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The centers of the spheres circumscribed and inscribed to a given tetrahedron coincide. Prove that the faces are congruent triangles.
This post has been edited 1 time. Last edited by parmenides51, Oct 16, 2021, 10:26 PM
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k a April Highlights and 2025 AoPS Online Class Information
jlacosta 0
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.
WOOT early bird pricing is in effect, don’t miss out! If you took MathWOOT Level 2 last year, no worries, it is all new problems this year! Our Worldwide Online Olympiad Training program is for high school level competitors. AoPS designed these courses to help our top students get the deep focus they need to succeed in their specific competition goals. Check out the details at this link for all our WOOT programs in math, computer science, chemistry, and physics.
Looking for summer camps in math and language arts? Be sure to check out the video-based summer camps offered at the Virtual Campus that are 2- to 4-weeks in duration. 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 events:
[list][*]April 3rd (Webinar), 4pm PT/7:00pm ET, Learning with AoPS: Perspectives from a Parent, Math Camp Instructor, and University Professor
[*]April 8th (Math Jam), 4:30pm PT/7:30pm ET, 2025 MATHCOUNTS State Discussion
April 9th (Webinar), 4:00pm PT/7:00pm ET, Learn about Video-based Summer Camps at the Virtual Campus
[*]April 10th (Math Jam), 4:30pm PT/7:30pm ET, 2025 MathILy and MathILy-Er Math Jam: Multibackwards Numbers
[*]April 22nd (Webinar), 4:00pm PT/7:00pm ET, Competitive Programming at AoPS (USACO).[/list]
Our full course list for upcoming classes is below:
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Mon, Tue, Wed & Thurs, Jun 23 - Jun 26 (meets every day of the week!)
WOOT early bird pricing is in effect, don’t miss out! If you took MathWOOT Level 2 last year, no worries, it is all new problems this year! Our Worldwide Online Olympiad Training program is for high school level competitors. AoPS designed these courses to help our top students get the deep focus they need to succeed in their specific competition goals. Check out the details at this link for all our WOOT programs in math, computer science, chemistry, and physics.
Looking for summer camps in math and language arts? Be sure to check out the video-based summer camps offered at the Virtual Campus that are 2- to 4-weeks in duration. 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 events:
[list][*]April 3rd (Webinar), 4pm PT/7:00pm ET, Learning with AoPS: Perspectives from a Parent, Math Camp Instructor, and University Professor
[*]April 8th (Math Jam), 4:30pm PT/7:30pm ET, 2025 MATHCOUNTS State Discussion
April 9th (Webinar), 4:00pm PT/7:00pm ET, Learn about Video-based Summer Camps at the Virtual Campus
[*]April 10th (Math Jam), 4:30pm PT/7:30pm ET, 2025 MathILy and MathILy-Er Math Jam: Multibackwards Numbers
[*]April 22nd (Webinar), 4:00pm PT/7:00pm ET, Competitive Programming at AoPS (USACO).[/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
Sunday, Apr 13 - Aug 10
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
Sunday, Apr 13 - Aug 10
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Introduction to Algebra A Self-Paced
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Introduction to Counting & Probability Self-Paced
Introduction to Counting & Probability
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Introduction to Number Theory
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Intermediate: Grades 8-12
Intermediate Algebra
Monday, Apr 21 - Oct 13
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Thursday, Jul 24 - Jan 22
Intermediate Counting & Probability
Wednesday, May 21 - Sep 17
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Intermediate Number Theory
Friday, Apr 11 - Jun 27
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Wednesday, Jun 18 - Sep 3
Precalculus
Wednesday, Apr 9 - Sep 3
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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
Wednesday, Apr 16 - Jul 2
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
Friday, Apr 11 - Jun 27
Sunday, May 11 - Aug 10
Tuesday, May 27 - Aug 12
Wednesday, Jun 11 - Aug 27
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Tues & Thurs, Jul 8 - Aug 14 (meets twice a week!)
AMC 10 Problem Series
Friday, May 9 - Aug 1
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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
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AMC 12 Problem Series
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Sunday, Jun 22 - Sep 21
Wednesday, Aug 6 - Oct 22
AMC 12 Final Fives
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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
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USACO Bronze Problem Series
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Sunday, Jun 22 - Sep 1
Physics
Introduction to Physics
Wednesday, May 21 - Aug 6
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Monday, Jun 23 - Sep 15
Physics 1: Mechanics
Thursday, May 22 - Oct 30
Monday, Jun 23 - Dec 15
Relativity
Sat & Sun, Apr 26 - Apr 27 (4:00 - 7:00 pm ET/1:00 - 4:00pm PT)
Mon, Tue, Wed & Thurs, Jun 23 - Jun 26 (meets every day of the week!)
0 replies
Problems for v_p(n)
xytunghoanh 1
N
by whwlqkd
Hello everyone. I need some easy problems for 
(use in a problem for junior) to practice. Can anyone share to me?
Thanks :>
(use in a problem for junior) to practice. Can anyone share to me?Thanks :>
1 reply
1 viewing
Parallel Lines and Q Point
taptya17 13
N
by everythingpi3141592
Source: India EGMO TST 2025 Day 1 P3
Let 
be an acute angled scalene triangle with circumcircle 
. Let 
and 
be the circumcenter and orthocenter of 
respectively. Let 
and 
be points on segments 
and , respectively, such that
Prove that 
and 
intersect on the circumcircle of 
.
