Olympiad problems

2003 District Olympiad, 1

Tags: geometry , angle , plane , 3d geometry , equilateral

Let $ABC$ be an equilateral triangle. On the plane $(ABC)$ rise the perpendiculars $AA'$ and $BB'$ on the same side of the plane, so that $AA' = AB$ and $BB' =\frac12 AB$. Determine the measure the angle between the planes $(ABC)$ and $(A'B'C')$.

2020 Tournament Of Towns, 3

Tags: geometry , square , 3d geometry , plane

Is it possible that two cross-sections of a tetrahedron by two different cutting planes are two squares, one with a side of length no greater than $1$ and another with a side of length at least $100$? Mikhail Evdokimov

2010 District Olympiad, 3

Tags: geometry , 3d geometry , cube , distance , plane

Consider the cube $ABCDA'B'C'D'$. The bisectors of the angles $\angle A' C'A$ and $\angle A' AC'$ intersect $AA'$ and $A'C$ in the points $P$, respectively $S$. The point $M$ is the foot of the perpendicular from $A'$ on $CP$ , and $N$ is the foot of the perpendicular from $A'$ to $AS$. Point $O$ is the center of the face $ABB'A'$ a) Prove that the planes $(MNO)$ and $(AC'B)$ are parallel. b) Calculate the distance between these planes, knowing that $AB = 1$.

1992 All Soviet Union Mathematical Olympiad, 575

Tags: 3d geometry , plane , fixed , product , sphere , geometry

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 $p$.

1952 Moscow Mathematical Olympiad, 220

Tags: geometry , 3d geometry , trihedral angle , plane , perpendicular

A sphere with center at $O$ is inscribed in a trihedral angle with vertex $S$. Prove that the plane passing through the three tangent points is perpendicular to $OS$.

1953 Czech and Slovak Olympiad III A, 1

Tags: complex number , 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.

1956 Putnam, B2

Tags: plane , subset

Suppose that each set $X$ of points in the plane has an associated set $\overline{X}$ of points called its cover. Suppose further that (1) $\overline{X\cup Y} \supset \overline{\overline{X}} \cup \overline{Y} \cup Y$ for all sets $X,Y$ . Show that i) $\overline{X} \supset X$, ii) $\overline{\overline{X}}=\overline{X}$ and iii) $X\supset Y \Rightarrow \overline{X} \supset \overline{Y}.$ Prove also that these three statements imply (1).

2000 Tuymaada Olympiad, 2

Tags: combinatorial geometry , combinatorics , 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?

1990 Czech and Slovak Olympiad III A, 3

Tags: geometry , 3d geometry , tetrahedron , obtuse , triangle , plane

Let $ABCDEFGH$ be a cube. Consider a plane whose intersection with the tetrahedron $ABDE$ is a triangle with an obtuse angle $\varphi.$ Determine all $\varphi>\pi/2$ for which there is such a plane.