staircase with height 1, width 1,2,...,2019 (2019 Lomonosov Tournament R1)

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0 replies

jlacosta

Apr 2, 2025

0 replies

Basic geometry

AlexCenteno2007 6

N 31 minutes ago by vanstraelen

Given an isosceles triangle ABC with AB=BC, the inner bisector of Angle BAC And cut next to it BC in D. A point E is such that AE=DC. The inner bisector of the AED angle cuts to the AB side at the point F. Prove that the angle AFE= angle DFE

6 replies

AlexCenteno2007

Feb 9, 2025

vanstraelen

31 minutes ago

Algebraic Manipulation

Darealzolt 1

N 42 minutes ago by Soupboy0

Find the number of pairs of real numbers $a, b, c$ that satisfy the equation $a^4 + b^4 + c^4 + 1 = 4abc$ .

1 reply

Darealzolt

2 hours ago

Soupboy0

42 minutes ago

BrUMO 2025 Team Round Problem 13

lpieleanu 1

N 43 minutes ago by vanstraelen

Let $\omega$ be a circle, and let a line $\ell$ intersect $\omega$ at two points, $P$ and $Q.$ Circles $\omega_1$ and $\omega_2$ are internally tangent to $\omega$ at points $X$ and $Y,$ respectively, and both are tangent to $\ell$ at a common point $D.$ Similarly, circles $\omega_3$ and $\omega_4$ are externally tangent to $\omega$ at $X$ and $Y,$ respectively, and are tangent to $\ell$ at points $E$ and $F,$ respectively.

Given that the radius of $\omega$ is $13,$ the segment $\overline{PQ}$ has a length of $24,$ and $YD=YE,$ find the length of segment $\overline{YF}.$

1 reply

lpieleanu

Apr 27, 2025

vanstraelen

43 minutes ago

Inequlities

sqing 33

N an hour ago by sqing

Let $a,b,c\geq 0$ and $a^2+ab+bc+ca=3 .$ Prove that $\frac{1}{1+a^2}+ \frac{1}{1+b^2}+ \frac{1}{1+c^2} \geq \frac{3}{2}$ $\frac{1}{1+a^2}+ \frac{1}{1+b^2}+ \frac{1}{1+c^2}-bc \geq -\frac{3}{2}$

33 replies

sqing

Jul 19, 2024

sqing

an hour ago

No more topics!

staircase with height 1, width 1,2,...,2019 (2019 Lomonosov Tournament R1)

parmenides51 1

N Apr 20, 2021 by natmath

The height of each step of the “stairs” (see figure) is $1$ , and the width of each step increases from one to $2019$ . Is it true that the segment from the lower left point of the stairs to the upper right point of the stairs doesn’t cross the stairs?
IMAGE

1 reply

parmenides51

Apr 20, 2021

natmath

Apr 20, 2021

staircase with height 1, width 1,2,...,2019 (2019 Lomonosov Tournament R1)

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Lomonosov Tournament

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parmenides51

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parmenides51

#1 h Apr 20, 2021, 4:29 AM • 1 Y

Y by Mango247

https://2.bp.blogspot.com/-SSX2zt_XeUI/XpyWHnyB6BI/AAAAAAAAL0s/GQF-HTSNUwUfoZafwvW5wn_DQBKCKBWvACK4BGAYYCw/s1600/2019lom.png

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natmath

8219 posts

natmath

#2 h Apr 20, 2021, 5:43 AM • 1 Y

Y by Mango247

solution

The upper right point of a step is the lower left point of the next step, so we can just look at the lower left point of all steps.
If the bottom of the staircase is the origin than the lower left points lie on the curve with parametrization $(\binom{t+1}{2},t)$ , which has equation
$2x=y^2+y$ $y^2+y-2x=0$ $y=\frac{-1+\sqrt{1+8x}}{2}$ Since this function's concavity is strictly negative, a segment connecting 2 points on the curve does not intersect the curve again because of Jensen's.

I think this question is ~~Trivial by Jensen's~~

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