Olympiad problems

2022 AMC 10, 5

Tags: square , geometry

Square $ABCD$ has side length $1$. Point $P$, $Q$, $R$, and $S$ each lie on a side of $ABCD$ such that $APQCRS$ is an equilateral convex hexagon with side length $s$. What is $s$? $\textbf{(A) } \frac{\sqrt{2}}{3} \qquad \textbf{(B) } \frac{1}{2} \qquad \textbf{(C) } 2-\sqrt{2} \qquad \textbf{(D) } 1-\frac{\sqrt{2}}{4} \qquad \textbf{(E) } \frac{2}{3}$

2010 Dutch IMO TST, 4

Tags: circles , equal segments , geometry , square

Let $ABCD$ be a square with circumcircle $\Gamma_1$. Let $P$ be a point on the arc $AC$ that also contains $B$. A circle $\Gamma_2$ touches $\Gamma_1$ in $P$ and also touches the diagonal $AC$ in $Q$. Let $R$ be a point on $\Gamma_2$ such that the line $DR$ touches $\Gamma_2$. Proof that $|DR| = |DA|$.

2020 Novosibirsk Oral Olympiad in Geometry, 1

Tags: geometry , square

Three squares of area $4, 9$ and $36$ are inscribed in the triangle as shown in the figure. Find the area of the big triangle [img]https://cdn.artofproblemsolving.com/attachments/9/7/3e904a9c78307e1be169ec0b95b1d3d24c1aa2.png[/img]

2009 Denmark MO - Mohr Contest, 4

Tags: square , equal segments , geometry

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 $DF = AG$.

2019 May Olympiad, 4

Tags: combinatorics , geometry , cutting the paper , paper , obtuse triangle , square

You have to divide a square paper into three parts, by two straight cuts, so that by locating these parts properly, without gaps or overlaps, an obtuse triangle is formed. Indicate how to cut the square and how to assemble the triangle with the three parts.

2002 BAMO, 1

Tags: square , angle , right triangle , geometry

Let $ABC$ be a right triangle with right angle at $B$. Let $ACDE$ be a square drawn exterior to triangle $ABC$. If $M$ is the center of this square, find the measure of $\angle MBC$.

2022 Durer Math Competition Finals, 1

Tags: equilateral , square , tangent , geometry

To the exterior of side $AB$ of square $ABCD$, we have drawn the regular triangle $ABE$. Point $A$ reflected on line $BE$ is $F$, and point $E$ reflected on line $BF$ is $G$. Let the perpendicular bisector of segment $FG$ meet segment $AD$ at $X$. Show that the circle centered at $X$ with radius $XA$ touches line$ FB$.

1996 Mexico National Olympiad, 5

Tags: combinatorics , square , perfect cube , sum

The numbers $1$ to $n^2$ are written in an n×n squared paper in the usual ordering. Any sequence of right and downwards steps from a square to an adjacent one (by side) starting at square $1$ and ending at square $n^2$ is called a path. Denote by $L(C)$ the sum of the numbers through which path $C$ goes. (a) For a fixed $n$, let $M$ and $m$ be the largest and smallest $L(C)$ possible. Prove that $M-m$ is a perfect cube. (b) Prove that for no $n$ can one find a path $C$ with $L(C ) = 1996$.

Novosibirsk Oral Geo Oly VII, 2023.2

Tags: area , square , geometry

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]

1977 IMO Shortlist, 12

Tags: polygon , square , geometry , complex number

In the interior of a square $ABCD$ we construct the equilateral triangles $ABK, BCL, CDM, DAN.$ Prove that the midpoints of the four segments $KL, LM, MN, NK$ and the midpoints of the eight segments $AK, BK, BL, CL, CM, DM, DN, AN$ are the 12 vertices of a regular dodecagon.

Kyiv City MO 1984-93 - geometry, 1992.9.2

Tags: square , geometry , congruence

Two lines divide a square into $4$ figures of the same area. Prove that all these figures are congruent.

1999 Switzerland Team Selection Test, 7

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

A square is dissected into rectangles with sides parallel to the sides of the square. For each of these rectangles, the ratio of its shorter side to its longer side is considered. Show that the sum of all these ratios is at least $1$.

Russian TST 2016, P1

Tags: geometry , square

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 $AC\perp{}XY$.

