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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
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0 replies
jwelsh
Jul 1, 2025
0 replies
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If OAB and OAC share equal angles and sides, why aren't they congruent?
Merkane   0
an hour ago

Problem 1.39 (CGMO 2012/5). Let ABC be a triangle. The incircle of ABC is tangent
to AB and AC at D and E respectively. Let O denote the circumcenter of BCI .
Prove that ∠ODB = ∠OEC. Hints: 643 89 Sol: p.241

While I have solved the problem, I encountered a step that seems logically sound but leads to a contradiction, and I would like help identifying the flaw.

Here is the reasoning I followed:

The quadrilateral ABOC is cyclic.

OB = OC.

∠OAB = ∠OCB.
Similarly, ∠OAC = ∠OBC.

From symmetry and the above, it seems that ∠OAB = ∠OAC.

Since OA is a shared side, I concluded that triangle OAB ≅ triangle OAC.


But clearly, OAB and OAC are not congruent.
Where exactly is the logical error in this argument?
0 replies
Merkane
an hour ago
0 replies
J
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Geometry Problem
Rice_Farmer   0
2 hours ago
Let $w_1$ ad $w_2$ be two circles intersecting at $P$ and $Q.$ The tangent like closer to $Q$ touches $w_1$ and $w_2$ at $M$ and $N$ respectively. If $PQ=3,NQ=2,$ and $MN=PN,$ find $QM.$
0 replies
Rice_Farmer
2 hours ago
0 replies
J
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A writing game
Ecrin_eren   2
N 4 hours ago by Ecrin_eren


There is an integer greater than 1 written on the board in A’s house. Every morning when A wakes up, he erases the number n on the board and does the following:

If there is a positive integer m such that m^3= n, then he writes m on the board.

Otherwise, he writes 2n+1 on the board.


Since A repeats this process infinitely many times, prove that among all the numbers A has written and will write on the board, there are infinitely many greater than 10^100.





2 replies
Ecrin_eren
Jul 28, 2025
Ecrin_eren
4 hours ago
J
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Perfect square
Ecrin_eren   2
N 4 hours ago by Ecrin_eren


Find all integer values of n such that for every pair of integers (a, b) satisfying:

n·a² + a = (n + 1)·b² + b

the number |a − b| is always a perfect square.




