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k a April Highlights and 2025 AoPS Online Class Information
jlacosta   0
Apr 2, 2025
Spring is in full swing and summer is right around the corner, what are your plans? At AoPS Online our schedule has new classes starting now through July, so be sure to keep your skills sharp and be prepared for the Fall school year! Check out the schedule of upcoming classes below.

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[*]April 22nd (Webinar), 4:00pm PT/7:00pm ET, Competitive Programming at AoPS (USACO).[/list]
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jlacosta
Apr 2, 2025
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Inspired by bamboozled
sqing   0
5 minutes ago
Source: Own
Let $ a,b,c $ be reals such that $(a^2+1)(b^2+1)(c^2+1) = 27. $Prove that $$1-3\sqrt 3\leq ab + bc + ca\leq 6$$
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sqing
5 minutes ago
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Range of ab + bc + ca
bamboozled   1
N 13 minutes ago by sqing
Let $(a^2+1)(b^2+1)(c^2+1) = 9$, where $a, b, c \in R$, then the number of integers in the range of $ab + bc + ca$ is __
1 reply
+1 w
bamboozled
28 minutes ago
sqing
13 minutes ago
J
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Functional Equation
AnhQuang_67   4
N 14 minutes ago by AnhQuang_67
Find all functions $f: \mathbb{R} \to \mathbb{R}$ satisfying $$2\cdot f\Big(\dfrac{-xy}{2}+f(x+y)\Big)=xf(y)+yf(x), \forall x, y \in \mathbb{R} $$
4 replies
AnhQuang_67
Yesterday at 4:50 PM
AnhQuang_67
14 minutes ago
J
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Inradius and ex-radii
bamboozled   0
22 minutes ago
Let $ABC$ be a triangle and $r, r_1, r_2, r_3$ denote its inradius and ex-radii opposite to the vertices $A, B, C$ respectively. If $a> r_1, b > r_2$ and $c > r_3$, then which of the following is/are true?
(A) $\angle{B}$ is obtuse
(B) $\angle{A}$ is acute
(C) $3r > s$, where $s$ is semi perimeter
(D) $3r < s$, where $s$ is semi perimeter
0 replies
bamboozled
22 minutes ago
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feet of altitudes wanted, 3 pairs of equal angles given
parmenides51   2
N Jun 28, 2020 by Nersik.883
Source: All-Siberian Open School Olympiad 2010-11 10.4
On the sides of $AB, BC$ and $AC$ of an acute-angled triangle $ABC$ marked points $P,Q$ and $R$, respectively, so that $\angle APR = \angle BPQ$, $\angle BQP = \angle CQR$, $\angle CRQ= \angle ARP$. Prove that $P,Q$ and $R$ are the feet of the altitudes of the triangle $ABC$.
IMAGE
2 replies
parmenides51
Jun 28, 2020
Nersik.883
Jun 28, 2020
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feet of altitudes wanted, 3 pairs of equal angles given
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Reveal topic tags
geometry
altitudes
equal angles
altitude
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Source: All-Siberian Open School Olympiad 2010-11 10.4
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parmenides51
30629 posts
parmenides51
#1 h  Jun 28, 2020, 8:43 PM • 1 Y
Y by Mango247
On the sides of $AB, BC$ and $AC$ of an acute-angled triangle $ABC$ marked points $P,Q$ and $R$, respectively, so that $\angle APR = \angle BPQ$, $\angle BQP = \angle CQR$, $\angle CRQ= \angle ARP$. Prove that $P,Q$ and $R$ are the feet of the altitudes of the triangle $ABC$.
https://cdn.artofproblemsolving.com/attachments/a/6/dd7661f0f77bb86b5397e950a3618c61b50ed4.png
This post has been edited 2 times. Last edited by parmenides51, Jun 21, 2022, 2:57 AM
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parmenides51
30629 posts
parmenides51
#2 h  Jun 28, 2020, 8:45 PM
Y by
posted for the image link
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Nersik.883
11 posts
Nersik.883
#3 h  Jun 28, 2020, 8:53 PM
Y by
$AB$ is the external bisector of $\angle RPQ$ and $BC$ is the external bisector of $\angle PQR$ so bisector of $\angle PRQ$ passes through $B$ and $\angle BRP$=$\angle BRQ$, $\angle BRP+\angle PRA=\angle BRQ+\angle QRC$, $\angle BRA=\angle BRC$
This post has been edited 3 times. Last edited by Nersik.883, Jun 28, 2020, 9:07 PM
Reason: i wrote with mistakes
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