Found problems: 467
1955 Moscow Mathematical Olympiad, 291
Find all rectangles that can be cut into $13$ equal squares.
2013 Swedish Mathematical Competition, 2
The paper folding art origami is usually performed with square sheets of paper. Someone folds the sheet once along a line through the center of the sheet in orde to get a nonagon. Let $p$ be the perimeter of the nonagon minus the length of the fold, i.e. the total length of the eight sides that are not folds, and denote by s the original side length of the square. Express the area of the nonagon in terms of $p$ and $s$.
Kyiv City MO Juniors Round2 2010+ geometry, 2013.7.3
In the square $ABCD$ on the sides $AD$ and $DC$, the points $M$ and $N$ are selected so that $\angle BMA = \angle NMD = 60 { } ^ \circ $. Find the value of the angle $MBN$.
Ukrainian From Tasks to Tasks - geometry, 2014.4
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 $KB$.
2022 Cyprus JBMO TST, 2
Let $ABCD$ be a square. Let $E, Z$ be points on the sides $AB, CD$ of the square respectively, such that $DE\parallel BZ$. Assume that the triangles $\triangle EAD, \triangle ZCB$ and the parallelogram $BEDZ$ have the same area.
If the distance between the parallel lines $DE$ and $BZ$ is equal to $1$, determine the area of the square.
1997 All-Russian Olympiad Regional Round, 8.5
Segments $AB$, $BC$ and $CA$ are, respectively, diagonals of squares $K_1$, $K_2$, $K3$. Prove that if triangle $ABC$ is acute, then it completely covered by squares $K_1$, $K_2$ and $K_3$.
2003 Dutch Mathematical Olympiad, 2
Two squares with side $12$ lie exactly on top of each other.
One square is rotated around a corner point through an angle of $30$ degrees relative to the other square.
Determine the area of the common piece of the two squares.
[asy]
unitsize (2 cm);
pair A, B, C, D, Bp, Cp, Dp, P;
A = (0,0);
B = (-1,0);
C = (-1,1);
D = (0,1);
Bp = rotate(-30)*(B);
Cp = rotate(-30)*(C);
Dp = rotate(-30)*(D);
P = extension(C, D, Bp, Cp);
fill(A--Bp--P--D--cycle, gray(0.8));
draw(A--B--C--D--cycle);
draw(A--Bp--Cp--Dp--cycle);
label("$30^\circ$", (-0.5,0.1), fontsize(10));
[/asy]
Cono Sur Shortlist - geometry, 2018.G5
We say that a polygon $P$ is inscribed in another polygon $Q$ when all the vertices of $P$ belong to the perimeter of $Q$. We also say in this case that $Q$ is circumscribed to $P$. Given a triangle $T$, let $\ell$ be the largest side of a square inscribed in $T$ and $L$ is the shortest side of a square circumscribed to $T$ . Find the smallest possible value of the ratio $L/\ell$ .
1962 Swedish Mathematical Competition, 2
$ABCD$ 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 $PQR$?
May Olympiad L1 - geometry, 2004.4
In a square $ABCD$ of diagonals $AC$ and $BD$, we call $O$ at the center of the square. A square $PQRS$ is constructed with sides parallel to those of $ABCD$ with $P$ in segment $AO, Q$ in segment $BO, R$ in segment $CO, S$ in segment $DO$. If area of $ABCD$ equals two times the area of $PQRS$, and $M$ is the midpoint of the $AB$ side, calculate the measure of the angle $\angle AMP$.
1998 Croatia National Olympiad, Problem 3
Points $E$ and $F$ are chosen on the sides $AB$ and $BC$ respectively of a square $ABCD$ such that $BE=BF$. Let $BN$ be an altitude of the triangle $BCE$. Prove that the triangle $DNF$ is right-angled.
Revenge EL(S)MO 2024, 3
Find all solutions to
\[ (abcde)^2 = a^2+b^2+c^2+d^2+e^2+f^2. \]
in integers.
Proposed by [i]Seongjin Shim[/i]
1982 Bulgaria National Olympiad, Problem 6
Find the locus of centroids of equilateral triangles whose vertices lie on sides of a given square $ABCD$.
2012 Sharygin Geometry Olympiad, 3
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)
1999 Czech And Slovak Olympiad IIIA, 5
Given an acute angle $APX$ in the plane, construct a square $ABCD$ such that $P$ lies on the side $BC$ and ray $PX$ meets $CD$ in a point $Q$ such that $AP$ bisects the angle $BAQ$.
2014 Tournament of Towns., 3
A square table is covered with a square cloth (may be of a different size) without folds and wrinkles. All corners of the table are left uncovered and all four hanging parts are triangular. Given that two adjacent hanging parts are equal prove that two other parts are also equal.
2017 OMMock - Mexico National Olympiad Mock Exam, 3
Let $x, y, z$ be positive integers such that $xy=z^2+2$. Prove that there exist integers $a, b, c, d$ such that the following equalities are satisfied:
\begin{eqnarray*} x=a^2+2b^2\\
y=c^2+d^2\\
z=ac+2bd\\
\end{eqnarray*}
[i]Proposed by Isaac Jiménez[/i]
Russian TST 2016, P1
The squares $ABCD$ and $AXYZ$ are given. It turns out that $CDXY$ is a cyclic quadrilateral inscribed in the circle $\Omega$, and the points $A, B$ and $Z{}$ lie inside this circle. Prove that either $AB = AX$ or $AC\perp{}XY$.
1985 IMO Shortlist, 15
Let $K$ and $K'$ be two squares in the same plane, their sides of equal length. Is it possible to decompose $K$ into a finite number of triangles $T_1, T_2, \ldots, T_p$ with mutually disjoint interiors and find translations $t_1, t_2, \ldots, t_p$ such that
\[K'=\bigcup_{i=1}^{p} t_i(T_i) \ ? \]
Durer Math Competition CD 1st Round - geometry, 2008.C3
Given the squares $ABCD$ and $DEFG$, whose only common point is $D$. Let the midpoints of segments $AG$, $GE$, $EC$, and $CA$ be $H, I, J$, and $K$ respectively . Prove that $HIJK$ is a square.
[img]https://cdn.artofproblemsolving.com/attachments/f/d/c3313e5bbf581977a74ea2b114d14950e38605.png[/img]
2010 Abels Math Contest (Norwegian MO) Final, 1b
The edges of the square in the figure have length $1$.
Find the area of the marked region in terms of $a$, where $0 \le a \le 1$.
[img]https://cdn.artofproblemsolving.com/attachments/2/2/f2b6ca973f66c50e39124913b3acb56feff8bb.png[/img]
1980 All Soviet Union Mathematical Olympiad, 295
Some squares of the infinite sheet of cross-lined paper are red. Each $2\times 3$ rectangle (of $6$ squares) contains exactly two red squares. How many red squares can be in the $9\times 11$ rectangle of $99$ squares?
Novosibirsk Oral Geo Oly VIII, 2023.2
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]
2019 Novosibirsk Oral Olympiad in Geometry, 7
Cut a square into eight acute-angled triangles.
1974 Bundeswettbewerb Mathematik, 2
Seven polygons of area $1$ lie in the interior of a square with side length $2$. Show that there are two of these polygons whose intersection has an area of at least $1\slash 7.$