Olympiad problems

1993 Bundeswettbewerb Mathematik, 2

For the real number $a$ it holds that there is exactly one square whose vertices are all on the graph with the equation $y = x^3 + ax$. Find the side length of this square.

2000 Poland - Second Round, 6

Tags: square , algebra , polynomial

Polynomial $w(x)$ of second degree with integer coefficients takes for integer arguments values, which are squares of integers. Prove that polynomial $w(x)$ is a square of a polynomial.

Indonesia Regional MO OSP SMA - geometry, 2020.1

Tags: square , geometry , rectangle

In the figure, point $P, Q,R,S$ lies on the side of the rectangle $ABCD$. [img]https://1.bp.blogspot.com/-Ff9rMibTuHA/X9PRPbGVy-I/AAAAAAAAMzA/2ytG0aqe-k0fPL3hbSp_zHrMYAfU-1Y_ACLcBGAsYHQ/s426/2020%2BIndonedia%2BMO%2BProvince%2BP2%2Bq1.png[/img] If it is known that the area of the small square is $1$ unit, determine the area of the rectangle $ABCD$.

2017 Greece Junior Math Olympiad, 1

Tags: square , geometry , area of a triangle , area

Let $ABCD$ be a square of side $a$. On side $AD$ consider points $E$ and $Z$ such that $DE=a/3$ and $AZ=a/4$. If the lines $BZ$ and $CE$ intersect at point $H$, calculate the area of the triangle $BCH$ in terms of $a$.

2013 BMT Spring, 4

Tags: semicircle , square , geometry

Let $ABCD$ be a square with side length $2$, and let a semicircle with flat side $CD$ be drawn inside the square. Of the remaining area inside the square outside the semi-circle, the largest circle is drawn. What is the radius of this circle?

2021 Brazil National Olympiad, 5

Tags: number theory , equation , square , triplets

Find all triples of non-negative integers $(a, b, c)$ such that \[a^{2}+b^{2}+c^{2} = a b c+1.\]

1995 Romania Team Selection Test, 3

Tags: geometry , square

Let $M, N, P, Q$ be points on sides $AB, BC, CD, DA$ of a convex quadrilateral $ABCD$ such that $AQ = DP = CN = BM$. Prove that if $MNPQ$ is a square, then $ABCD$ is also a square.

2023 Novosibirsk Oral Olympiad in Geometry, 7

Tags: geometry , square , parallel lines

A square with side $1$ is intersected by two parallel lines as shown in the figure. Find the sum of the perimeters of the shaded triangles if the distance between the lines is also $1$. [img]https://cdn.artofproblemsolving.com/attachments/9/e/4e70610b80871325a72e923a0909eff06aebfa.png[/img]

2008 District Olympiad, 2

Tags: geometry , square , perpendicular

Consider the square $ABCD$ and $E \in (AB)$. The diagonal $AC$ intersects the segment $[DE]$ at point $P$. The perpendicular taken from point $P$ on $DE$ intersects the side $BC$ at point $F$. Prove that $EF = AE + FC$.

2009 Kyiv Mathematical Festival, 4

Tags: geometry , convex polygon , perimeter , square

Two convex polygons can be placed into a square with the side $1$ without intersection. Prove that at least one polygon has the perimeter that is less than or equal to $3,5$ .

Kyiv City MO 1984-93 - geometry, 1992.9.2

Tags: geometry , square , congruence

Two lines divide a square into $4$ figures of the same area. Prove that all these figures are congruent.

2013 Tournament of Towns, 2

Tags: square , geometry , right angle , right triangle

Let $C$ be a right angle in triangle $ABC$. On legs $AC$ and$BC$ the squares $ACKL, BCMN$ are constructed outside of triangle. If $CE$ is an altitude of the triangle prove that $LEM$ is a right angle.

