Found problems: 23
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).
A right-angled triangle made of paper is folded along a straight line so that the vertex at the right angle coincides with one of the other vertices of the triangle and a quadrilateral is obtained .
(a) What is the ratio into which the diagonals of this quadrilateral divide each other?
(b) This quadrilateral is cut along its longest diagonal. Find the area of the smallest piece of paper thus obtained if the area of the original triangle is $1$ .
(A Shapovalov)
Let $ABC$ be an acute-angled triangle. Construct three semi-circles, each having a different side of ABC as diameter, and outside $ABC$. The perpendiculars dropped from $A,B,C$ to the opposite sides intersect these semi-circles in points $E,F,G$, respectively. Prove that the hexagon $AGBECF$ can be folded so as to form a pyramid having $ABC$ as base.
The figure below, composed of four regular pentagons with a side length of $1$, was glued in space as follows. First, it was folded along the broken sections, by combining the bold sections, and then formed in such a way that colored sections formed a square. Find the length of the segment $AB$ created in this way.
[img]https://cdn.artofproblemsolving.com/attachments/0/7/bddad6449f74cbc7ea2623957ef05b3b0d2f19.png[/img]
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.
A rectangular sheet of paper whose dimensions are $12 \times 18$ is folded along a diagonal, creating the $M$-shaped region drawn in the picture (see below). Find the area of the shaded region.
[img]https://cdn.artofproblemsolving.com/attachments/4/7/d82cde3e91ab83fa14cd6cefe9bba28462dde1.png[/img]
A square sheet of paper $ABCD$ is folded straight in such a way that point $B$ hits to the midpoint of side $CD$. In what ratio does the fold line divide side $BC$?
Given a rectangle paper of size $15$ cm $\times$ $20$ cm, fold it along a diagonal.
Determine the area of the common part of two halfs of the paper?
A rectangular sheet with sides $a$ and $b$ is fold along a diagonal. Compute the area of the overlapping triangle.
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]
A sheet of rectangular $ABCD$ paper, of area $1$, is folded along its diagonal $AC$ and then unfolded, then it is bent so that vertex $A$ coincides with vertex $C$ and then unfolded, leaving the crease $MN$, as shown below.
a) Show that the quadrilateral $AMCN$ is a rhombus.
b) If the diagonal $AC$ is twice the width $AD$, what is the area of the rhombus $AMCN$?
[img]https://2.bp.blogspot.com/-TeQ0QKYGzOQ/Xp9lQcaLbsI/AAAAAAAAL2E/JLXwEIPSr4U79tATcYzmcJjK5bGA6_RqACK4BGAYYCw/s400/2001%2Baomb%2Bl2.png[/img]
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.
A paper square is bent along the line $\ell$, which passes through its center, so that a non-convex hexagon is formed. Investigate the question of the circle of largest radius that can be placed in such a hexagon.
An infinite rectangular stripe of width $3$ cm is folded along a line. What is the minimum possible area of the region of overlapping?
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 $s$.
Let $ABC$ be an acute-angled triangle. Construct three semi-circles, each having a different side of ABC as diameter, and outside $ABC$. The perpendiculars dropped from $A,B,C$ to the opposite sides intersect these semi-circles in points $E,F,G$, respectively. Prove that the hexagon $AGBECF$ can be folded so as to form a pyramid having $ABC$ as base.
A square sheet of paper $ABCD$ is folded straight in such a way that point $B$ hits to the midpoint of side $CD$. In what ratio does the fold line divide side $BC$?
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 $A'BE$.
On a rectangular piece of paper there are
(a) several marked points all on one straight line,
(b) three marked points (not necessarily on a straight line).
We are allowed to fold the paper several times along a straight line not containing marked points and then puncture the folded paper with a needle. Show that this can be done so that after the paper has been unfolded all the marked points are punctured and there are no extra holes.
(A Shapovalov)
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 $1$?
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]
A semicircular paper is folded along a chord such that the folded circular arc is tangent to the diameter of the semicircle. The radius of the semicircle is $4$ units and the point of tangency divides the diameter in the ratio $7 :1$. If the length of the crease (the dotted line segment in the figure) is $\ell$ then determine $ \ell^2$.
[img]https://cdn.artofproblemsolving.com/attachments/5/6/63fed83742c8baa92d9e63962a77a57d43556f.png[/img]
A paper rectangle $ABCD $ of area $1$ is folded along a straight line so that $C$ coincides with $A$. Prove that the area of the pentagon thus obtained is less than $3/4$.