R <= (AB + BC + CA)/4 in acute triangle

May is an exciting month! National MATHCOUNTS is the second week of May in Washington D.C. and our Founder, Richard Rusczyk will be presenting a seminar, Preparing Strong Math Students for College and Careers, on May 11th.

Are you interested in working towards MATHCOUNTS and don’t know where to start? We have you covered! If you have taken Prealgebra, then you are ready for MATHCOUNTS/AMC 8 Basics. Already aiming for State or National MATHCOUNTS and harder AMC 8 problems? Then our MATHCOUNTS/AMC 8 Advanced course is for you.

Summer camps are starting next month at the Virtual Campus in math and language arts that are 2 - to 4 - weeks in duration. Spaces are still available - don’t miss your chance to have an enriching summer experience. There are middle and high school competition math camps as well as Math Beasts camps that review key topics coupled with fun explorations covering areas such as graph theory (Math Beasts Camp 6), cryptography (Math Beasts Camp 7-8), and topology (Math Beasts Camp 8-9)!

Be sure to mark your calendars for the following upcoming events:
[list][*]May 9th, 4:30pm PT/7:30pm ET, Casework 2: Overwhelming Evidence — A Text Adventure, a game where participants will work together to navigate the map, solve puzzles, and win! All are welcome.
[*]May 19th, 4:30pm PT/7:30pm ET, What's Next After Beast Academy?, designed for students finishing Beast Academy and ready for Prealgebra 1.
[*]May 20th, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 1 Math Jam, Problems 1 to 4, join the Canada/USA Mathcamp staff for this exciting Math Jam, where they discuss solutions to Problems 1 to 4 of the 2025 Mathcamp Qualifying Quiz!
[*]May 21st, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 2 Math Jam, Problems 5 and 6, Canada/USA Mathcamp staff will discuss solutions to Problems 5 and 6 of the 2025 Mathcamp Qualifying Quiz![/list]
Our full course list for upcoming classes is below:
All classes run 7:30pm-8:45pm ET/4:30pm - 5:45pm PT unless otherwise noted.

Introductory: Grades 5-10

Prealgebra 1 Self-Paced

Prealgebra 1
Tuesday, May 13 - Aug 26
Thursday, May 29 - Sep 11
Sunday, Jun 15 - Oct 12
Monday, Jun 30 - Oct 20
Wednesday, Jul 16 - Oct 29

Prealgebra 2 Self-Paced

Prealgebra 2
Wednesday, May 7 - Aug 20
Monday, Jun 2 - Sep 22
Sunday, Jun 29 - Oct 26
Friday, Jul 25 - Nov 21

Introduction to Algebra A Self-Paced

Introduction to Algebra A
Sunday, May 11 - Sep 14 (1:00 - 2:30 pm ET/10:00 - 11:30 am PT)
Wednesday, May 14 - Aug 27
Friday, May 30 - Sep 26
Monday, Jun 2 - Sep 22
Sunday, Jun 15 - Oct 12
Thursday, Jun 26 - Oct 9
Tuesday, Jul 15 - Oct 28

Introduction to Counting & Probability Self-Paced

Introduction to Counting & Probability
Thursday, May 15 - Jul 31
Sunday, Jun 1 - Aug 24
Thursday, Jun 12 - Aug 28
Wednesday, Jul 9 - Sep 24
Sunday, Jul 27 - Oct 19

Introduction to Number Theory
Friday, May 9 - Aug 1
Wednesday, May 21 - Aug 6
Monday, Jun 9 - Aug 25
Sunday, Jun 15 - Sep 14
Tuesday, Jul 15 - Sep 30

Introduction to Algebra B Self-Paced

Introduction to Algebra B
Tuesday, May 6 - Aug 19
Wednesday, Jun 4 - Sep 17
Sunday, Jun 22 - Oct 19
Friday, Jul 18 - Nov 14

Introduction to Geometry
Sunday, May 11 - Nov 9
Tuesday, May 20 - Oct 28
Monday, Jun 16 - Dec 8
Friday, Jun 20 - Jan 9
Sunday, Jun 29 - Jan 11
Monday, Jul 14 - Jan 19

Paradoxes and Infinity
Mon, Tue, Wed, & Thurs, Jul 14 - Jul 16 (meets every day of the week!)

