This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 27

2000 Argentina National Olympiad, 6
Tags: paper , min , folding , geometry , area
You have an equilateral paper triangle of area $9$ and fold it in two, following a straight line that passes through the center of the triangle and does not contain any vertex of the triangle. Thus there remains a quadrilateral in which the two pieces overlap, and three triangles without overlaps. Determine the smallest possible value of the quadrilateral area of the overlay.

2007 May Olympiad, 5
Tags: geometry , paper , rectangle , pentagon
You have a paper pentagon, $ABCDE$, such that $AB = BC = 3$ cm, $CD = DE= 5$ cm, $EA = 4$ cm, $\angle ABC = 100^o$ ,$ \angle CDE = 80^o$. You have to divide the pentagon into four triangles, by three straight cuts, so that with the four triangles assemble a rectangle, without gaps or overlays. (The triangles can be rotated and / or turned around.)

May Olympiad L2 - geometry, 2023.4
Tags: paper , rectangle , folding , geometry
Matías has a rectangular sheet of paper $ABCD$, with $AB<AD$.Initially, he folds the sheet along a straight line $AE$, where $E$ is a point on the side $DC$ , so that vertex $D$ is located on side $BC$, as shown in the figure. Then folds the sheet again along a straight line $AF$, where $F$ is a point on side $BC$, so that vertex $B$ lies on the line $AE$; and finally folds the sheet along the line $EF$. Matías observed that the vertices $B$ and $C$ were located on the same point of segment $AE$ after making the folds. Calculate the measure of the angle $\angle DAE$. [img]https://cdn.artofproblemsolving.com/attachments/0/9/b9ab717e1806c6503a9310ee923f20109da31a.png[/img]

2011 Sharygin Geometry Olympiad, 2
Tags: construction , paper , geometry , rectangle
Peter made a paper rectangle, put it on an identical rectangle and pasted both rectangles along their perimeters. Then he cut the upper rectangle along one of its diagonals and along the perpendiculars to this diagonal from two remaining vertices. After this he turned back the obtained triangles in such a way that they, along with the lower rectangle form a new rectangle. Let this new rectangle be given. Restore the original rectangle using compass and ruler.

2006 Sharygin Geometry Olympiad, 5
Tags: combinatorial geometry , geometry , paper , folding , combinatorics , square
a) Fold a $10 \times 10$ square from a $1 \times 118$ rectangular strip. b) Fold a $10 \times 10$ square from a $1 \times (100+9\sqrt3)$ rectangular strip (approximately $1\times 115.58$). The strip can be bent, but not torn.

May Olympiad L1 - geometry, 2019.4
Tags: paper , cutting the paper , obtuse triangle , geometry , square , combinatorics
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.

2013 Swedish Mathematical Competition, 2
Tags: paper , folding , nonagon , square , geometry
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 $s$.

May Olympiad L1 - geometry, 2012.3
Tags: geometry , paper , area
From a paper quadrilateral like the one in the figure, you have to cut out a new quadrilateral whose area is equal to half the area of the original quadrilateral.You can only bend one or more times and cut by some of the lines of the folds. Describe the folds and cuts and justify that the area is half. [img]https://2.bp.blogspot.com/-btvafZuTvlk/XNY8nba0BmI/AAAAAAAAKLo/nm4c21A1hAIK3PKleEwt6F9cd6zv4XffwCK4BGAYYCw/s400/may%2B2012%2Bl1.png[/img]

May Olympiad L1 - geometry, 2005.4
Tags: geometry , rectangle , equilateral triangle , paper
There are two paper figures: an equilateral triangle and a rectangle. The height of rectangle is equal to the height of the triangle and the base of the rectangle is equal to the base of the triangle. Divide the triangle into three parts and the rectangle into two, using straight cuts, so that with the five pieces can be assembled, without gaps or overlays, a equilateral triangle. To assemble the figure, each part can be rotated and / or turned around.

2010 Oral Moscow Geometry Olympiad, 2
Tags: geometry , paper , folding , distance , square
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).

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.

2015 Sharygin Geometry Olympiad, P1
Tags: geometry , paper , convex polygon
Tanya cut out a convex polygon from the paper, fold it several times and obtained a two-layers quadrilateral. Can the cutted polygon be a heptagon?

2016 Sharygin Geometry Olympiad, 5
Tags: paper , equilateral triangle , geometry
Three points are marked on the transparent sheet of paper. Prove that the sheet can be folded along some line in such a way that these points form an equilateral triangle. by A.Khachaturyan

2006 May Olympiad, 2
Tags: rectangle , geometry , pentagon , paper , area
A rectangle of paper of $3$ cm by $9$ cm is folded along a straight line, making two opposite vertices coincide. In this way a pentagon is formed. Calculate it's area.

