Olympiad problems

2017 Sharygin Geometry Olympiad, 5

A square $ABCD$ is given. Two circles are inscribed into angles $A$ and $B$, and the sum of their diameters is equal to the sidelength of the square. Prove that one of their common tangents passes through the midpoint of $AB$.

1976 Chisinau City MO, 133

Tags: geometry , geometric inequality , square , parallelogram , area

A triangle with a parallelogram inside was placed in a square. Prove that the area of a parallelogram is not more than a quarter of a square.

2002 Germany Team Selection Test, 2

Tags: geometry , triangle , square

Let $A_1$ be the center of the square inscribed in acute triangle $ABC$ with two vertices of the square on side $BC$. Thus one of the two remaining vertices of the square is on side $AB$ and the other is on $AC$. Points $B_1,\ C_1$ are defined in a similar way for inscribed squares with two vertices on sides $AC$ and $AB$, respectively. Prove that lines $AA_1,\ BB_1,\ CC_1$ are concurrent.

Novosibirsk Oral Geo Oly VII, 2019.7

Tags: geometry , acute , square

Cut a square into eight acute-angled triangles.

2000 Abels Math Contest (Norwegian MO), 3

Tags: geometry , perimeter , coloring , square

a) Each point, on the perimeter of a square, is colored either red, or blue. Show that, there is a right-angled triangle where all the corners are on the square of the square and so that all the corners are on points of the same color. b) Show that, it is possible to color each point on the perimeter of one square, red, white, or blue so that, there is not a right-angled triangle where all the three corners are at points of same color.

1996 Singapore Team Selection Test, 1

Tags: geometry , square , perpendicular , equal segments

Let $P$ be a point on the side $AB$ of a square $ABCD$ and $Q$ a point on the side $BC$. Let $H$ be the foot of the perpendicular from $B$ to $PC$. Suppose that $BP = BQ$. Prove that $QH$ is perpendicular to $HD$.

1995 Tournament Of Towns, (443) 3

Tags: square , geometry , angle

Suppose $L$ is the circle inscribed in the square $T_1$, and $T_2$ is the square inscribed in $L$, so that vertices of $T_1$ lie on the straight lines containing the sides of $T_2$. Find the angles of the convex octagon whose vertices are at the tangency points of $L$ with the sides of $T_1$ and at the vertices of $T_2$. (S Markelov)

Kyiv City MO Juniors 2003+ geometry, 2018.9.51

Tags: geometry , square , angle

Given a circle $\Gamma$ with center at point $O$ and diameter $AB$. $OBDE$ is square, $F$ is the second intersection point of the line $AD$ and the circle $\Gamma$, $C$ is the midpoint of the segment $AF$. Find the value of the angle $OCB$.

2001 Denmark MO - Mohr Contest, 5

Tags: geometry , equilateral , square

Is it possible to place within a square an equilateral triangle whose area is larger than $9/ 20$ of the area of the square?

2022 Paraguay Mathematical Olympiad, 5

Tags: square , geometry , area

In the figure, there is a circle of radius $1$ such that the segment $AG$ is diameter and that line $AF$ is perpendicular to line $DC$. There are also two squares $ABDC$ and $DEGF$, where $B$ and $E$ are points on the circle, and the points $A$, $D$ and $E$ are collinear. What is the area of square $DEGF$? [img]https://cdn.artofproblemsolving.com/attachments/1/e/794da3bc38096ef5d5daaa01d9c0f8c41a6f84.png[/img]

1967 IMO Shortlist, 4

Tags: geometry , perimeter , square , dissection , triangulation

The square $ABCD$ has to be decomposed into $n$ triangles (which are not overlapping) and which have all angles acute. Find the smallest integer $n$ for which there exist a solution of that problem and for such $n$ construct at least one decomposition. Answer whether it is possible to ask moreover that (at least) one of these triangles has the perimeter less than an arbitrarily given positive number.

2007 Bosnia and Herzegovina Junior BMO TST, 3

Tags: combinatorial geometry , square , geometry , circles

Is it possible to place some circles inside a square side length $1$, such that no two circles intersect and the sum of their radii is $2007$?