2020 Yasinsky Geometry Olympiad, 6

Tags: geometry , incenter , square , angle bisector

Let $ABCD$ be a square, point $E$ be the midpoint of the side $BC$. The point $F$ belongs to the side $AB$, and $DE \perp EF$. The point $G$ lies inside the square, and $GF = FE$ and $GF \perp FE$. Prove that: a) $DE$ is the bisector of the $\angle FDC$ b) $FG$ is the bisector of the $\angle AFD$ c) the point $G$ is the center of the circle inscribed in the triangle $ADF$. (Ercole Suppa, Italy)

2021 Malaysia IMONST 1, 1

Tags: square , pentagon , geometry

Dinesh has several squares and regular pentagons, all with side length $ 1$. He wants to arrange the shapes alternately to form a closed loop (see diagram). How many pentagons would Dinesh need to do so? [img]https://cdn.artofproblemsolving.com/attachments/8/9/6345d7150298fe26cfcfba554656804ed25a6d.jpg[/img]

2020 Ukrainian Geometry Olympiad - April, 4

Tags: geometry , equal segments , equal angles , square , angle

On the sides $AB$ and $AD$ of the square $ABCD$, the points $N$ and $P$ are selected respectively such that $NC=NP$. The point $Q$ is chosen on the segment $AN$ so that $\angle QPN = \angle NCB$. Prove that $2\angle BCQ = \angle AQP$.

Kyiv City MO Juniors 2003+ geometry, 2015.9.3

Tags: geometry , trapezoid , construction , square

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)

1979 Chisinau City MO, 177

Tags: square , combinatorial geometry , geometry , tiling

Is it possible to cut a square into five squares?

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.

2022 AMC 10, 5

Tags: geometry , square

Square $ABCD$ has side length $1$. Point $P$, $Q$, $R$, and $S$ each lie on a side of $ABCD$ such that $APQCRS$ is an equilateral convex hexagon with side length $s$. What is $s$? $\textbf{(A) } \frac{\sqrt{2}}{3} \qquad \textbf{(B) } \frac{1}{2} \qquad \textbf{(C) } 2-\sqrt{2} \qquad \textbf{(D) } 1-\frac{\sqrt{2}}{4} \qquad \textbf{(E) } \frac{2}{3}$

2009 Swedish Mathematical Competition, 1

Tags: square , geometry , area

Five square carpets have been bought for a square hall with a side of $6$ m , two with the side $2$ m, one with the side $2.1$ m and two with the side $2.5$ m. Is it possible to place the five carpets so that they do not overlap in any way each other? The edges of the carpets do not have to be parallel to the cradles in the hall.

2021 Francophone Mathematical Olympiad, 3

Tags: geometry , square , incircle , tangent

Let $ABCD$ be a square with incircle $\Gamma$. Let $M$ be the midpoint of the segment $[CD]$. Let $P \neq B$ be a point on the segment $[AB]$. Let $E \neq M$ be the point on $\Gamma$ such that $(DP)$ and $(EM)$ are parallel. The lines $(CP)$ and $(AD)$ meet each other at $F$. Prove that the line $(EF)$ is tangent to $\Gamma$

1983 Tournament Of Towns, (044) 1

Tags: geometry , square

Inside square $ABCD$ consider a point $M$. Prove that the points of intersection of the medians of triangles $ABM, BCM, CDM$ and $DAM$ form a square. (V Prasolov)

2003 Austria Beginners' Competition, 4

Tags: square , rectangle , geometry

Prove that every rectangle circumscribed by a square is itself a square. (A rectangle is circumscribed by a square if there is exactly one corner point of the square on each side of the rectangle.)

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 $|DR| = |DA|$.

1977 IMO, 1

Tags: complex number , geometry , square , polygon

In the interior of a square $ABCD$ we construct the equilateral triangles $ABK, BCL, CDM, DAN.$ Prove that the midpoints of the four segments $KL, LM, MN, NK$ and the midpoints of the eight segments $AK, BK, BL, CL, CM, DM, DN, AN$ are the 12 vertices of a regular dodecagon.