Novosibirsk Oral Geo Oly VII, 2019.4

Tags: geometry , square , collinear , isosceles

Two squares and an isosceles triangle are positioned as shown in the figure (the up left vertex of the large square lies on the side of the triangle). Prove that points $A, B$ and $C$ are collinear. [img]https://cdn.artofproblemsolving.com/attachments/d/c/03515e40f74ced1f8243c11b3e610ef92137ac.png[/img]

2002 District Olympiad, 4

Tags: rectangle , square , geometry , equal angles

Given the rectangle $ABCD$. The points $E ,F$ lie on the segments $(BC) , (DC)$ respectively, such that $\angle DAF = \angle FAE$. Proce that if $DF + BE = AE$ then $ABCD$ is square.

Kyiv City MO Juniors 2003+ geometry, 2017.8.4

Tags: collinear , square , geometry

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.

Novosibirsk Oral Geo Oly IX, 2016.3

Tags: angle , geometry , square

A square is drawn on a sheet of grid paper on the sides of the cells $ABCD$ with side $8$. Point $E$ is the midpoint of side $BC$, $Q$ is such a point on the diagonal $AC$ such that $AQ: QC = 3: 1$. Find the angle between straight lines $AE$ and $DQ$.

2009 Oral Moscow Geometry Olympiad, 2

Tags: square , rectangle , geometry

A square and a rectangle of the same perimeter have a common corner. Prove that the intersection point of the diagonals of the rectangle lies on the diagonal of the square. (Yu. Blinkov)

Novosibirsk Oral Geo Oly IX, 2020.2

Tags: geometry , ruler , construction , square

A $2 \times 2$ square was cut out of a sheet of grid paper. Using only a ruler without divisions and without going beyond the square, divide the diagonal of the square into $6$ equal parts.

1963 German National Olympiad, 5

Tags: midpoint , locus , square , geometry

Given is a square with side length $a$. A distance $PQ$ of length $p$, where $p < a$, moves so that its end points are always on the sides of the square. What is the geometric locus of the midpoints of the segments $PQ$?

2013 NZMOC Camp Selection Problems, 6

Tags: geometry , square , cyclic , tangential

$ABCD$ is a quadrilateral having both an inscribed circle (one tangent to all four sides) with center $I,$ and a circumscribed circle with center $O$. Let $S$ be the point of intersection of the diagonals of $ABCD$. Show that if any two of $S, I$ and $O$ coincide, then $ABCD$ is a square (and hence all three coincide).

2020 Costa Rica - Final Round, 2

Tags: circles , geometry , square

Consider a square $ABCD$. Let $M$ be the midpoint of segment $AB$, $\Gamma_1$ be the circle tangent to $\overline{AD}$, $\overline{AM}$ and $\overline{MC}$ with radius $r > 0$ and let $\Gamma_2$ be the circle tangent to $\overline{AD}$, $\overline{DC}$ and $\overline{MC}$ with radius $R > 0$. Prove that $R =\frac{2r}{r+1}$.

Ukraine Correspondence MO - geometry, 2016.11

Tags: trapezoid , angle , geometry , square

Inside the square $ABCD$ mark the point $P$, for which $\angle BAP = 30^o$ and $\angle BCP = 15^o$. The point $Q$ was chosen so that $APCQ$ is an isosceles trapezoid ($PC\parallel AQ$). Find the angles of the triangle $CAM$, where $M$ is the midpoint of $PQ$.

1989 Dutch Mathematical Olympiad, 2

Tags: geometry , tangent , square

Given is a square $ABCD$ with $E \in BC$, arbitrarily. On $CD$ lies the point $F$ is such that $\angle EAF = 45^o$. Prove that $EF$ is tangent to the circle with center $A$ and radius $AB$.

Estonia Open Junior - geometry, 2000.1.3

Tags: ratio , square , area

Consider a shape obtained from two equal squares with the same center. Prove that the ratio of the area of this shape to the perimeter does not change when the squares are rotated around their center. [img]http://4.bp.blogspot.com/-1AI4FxsNSr4/XovZWkvAwiI/AAAAAAAALvY/-kIzOgXB5rk3iIqGbpoKRCW9rwJPcZ3uQCK4BGAYYCw/s400/estonia%2B2000%2Bo.j.1.3.png[/img]