Found problems: 297
Kyiv City MO Juniors 2003+ geometry, 2015.9.3
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)
2020 Indonesia MO, 5
A set $A$ contains exactly $n$ integers, each of which is greater than $1$ and every of their prime factors is less than $10$. Determine the smallest $n$ such that $A$ must contain at least two distinct elements $a$ and $b$ such that $ab$ is the square of an integer.
Ukrainian TYM Qualifying - geometry, 2018.16
Let $K, T$ be the points of tangency of inscribed and exscribed circles to the side $BC$ triangle $ABC$, $M$ is the midpoint of the side $BC$. Using a compass and a ruler, construct triangle ABC given rays $AK$ and $AT$ (points $K, T$ are not marked on them) and point $M$.
2022 Yasinsky Geometry Olympiad, 3
Reconstruct the triangle$ ABC$, in which $\angle B - \angle C = 90^o$ , by the orthocenter $H$ and points $M_1$ and $L_1$ the feet of the median and angle bisector drawn from vertex $A$, respectively.
(Gryhoriy Filippovskyi)
2017 Yasinsky Geometry Olympiad, 3
Given circle arc, whose center is an inaccessible point. $A$ is a point on this arc (see fig.). How to construct using compass and ruler without divisions, a tangent to given circle arc at point $A$ ?
[img]https://1.bp.blogspot.com/-7oQBNJGLsVw/W6dYm4Xw7bI/AAAAAAAAJH8/sJ-rgAQZkW0kvlPOPwYiGjnOXGQZuDnRgCK4BGAYYCw/s1600/Yasinsky%2B2017%2BVIII-IX%2Bp3.png[/img]
1960 Czech and Slovak Olympiad III A, 3
Two different points $A, M$ are given in a plane, $AM = d > 0$. Let a number $v > 0$ be given. Construct a rhombus $ABCD$ with the height of length $v$ and $M$ being a midpoint of $BC$. Discuss conditions of solvability and determine number of solutions. Can the resulting quadrilateral $ABCD$ be a square?
1996 Tuymaada Olympiad, 4
Given a segment of length $7\sqrt3$ .
Is it possible to use only compass to construct a segment of length $\sqrt7$?
1969 IMO Longlists, 50
$(NET 5)$ The bisectors of the exterior angles of a pentagon $B_1B_2B_3B_4B_5$ form another pentagon $A_1A_2A_3A_4A_5.$ Construct $B_1B_2B_3B_4B_5$ from the given pentagon $A_1A_2A_3A_4A_5.$
2022 Kyiv City MO Round 1, Problem 1
Represent $\frac{1}{2021}$ as a difference of two irreducible fractions with smaller denominators.
[i](Proposed by Bogdan Rublov)[/i]
1971 IMO Longlists, 27
Let $n \geq 2$ be a natural number. Find a way to assign natural numbers to the vertices of a regular $2n$-gon such that the following conditions are satisfied:
(1) only digits $1$ and $2$ are used;
(2) each number consists of exactly $n$ digits;
(3) different numbers are assigned to different vertices;
(4) the numbers assigned to two neighboring vertices differ at exactly one digit.
2006 Oral Moscow Geometry Olympiad, 1
An arbitrary triangle $ABC$ is given. Construct a line that divides it into two polygons, which have equal radii of the circumscribed circles.
(L. Blinkov)
2006 Sharygin Geometry Olympiad, 8.1
Inscribe the equilateral triangle of the largest perimeter in a given semicircle.
1999 Austrian-Polish Competition, 4
Three lines $k, l, m$ are drawn through a point $P$ inside a triangle $ABC$ such that $k$ meets $AB$ at $A_1$ and $AC$ at $A_2 \ne A_1$ and $PA_1 = PA_2$, $l $ meets $BC$ at $B_1$ and $BA$ at $B_2 \ne B_1$ and $PB_1 = PB_2$, $m$ meets $CA$ at $C_1$ and $CB$ at $C_2\ne C_1$ and $PC_1=PC_2$. Prove that the lines $k,l,m$ are uniquely determined by these conditions. Find point $P$ for which the triangles $AA_1A_2, BB_1B_2, CC_1C_2$ have the same area and show that this point is unique.
Cono Sur Shortlist - geometry, 1993.3
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.
2021 Yasinsky Geometry Olympiad, 5
A circle is circumscribed around an isosceles triangle $ABC$ with base $BC$. The bisector of the angle $C$ and the bisector of the angles $A$ intersect the circle at the points $E$ and $D$, respectively, and the segment $DE$ intersects the sides $BC$ and $AB$ at the points $P$ and $Q$, respectively. Reconstruct $\vartriangle ABC$ given points $D, P, Q$, if it is known in which half-plane relative to the line $DQ$ lies the vertex $A$.
(Maria Rozhkova)
2016 Sharygin Geometry Olympiad, P11
Restore a triangle $ABC$ by vertex $B$, the centroid and the common point of the symmedian from $B$ with the circumcircle.
Ukrainian From Tasks to Tasks - geometry, 2014.15
Construct a right triangle given the hypotenuse and the median drawn to the leg.
2009 Oral Moscow Geometry Olympiad, 4
Construct a triangle given a side, the radius of the inscribed circle, and the radius of the exscribed circle tangent to that side. (Research is not required.)
1962 Dutch Mathematical Olympiad, 1
Given a triangle $ABC$ with $\angle C = 90^o$.
a) Construct the circle with center $C$, so that one of the tangents from $A$ to that circle is parallel to one of the tangents from $B$ to that circle.
b) A circle with center $C$ has two parallel tangents passing through A and go respectively. If $AC = b$ and $BC = a$, express the radius of the circle in terms of $a$ and $b$.
1953 Poland - Second Round, 6
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 $.
1995 Tournament Of Towns, (476) 4
Three different points $A$, $B$ and $C$ are placed in the plane. Construct a line $m$ through $C$ so that the product of the distances from $A$ and $B$ to $m$ has the maximal value. Is $m$ unique for every triple $A$, $B$ and $C$?
(NB Vassiliev)
1991 IMO, 3
An infinite sequence $ \,x_{0},x_{1},x_{2},\ldots \,$ of real numbers is said to be [b]bounded[/b] if there is a constant $ \,C\,$ such that $ \, \vert x_{i} \vert \leq C\,$ for every $ \,i\geq 0$. Given any real number $ \,a > 1,\,$ construct a bounded infinite sequence $ x_{0},x_{1},x_{2},\ldots \,$ such that
\[ \vert x_{i} \minus{} x_{j} \vert \vert i \minus{} j \vert^{a}\geq 1
\]
for every pair of distinct nonnegative integers $ i, j$.
2024 Kyiv City MO Round 2, Problem 3
$2024$ ones and $2024$ twos are arranged in a circle in some order. Is it always possible to divide the circle into
[b]a)[/b] two (contiguous) parts with equal sums?
[b]b)[/b] three (contiguous) parts with equal sums?
[i]Proposed by Fedir Yudin[/i]
2020 Indonesia MO, 8
Determine the smallest natural number $n > 2$, or show that no such natural numbers $n$ exists, that satisfy the following condition: There exists natural numbers $a_1, a_2, \dots, a_n$ such that
\[ \gcd(a_1, a_2, \dots, a_n) = \sum_{k = 1}^{n - 1} \underbrace{\left( \frac{1}{\gcd(a_k, a_{k + 1})} + \frac{1}{\gcd(a_k, a_{k + 2})} + \dots + \frac{1}{\gcd(a_k, a_n)} \right)}_{n - k \ \text{terms}} \]
2019 India Regional Mathematical Olympiad, 2
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 $l$.