Intermediate: Grades 8-12

Intermediate Algebra
Sunday, Jun 1 - Nov 23
Tuesday, Jun 10 - Nov 18
Wednesday, Jun 25 - Dec 10
Sunday, Jul 13 - Jan 18
Thursday, Jul 24 - Jan 22

Intermediate Counting & Probability
Wednesday, May 21 - Sep 17
Sunday, Jun 22 - Nov 2

Intermediate Number Theory
Sunday, Jun 1 - Aug 24
Wednesday, Jun 18 - Sep 3

Precalculus
Friday, May 16 - Oct 24
Sunday, Jun 1 - Nov 9
Monday, Jun 30 - Dec 8

Advanced: Grades 9-12

Olympiad Geometry
Tuesday, Jun 10 - Aug 26

Calculus
Tuesday, May 27 - Nov 11
Wednesday, Jun 25 - Dec 17

Group Theory
Thursday, Jun 12 - Sep 11

Contest Preparation: Grades 6-12

MATHCOUNTS/AMC 8 Basics
Friday, May 23 - Aug 15
Monday, Jun 2 - Aug 18
Thursday, Jun 12 - Aug 28
Sunday, Jun 22 - Sep 21
Tues & Thurs, Jul 8 - Aug 14 (meets twice a week!)

MATHCOUNTS/AMC 8 Advanced
Sunday, May 11 - Aug 10
Tuesday, May 27 - Aug 12
Wednesday, Jun 11 - Aug 27
Sunday, Jun 22 - Sep 21
Tues & Thurs, Jul 8 - Aug 14 (meets twice a week!)

AMC 10 Problem Series
Friday, May 9 - Aug 1
Sunday, Jun 1 - Aug 24
Thursday, Jun 12 - Aug 28
Tuesday, Jun 17 - Sep 2
Sunday, Jun 22 - Sep 21 (1:00 - 2:30 pm ET/10:00 - 11:30 am PT)
Monday, Jun 23 - Sep 15
Tues & Thurs, Jul 8 - Aug 14 (meets twice a week!)

AMC 10 Final Fives
Sunday, May 11 - Jun 8
Tuesday, May 27 - Jun 17
Monday, Jun 30 - Jul 21

AMC 12 Problem Series
Tuesday, May 27 - Aug 12
Thursday, Jun 12 - Aug 28
Sunday, Jun 22 - Sep 21
Wednesday, Aug 6 - Oct 22

AMC 12 Final Fives
Sunday, May 18 - Jun 15

AIME Problem Series A
Thursday, May 22 - Jul 31

AIME Problem Series B
Sunday, Jun 22 - Sep 21

F=ma Problem Series
Wednesday, Jun 11 - Aug 27

WOOT Programs
Visit the pages linked for full schedule details for each of these programs!

MathWOOT Level 1
MathWOOT Level 2
ChemWOOT
CodeWOOT
PhysicsWOOT

Programming

Introduction to Programming with Python
Thursday, May 22 - Aug 7
Sunday, Jun 15 - Sep 14 (1:00 - 2:30 pm ET/10:00 - 11:30 am PT)
Tuesday, Jun 17 - Sep 2
Monday, Jun 30 - Sep 22

Intermediate Programming with Python
Sunday, Jun 1 - Aug 24
Monday, Jun 30 - Sep 22

USACO Bronze Problem Series
Tuesday, May 13 - Jul 29
Sunday, Jun 22 - Sep 1

Physics

Introduction to Physics
Wednesday, May 21 - Aug 6
Sunday, Jun 15 - Sep 14
Monday, Jun 23 - Sep 15

Physics 1: Mechanics
Thursday, May 22 - Oct 30
Monday, Jun 23 - Dec 15

Relativity
Mon, Tue, Wed & Thurs, Jun 23 - Jun 26 (meets every day of the week!)

0 replies

jlacosta

May 1, 2025

0 replies

Converse of a classic orthocenter problem

spartacle 43

N 5 minutes ago by ihategeo_1969

Source: USA TSTST 2020 Problem 6

Let $A$ , $B$ , $C$ , $D$ be four points such that no three are collinear and $D$ is not the orthocenter of $ABC$ . Let $P$ , $Q$ , $R$ be the orthocenters of $\triangle BCD$ , $\triangle CAD$ , $\triangle ABD$ , respectively. Suppose that the lines $AP$ , $BQ$ , $CR$ are pairwise distinct and are concurrent. Show that the four points $A$ , $B$ , $C$ , $D$ lie on a circle.

