Olympiad problems

2011 Saudi Arabia BMO TST, 1

Tags: square , geometry

Let $ABCD$ be a square of center $O$. The parallel to $AD$ through $O$ intersects $AB$ and $CD$ at $M$ and $N$ and a parallel to $AB$ intersects diagonal $AC$ at $P$. Prove that $$OP^4 + \left(\frac{MN}{2} \right)^4 = MP^2 \cdot NP^2$$

2015 Balkan MO Shortlist, N1

Tags: number theory , sum of squares , square , combinatorics

Let $d$ be an even positive integer. John writes the numbers $1^2 ,3^2 ,\ldots,(2n-1)^2 $ on the blackboard and then chooses three of them, let them be ${a_1}, {a_2}, {a_3}$, erases them and writes the number $1+ \displaystyle\sum_{1\le i<j\leq 3} |{a_i} -{a_j}|$ He continues until two numbers remain written on on the blackboard. Prove that the sum of squares of those two numbers is different than the numbers $1^2 ,3^2 ,\ldots,(2n-1)^2$. (Albania)

2017 Hanoi Open Mathematics Competitions, 11

Tags: geometry , square

Let $S$ denote a square of the side-length $7$, and let eight squares of the side-length $3$ be given. Show that $S$ can be covered by those eight small squares.

1967 German National Olympiad, 1

Tags: locus , equilateral , square , geometry

In a plane, a square $ABCD$ and a point $P$ located inside it are given. Let a point $ Q$ pass through all sides of the square. Describe the set of all those points $R$ in for which the triangle $PQR$ is equilateral.

1991 All Soviet Union Mathematical Olympiad, 555

Tags: circumcircle , square , geometry

$ABCD$ is a square. The points $X$ on the side $AB$ and $Y$ on the side $AD$ are such that $AX\cdot AY = 2 BX\cdot DY$. The lines $CX$ and $CY$ meet the diagonal $BD$ in two points. Show that these points lie on the circumcircle of $AXY$.

Estonia Open Junior - geometry, 2005.2.3

Tags: regular , octagon , geometry , square

The vertices of the square $ABCD$ are the centers of four circles, all of which pass through the center of the square. Prove that the intersections of the circles on the square $ABCD$ sides are vertices of a regular octagon.

2012 Denmark MO - Mohr Contest, 2

Tags: rectangle , square , geometry , combinatorial geometry

It is known about a given rectangle that it can be divided into nine squares which are situated relative to each other as shown. The black rectangle has side length $1$. Are there more than one possibility for the side lengths of the rectangle? [img]https://cdn.artofproblemsolving.com/attachments/1/0/af6bc5b867541c04586e4b03db0a7f97f8fe87.png[/img]

2016 Novosibirsk Oral Olympiad in Geometry, 4

Tags: midpoint , square , geometry

The two angles of the squares are adjacent, and the extension of the diagonals of one square intersect the diagonal of another square at point $O$ (see figure). Prove that $O$ is the midpoint of $AB$. [img]https://cdn.artofproblemsolving.com/attachments/7/8/8daaaa55c38e15c4a8ac7492c38707f05475cc.png[/img]

1962 Swedish Mathematical Competition, 2

Tags: radius , geometry , square

$ABCD$ is a square side $1$. $P$ and $Q$ lie on the side $AB$ and $R$ lies on the side $CD$. What are the possible values for the circumradius of $PQR$?

2002 Portugal MO, 5

Tags: perfect square , square , geometry

Consider the three squares indicated in the figure. Show that if the lengths of the sides of the smaller square and the square greater are integers, then adding to the area of the smallest square the area of the inclined square, a perfect square is obtained. [img]https://1.bp.blogspot.com/-B0QdvZIjOLw/X4URvs3C0ZI/AAAAAAAAMmw/S5zMpPBXBn8Jj39d-OZVtMRUDn3tXbyWgCLcBGAsYHQ/s0/2002%2Bportugal%2Bp5.png[/img]

2012 Sharygin Geometry Olympiad, 3

Tags: geometry , square , inradius , incircle

A paper square was bent by a line in such way that one vertex came to a side not containing this vertex. Three circles are inscribed into three obtained triangles (see Figure). Prove that one of their radii is equal to the sum of the two remaining ones. (L.Steingarts)

2017 Hanoi Open Mathematics Competitions, 15

Tags: geometry , square , combinatorial geometry

Let $S$ denote a square of side-length $7$, and let eight squares with side-length $3$ be given. Show that it is impossible to cover $S$ by those eight small squares with the condition: an arbitrary side of those (eight) squares is either coincided, parallel, or perpendicular to others of $S$ .

