This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 5

2013 Oral Moscow Geometry Olympiad, 2
Tags: ruler , construction , compass , geometry
With a compass and a ruler, split a triangle into two smaller triangles with the same sum of squares of sides.

2021/2022 Tournament of Towns, P2
Tags: analytic geometry , function , ruler , compass , geometry
On a blank paper there were drawn two perpendicular axes $x$ and $y$ with the same scale. The graph of a function $y=f(x)$ was drawn in this coordinate system. Then the $y$ axis and all the scale marks on the $x$ axis were erased. Provide a way how to draw again the $y$ axis using pencil, ruler and compass: (a) $f(x)= 3^x$; (b) $f(x)= \log_a x$, where $a>1$ is an unknown number.

Ukraine Correspondence MO - geometry, 2009.3
Tags: geometry , compass , right triangle , inradius
A right triangle is drawn on the plane. How to use only a compass to mark two points, such that the distance between them is equal to the diameter of the circle inscribed in this triangle?

2022 Moscow Mathematical Olympiad, 2
Tags: analytic geometry , ruler , compass , geometry
In a Cartesian coordinate system (with the same scale on the x and y axes)there is a graph of the exponential function $y=3^x$. Then the y-axis and all marks on the x-axis erased. Only the graph of the function and the x-axis remained without a scale and a mark of $0$. How can you restore the y-axis using a compass and ruler?

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 $\sqrt7$?