2023 Novosibirsk Oral Olympiad in Geometry, 7

Tags: square , geometry , college

Squares $ABCD$ and $BEFG$ are located as shown in the figure. It turned out that points $A, G$ and $E$ lie on the same straight line. Prove that then the points $D, F$ and $E$ also lie on the same line. [img]https://cdn.artofproblemsolving.com/attachments/4/2/9faf29a399d3a622c84f5d4a3cfcf5e99539c0.png[/img]

1975 Czech and Slovak Olympiad III A, 5

Tags: triangle , geometry , locus , isosceles , square

Let a square $\mathbf P=P_1P_2P_3P_4$ be given in the plane. Determine the locus of all vertices $A$ of isosceles triangles $ABC,AB=BC$ such that the vertices $B,C$ are points of the square $\mathbf P.$

Mathematical Minds 2024, P2

Tags: square , geometry , olympiad geometry

Let $ABCD$ be a square and $E$ a point on side $CD$ such that $\angle DAE = 30^{\circ}$. The bisector of angle $\angle AEC$ intersects line $BD$ at point $F$. Lines $FC$ and $AE$ intersect at $S$. Find $\angle SDC$. [i]Proposed by Ana Boiangiu[/i]

1941 Moscow Mathematical Olympiad, 081

Tags: square , rectangle , combinatorial geometry , impossible , geometry

a) Prove that it is impossible to divide a rectangle into five squares of distinct sizes. b) Prove that it is impossible to divide a rectangle into six squares of distinct sizes.

2003 Paraguay Mathematical Olympiad, 5

Tags: geometry , square , area

In a square $ABCD$, $E$ is the midpoint of side $BC$. Line $AE$ intersects line $DC$ at $F$ and diagonal $BD$ at $G$. If the area $(EFC) = 8$, determine the area $(GBE)$.

Estonia Open Senior - geometry, 1999.2.5

Tags: square , equal area , geometry , area

Inside the square $ABCD$ there is the square $A'B' C'D'$ so that the segments $AA', BB', CC'$ and $DD'$ do not intersect each other neither the sides of the smaller square (the sides of the larger and the smaller square do not need to be parallel). Prove that the sum of areas of the quadrangles $AA'B' B$ and $CC'D'D$ is equal to the sum of areas of the quadrangles $BB'C'C$ and $DD'A'A$.

1954 Putnam, A2

Tags: square , distance

Consider any five points in the interior of square $S$ of side length $1$. Prove that at least one of the distances between these points is less than $\sqrt{2} \slash 2.$ Can this constant be replaced by a smaller number?

1996 Tournament Of Towns, (498) 5

Tags: square , geometry , area

The squares $ABMN$, $BCKL$ and $ACPQ$ are constructed outside triangle $ABC$. The difference between the areas of $AB MN$ and $BCKL$ is $d$. Find the difference between the areas of the squares with sides $NQ$ and $PK$ respectively, if $\angle ABC$ is (a) a right angle; (b) not necessarily a right angle. (A Gerko)

1954 Moscow Mathematical Olympiad, 264

Tags: cube , unfolding , cut , square

* Cut out of a $3 \times 3$ square an unfolding of the cube with edge $1$.

1986 All Soviet Union Mathematical Olympiad, 438

Tags: square , geometry , geometric inequality , area , circles

A triangle and a square are circumscribed around the unit circle. Prove that the intersection area is more than $3.4$. Is it possible to assert that it is more than $3.5$?

1983 Spain Mathematical Olympiad, 5

Tags: analytic geometry , square

Find the coordinates of the vertices of a square $ABCD$, knowing that $A$ is on the line $y -2x -6 = 0$, $C$ at $x = 0$ and $B$ is the point $(a, 0)$ , being $a = \log_{2/3}(16/81)$.

May Olympiad L2 - geometry, 2002.3

Tags: right triangle , isosceles , square , geometry

In a triangle $ABC$, right in $A$ and isosceles, let $D$ be a point on the side $AC$ ($A \ne D \ne C$) and $E$ be the point on the extension of $BA$ such that the triangle $ADE$ is isosceles. Let $P$ be the midpoint of segment $BD$, $R$ be the midpoint of the segment $CE$ and $Q$ the intersection point of $ED$ and $BC$. Prove that the quadrilateral $ARQP$ is a square.

2000 Tuymaada Olympiad, 6

Tags: square , geometry , circumcircle , right triangle

Let $O$ be the center of the circle circumscribed around the the triangle $ABC$. The centers of the circles circumscribed around the squares $OAB,OBC,OCA$ lie at the vertices of a regular triangle. Prove that the triangle $ABC$ is right.