Kharkiv City MO Seniors - geometry, 2014.10.4

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

Let $ABCD$ be a square. The points $N$ and $P$ are chosen on the sides $AB$ and $AD$ respectively, such that $NP=NC$. The point $Q$ on the segment $AN$ is such that that $\angle QPN=\angle NCB$. Prove that $\angle BCQ=\frac{1}{2}\angle AQP$.

Novosibirsk Oral Geo Oly IX, 2019.3

Tags: geometry , area , square

The circle touches the square and goes through its two vertices as shown in the figure. Find the area of the square. (Distance in the picture is measured horizontally from the midpoint of the side of the square.) [img]https://cdn.artofproblemsolving.com/attachments/7/5/ab4b5f3f4fb4b70013e6226ce5189f3dc2e5be.png[/img]

2013 Czech-Polish-Slovak Junior Match, 6

Tags: geometry , equal circles , square , circles

There is a square $ABCD$ in the plane with $|AB|=a$. Determine the smallest possible radius value of three equal circles to cover a given square.

Ukrainian From Tasks to Tasks - geometry, 2014.4

Tags: right triangle , geometry , square

In the triangle $ABC$ it is known that $AC = 21$ cm, $BC = 28$ cm and $\angle C = 90^o$. On the hypotenuse $AB$, we construct a square $ABMN$ with center $O$ such that the segment $CO$ intersects the hypotenuse $AB$ at the point $K$. Find the lengths of the segments $AK$ and $KB$.

2013 Estonia Team Selection Test, 4

Tags: right triangle , cyclic quadrilateral , concyclic , square , geometry

Let $D$ be the point different from $B$ on the hypotenuse $AB$ of a right triangle $ABC$ such that $|CB| = |CD|$. Let $O$ be the circumcenter of triangle $ACD$. Rays $OD$ and $CB$ intersect at point $P$, and the line through point $O$ perpendicular to side AB and ray $CD$ intersect at point $Q$. Points $A, C, P, Q$ are concyclic. Does this imply that $ACPQ$ is a square?

1954 Putnam, A2

Tags: distance , square

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?

1980 Tournament Of Towns, (005) 5

Tags: combinatorial geometry , area , combinatorics , 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)

2012 NZMOC Camp Selection Problems, 1

Tags: regular , octagon , area , square , geometry

From a square of side length $1$, four identical triangles are removed, one at each corner, leaving a regular octagon. What is the area of the octagon?

1979 All Soviet Union Mathematical Olympiad, 277

Tags: area , square , combinatorial geometry , geometry

Given some square carpets with the total area $4$. Prove that they can fully cover the unit square.

2003 Austria Beginners' Competition, 4

Tags: geometry , rectangle , square

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.)

2019 Novosibirsk Oral Olympiad in Geometry, 7

Tags: acute , square , geometry

The square was cut into acute -angled triangles. Prove that there are at least eight of them.

1970 Dutch Mathematical Olympiad, 1

Tags: square , construction , geometry

Four different points $A,B,C$ and $D$ lie in a plane. No three of these points lie on a single straight line. Describe the construction of a square $PQRS$ such that on each of the sides of $PQRS$, or the extensions , lies one of the points $A, B, C$ and $D$.

1942 Putnam, B1

Tags: conic , square

A square of side $2a$, lying always in the first quadrant of the $xy$-plane, moves so that two consecutive vertices are always on the $x$- and $y$-axes respectively. Prove that a point within or on the boundary of the square will in general describe a portion of a conic. For what points of the square does this locus degenerate?