Olympiad problems

1976 Chisinau City MO, 133

A triangle with a parallelogram inside was placed in a square. Prove that the area of a parallelogram is not more than a quarter of a square.

2014 Romania National Olympiad, 4

Tags: geometry , square , perpendicular , isosceles , right triangle

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 $MQ \perp CF$.

1979 Bundeswettbewerb Mathematik, 2

Tags: geometry , square , concurrency

The squares $OABC$ and $OA_1B_1C_1$ are situated in the same plane and are directly oriented. Prove that the lines $AA_1$ , $BB_1$, and $CC_1$ are concurrent.

2007 Postal Coaching, 1

Tags: sum , geometry , constant , circumcircle , square

Let $P$ be a point on the circumcircle of a square $ABCD$. Find all integers $n > 0$ such that the sum $$S_n(P) = |PA|^n + |PB|^n + |PC|^n + |PD|^n$$ is constant with respect to the point $P$.

1960 Kurschak Competition, 3

Tags: square , geometry

$E$ is the midpoint of the side $AB$ of the square $ABCD$, and $F, G$ are any points on the sides $BC$, $CD$ such that $EF$ is parallel to $AG$. Show that $FG$ touches the inscribed circle of the square.

1997 Denmark MO - Mohr Contest, 2

Tags: geometry , square , area

Two squares, both with side length $1$, are arranged so that one has one vertex in the center of the other. Determine the area of the gray area. [img]https://1.bp.blogspot.com/-xt3pe0rp1SI/XzcGLgEw1EI/AAAAAAAAMYM/vFKxvvVuLvAJ5FO_yX315X3Fg_iFaK2fACLcBGAsYHQ/s0/1997%2BMohr%2Bp2.png[/img]

2013 May Olympiad, 3

Tags: geometry , square , paper , perimeter

Let $ABCD$ be a square of side paper $10$ and $P$ a point on side $BC$. By folding the paper along the $AP$ line, point $B$ determines the point $Q$, as seen in the figure. The line $PQ$ cuts the side $CD$ at $R$. Calculate the perimeter of the triangle $ PCR$ [img]https://3.bp.blogspot.com/-ZSyCUznwutE/XNY7cz7reQI/AAAAAAAAKLc/XqgQnjm8DQYq6Q7fmCAKJwKt3ihoL8AuQCK4BGAYYCw/s400/may%2B2013%2Bl1.png[/img]

Novosibirsk Oral Geo Oly VII, 2023.3

Tags: geometry , rectangle , square

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]

2016 Latvia Baltic Way TST, 11

Tags: square , combinatorial geometry , rectangle , geometry , combinatorics

Is it possible to cut a square with side $\sqrt{2015}$ into no more than five pieces so that these pieces can be rearranged into a rectangle with sides of integer length? (The cuts should be made using straight lines, and flipping of the pieces is disallowed.)

2017 Singapore Junior Math Olympiad, 1

Tags: geometry , rectangle , ratio , geometric inequalities , inequalities , square , sidelenghts

A square is cut into several rectangles, none of which is a square, so that the sides of each rectangle are parallel to the sides of the square. For each rectangle with sides $a, b,a<b$, compute the ratio $a/b$. Prove that sum of these ratios is at least $1$.

Kharkiv City MO Seniors - geometry, 2014.10.4

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

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 $\angle BCQ=\frac{1}{2}\angle AQP$.

1977 Dutch Mathematical Olympiad, 2

Tags: geometry , square , 3d geometry

Four masts stand on a flat horizontal piece of land at the vertices of a square $ABCD$. The height of the mast on $A$ is $7$ meters, of the mast on $B$ $13$ meters, and of the mast on $C$ $15$ meters. Within the square there is a point $P$ on the ground equidistant from each of the tops of these three masts. (a) What length must the sides of the square be at least for this to be possible? (b) The distance from $P$ to the top of the mast on $D$ is equal to the distance from$ P$ to each of the tops of the three other masts. Calculate the height of the mast at $D$.

2014 Swedish Mathematical Competition, 4

Tags: geometry , combinatorial geometry , square , geometric inequalities

A square is cut into a finitely number of triangles in an arbitrary way. Show the sum of the diameters of the inscribed circles in these triangles is greater than the side length of the square.

