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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.

Our full course list for upcoming classes is below:
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0 replies
jwelsh
Jul 1, 2025
0 replies
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Peru IMO TST 2022
diegoca1   0
2 minutes ago
Source: Peru IMO TST 2022 D1 P4
Let $\Omega$ be the circumcircle of triangle $ABC$, with $\angle BAC > 90^\circ $ and $ AB > AC $. The tangents to $\Omega$ at points $B$ and $C$ intersect at $D$. The tangent to $\Omega$ at point $A$ intersects line $BC$ at $E$. The line through $D$ parallel to $AE$ intersects line $BC$ at $F$. The circumference $\Gamma$ with diameter $EF$ intersects line $AB$ at points $P$ and $Q$, and line $AC$ at points $X$ and $Y$.
Prove that one of the angles $\angle AEB$, $\angle PEQ$, $\angle XEY$ is equal to the sum of the other two.
0 replies
diegoca1
2 minutes ago
0 replies
J
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number theory
Hoapham235   5
N 2 minutes ago by Jjesus
Let $x > y$ be positive integer such that \[ \text{LCM}(x+2, y+2)+\text{LCM}(x, y)=2\text{LCM}(x+1, y+1).\]Prove that $x$ is divisible by $y$.
5 replies
Hoapham235
Wednesday at 4:51 AM
Jjesus
2 minutes ago
J
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centroid lies outside of triangle (not clickbait)
Scilyse   2
N 10 minutes ago by EthanWYX2009
Source: 数之谜 January (CHN TST Mock) Problem 5
Let $P$ be a convex polygon with centroid $G$, and let $\mathcal P$ be the set of vertices of $P$. Let $\mathcal X$ be the set of triangles with vertices all in $\mathcal P$. We sort the elements $\triangle ABC$ of $\mathcal X$ into the following three types:
[list]
[*] (Type 1) $G$ lies in the strict interior of $\triangle ABC$; let $\mathcal A$ be the set of triangles of this type.
[*] (Type 2) $G$ lies in the strict exterior of $\triangle ABC$; let $\mathcal B$ be the set of triangles of this type.
[*] (Type 3) $G$ lies on the boundary of $\triangle ABC$.
[/list]
For any triangle $T$, denote by $S_T$ the area of $T$. Prove that \[\sum_{T \in \mathcal A} S_T \geq \sum_{T \in \mathcal B} S_T.\]
2 replies
Scilyse
Jan 26, 2025
EthanWYX2009
10 minutes ago
J
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Peru IMO TST 2022
diegoca1   0
16 minutes ago
Source: Peru IMO TST 2022 D1 P3
Consider an $n$-gon (a polygon with $n$ sides) with $n \geq 3$. Distinct non-negative integers are assigned to each side and diagonal of the n-gon in such a way that, for any triangle whose vertices are vertices of the n-gon, the numbers assigned to its three sides form an arithmetic progression. Determine the maximum value of $n$ for which this is possible
0 replies
diegoca1
16 minutes ago
0 replies
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No more topics!
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Combinatorial Double Sum
P162008   0
Apr 24, 2025
Compute $\sum_{r=0}^{n} \sum_{k=0}^{r} (-1)^k (k + 1)(k + 2) \binom {n + 5}{r - k}$
0 replies
P162008
Apr 24, 2025
0 replies
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Combinatorial Double Sum
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combinatorics
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P162008
#1 h  Apr 24, 2025, 1:03 AM
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Compute $\sum_{r=0}^{n} \sum_{k=0}^{r} (-1)^k (k + 1)(k + 2) \binom {n + 5}{r - k}$
This post has been edited 1 time. Last edited by P162008, Apr 25, 2025, 3:25 AM
Reason: Typo
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