

1992 All Soviet Union Mathematical Olympiad, 575

Tags: 3D geometry , geometry , Plane , fixed , Product , sphere

A plane intersects a sphere in a circle $C$. The points $A$ and $B$ lie on the sphere on opposite sides of the plane. The line joining $A$ to the center of the sphere is normal to the plane. Another plane $p$ intersects the segment $AB$ and meets $C$ at $P$ and $Q$. Show that $BP\cdot BQ$ is independent of the choice of $p$.

1998 Israel National Olympiad, 1

Tags: combinatorial geometry , Plane , geometry , angles

In space are given $n$ segments $A_iB_i$ and a point $O$ not lying on any segment, such that the sum of the angles $A_iOB_i$ is less than $180^o$ . Prove that there exists a plane passing through $O$ and not intersecting any of the segments.

2000 Tuymaada Olympiad, 2

Tags: combinatorics , combinatorial geometry , Coloring , Plane , Color problem

Is it possible to paint the plane in $4$ colors so that inside any circle are the dots of all four colors?

2018 Polish Junior MO First Round, 7

Tags: 3D geometry , square , Volume , Poland , Plane , geometry

Square $ABCD$ with sides of length $4$ is a base of a cuboid $ABCDA'B'C'D'$. Side edges $AA'$, $BB'$, $CC'$, $DD'$ of this cuboid have length $7$. Points $K, L, M$ lie respectively on line segments $AA'$, $BB'$, $CC'$, and $AK = 3$, $BL = 2$, $CM = 5$. Plane passing through points $K, L, M$ cuts cuboid on two blocks. Calculate volumes of these blocks.

2011 IFYM, Sozopol, 4

Tags: geometry , Plane , points , lines

There are $n$ points in a plane. Prove that there exist a point $O$ (not necessarily from the given $n$) such that on each side of an arbitrary line, through $O$, lie at least $\frac{n}{3}$ points (including the points on the line).

2017 EGMO, 3

Tags: Plane , Line , combinatorics , EGMO , combinatorial geometry , EGMO 2017

There are $2017$ lines in the plane such that no three of them go through the same point. Turbo the snail sits on a point on exactly one of the lines and starts sliding along the lines in the following fashion: she moves on a given line until she reaches an intersection of two lines. At the intersection, she follows her journey on the other line turning left or right, alternating her choice at each intersection point she reaches. She can only change direction at an intersection point. Can there exist a line segment through which she passes in both directions during her journey?

2010 Sharygin Geometry Olympiad, 7

Tags: geometry , polyhedron , cut , Plane

Each of two regular polyhedrons $P$ and $Q$ was divided by the plane into two parts. One part of $P$ was attached to one part of $Q$ along the dividing plane and formed a regular polyhedron not equal to $P$ and $Q$. How many faces can it have?

2019 Dürer Math Competition (First Round), P3

Tags: combinatorics , Plane

Anne has thought of a finite set $A \subseteq \mathbb{R}^2 $ . Bob does not know how many elements $A$ has, but his goal is to completely determine $A$. To achieve this, Bob can chooseany point $b \in \mathbb{R}^2 $ and ask Anne how far it is from$ A$ . Anne replies with the distance, defined as $min \{d(a, b) | a \in A\}$. (Here $d(a, b)$ denotes the distance between points $a, b \in \mathbb{R}$ .) Bob can ask as many questions of this type as he wants, until he can determine A with certainty. a) Can Bob achieve his goal with finitely many questions? b) What if Anne tells Bob in advance that all points of A have both coordinates in the interval$\ [0, 1]\ $? Note: $\mathbb{R}^2$ is the set of points in the plane.

2016 IFYM, Sozopol, 7

Tags: Plane , combinatorics

A grasshopper hops from an integer point to another integer point in the plane, where every even jump has a length $\sqrt{98}$ and every odd one - $\sqrt{149}$. What’s the least number of jumps the grasshopper has to make in order to return to its starting point after odd number of jumps?

IV Soros Olympiad 1997 - 98 (Russia), 10.11

Tags: geometry , 3D geometry , cube , Plane , geometric inequality

A plane intersecting a unit cube divides it into two polyhedra. It is known that for each polyhedron the distance between any two points of it does not exceeds $\frac32$ m. What can be the cross-sectional area of a cube drawn by a plane?

1952 Moscow Mathematical Olympiad, 227

Tags: combinatorial geometry , Plane , lines in plane

$99$ straight lines divide a plane into $n$ parts. Find all possible values of $n$ less than $199$.

1953 Czech and Slovak Olympiad III A, 1

Tags: complex numbers , Plane

Find the locus of all numbers $z\in\mathbb C$ in complex plane satisfying $$z+\bar z=a\cdot|z|,$$ where $a\in\mathbb R$ is given.

1994 ITAMO, 5

Tags: area , minimum , maximum , cube , Plane , geometry

Let $OP$ be a diagonal of a unit cube. Find the minimum and the maximum value of the area of the intersection of the cube with a plane through $OP$.

1993 Bundeswettbewerb Mathematik, 2

Tags: geometry , combinatorics , combinatorial geometry , Equilateral , Plane

Let $M$ be a finite subset of the plane such that for any two different points $A,B\in M$ there is a point $C\in M$ such that $ABC$ is equilateral. What is the maximal number of points in $M?$

2007 Bulgarian Autumn Math Competition, Problem 12.2

Tags: 3D geometry , geometry , pyramid , trapezoid , Plane

All edges of the triangular pyramid $ABCD$ are equal in length. Let $M$ be the midpoint of $DB$, $N$ is the point on $\overline{AB}$, such that $2NA=NB$ and $N\not\in AB$ and $P$ is a point on the altitude through point $D$ in $\triangle BCD$. Find $\angle MPD$ if the intersection of the pyramid with the plane $(NMP)$ is a trapezoid.

1980 All Soviet Union Mathematical Olympiad, 293

Tags: vector , Plane , combinatorial geometry , geometry

Given $1980$ vectors in the plane, and there are some non-collinear among them. The sum of every $1979$ vectors is collinear to the vector not included in that sum. Prove that the sum of all vectors equals to the zero vector.

2014 Bosnia And Herzegovina - Regional Olympiad, 4

Tags: combinatorics , Plane , geometry

Determine the set $S$ with minimal number of points defining $7$ distinct lines

2023 Tuymaada Olympiad, 6

Tags: inequalities , geometry , Plane , Tuymaada , Geometry inequality

In the plane $n$ segments with lengths $a_1, a_2, \dots , a_n$ are drawn. Every ray beginning at the point $O$ meets at least one of the segments. Let $h_i$ be the distance from $O$ to the $i$-th segment (not the line!) Prove the inequality \[\frac{a_1}{h_1}+\frac{a_2}{h_2} + \ldots + \frac{a_i}{h_i} \geqslant 2 \pi.\]

1984 Tournament Of Towns, (061) O2

Tags: geometry , 3D geometry , tetrahedron , parallel , Plane , altitude

Six altitudes are constructed from the three vertices of the base of a tetrahedron to the opposite sides of the three lateral faces. Prove that all three straight lines joining two base points of the altitudes in each lateral face are parallel to a certain plane. (IF Sharygin, Moscow)