Found problems: 467
1986 All Soviet Union Mathematical Olympiad, 419
Two equal squares, one with red sides, another with blue ones, give an octagon in intersection. Prove that the sum of red octagon sides lengths is equal to the sum of blue octagon sides lengths.
May Olympiad L2 - geometry, 1996.4
Let $ABCD$ be a square and let point $F$ be any point on side $BC$. Let the line perpendicular to $DF$, that passes through $B$, intersect line $DC$ at $Q$. What is value of $\angle FQC$?
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.
1966 Czech and Slovak Olympiad III A, 3
A square $ABCD,AB=s=1$ is given in the plane with its center $S$. Furthermore, points $E,F$ are given on the rays opposite to $CB,DA$, respectively, $CE=a,DF=b$. Determine all triangles $XYZ$ such that $X,Y,Z$ lie in this order on segments $CD,AD,BC$ and $E,S,F$ lie on lines $XY,YZ,ZX$ respectively. Discuss conditions of solvability in terms of $a,b,s$ and unknown $x=CX$.
2016 Bulgaria JBMO TST, 2
The vertices of the pentagon $ABCDE$ are on a circle, and the points $H_1, H_2, H_3,H_4$ are the orthocenters of the triangles $ABC, ABE, ACD, ADE$ respectively . Prove that the quadrilateral determined by the four orthocenters is square if and only if $BE \parallel CD$ and the distance between them is $\frac{BE + CD}{2}$.
2007 Estonia Math Open Junior Contests, 7
The center of square $ABCD$ is $K$. The point $P$ is chosen such that $P \ne K$ and the angle $\angle APB$ is right . Prove that the line $PK$ bisects the angle between the lines $AP$ and $BP$.
Denmark (Mohr) - geometry, 2009.4
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 $DF = AG$.
2017 BMT Spring, 2
Barack is an equilateral triangle and Michelle is a square. If Barack and Michelle each have perimeter $ 12$, find the area of the polygon with larger area.
2010 Saudi Arabia BMO TST, 2
Let $ABC$ be an acute triangle and let $MNPQ$ be a square inscribed in the triangle such that $M ,N \in BC$, $P \in AC$, $Q \in AB$. Prove that $area \, [MNPQ] \le \frac12 area\, [ABC]$.
1991 All Soviet Union Mathematical Olympiad, 548
A polygon can be transformed into a new polygon by making a straight cut, which creates two new pieces each with a new edge. One piece is then turned over and the two new edges are reattached. Can repeated transformations of this type turn a square into a triangle?
2010 Oral Moscow Geometry Olympiad, 2
Given a square sheet of paper with side $1$. Measure on this sheet a distance of $ 5/6$. (The sheet can be folded, including, along any segment with ends at the edges of the paper and unbend back, after unfolding, a trace of the fold line remains on the paper).
1963 Swedish Mathematical Competition., 2
The squares of a chessboard have side $4$. What is the circumference of the largest circle that can be drawn entirely on the black squares of the board?
2023 Iranian Geometry Olympiad, 3
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]
Kvant 2020, M2598
Is it possible that two cross-sections of a tetrahedron by two different cutting planes are two squares, one with a side of length no greater than $1$ and another with a side of length at least $100$?
Mikhail Evdokimov
2012 Junior Balkan Team Selection Tests - Romania, 2
From an $n \times n $ square, $n \ge 2,$ the unit squares situated on both odd numbered rows and odd numbers columns are removed. Determine the minimum number of rectangular tiles needed to cover the remaining surface.
2018 Portugal MO, 2
In the figure, $[ABCD]$ is a square of side $1$. The points $E, F, G$ and $H$ are such that $[AFB], [BGC], [CHD]$ and $[DEA]$ are right-angled triangles. Knowing that the circles inscribed in each of these triangles and the circle inscribed in the square $[EFGH]$ has all the same radius, what is the measure of the radius of the circles?
[img]https://1.bp.blogspot.com/-l37AEXa7_-c/X4KaJwe6HQI/AAAAAAAAMk4/14wvIipf26cRge_GqKSRwH32bp291vX4QCLcBGAsYHQ/s0/2018%2Bportugal%2Bp2.png[/img]
1974 All Soviet Union Mathematical Olympiad, 195
Given a square $ABCD$. Points $P$ and $Q$ are in the sides $[AB]$ and $[BC]$ respectively. $|BP|=|BQ|$. Let $H$ be the foot of the perpendicular from the point $B$ to the segment $[PC]$. Prove that the $\angle DHQ =90^o$ .
2016 Regional Olympiad of Mexico Northeast, 4
Let $ABCD$ be a square. Let $P$ be a point on the semicircle of diameter $AB$ outside the square. Let $M$ and $N$ be the intersections of $PD$ and $PC$ with $AB$, respectively. Prove that $MN^2 = AM \cdot BN$.
1995 Romania Team Selection Test, 3
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.
2017 Puerto Rico Team Selection Test, 6
Miguel has a square piece of paper $ABCD$ that he folded along a line $EF$, $E$ on $AB$, and $F$ on $CD$. This fold sent $A$ to point $A'$ on $BC$, distinct from $B$ and $C$. Also, it brought $D$ to point $D'$. $G$ is the intersection of $A'D'$ and $DC$. Prove that the inradius of $GCA'$ is equal to the sum of the inradius of $D'GF$ and $A'BE$.
2012 Sharygin Geometry Olympiad, 8
A square is divided into several (greater than one) convex polygons with mutually different numbers of sides. Prove that one of these polygons is a triangle.
(A.Zaslavsky)
2025 Kyiv City MO Round 1, Problem 2
Prove that the number
\[
3 \underbrace{99\ldots9}_{2025} \underbrace{60\ldots01}_{2025}
\]
is a square of a positive integer.
2014 Romania National Olympiad, 4
Outside the square $ABCD$ is constructed the right isosceles triangle $ABD$ with hypotenuse $[AB]$. Let $N$ be the midpoint of the side $[AD]$ and ${M} = CE \cap AB$, ${P} = CN \cap AB$ , ${F} = PE \cap MN$. On the line $FP$ the point $Q$ is considered such that the $[CE$ is the bisector of the angle $QCB$. Prove that $MQ \perp CF$.
2005 Chile National Olympiad, 1
In the center of the square of side $1$ shown in the figure is an ant. At one point the ant starts walking until it touches the left side $(a)$, then continues walking until it reaches the bottom side $(b)$, and finally returns to the starting point. Show that, regardless of the path followed by the ant, the distance it travels is greater than the square root of $2$.
[asy]
unitsize(2 cm);
draw((0,0)--(1,0)--(1,1)--(0,1)--cycle);
label("$a$", (0,0.5), W);
label("$b$", (0.5,0), S);
dot((0.5,0.5));
[/asy]
2022 Kurschak Competition, 2
Let $p$ and $q$ be prime numbers of the form $4k+3$. Suppose that there exist integers $x$ and $y$ such that $x^2-pqy^2=1$. Prove that there exist positive integers $a$ and $b$ such that $|pa^2-qb^2|=1$.