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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
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[20th PMO Qualifying Stage] 1.15
pensive   1
N 22 minutes ago by pensive
Suppose that $\{a_n\}$ is a nonconstant arithmetic sequence such that $a_1 = 1$ and the terms $a_3, a_{15}, a_{24}$ form a geometric sequence in that order. Find the smallest index $n$ for which $a_n < 0$.

Answer
1 reply
pensive
23 minutes ago
pensive
22 minutes ago
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Inequalities
sqing   27
N 43 minutes ago by DAVROS
Let $ x,y,z $ be real numbers such that $ x^2+2y^2+z^2+xy+yz+zx=1 $.Prove that$$-\sqrt{\frac 75}\leq x+y+z \leq \sqrt{\frac 75}$$
27 replies
sqing
Aug 31, 2024
DAVROS
43 minutes ago
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Padres vs Dodgers expected wins
ehz2701   0
2 hours ago
Suppose the Padres have 6 pitchers of skill level $\{2, 4, 6, 8, 10, 12\}$. The dodgers have 6 pitchers of skill level $\{1,3,5,7,9, 11\}$. When a Padres pitcher with skill level $a$ is against a Dodgers pitcher with skill level $b$, the probability that the Padres win that game is $\frac{a}{a+b}$. The Padres have already determined that they will start their pitchers in the increasing order of skill level. In what order should the dodgers start their pitchers to maximize their expected wins? Give your answer as a sequence of six integers.

***DISCLAIMER: (1) If you are a hitter, I know you are very vital and important to the game, and pitching does not determine everything. (2) Yes, there is no such thing as a one-all stat like skill level. ERA, WHIP, FIP, and many other stats govern just fine. WAR even. (3) I am NOT making a statement on whether the dodgers and padres pitching staff is better.
0 replies
ehz2701
2 hours ago
0 replies
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(a+sqrt(b)^p when a-sqrt(b) is actually larger than 1
ehz2701   0
3 hours ago
We’ve all seen problems like what is the one’s place of $(\sqrt{2}+1)^{2025}$? (It’s 5 btw). In any such radical $a+\sqrt{b}$ for some integers $a,b$, the key is that $a-\sqrt{b} < 1$ such that $(a-\sqrt{b})^{p}$ is negligibly small. And adding this to $(a+\sqrt{b})^{p}$ squares away the square roots and cancels away the roots bits, leaving only integers behind. From this answer extraction is typically easy, perhaps a little module arithmetic is needed in combination with the binomial theorem.

Now, what if the conjugate $a-\sqrt{b}>1$? For example, what is the ones digit of $(3+\sqrt{2})^{2025}$? Is this feasible? answer according to wolfram alpha
0 replies
ehz2701
3 hours ago
0 replies
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No more topics!
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Domain of Nested Functions
pensive   1
N Jun 22, 2025 by pensive
Given $f(x) = \log_2 (x) - 1$ and $g(x) = -x^2+x+2$, for what integer $x$ is $f(g(f(x)))$ positive?
Answer
1 reply
pensive
Jun 22, 2025
pensive
Jun 22, 2025
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Domain of Nested Functions
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pensive
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pensive
#1 h  Jun 22, 2025, 3:09 PM
Y by
Given $f(x) = \log_2 (x) - 1$ and $g(x) = -x^2+x+2$, for what integer $x$ is $f(g(f(x)))$ positive?
Answer
This post has been edited 1 time. Last edited by pensive, Jun 22, 2025, 3:10 PM
Reason: Misleading text
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32 posts
pensive
#2 h  Jun 22, 2025, 3:35 PM
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Solution
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