Olympiad problems

2013 Balkan MO Shortlist, C2

Tags: rectangle , combinatorics , center , chessboard , square

Some squares of an $n \times n$ chessboard have been marked ($n \in N^*$). Prove that if the number of marked squares is at least $n\left(\sqrt{n} + \frac12\right)$, then there exists a rectangle whose vertices are centers of marked squares.

2016 BMT Spring, 14

Tags: equal circles , circles , square , geometry

Three circles of radius $1$ are inscribed in a square of side length $s$, such that the circles do not overlap or coincide with each other. What is the minimum $s$ where such a configuration is possible?

2000 Poland - Second Round, 6

Tags: polynomial , square , algebra

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.

1978 Bundeswettbewerb Mathematik, 2

Tags: geometry , square , area , vertices , triangle

Seven distinct points are given inside a square with side length $1.$ Together with the square's vertices, they form a set of $11$ points. Consider all triangles with vertices in $M.$ a) Show that at least one of these triangles has an area not exceeding $1\slash 16.$ b) Give an example in which no four of the seven points are on a line and none of the considered triangles has an area of less than $1\slash 16.$

1979 Austrian-Polish Competition, 1

Tags: square , right angle , equal segments , geometry

On sides $AB$ and $BC$ of a square $ABCD$ the respective points $E$ and $F$ have been chosen so that $BE = BF$. Let $BN$ be the altitude in triangle $BCE$. Prove that $\angle DNF = 90$.

2019 Oral Moscow Geometry Olympiad, 1

Tags: geometry , square , incircle , angle

Circle inscribed in square $ABCD$ , is tangent to sides $AB$ and $CD$ at points $M$ and $K$ respectively. Line $BK$ intersects this circle at the point $L, X$ is the midpoint of $KL$. Find the angle $\angle MXK $.

1998 May Olympiad, 4

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

$ABCD$ is a square of center $O$. On the sides $DC$ and $AD$ the equilateral triangles DAF and DCE have been constructed. Decide if the area of the $EDF$ triangle is greater, less or equal to the area of the $DOC$ triangle. [img]https://4.bp.blogspot.com/-o0lhdRfRxl0/XNYtJgpJMmI/AAAAAAAAKKg/lmj7KofAJosBZBJcLNH0JKjW3o17CEMkACK4BGAYYCw/s1600/may4_2.gif[/img]

2019 AMC 8, 20

Tags: square

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

Indonesia Regional MO OSP SMA - geometry, 2020.1

Tags: geometry , rectangle , square

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 $ABCD$.

1996 Tournament Of Towns, (498) 5

Tags: area , geometry , square

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)

2023 Novosibirsk Oral Olympiad in Geometry, 2

Tags: geometry , square , area

In the square, the midpoints of the two sides were marked and the segments shown in the figure on the left were drawn. Which of the shaded quadrilaterals has the largest area? [img]https://cdn.artofproblemsolving.com/attachments/d/f/2be7bcda3fa04943687de9e043bd8baf40c98c.png[/img]

2013 Estonia Team Selection Test, 4

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

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?

2019 Romania National Olympiad, 2

Tags: equal segments , geometry , square , perpendicular

Let $ABCD$ be a square and $E$ a point on the side $(CD)$. Squares $ENMA$ and $EBQP$ are constructed outside the triangle $ABE$. Prove that: a) $ND = PC$ b) $ND\perp PC$.

2010 Singapore Junior Math Olympiad, 1

Tags: geometry , inradius , square , tangential

Let the diagonals of the square $ABCD$ intersect at $S$ and let $P$ be the midpoint of $AB$. Let $M$ be the intersection of $AC$ and $PD$ and $N$ the intersection of $BD$ and $PC$. A circle is incribed in the quadrilateral $PMSN$. Prove that the radius of the circle is $MP- MS$.