2023 May Olympiad, 4
Tags: rectangle , paper , folding , geometry
Matías has a rectangular sheet of paper $ABCD$, with $AB<AD$.Initially, he folds the sheet along a straight line $AE$, where $E$ is a point on the side $DC$ , so that vertex $D$ is located on side $BC$, as shown in the figure. Then folds the sheet again along a straight line $AF$, where $F$ is a point on side $BC$, so that vertex $B$ lies on the line $AE$; and finally folds the sheet along the line $EF$. Matías observed that the vertices $B$ and $C$ were located on the same point of segment $AE$ after making the folds. Calculate the measure of the angle $\angle DAE$. [img]https://cdn.artofproblemsolving.com/attachments/0/9/b9ab717e1806c6503a9310ee923f20109da31a.png[/img]

2016 Oral Moscow Geometry Olympiad, 6
Tags: paper , segment , folding , square , geometry
Given a square sheet of paper with a side of $2016$. Is it possible to bend its not more than ten times, construct a segment of length $1$?

2012 May Olympiad, 3
Tags: geometry , paper , area
From a paper quadrilateral like the one in the figure, you have to cut out a new quadrilateral whose area is equal to half the area of the original quadrilateral.You can only bend one or more times and cut by some of the lines of the folds. Describe the folds and cuts and justify that the area is half. [img]https://2.bp.blogspot.com/-btvafZuTvlk/XNY8nba0BmI/AAAAAAAAKLo/nm4c21A1hAIK3PKleEwt6F9cd6zv4XffwCK4BGAYYCw/s400/may%2B2012%2Bl1.png[/img]

May Olympiad L1 - geometry, 2007.5
Tags: geometry , pentagon , paper , rectangle
You have a paper pentagon, $ABCDE$, such that $AB = BC = 3$ cm, $CD = DE= 5$ cm, $EA = 4$ cm, $\angle ABC = 100^o$ ,$ \angle CDE = 80^o$. You have to divide the pentagon into four triangles, by three straight cuts, so that with the four triangles assemble a rectangle, without gaps or overlays. (The triangles can be rotated and / or turned around.)

2005 May Olympiad, 4
Tags: rectangle , equilateral triangle , paper , geometry
There are two paper figures: an equilateral triangle and a rectangle. The height of rectangle is equal to the height of the triangle and the base of the rectangle is equal to the base of the triangle. Divide the triangle into three parts and the rectangle into two, using straight cuts, so that with the five pieces can be assembled, without gaps or overlays, a equilateral triangle. To assemble the figure, each part can be rotated and / or turned around.

May Olympiad L1 - geometry, 2006.2
Tags: geometry , rectangle , pentagon , paper , area
A rectangle of paper of $3$ cm by $9$ cm is folded along a straight line, making two opposite vertices coincide. In this way a pentagon is formed. Calculate it's area.

2013 May Olympiad, 3
Tags: square , paper , perimeter , geometry
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]

2010 Sharygin Geometry Olympiad, 7
Tags: polygon , regular polygon , paper , geometry
Each of two regular polygons $P$ and $Q$ was divided by a line into two parts. One part of $P$ was attached to one part of $Q$ along the dividing line so that the resulting polygon was regular and not congruent to $P$ or $Q$. How many sides can it have?

2002 May Olympiad, 2
Tags: geometry , rectangle , paper , area
A rectangular sheet of paper (white on one side and gray on the other) was folded three times, as shown in the figure: Rectangle $1$, which was white after the first fold, has $20$ cm more perimeter than rectangle $2$, which was white after the second fold, and this in turn has $16$ cm more perimeter than rectangle $3$, which was white after the third fold. Determine the area of the sheet. [img]https://cdn.artofproblemsolving.com/attachments/d/f/8e363b40654ad0d8e100eac38319ee3784a7a7.png[/img]

2023 Oral Moscow Geometry Olympiad, 2
Tags: geometry , rectangle , ratio , paper
There is a square sheet of paper. How to get a rectangular sheet of paper with an aspect ratio equal to $\sqrt2$? (There are no tools, the sheet can only be bent.)

May Olympiad L1 - geometry, 2002.2
Tags: geometry , rectangle , area , paper
A rectangular sheet of paper (white on one side and gray on the other) was folded three times, as shown in the figure: Rectangle $1$, which was white after the first fold, has $20$ cm more perimeter than rectangle $2$, which was white after the second fold, and this in turn has $16$ cm more perimeter than rectangle $3$, which was white after the third fold. Determine the area of the sheet. [img]https://cdn.artofproblemsolving.com/attachments/d/f/8e363b40654ad0d8e100eac38319ee3784a7a7.png[/img]