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k a May Highlights and 2025 AoPS Online Class Information
jlacosta   0
May 1, 2025
May is an exciting month! National MATHCOUNTS is the second week of May in Washington D.C. and our Founder, Richard Rusczyk will be presenting a seminar, Preparing Strong Math Students for College and Careers, on May 11th.

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[*]May 21st, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 2 Math Jam, Problems 5 and 6, Canada/USA Mathcamp staff will discuss solutions to Problems 5 and 6 of the 2025 Mathcamp Qualifying Quiz![/list]
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0 replies
jlacosta
May 1, 2025
0 replies
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2 var inquality
Iveela   19
N 16 minutes ago by sqing
Source: Izho 2025 P1
Let $a, b$ be positive reals such that $a^3 + b^3 = ab + 1$. Prove that \[(a-b)^2 + a + b \geq 2\]
19 replies
Iveela
Jan 14, 2025
sqing
16 minutes ago
J
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Set of perfect powers is irreducible
Assassino9931   0
17 minutes ago
Source: Al-Khwarizmi International Junior Olympiad 2025 P4
For two sets of integers $X$ and $Y$ we define $X\cdot Y$ as the set of all products of an element of $X$ and an element of $Y$. For example, if $X=\{1, 2, 4\}$ and $Y=\{3, 4, 6\}$ then $X\cdot Y=\{3, 4, 6, 8, 12, 16, 24\}.$ We call a set $S$ of positive integers good if there do not exist sets $A,B$ of positive integers, each with at least two elements and such that the sets $A\cdot B$ and $S$ are the same. Prove that the set of perfect powers greater than or equal to $2025$ is good.

(In any of the sets $A$, $B$, $A\cdot B$ no two elements are equal, but any two or three of these sets may have common elements. A perfect power is an integer of the form $n^k$, where $n>1$ and $k > 1$ are integers.)

Lajos Hajdu and Andras Sarkozy, Hungary
0 replies
Assassino9931
17 minutes ago
0 replies
J
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Al-Khwarizmi birth year in a combi process
Assassino9931   0
19 minutes ago
Source: Al-Khwarizmi International Junior Olympiad 2025 P3
On a circle are arranged $100$ baskets, each containing at least one candy. The total number of candies is $780$. Asad and Sevinch make moves alternatingly, with Asad going first. On one move, Asad can take all the candies from $9$ consecutive non-empty baskets, while Sevinch can take all the candies from a single non-empty basket that has at least one empty neighboring basket. Prove that Asad can take overall at least $700$ candies, regardless of the initial distribution of candies and Sevinch's actions.

Shubin Yakov, Russia
0 replies
Assassino9931
19 minutes ago
0 replies
J
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Grouping angles in a pentagon with bisectors
Assassino9931   0
21 minutes ago
Source: Al-Khwarizmi International Junior Olympiad 2025 P2
Let $ABCD$ be a convex quadrilateral with \[\angle ADC = 90^\circ, \ \ \angle BCD = \angle ABC > 90^\circ, \mbox{ and } AB = 2CD.\]The line through \(C\), parallel to \(AD\), intersects the external angle bisector of \(\angle ABC\) at point \(T\). Prove that the angles $\angle ATB$, $\angle TBC$, $\angle BCD$, $\angle CDA$, $\angle DAT$ can be divided into two groups, so that the angles in each group have a sum of $270^{\circ}$.

Miroslav Marinov, Bulgaria
0 replies
Assassino9931
21 minutes ago
0 replies
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Problem 2, Grade 12th RMO Shortlist - Year 2002
sticknycu   3
N Yesterday at 11:59 PM by RobertRogo
Let $A \in M_2(C), A \neq O_2, A \neq I_2, n \in \mathbb{N}^*$ and $S_n = \{ X \in M_2(C) | X^n = A \}$.
Show:
a) $S_n$ with multiplication of matrixes operation is making an isomorphic-group structure with $U_n$.
b) $A^2 = A$.

Marian Andronache
3 replies
sticknycu
Jan 3, 2020
RobertRogo
Yesterday at 11:59 PM
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2024 Putnam A1
KevinYang2.71   21
N Yesterday at 10:01 PM by KAME06
Determine all positive integers $n$ for which there exists positive integers $a$, $b$, and $c$ satisfying
\[ 2a^n+3b^n=4c^n. \]
21 replies
KevinYang2.71
Dec 10, 2024
KAME06
Yesterday at 10:01 PM
J
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Miklos Schweitzer 1968_9
ehsan2004   1
N Yesterday at 7:52 PM by pi_quadrat_sechstel
Let $ f(x)$ be a real function such that
\[ \lim_{x \rightarrow +\infty} \frac{f(x)}{e^x}=1\]
and $ |f''(x)|\leq c|f'(x)|$ for all sufficiently large $ x$. Prove that \[ \lim_{x \rightarrow +\infty} \frac{f'(x)}{e^x}=1.\]

P. Erdos
1 reply
ehsan2004
Oct 8, 2008
pi_quadrat_sechstel
Yesterday at 7:52 PM
J
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Putnam 1956 B7
sqrtX   7
N Yesterday at 7:08 PM by bjump
Source: Putnam 1956
The polynomials $P(z)$ and $Q(z)$ with complex coefficients have the same set of numbers for their zeros but possibly different multiplicities. The same is true for the polynomials
$$P(z)+1 \;\; \text{and} \;\; Q(z)+1.$$Prove that $P(z)=Q(z).$
7 replies
sqrtX
Jul 5, 2022
bjump
Yesterday at 7:08 PM
J
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Linear Space Decomposition
Suan_16   1
N Yesterday at 6:09 PM by loup blanc
Let $A$ be a linear transformation on linear space $V$ satisfying:$$A^l=0$$but $$A^{l-1} \neq 0$$, and $V_0$ is the eigensubspace of eigenvalue $0$. Prove that $V$ can be decomposed to $dim V_0$ $A$-cyclic subspace's direct sum.

