paperfolding a square paper of side 1 to measure a distance 5/6

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0 replies

jlacosta

Apr 2, 2025

0 replies

Find the angle

Alfombraking 0

13 minutes ago

Inside a right triangle ABC at , point Q is located, which belongs to the bisector of angle C. On the extension of BQ, point P is located from which PM⊥CQ(M en CQ) is drawn, such that BP=2(MC). If AQ=BC, then the measure of angle BAQ is.

0 replies

Alfombraking

13 minutes ago

0 replies

IMO Problem 4

iandrei 105

N 38 minutes ago by cj13609517288

Source: IMO ShortList 2003, geometry problem 1

Let $ABCD$ be a cyclic quadrilateral. Let $P$ , $Q$ , $R$ be the feet of the perpendiculars from $D$ to the lines $BC$ , $CA$ , $AB$ , respectively. Show that $PQ=QR$ if and only if the bisectors of $\angle ABC$ and $\angle ADC$ are concurrent with $AC$ .

105 replies

iandrei

Jul 14, 2003

cj13609517288

38 minutes ago

Inspired by old results

sqing 2

N 38 minutes ago by sqing

Source: Own

Let $a,b$ be reals such that $a^2+b^2=2$ . Prove that
$(a+b)\left( \frac{1}{a+1} + \frac{1}{b+1}\right)\leq 2$ $(a+b)\left( \frac{a}{b+1} + \frac{b}{a+1}\right)\leq 2$

2 replies

sqing

an hour ago

sqing

38 minutes ago

NEPAL TST 2025 DAY 2

Tony_stark0094 5

N an hour ago by iStud

Consider an acute triangle $\Delta ABC$ . Let $D$ and $E$ be the feet of the altitudes from $A$ to $BC$ and from $B$ to $AC$ respectively.

Define $D_1$ and $D_2$ as the reflections of $D$ across lines $AB$ and $AC$ , respectively. Let $\Gamma$ be the circumcircle of $\Delta AD_1D_2$ . Denote by $P$ the second intersection of line $D_1B$ with $\Gamma$ , and by $Q$ the intersection of ray $EB$ with $\Gamma$ .

If $O$ is the circumcenter of $\Delta ABC$ , prove that $O$ , $D$ , and $Q$ are collinear if and only if quadrilateral $BCQP$ can be inscribed within a circle.

$\textbf{Proposed by Kritesh Dhakal, Nepal.}$

5 replies

Tony_stark0094

Yesterday at 8:40 AM

iStud

an hour ago

No more topics!

paperfolding a square paper of side 1 to measure a distance 5/6

parmenides51 1

N Aug 16, 2020 by jchang0313

Source: 2010 Oral Moscow Geometry Olympiad grades 8-9 p2

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).

1 reply

parmenides51

Aug 16, 2020

jchang0313

Aug 16, 2020

paperfolding a square paper of side 1 to measure a distance 5/6

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Source: 2010 Oral Moscow Geometry Olympiad grades 8-9 p2

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parmenides51

30630 posts

parmenides51

#1 h Aug 16, 2020, 9:30 AM

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jchang0313

54 posts

jchang0313

#2 h Aug 16, 2020, 9:38 AM

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1. fold into half upper and lower.
2.lower left-->right middle. Mark the intersection of the left side and top side with a fold.(its 1/3 from left)
3. fold the same way with upper left-->right middle. Mark intersection.
4. fold the line with both marks(it should be parallel to left and right side)
5. fold the leftmost 1/3 part into half :-D

I hope this is clear!

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