Olympiad problems

1981 Brazil National Olympiad, 3

Tags: geometry , construction

Given a sheet of paper and the use of a rule, compass and pencil, show how to draw a straight line that passes through two given points, if the length of the ruler and the maximum opening of the compass are both less than half the distance between the two points. You may not fold the paper.

1996 German National Olympiad, 5

Tags: geometry , construction , geometric construction

Given two non-intersecting chords $AB$ and $CD$ of a circle $k$ and a length $a <CD$. Determine a point $X$ on $k$ with the following property: If lines $XA$ and $XB$ intersect $CD$ at points $P$ and $Q$ respectively, then $PQ = a$. Show how to construct all such points $X$ and prove that the obtained points indeed have the desired property.

2008 Oral Moscow Geometry Olympiad, 1

Tags: geometry , analytic geometry , construction

A coordinate system was drawn on the board and points $A (1,2)$ and $B (3,1)$ were marked. The coordinate system was erased. Restore it by the two marked points.

1966 IMO Shortlist, 19

Tags: geometry , excircle , triangle , construction

Construct a triangle given the radii of the excircles.

2014 Sharygin Geometry Olympiad, 4

Tags: construction , geometry

Tanya has cut out a triangle from checkered paper as shown in the picture. The lines of the grid have faded. Can Tanya restore them without any instruments only folding the triangle (she remembers the triangle sidelengths)? (T. Kazitsyna)

1996 Tuymaada Olympiad, 4

Tags: construction , compass , geometry

Given a segment of length $7\sqrt3$ . Is it possible to use only compass to construct a segment of length $\sqrt7$?

2014 Sharygin Geometry Olympiad, 5

Tags: construction , geometry

The altitude from one vertex of a triangle, the bisector from the another one and the median from the remaining vertex were drawn, the common points of these three lines were marked, and after this everything was erased except three marked points. Restore the triangle. (For every two erased segments, it is known which of the three points was their intersection point.) (A. Zaslavsky)

1985 IMO Shortlist, 16

Tags: construction , geometry , triangle , circles

If possible, construct an equilateral triangle whose three vertices are on three given circles.

2017 India Regional Mathematical Olympiad, 1

Tags: geometry , construction

Let $AOB$ be a given angle less than $180^{\circ}$ and let $P$ be an interior point of the angular region determined by $\angle AOB$. Show, with proof, how to construct, using only ruler and compass, a line segment $CD$ passing through $P$ such that $C$ lies on the way $OA$ and $D$ lies on the ray $OB$, and $CP:PD=1:2$.

Kyiv City MO 1984-93 - geometry, 1991.8.4

Tags: construction , geometry , square

Construct a square, if you know its center and two points that lie on adjacent sides.

Kyiv City MO Juniors 2003+ geometry, 2015.9.3

Tags: construction , geometry , trapezoid , square

It is known that a square can be inscribed in a given right trapezoid so that each of its vertices lies on the corresponding side of the trapezoid (none of the vertices of the square coincides with the vertex of the trapezoid). Construct this inscribed square with a compass and a ruler. (Maria Rozhkova)

1977 Bundeswettbewerb Mathematik, 2

Tags: construction , geometry , parallelogram , disks

On a plane are given three non-collinear points $A, B, C$. We are given a disk of diameter different from that of the circle passing through $A, B, C$ large enough to cover all three points. Construct the fourth vertex of the parallelogram $ABCD$ using only this disk (The disk is to be used as a circular ruler, for constructing a circle passing through two given points).

2017 Thailand TSTST, 4

Tags: construction , combinatorics , coloring , table , cell

The cells of a $8 \times 8$ table are colored either black or white so that each row has a different number of black squares, and each column has a different number of black squares. What is the maximum number of pairs of adjacent cells of different colors?