Proposed by Antareep Nath
be an acute angled scalene triangle with circumcircle 
. Let 
and 
be the circumcenter and orthocenter of 
respectively. Let 
and 
be points on segments 
and , respectively, such that
Prove that 
and 
intersect on the circumcircle of 
.Proposed by Antareep Nath
13 replies
+1 wCool combinatorial problem (grid)
Anto0110 0
Suppose you have an 
grid with 
rows and 
columns, and each square of the grid is filled with a non-negative integer. Let 
be the average of all the numbers in the grid. Prove that if 
the there exist two identical rows (meaning same numbers in the same order).
grid with 
rows and 
columns, and each square of the grid is filled with a non-negative integer. Let 
be the average of all the numbers in the grid. Prove that if 
the there exist two identical rows (meaning same numbers in the same order).
0 replies
Cards and combi
AlephG_64 0
Source: 2025 Finals Portuguese Mathematical Olympiad P6
Maria wants to solve a challenge with a deck of cards, each with a different figure. Initially, the cards are distributed randomly into two piles, not necessarily in equal parts. Maria's goal is to get all the cards into the same pile.
On each turn, Maria takes the top card from each pile and compares them. In the rule book, there's a table that indicates, for each card match, which of the two wins. Both cards are then placed on the bottom of the winning card in the order Maria chooses. The challenge ends when all the cards are in one pile.
Show that it is always possible for Maria to solve the challenge. Regardless of the initial distribution of the cards and the table in the rule book.
On each turn, Maria takes the top card from each pile and compares them. In the rule book, there's a table that indicates, for each card match, which of the two wins. Both cards are then placed on the bottom of the winning card in the order Maria chooses. The challenge ends when all the cards are in one pile.
Show that it is always possible for Maria to solve the challenge. Regardless of the initial distribution of the cards and the table in the rule book.
0 replies
that statement is true
pennypc123456789 3
N
by sqing
we have 
and 
, then we can deduce 
\le 2 $ ?
and 
, then we can deduce 
\le 2 $ ?
3 replies
Distance vs time swimming problem
smalkaram_3549 1
N
by Lankou
How should I approach a problem where we deal with velocities becoming negative and stuff. I know that they both travel 3 Lengths of the pool before meeting a second time.
1 reply
.problem.
Cobedangiu 4
N
by Lankou
Find the integer coefficients after expanding Newton's binomial:
4 replies
inequalities - 5/4
pennypc123456789 2
N
by sqing
Given real numbers 
satisfying 
. Prove that:
![\[
0 \le (x^2 + 1)(y^2 + 1) + 4(x - 1)(y - 1) \le 164.
\]](//latex.artofproblemsolving.com/4/2/d/42d0b88388462f16dd99d928b94024e689ba7c2e.png)
satisfying 
. Prove that:![\[
0 \le (x^2 + 1)(y^2 + 1) + 4(x - 1)(y - 1) \le 164.
\]](http://latex.artofproblemsolving.com/4/2/d/42d0b88388462f16dd99d928b94024e689ba7c2e.png)
2 replies
Source of a combinatorics problem
isodynamicappolonius2903 0
Does anyone know exactly the source of this problem?? I just remember that it from a combinatorics book of Russia.
0 replies
KSEA NMSC Mock Contest Group B (Last Problem)
Shiyul 5
N
by Shiyul
Let 
be a sequence defined by 
. Then the product of four consecutive terms in can be written as the product of two terms in . Find 
if 
.
be a sequence defined by 
. Then the product of four consecutive terms in can be written as the product of two terms in . Find 
if 
.
5 replies
Regarding IMO prepartion
omega2007 1
N
by omega2007
<Hey Everyone'>
I'm 10 grader student and Im starting prepration for maths olympiad..>>> From scratch (not 2+2=4 )
Do you haves compilled resources of Handouts,
PDF,
Links,
List of books topic wise
which are shared on AOPS (and from your prespective) for maths olympiad and any useful thing, which will help me in boosting Maths olympiad prepration.
I'm 10 grader student and Im starting prepration for maths olympiad..>>> From scratch (not 2+2=4 )
Do you haves compilled resources of Handouts,
PDF,
Links,
List of books topic wise
which are shared on AOPS (and from your prespective) for maths olympiad and any useful thing, which will help me in boosting Maths olympiad prepration.
1 reply
School Math Problem
math_cool123 6
N
by anduran
Find all ordered pairs of nonzero integers 
that satisfy 
that satisfy 
6 replies
New geometry problem
titaniumfalcon 3
N
by mathprodigy2011
Post any solutions you have, with explanation or proof if possible, good luck!
3 replies
Geo Mock #10
Bluesoul 2
N
by Bluesoul
Consider acute 
with 
, 
and area 
. The circle with diameter 
meets 
at 
. Let the orthocenter of the triangle be , connect 
and extend to meet at 
such that 
and 
is the diameter of . Draw the circumcircle 
of 
, chord 
of is tangent to and it passes through , compute .
with 
, 
and area 
. The circle with diameter 
meets 
at 
. Let the orthocenter of the triangle be , connect 
and extend to meet at 
such that 
and 
is the diameter of . Draw the circumcircle 
of 
, chord 
of is tangent to and it passes through , compute .
2 replies
k congruent faces if centers coincide, circumsphere , insphere of tetrahedron
parmenides51 1
N
by parmenides51
Source: 1998 Argentina IberoAmerican Training List p25 https://artofproblemsolving.com/community/c2434876_argentina_training_lists_ibero
The centers of the spheres circumscribed and inscribed to a given tetrahedron coincide. Prove that the faces are congruent triangles.
1 reply
congruent faces if centers coincide, circumsphere , insphere of tetrahedron
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H
Source: 1998 Argentina IberoAmerican Training List p25 https://artofproblemsolving.com/community/c2434876_argentina_training_lists_ibero
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#2
Dec 22, 2022, 2:20 AM
Y by
solved in an older source
Austrian-Polish 1979
Austrian-Polish 1979
This post has been edited 1 time. Last edited by parmenides51, Dec 22, 2022, 2:24 AM
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Y
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