2022 Novosibirsk Oral Olympiad in Geometry, 1

Tags: geometry , square

Cut a square with three straight lines into three triangles and four quadrilaterals.

Swiss NMO - geometry, 2008.5

Tags: geometry , locus , square

Let $ABCD$ be a square with side length $1$. Find the locus of all points $P$ with the property $AP\cdot CP + BP\cdot DP = 1$.

1973 Chisinau City MO, 66

Tags: symmetry , geometry , combinatorics , square

If $A$ and $B$ are points of the plane, then by $A * B$ we denote a point symmetric to $A$ with respect to $B$. Is it possible, by applying the operation $*$ several times, to obtain from the three vertices of a given square its fourth vertex?

Estonia Open Senior - geometry, 2000.2.4

Tags: tangential , inradius , geometry , square , midpoint , incircle , diagonal

The diagonals of the square $ABCD$ intersect at $P$ and the midpoint of the side $AB$ is $E$. Segment $ED$ intersects the diagonal $AC$ at point $F$ and segment $EC$ intersects the diagonal $BD$ at $G$. Inside the quadrilateral $EFPG$, draw a circle of radius $r$ tangent to all the sides of this quadrilateral. Prove that $r = | EF | - | FP |$.

1974 Czech and Slovak Olympiad III A, 6

Tags: geometry , geometric transformation , rotation , locus , square , area

Let a unit square $\mathcal D$ be given in the plane. For any point $X$ in the plane denote $\mathcal D_X$ the image of $\mathcal D$ in rotation with respect to origin $X$ by $+90^\circ.$ Find the locus of all $X$ such that the area of union $\mathcal D\cup\mathcal D_X$ is at most 1.5.

2010 Oral Moscow Geometry Olympiad, 5

Tags: square , equal segments , angle

Points $K$ and $M$ are taken on the sides $AB$ and $CD$ of square $ABCD$, respectively, and on the diagonal $AC$ - point $L$ such that $ML = KL$. Let $P$ be the intersection point of the segments $MK$ and $BD$. Find the angle $\angle KPL$.

1981 Czech and Slovak Olympiad III A, 3

Tags: geometry , locus , square , equilateral , triangle

Let $ABCD$ be a unit square. Consider an equilateral triangle $XYZ$ with $X,Y$ as (inner or boundary) points of the square. Determine the locus $M$ of vertices $Z$ of all these triangles $XYZ$ and compute the area of $M.$

2018 Hanoi Open Mathematics Competitions, 8

Tags: geometry , square , angle

Let $P$ be a point inside the square $ABCD$ such that $\angle PAC = \angle PCD = 17^o$ (see Figure 1). Calculate $\angle APB$? [img]https://cdn.artofproblemsolving.com/attachments/d/0/0b20ebee1fe28e9c5450d04685ac8537acda07.png[/img]

Kyiv City MO Juniors 2003+ geometry, 2020.7.4

Tags: geometry , rectangle , square

Given a square $ABCD$ with side $10$. On sides BC and $AD$ of this square are selected respectively points $E$ and $F$ such that formed a rectangle $ABEF$. Rectangle $KLMN$ is located so that its the vertices $K, L, M$ and $N$ lie one on each segments $CD, DF, FE$ and $EC$, respectively. It turned out that the rectangles $ABEF$ and $KLMN$ are equal with $AB = MN$. Find the length of segment $AL$.

Denmark (Mohr) - geometry, 2009.4

Tags: geometry , square , equal segments

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 $DF = AG$.

1988 Tournament Of Towns, (189) 2

Tags: geometry , equal angles , square , angle

A point $M$ is chosen inside the square $ABCD$ in such a way that $\angle MAC = \angle MCD = x$ . Find $\angle ABM$.

Kyiv City MO Juniors 2003+ geometry, 2017.8.4

Tags: geometry , square , collinear

On the sides $BC$ and $CD$ of the square $ABCD$, the points $M$ and $N$ are selected in such a way that $\angle MAN= 45^o$. Using the segment $MN$, as the diameter, we constructed a circle $w$, which intersects the segments $AM$ and $AN$ at points $P$ and $Q$, respectively. Prove that the points $B, P$ and $Q$ lie on the same line.