1968 Putnam, B3

Tags: geometry , construction

Given that a $60^{\circ}$ angle cannot be trisected with ruler and compass, prove that a $\frac{120^{\circ}}{n}$ angle cannot be trisected with ruler and compass for $n=1,2,\ldots$

Ukraine Correspondence MO - geometry, 2019.7

Tags: geometry , construction , equal angles

Given a triangle $ABC$. Construct a point $D$ on the side $AB$ and point $E$ on the side $AC$ so that $BD = CE$ and $\angle ADC = \angle BEC$

1983 Spain Mathematical Olympiad, 2

Tags: construction , geometry

Construct a triangle knowing an angle, the ratio of the sides that form it and the radius of the inscribed circle.

1955 Poland - Second Round, 3

Tags: construction , geometry

What should the angle at the vertex of an isosceles triangle be so that it is possible to construct a triangle with sides equal to the height, base, and one of the other sides of the isosceles triangle?

2018 Thailand TSTST, 2

Tags: line , construction , combinatorics

$9$ horizontal and $9$ vertical lines are drawn through a square. Prove that it is possible to select $20$ rectangles so that the sides of each rectangle is a segment of one of the given lines (including the sides of the square), and for any two of the $20$ rectangles, it is possible to cover one of them with the other (rotations are allowed).

Cono Sur Shortlist - geometry, 1993.3

Tags: angle bisector , geometry , construction

Justify the following construction of the bisector of an angle with an inaccessible vertex: [img]https://cdn.artofproblemsolving.com/attachments/9/d/be4f7799d58a28cab3b4c515633b0e021c1502.png[/img] $M \in a$ and $N \in b$ are taken, the $4$ bisectors of the $4$ internal angles formed by $MN$ are traced with $a$ and $ b$. Said bisectors intersect at $P$ and $Q$, then $PQ$ is the bisector sought.

1993 Czech And Slovak Olympiad IIIA, 3

Tags: geometry , isosceles trapezoid , trapezoid , geometric construction , construction , concurrency

Let $AKL$ be a triangle such that $\angle ALK > 90^o +\angle LAK$. Construct an isosceles trapezoid $ABCD$ with $AB \parallel CD$ such that $K$ lies on the side $BC, L$ on the diagonal $AC$ and the lines $AK$ and $BL$ intersect at the circumcenter of the trapezoid.

1953 Poland - Second Round, 6

Tags: geometry , construction

Given a circle and two tangents to this circle. Draw a third tangent to the circle in such a way that its segment contained by the given tangents has the given length $ d $.

2019 Oral Moscow Geometry Olympiad, 3

Tags: geometry , circumcircle , construction

Restore the acute triangle $ABC$ given the vertex $A$, the foot of the altitude drawn from the vertex $B$ and the center of the circle circumscribed around triangle $BHC$ (point $H$ is the orthocenter of triangle $ABC$).

Kyiv City MO Juniors 2003+ geometry, 2007.9.3

Tags: geometry , circumcircle , incircle , construction

On a straight line $4$ points are successively set , $A, P, Q,W $, which are the points of intersection of the bisector $AL $ of the triangle $ABC$ with the circumscribed and inscribed circle. Knowing only these points, construct a triangle $ABC $.

2021 Taiwan TST Round 1, 6

Tags: construction , combinatorics

Let $n$ be a positive integer and $N=n^{2021}$. There are $2021$ concentric circles centered at $O$, and $N$ equally-spaced rays are emitted from point $O$. Among the $2021N$ intersections of the circles and the rays, some are painted red while the others remain unpainted. It is known that, no matter how one intersection point from each circle is chosen, there is an angle $\theta$ such that after a rotation of $\theta$ with respect to $O$, all chosen points are moved to red points. Prove that the minimum number of red points is $2021n^{2020}$. [I]Proposed by usjl.[/i]

2005 Tournament of Towns, 2

Tags: geometry , construction

A segment of length $\sqrt2 + \sqrt3 + \sqrt5$ is drawn. Is it possible to draw a segment of unit length using a compass and a straightedge? [i](3 points)[/i]