2 replies
Ecrin_eren
Jul 28, 2025
Ecrin_eren
4 hours ago
J
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Bijection on set of maps
enter16180   2
N Yesterday at 6:32 PM by GreenKeeper
Source: IMC 2025, Problem 5
For a positive integer $n$, let $[n]=\{1,2, \ldots, n\}$. Denote by $S_n$ the set of all bijections from $[n]$ to $[n]$, and let $T_n$ be the set of all maps from $[n]$ to $[n]$. Define the order $\operatorname{ord}(\tau)$ of a map $\tau \in T_n$ as the number of distinct maps in the set $\{\tau, \tau \circ \tau, \tau \circ \tau \circ \tau, \ldots\}$ where o denotes composition. Finally, let
$$ f(n)=\max _{\tau \in S_n} \operatorname{ord}(\tau) \quad \text { and } \quad g(n)=\max _{\tau \in T_n} \operatorname{ord}(\tau) . $$Prove that $g(n)<f(n)+n^{0.501}$ for sufficiently large $n$.
2 replies
enter16180
Yesterday at 11:23 AM
GreenKeeper
Yesterday at 6:32 PM
J
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new inequality
P0tat0b0y   0
Yesterday at 4:56 PM
Source: Own
Prove $\frac{2\cdot 4\cdot 6\cdot ...\cdot (2n)}{1\cdot 3\cdot 5\cdot ...\cdot (2n-1)}<\sqrt{\pi n+\frac{8}{9}},\forall n\ge 1$
0 replies
P0tat0b0y
Yesterday at 4:56 PM
0 replies
J
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Floor function
enter16180   2
N Yesterday at 4:11 PM by Assassino9931
Source: IMC 2025, Problem 4
Let $a$ be an even positive integer. Find all real numbers $x$ such that
$$ \left\lfloor\sqrt[a]{b^a+x} \cdot b^{a-1}\right\rfloor=b^a+\lfloor x / a\rfloor $$holds for every positive integer $b$.
(Here $\lfloor x\rfloor$ denotes the largest integer that is no greater than $x$.)
2 replies
enter16180
Yesterday at 11:18 AM
Assassino9931
Yesterday at 4:11 PM
J
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Application of Derivatives
prtoi   2
N Yesterday at 3:44 PM by prtoi
Source: Jee Advanced 2020
can someone tell me a solution that uses only algebra, since all the solutions i have seen involve some inference from graphs
2 replies
prtoi
Jul 20, 2025
prtoi
Yesterday at 3:44 PM
J
H
IMC 2021 P8: Maximum number of vectors such that for any 3, 2 are orthogonal
Sumgato   19
N Yesterday at 2:38 PM by Tintarn
Source: IMC 2021 P8
Let $n$ be a positive integer. At most how many distinct unit vectors can be selected in $\mathbb{R}^n$ such that from any three of them, at least two are orthogonal?
19 replies
Sumgato
Aug 5, 2021
Tintarn
Yesterday at 2:38 PM
J
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Polynomial
enter16180   4
N Yesterday at 1:47 PM by grupyorum
Source: IMC 2025, Problem 1
Let $P \in \mathbb{R}[x]$ be a polynomial with real coefficients, and suppose $\operatorname{deg}(P) \geq 2$. For every $x \in \mathbb{R}$, let $\ell_x \subset \mathbb{R}^2$ denote the line tangent to the graph of $P$ at the point ( $x, P(x)$ ).
a) Suppose that the degree of $P$ is odd. Show that $\bigcup_{x \in \mathbb{R}} \ell_x=\mathbb{R}^2$.
b) Does there exist a polynomial of even degree for which the above equality still holds?
4 replies
enter16180
Yesterday at 11:12 AM
grupyorum
Yesterday at 1:47 PM
J
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Twice continuously differntiable function
enter16180   5
N Yesterday at 1:40 PM by bsf714
Source: IMC 2025, Problem 2
Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a twice continuously differentiable function, and suppose that $\int_{-1}^1 f(x) \mathrm{d} x=0$ and $f(1)=f(-1)=1$. Prove that
$$ \int_{-1}^1\left(f^{\prime \prime}(x)\right)^2 \mathrm{~d} x \geq 15 $$and find all such functions for which equality holds.
5 replies
enter16180
Yesterday at 11:14 AM
bsf714
Yesterday at 1:40 PM
J
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Real symmetric matrix of rank 1
enter16180   3
N Yesterday at 1:19 PM by grupyorum
Source: IMC 2025, Problem 3
Denote by $\mathcal{S}$ the set of all real symmetric $2025 \times 2025$ matrices of rank $1$ whose entries take values $-1$ or $+1$ . Let $A, B \in \mathcal{S}$ be matrices chosen independently uniformly at random. Find the probability that $A$ and $B$ commute, i.e. $A B=B A$.
3 replies
enter16180
Yesterday at 11:16 AM
grupyorum
Yesterday at 1:19 PM
J
H
Trigonometric functional equation
Eul12   0
Yesterday at 11:18 AM
Source: Dubikajtis
Find all derivates functions f : IR---->IR such that
2*(f(x))^2 + f(pi/2 - 2*x) = 1
for all real x.
Wecome for any ideas
0 replies
Eul12
Yesterday at 11:18 AM
0 replies
J
H
Putnam 1972 B6
Kunihiko_Chikaya   4
N Yesterday at 7:52 AM by smileapple
Let $ n_1<n_2<n_3<\cdots <n_k$ be a set of positive integers. Prove that the polynomial $ 1+z^{n_1}+z^{n_2}+\cdots +z^{n_k}$ has no roots inside the circle $ |z|<\frac{\sqrt{5}-1}{2}$.
4 replies
Kunihiko_Chikaya
Jun 5, 2008
smileapple
Yesterday at 7:52 AM
J
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J
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Maximum value of function (with two variables)
Saucepan_man02   1
N May 22, 2025 by Saucepan_man02
If $f(\theta) = \min(|2x-7|+|x-4|+|x-2 -\sin \theta|)$, where $x, \theta \in \mathbb R$, then maximum value of $f(\theta)$.
1 reply
Saucepan_man02
May 22, 2025
Saucepan_man02
May 22, 2025
J
Maximum value of function (with two variables)
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Saucepan_man02
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Saucepan_man02
#1 h  May 22, 2025, 1:25 PM
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If $f(\theta) = \min(|2x-7|+|x-4|+|x-2 -\sin \theta|)$, where $x, \theta \in \mathbb R$, then maximum value of $f(\theta)$.
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1411 posts
Saucepan_man02
#2 h  May 22, 2025, 1:39 PM
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