Andrew Gu

43 replies

spartacle

Dec 14, 2020

ihategeo_1969

5 minutes ago

Symmetric points part 2

CyclicISLscelesTrapezoid 22

N 6 minutes ago by ihategeo_1969

Source: USA TSTST 2022/6

Let $O$ and $H$ be the circumcenter and orthocenter, respectively, of an acute scalene triangle $ABC$ . The perpendicular bisector of $\overline{AH}$ intersects $\overline{AB}$ and $\overline{AC}$ at $X_A$ and $Y_A$ respectively. Let $K_A$ denote the intersection of the circumcircles of triangles $OX_AY_A$ and $BOC$ other than $O$ .

Define $K_B$ and $K_C$ analogously by repeating this construction two more times. Prove that $K_A$ , $K_B$ , $K_C$ , and $O$ are concyclic.

Hongzhou Lin

22 replies

CyclicISLscelesTrapezoid

Jun 27, 2022

ihategeo_1969

6 minutes ago

Periodicity of factorials

Cats_on_a_computer 0

24 minutes ago

Source: Thrill and challenge of pre-college mathematics

Let a_k denote the first non zero digit of the decimal representation of k!. Does the sequence a_1, a_2, a_3, … eventually become periodic?

0 replies

Cats_on_a_computer

24 minutes ago

0 replies

Cyclic Quad. and Intersections

Thelink_20 11

N 34 minutes ago by americancheeseburger4281

Source: My Problem

Let $ABCD$ be a quadrilateral inscribed in a circle $\Gamma$ . Let $AC\cap BD=E$ , $AB\cap CD=F$ , $(AEF)\cap\Gamma=X$ , $(BEF)\cap\Gamma=Y$ , $(CEF)\cap\Gamma=Z$ , $(DEF)\cap\Gamma=W$ , $XZ\cap YW=M$ , $XY\cap ZW=N$ . Prove that $MN$ lies over $EF$ .

11 replies

1 viewing

Thelink_20

Oct 29, 2024

americancheeseburger4281

34 minutes ago

No more topics!

R <= (AB + BC + CA)/4 in acute triangle

parmenides51 1

N Dec 31, 2020 by Mateescu Constantin

Source: 2019 UNSW School Mathematics Competition S4

Let $\vartriangle ABC$ be a triangle and let every angle of $\vartriangle ABC$ be acute. Prove that $R \le \frac{ AB + BC + CA}{4}$ where $R$ is the radius of the circumscribed circle.

1 reply

parmenides51

Dec 31, 2020

Mateescu Constantin

Dec 31, 2020

R <= (AB + BC + CA)/4 in acute triangle

G H J

Reveal topic tags

geometry

geometric inequality

circumradius

G H BBookmark kLocked kLocked NReply

Source: 2019 UNSW School Mathematics Competition S4

The post below has been deleted. Click to close.

This post has been deleted. Click here to see post.

parmenides51

30653 posts

parmenides51

#1 h Dec 31, 2020, 8:20 AM • 1 Y

Y by Mango247

Let $\vartriangle ABC$ be a triangle and let every angle of $\vartriangle ABC$ be acute. Prove that $R \le \frac{ AB + BC + CA}{4}$ where $R$ is the radius of the circumscribed circle.

This post has been edited 1 time. Last edited by parmenides51, Dec 31, 2020, 8:37 AM

Z K Y

The post below has been deleted. Click to close.

This post has been deleted. Click here to see post.

Mateescu Constantin

1842 posts

Mateescu Constantin

#2 h Dec 31, 2020, 9:46 AM • 1 Y

Y by teomihai

That's fairly weak actually. Note that in any $\triangle ABC$ we have: $\prod\cos A = \frac {s^2 - (2R + r)^2}{4R^2},$ where $R, r$ and $s$ denote the circumradius, inradius and semiperimeter respectively. This means that in any acute-angled triangle it holds that: $\boxed{s > 2R + r}\, ,$ which one can rearrange as $R + \frac r2 < \frac s2$ or $R + \frac r2 < \frac {AB + BC + CA}{4}.$ This is clearly stronger than the proposed inequality.

$\text{\LaTeX}$ tip: use $\verb#\triangle#$ instead of $\verb#\vartriangle#$ .

Z K Y

N Quick Reply