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k a July Highlights and 2025 AoPS Online Class Information
jwelsh   0
Jul 1, 2025
We are halfway through summer, so be sure to carve out some time to keep your skills sharp and explore challenging topics at AoPS Online and our AoPS Academies (including the Virtual Campus)!

[list][*]Over 60 summer classes are starting at the Virtual Campus on July 7th - check out the math and language arts options for middle through high school levels.
[*]At AoPS Online, we have accelerated sections where you can complete a course in half the time by meeting twice/week instead of once/week, starting on July 8th:
[list][*]MATHCOUNTS/AMC 8 Basics
[*]MATHCOUNTS/AMC 8 Advanced
[*]AMC Problem Series[/list]
[*]Plus, AoPS Online has a special seminar July 14 - 17 that is outside the standard fare: Paradoxes and Infinity
[*]We are expanding our in-person AoPS Academy locations - are you looking for a strong community of problem solvers, exemplary instruction, and math and language arts options? Look to see if we have a location near you and enroll in summer camps or academic year classes today! New locations include campuses in California, Georgia, New York, Illinois, and Oregon and more coming soon![/list]

MOP (Math Olympiad Summer Program) just ended and the IMO (International Mathematical Olympiad) is right around the corner! This year’s IMO will be held in Australia, July 10th - 20th. Congratulations to all the MOP students for reaching this incredible level and best of luck to all selected to represent their countries at this year’s IMO! Did you know that, in the last 10 years, 59 USA International Math Olympiad team members have medaled and have taken over 360 AoPS Online courses. Take advantage of our Worldwide Online Olympiad Training (WOOT) courses
and train with the best! Please note that early bird pricing ends August 19th!
Are you tired of the heat and thinking about Fall? You can plan your Fall schedule now with classes at either AoPS Online, AoPS Academy Virtual Campus, or one of our AoPS Academies around the US.

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0 replies
jwelsh
Jul 1, 2025
0 replies
J
H
Number Theory (Divisibility)
AbdulWaheed   0
an hour ago
Find all triples (a, b, c) of natural numbers such that the numbers $a^2$ + bc, $b^2$ + ac,
and $c^2$ + ab are powers of 2. (source, Sozopol 2023, Grades 8-9)
0 replies
AbdulWaheed
an hour ago
0 replies
J
H
Finding the max of a expression
PhysicsIsChad   10
N 3 hours ago by PhysicsIsChad
If the maximum value of the expression
$$(x - 9)^2 + \left( 3 + \sqrt{10x - x^2 - 21} \right)^2$$is $k$, then the value of
$$\frac{k + 51}{20}$$is $\boxed{?}$.
Using Maxima Minima to solve this problem is lengthy , i would love to your hear insights and hints using other techniques .
(My insight : I interpreted this as a two variable function where $$y = \sqrt{10x - x^2 - 21} $$$$(x - 9)^2 + (3 + y)^2$$and this sort of becomes an equation for a distance from a point but i am not able to move ahead )
10 replies
PhysicsIsChad
Yesterday at 2:07 PM
PhysicsIsChad
3 hours ago
J
H
Trig time!
NamelyOrange   1
N 3 hours ago by mathprodigy2011
(a) The roots of $x^2-ax+b=0$ are $\tan \alpha$ and $\tan \beta$. Express $\cos^2 \alpha-\sin^2\beta$ in terms of $a$ and $b$.

(b) Use the Law of Cosines to prove that the sum of the squares of the diagonals of a parallelogram is twice the sum of the squares of the distinct sides.

(Source: 2022 National Taiwan University STEM Development Program Admissions Test, P8)
1 reply
NamelyOrange
4 hours ago
mathprodigy2011
3 hours ago
J
H
Relatively simple 3D geometry
NamelyOrange   0
4 hours ago
$ABCD$ is a parallelogram and $ADEF$ is a rectangle such that $AF$ is perpendicular to the plane containing $ABCD$. If $AB = AF$ and $FB = FC = FE$, find the measure of $\angle DAB$.

(Source: 2022 National Taiwan University STEM Development Program Admissions Test, P6)
0 replies
NamelyOrange
4 hours ago
0 replies
J
No more topics!
H
J
H
intermediate algebra
ANGEW   2
N Jan 28, 2025 by ANGEW
Find all pairs of real numbers (a, b) such that (x - a)^2 + (2x - b)^2 = (x - 3)^2 + (2x)^2 for all x.
The answer is (a,b)=(3,0) and (-9/5, 12/5)
By graphing y=2x and plotting these two points, we can see that the two points are reflections upon the line y=2x. Why?
2 replies
ANGEW
Jan 22, 2025
ANGEW
Jan 28, 2025
J
intermediate algebra
G H J
Reveal topic tags
Linear graphing
geometry
geometric transformation
reflection
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G H BBookmark kLocked kLocked NReply
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ANGEW
2 posts
ANGEW
#1 h  Jan 22, 2025, 8:45 AM
Y by
Find all pairs of real numbers (a, b) such that (x - a)^2 + (2x - b)^2 = (x - 3)^2 + (2x)^2 for all x.
The answer is (a,b)=(3,0) and (-9/5, 12/5)
By graphing y=2x and plotting these two points, we can see that the two points are reflections upon the line y=2x. Why?
Z K Y
The post below has been deleted. Click to close.
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vanstraelen
9163 posts
vanstraelen
#2 h  Jan 22, 2025, 9:33 PM
Y by
Points $A(3,0)$ and $B(-\frac{9}{5},\frac{12}{5})$.
Slope of the line $AB\ :\ m_{AB}=-\frac{1}{2}$, slope of the perpendicular $m_{l}=2$.
Midpoint $C(\frac{3}{5},\frac{6}{5})$.
Equation of the perpendicular $l$, through $C\ :\ y-\frac{6}{5}=2(x-\frac{3}{5})$ or $y=2x$.
Z K Y
The post below has been deleted. Click to close.
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ANGEW
2 posts
ANGEW
#3 h  Jan 28, 2025, 8:10 AM
Y by
Got it, thanks!
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