Olympiad problems

2014 Saudi Arabia GMO TST, 3

Turki has divided a square into finitely many white and green rectangles, each with sides parallel to the sides of the square. Within each white rectangle, he writes down its width divided by its height. Within each green rectangle, he writes down its height divided by its width. Finally, he calculates $S$, the sum of these numbers. If the total area of white rectangles equals the total area of green rectangles, determine the minimum possible value of $S$.

Durer Math Competition CD Finals - geometry, 2022.C3

Tags: geometry , tangent , equilateral , square

To the exterior of side $AB$ of square $ABCD$, we have drawn the regular triangle $ABE$. Point $A$ reflected on line $BE$ is $F$, and point $E$ reflected on line $BF$ is $G$. Let the perpendicular bisector of segment $FG$ meet segment $AD$ at $X$. Show that the circle centered at $X$ with radius $XA$ touches line$ FB$.

2015 BMT Spring, 17

Tags: geometry , square , area

A circle intersects square $ABCD$ at points $A, E$, and $F$, where $E$ lies on $AB$ and $F$ lies on $AD$, such that $AE + AF = 2(BE + DF)$. If the square and the circle each have area $ 1$, determine the area of the union of the circle and square.

Novosibirsk Oral Geo Oly VIII, 2019.3

Tags: ratio , paper folding , folding , geometry , square

A square sheet of paper $ABCD$ is folded straight in such a way that point $B$ hits to the midpoint of side $CD$. In what ratio does the fold line divide side $BC$?

2013 NZMOC Camp Selection Problems, 6

Tags: geometry , cyclic , tangential , square

$ABCD$ is a quadrilateral having both an inscribed circle (one tangent to all four sides) with center $I,$ and a circumscribed circle with center $O$. Let $S$ be the point of intersection of the diagonals of $ABCD$. Show that if any two of $S, I$ and $O$ coincide, then $ABCD$ is a square (and hence all three coincide).

2023 Iranian Geometry Olympiad, 3

Tags: combinatorics , combinatorial geometry , square , geometry

Let $ABCD$ be a square with side length $1$. How many points $P$ inside the square (not on its sides) have the property that the square can be cut into $10$ triangles of equal area such that all of them have $P$ as a vertex? [i]Proposed by Josef Tkadlec - Czech Republic[/i]

1993 Tournament Of Towns, (385) 3

Tags: square , midpoint , geometry

Three angles of a non-convex, non-self-intersecting quadrilateral are equal to $45$ degrees (i.e. the last equals $225$ degrees). Prove that the midpoints of its sides are vertices of a square. (V Proizvolov)

1988 All Soviet Union Mathematical Olympiad, 469

Tags: rational , square , number theory

If rationals $x, y$ satisfy $x^5 + y^5 = 2 x^2 y^2$, show that $1-x y$ is the square of a rational.

1979 Bundeswettbewerb Mathematik, 2

Tags: geometry , square , concurrency

The squares $OABC$ and $OA_1B_1C_1$ are situated in the same plane and are directly oriented. Prove that the lines $AA_1$ , $BB_1$, and $CC_1$ are concurrent.

2020 Adygea Teachers' Geometry Olympiad, 2

Tags: square , ratio , geometry , circles

The square $ABCD$ is inscribed in a circle. Points $E$ and $F$ are located on the side of the square, and points $G$ and $H$ are located on the smaller arc $AB$ of the circle so that the $EFGH$ is a square. Find the area ratio of these squares.

Estonia Open Senior - geometry, 2000.2.4

Tags: diagonal , tangential , inradius , geometry , square , midpoint , incircle

The diagonals of the square $ABCD$ intersect at $P$ and the midpoint of the side $AB$ is $E$. Segment $ED$ intersects the diagonal $AC$ at point $F$ and segment $EC$ intersects the diagonal $BD$ at $G$. Inside the quadrilateral $EFPG$, draw a circle of radius $r$ tangent to all the sides of this quadrilateral. Prove that $r = | EF | - | FP |$.

