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Cauchy-Schwarz 1
prtoi   3
N an hour ago by sqing
Source: Handout by Samin Riasat
$\frac{a}{b+c}+\frac{b}{c+d}+\frac{c}{d+a}+\frac{d}{a+b}\ge2$
3 replies
prtoi
Yesterday at 4:16 PM
sqing
an hour ago
J
H
AM-GM problem from a handout
prtoi   2
N an hour ago by sqing
Prove that:
$\frac{a}{b}+\frac{b}{c}+\frac{c}{a}+\frac{3(abc)^{1/3}}{a+b+c}\ge3+n$
2 replies
prtoi
Yesterday at 4:09 PM
sqing
an hour ago
J
H
Cauchy-Schwarz 6
prtoi   2
N an hour ago by sqing
Source: Handout by Samin Riasat
Let a, b, c > 0. Prove that
$\sum_{cyc}^{}\sqrt{\frac{2a}{b+c}}\le\sqrt{3(\frac{a}{b}+\frac{b}{c}+\frac{c}{a})}$
2 replies
prtoi
Yesterday at 4:30 PM
sqing
an hour ago
J
H
Another AM-GM problem
prtoi   2
N an hour ago by sqing
Source: Handout by Samin Riasat
Prove that:
$\frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{a}+\frac{3n}{a^2+b^2+c^2}\ge3+n$
2 replies
prtoi
Yesterday at 4:11 PM
sqing
an hour ago
J
H
Cauchy Schwarz 4
prtoi   3
N an hour ago by sqing
Source: Zhautykov Olympiad 2008
Let a, b, c be positive real numbers such that abc = 1.
Show that
$\frac{1}{b(a+b)}+\frac{1}{b(a+b)}+\frac{1}{b(a+b)}\ge\frac{3}{2}$
3 replies
prtoi
Yesterday at 4:25 PM
sqing
an hour ago
J
H
projection vector manipulation
RenheMiResembleRice   1
N an hour ago by RenheMiResembleRice
Source: Yanting Ji, Hanxue Dou
If $proj_{b}v=\left(3,11\right)$, find $proj_{b}\left(v+\left(-282,396\right)\right)$
1 reply
RenheMiResembleRice
2 hours ago
RenheMiResembleRice
an hour ago
J
H
Cauchy-Schwarz 2
prtoi   4
N an hour ago by sqing
Source: Handout by Samin Riasat
if $a^2+b^2+c^2+d^2=4$, prove that:
$\frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{d}+\frac{d^2}{a}\ge4$
4 replies
prtoi
Yesterday at 4:19 PM
sqing
an hour ago
J
H
Collinearity with orthocenter
liberator   179
N an hour ago by bjump
Source: IMO 2013 Problem 4
Let $ABC$ be an acute triangle with orthocenter $H$, and let $W$ be a point on the side $BC$, lying strictly between $B$ and $C$. The points $M$ and $N$ are the feet of the altitudes from $B$ and $C$, respectively. Denote by $\omega_1$ is the circumcircle of $BWN$, and let $X$ be the point on $\omega_1$ such that $WX$ is a diameter of $\omega_1$. Analogously, denote by $\omega_2$ the circumcircle of triangle $CWM$, and let $Y$ be the point such that $WY$ is a diameter of $\omega_2$. Prove that $X,Y$ and $H$ are collinear.

Proposed by Warut Suksompong and Potcharapol Suteparuk, Thailand
179 replies
liberator
Jan 4, 2016
bjump
an hour ago
J
H
Inspired by old results
sqing   2
N an hour ago by sqing
Source: Own
Let $a,b,c$ be real numbers.Prove that
$$\frac{ (a-b)(b-c)(c-a)}{ (a^2+1)(b^2+1)(c^2+1)}\leq\frac{3\sqrt 3}{8}$$$$\frac{ (a-b)(b-c)(c-a)}{ (a^2+2)(b^2+1)(c^2+2)}\leq\frac{3}{8}\sqrt{\frac{3}{2}}$$$$\frac{ (a-b)(b-c)(c-a)}{ (a^2+3)(b^2+1)(c^2+3)}\leq\frac{3}{8} $$$$\frac{ (a-b)(b-c)(c-a)}{ (a^2+3)(b^2+2)(c^2+3)}\leq\frac{3}{16} $$
2 replies
sqing
Yesterday at 12:17 PM
sqing
an hour ago
J
H
a_1 = 2025 implies a_k < 1/2025?
navi_09220114   6
N an hour ago by navi_09220114
Source: Own. Malaysian APMO CST 2025 P1
A sequence is defined as $a_1=2025$ and for all $n\ge 2$, $$a_n=\frac{a_{n-1}+1}{n}$$Determine the smallest $k$ such that $\displaystyle a_k<\frac{1}{2025}$.

Proposed by Ivan Chan Kai Chin
6 replies
navi_09220114
Feb 27, 2025
navi_09220114
an hour ago
J
a