Olympiad problems

2016 IFYM, Sozopol, 7

We are given a ruler with two marks at a distance 1. With its help we can do all possible constructions as with a ruler with no measurements, including one more: If there is a line $l$ and point $A$ on $l$, then we can construct points $P_1,P_2\in l$ for which $AP_1=AP_2=1$. By using this ruler, construct a perpendicular from a given point to a given line.

1987 Traian Lălescu, 1.4

Tags: constructive geometry , geometry , pure geometry

Through a given point inside a circle, construct two perpendicular chords such that the sum of their lengths would be: [b]a)[/b] maximum. [b]b)[/b] minimum.

1987 Traian Lălescu, 2.2

Tags: constructive geometry , geometry

Construct a convex quadrilateral given two opposite angles and sides.

2023 Sharygin Geometry Olympiad, 18

Tags: bicentral , arc midpoint , constructive geometry , geometry

Restore a bicentral quadrilateral $ABCD$ if the midpoints of the arcs $AB,BC,CD$ of its circumcircle are given.

Revenge EL(S)MO 2024, 7

Tags: incenter , feuerbach point , nagel point , constructive geometry , geometry

A scalene triangle $ABC$ was drawn, and Elmo marked its incenter $I$, Feuerbach point $X$, and Nagel point $N$. Sadly, after taking the abcdEfghijkLMnOpqrstuvwxyz, Elmo lost the triangle $ABC$. Can Elmo use only a ruler and compass to reconstruct the triangle? Proposed by [i]Karn Chutinan[/i]

2010 IFYM, Sozopol, 3

Tags: constructive geometry , geometry , construction

Two circles are intersecting in points $P$ and $Q$. Construct two points $A$ and $B$ on these circles so that $P\in AB$ and the product $AP.PB$ is maximal.

1957 Czech and Slovak Olympiad III A, 4

Tags: trapezoid , constructive geometry , geometry

Consider a non-zero convex angle $\angle POQ$ and its inner point $M$. Moreover, let $m>0$ be given. Construct a trapezoid $ABCD$ satisfying the following conditions: (1) vertices $A, D$ lie on ray $OP$ and vertices $B,C$ lie on ray $OQ$, (2) diagonals $AC$ and $BD$ intersect in $M$, (3) $AB=m$. Prove that your construction is correct and discuss conditions of solvability.

1978 Bundeswettbewerb Mathematik, 4

Tags: symmetry , triangle , constructive geometry , construction , geometry

In a triangle $ABC$, the points $A_1, B_1, C_1$ are symmetric to $A, B,C$ with respect to $B,C, A$, respectively. Given the points $A_1, B_1,C_1$ reconstruct the triangle $ABC$.

2019 Czech and Slovak Olympiad III A, 2

Tags: rectangle , geometry , constructive geometry

Let be $ABCD$ a rectangle with $|AB|=a\ge b=|BC|$. Find points $P,Q$ on the line $BD$ such that $|AP|=|PQ|=|QC|$. Discuss the solvability with respect to the lengths $a,b$.

1987 Czech and Slovak Olympiad III A, 1

Tags: circumcircle , geometry , trapezoid , constructive geometry , quadrilateral , cyclic quadrilateral

Given a trapezoid, divide it by a line into two quadrilaterals in such a way that both of them are cyclic with the same circumradius. Discuss conditions of solvability.

1956 Czech and Slovak Olympiad III A, 2

Tags: 3d geometry , constructive geometry , construction , parallelogram , geometry

In a given plane $\varrho$ consider a convex quadrilateral $ABCD$ and denote $E=AC\cap BD.$ Moreover, consider a point $V\notin\varrho$. On rays $VA,VB,VC,VD$ find points $A',B',C',D'$ respectively such that $E,A',B',C',D'$ are coplanar and $A'B'C'D'$ is a parallelogram. Discuss conditions of solvability.

1957 Czech and Slovak Olympiad III A, 2

Tags: geometry , computational geometry , constructive geometry , 3d geometry , pyramid

Consider a (right) square pyramid $ABCDV$ with the apex $V$ and the base (square) $ABCD$. Denote $d=AB/2$ and $\varphi$ the dihedral angle between planes $VAD$ and $ABC$. (1) Consider a line $XY$ connecting the skew lines $VA$ and $BC$, where $X$ lies on line $VA$ and $Y$ lies on line $BC$. Describe a construction of line $XY$ such that the segment $XY$ is of the smallest possible length. Compute the length of segment $XY$ in terms of $d,\varphi$. (2) Compute the distance $v$ between points $V$ and $X$ in terms of $d,\varphi.$

2019 India Regional Mathematical Olympiad, 2

Tags: geometry , construction , constructive geometry

Given a circle $\tau$, let $P$ be a point in its interior, and let $l$ be a line through $P$. Construct with proof using ruler and compass, all circles which pass through $P$, are tangent to $\tau$ and whose center lies on line $l$.