1980 Tournament Of Towns, (005) 5

Tags: combinatorial geometry , combinatorics , area , square , geometry

A finite set of line segments, of total length $18$, belongs to a square of unit side length (we assume that the square includes its boundary and that a line segment includes its end points). The line segments are parallel to the sides of the square and may intersect one another. Prove that among the regions into which the square is divided by the line segments, at least one of these must have area not less than $0.01$. (A Berzinsh, Riga)

2016 Auckland Mathematical Olympiad, 2

Tags: geometry , equal angles , square

In square $ABCD$, $\overline{AC}$ and $\overline{BD}$ meet at point $E$. Point $F$ is on $\overline{CD}$ and $\angle CAF = \angle FAD$. If $\overline{AF}$ meets $\overline{ED}$ at point $G$, and if $\overline{EG} = 24$ cm, then find the length of $\overline{CF}$.

2022 Iranian Geometry Olympiad, 5

Tags: equilateral triangle , square , geometry

a) Do there exist four equilateral triangles in the plane such that each two have exactly one vertex in common, and every point in the plane lies on the boundary of at most two of them? b) Do there exist four squares in the plane such that each two have exactly one vertex in common, and every point in the plane lies on the boundary of at most two of them? (Note that in both parts, there is no assumption on the intersection of interior of polygons.) [i]Proposed by Hesam Rajabzadeh[/i]

2020 Malaysia IMONST 1, 3

Tags: geometry , circles , square

Given a square with area $A$. A circle lies inside the square, such that the circle touches all sides of the square. Another square with area $B$ lies inside the circle, such that all its vertices lie on the circle. Find the value of $\frac{A}{B}.$

2023 Novosibirsk Oral Olympiad in Geometry, 3

Tags: square , geometry , rectangle

The rectangle is cut into $10$ squares as shown in the figure on the right. Find its sides if the side of the smallest square is $3$.[img]https://cdn.artofproblemsolving.com/attachments/e/5/1fe3a0e41b2d3182338a557d3d44ff5ef9385d.png[/img]

2009 Denmark MO - Mohr Contest, 4

Tags: geometry , square , equal segments

Let $E$ be an arbitrary point different from $A$ and $B$ on the side $AB$ of a square $ABCD$, and let $F$ and $G$ be points on the segment $CE$ so that $BF$ and $DG$ are perpendicular to $CE$. Prove that $DF = AG$.

Novosibirsk Oral Geo Oly VII, 2023.7

Tags: square , college , geometry

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]

Novosibirsk Oral Geo Oly VII, 2023.3

Tags: square , geometry , rectangle

The rectangle is cut into $10$ squares as shown in the figure on the right. Find its sides if the side of the smallest square is $3$.[img]https://cdn.artofproblemsolving.com/attachments/e/5/1fe3a0e41b2d3182338a557d3d44ff5ef9385d.png[/img]

2021 Malaysia IMONST 1, 16

Tags: geometry , octagon , square , area

Given a square $ABCD$ with side length $6$. We draw line segments from the midpoints of each side to the vertices on the opposite side. For example, we draw line segments from the midpoint of side $AB$ to vertices $C$ and $D$. The eight resulting line segments together bound an octagon inside the square. What is the area of this octagon?

2019 AMC 8, 20

Tags: square

How many different real numbers $x$ satisfy the equation $$(x^2-5)^2=16?$$ $\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }4\qquad\textbf{(E) }8$

2014 Sharygin Geometry Olympiad, 1

Tags: angle chasing , isosceles , geometry , square

The vertices and the circumcenter of an isosceles triangle lie on four different sides of a square. Find the angles of this triangle. (I. Bogdanov, B. Frenkin)

2007 Postal Coaching, 1

Tags: sum , geometry , constant , circumcircle , square

Let $P$ be a point on the circumcircle of a square $ABCD$. Find all integers $n > 0$ such that the sum $$S_n(P) = |PA|^n + |PB|^n + |PC|^n + |PD|^n$$ is constant with respect to the point $P$.

1995 Bundeswettbewerb Mathematik, 4

Tags: combinatorics , circles , square , combinatorial geometry

A number of unit discs are given inside a square of side $100$ such that (i) no two of the discs have a common interior point, and (ii) every segment of length $10$, lying entirely within the square, meets at least one disc. Prove that there are at least $400$ discs in the square.

2010 Dutch IMO TST, 4

Tags: geometry , square , circles , equal segments

Let $ABCD$ be a square with circumcircle $\Gamma_1$. Let $P$ be a point on the arc $AC$ that also contains $B$. A circle $\Gamma_2$ touches $\Gamma_1$ in $P$ and also touches the diagonal $AC$ in $Q$. Let $R$ be a point on $\Gamma_2$ such that the line $DR$ touches $\Gamma_2$. Proof that $|DR| = |DA|$.