Click to reveal hidden text
1 reply
Suan_16
Apr 18, 2025
loup blanc
Yesterday at 6:09 PM
J
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Romanian National Olympiad 1997 - Grade 11 - Problem 2
Filipjack   1
N Yesterday at 5:05 PM by loup blanc
Source: Romanian National Olympiad 1997 - Grade 11 - Problem 2
Let $A$ be a square matrix of odd order (at least $3$) whose entries are odd integers. Prove that if $A$ is invertible, then it is not possible for all the minors of the entries of a row to have equal absolute values.
1 reply
Filipjack
Apr 6, 2025
loup blanc
Yesterday at 5:05 PM
J
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Serious qustion
Thayaden   2
N Yesterday at 4:54 PM by ReticulatedPython
Let $F_n$ be then $n$-th fibbiance number. As $n$ gets bigger and bigger, we have,
$$\frac{F_{n+1}}{F_n}\approx\varphi,$$my question is dose,
$$\lim_{n\rightarrow \infty}\frac{F_{n+1}}{F_n}=\varphi.$$My reservations about this is that $\varphi\in\mathbb{R}\setminus\mathbb{Q}$ and $F_n\in\mathbb{Z}^+$ so $\frac{F_{n+1}}{F_n}\in\mathbb{Q}$. So, if the limit holds, does that mean that if $S$ is a set and $P$ is a set, for each $s\in S$ that $s\not\in P$ we can have, for $\text{Range}(f)=S$ we can have,
$$\lim_{x\rightarrow n}f(x)\in P,$$for some $n$?
2 replies
Thayaden
Yesterday at 4:40 PM
ReticulatedPython
Yesterday at 4:54 PM
J
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Putnam 2010 B5
Kent Merryfield   25
N Yesterday at 2:59 PM by Rohit-2006
Is there a strictly increasing function $f:\mathbb{R}\to\mathbb{R}$ such that $f'(x)=f(f(x))$ for all $x?$
25 replies
Kent Merryfield
Dec 6, 2010
Rohit-2006
Yesterday at 2:59 PM
J
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Determinant problem
Entrepreneur   3
N Yesterday at 2:49 PM by Entrepreneur
Source: Hall & Knight
If a determinant is of $n^{\text{th}}$ order, and if the constituents of its first, second, ..., $n^{\text{th}}$ rows are the first $n$ figurate numbers of the first, second, ..., $n^{\text{th}}$ orders respectively, show that it's value is $1.$
3 replies
Entrepreneur
May 5, 2025
Entrepreneur
Yesterday at 2:49 PM
J
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AB=BA if A-nilpotent
KevinDB17   2
N Yesterday at 1:01 PM by loup blanc
Let A,B 2 complex n*n matrices such that AB+I=A+B+BA
If A is nilpotent prove that AB=BA
2 replies
KevinDB17
Mar 30, 2025
loup blanc
Yesterday at 1:01 PM
J
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J
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Non-negative real variables inequality
KhuongTrang   2
N Apr 29, 2025 by NguyenVanHoa29
Source: own
Problem. Let $a,b,c\ge 0: ab+bc+ca>0.$ Prove that$$\color{blue}{\frac{\left(2ab+ca+cb\right)^{2}}{a^{2}+4ab+b^{2}}+\frac{\left(2bc+ab+ac\right)^{2}}{b^{2}+4bc+c^{2}}+\frac{\left(2ca+bc+ba\right)^{2}}{c^{2}+4ca+a^{2}}\ge \frac{8(ab+bc+ca)}{3}.}$$
2 replies
KhuongTrang
Apr 24, 2025
NguyenVanHoa29
Apr 29, 2025
J
Non-negative real variables inequality
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Reveal topic tags
inequalities
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KhuongTrang
731 posts
KhuongTrang
#1 h  Apr 24, 2025, 2:52 PM • 3 Y
Y by math90, NguyenVanHoa29, Zuyong
Problem. Let $a,b,c\ge 0: ab+bc+ca>0.$ Prove that$$\color{blue}{\frac{\left(2ab+ca+cb\right)^{2}}{a^{2}+4ab+b^{2}}+\frac{\left(2bc+ab+ac\right)^{2}}{b^{2}+4bc+c^{2}}+\frac{\left(2ca+bc+ba\right)^{2}}{c^{2}+4ca+a^{2}}\ge \frac{8(ab+bc+ca)}{3}.}$$
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Quantum-Phantom
272 posts
Quantum-Phantom
#2 h  Apr 25, 2025, 3:31 AM
Y by
Are there any easier methods?

After multiplying both sides by \(\prod\limits_{\rm cyc}\left(a^2+4ab+b^2\right)\), we need to show that
\[\frac13\sum_{\rm cyc}a^2b^2(a+3b)(3a+b)(a-b)^2+abc\cdot f(a,b,c)\ge0,\]where $f(a,b,c)$ is a fifth degree polynomial:
\[\sum_{\rm cyc}\left(2a^5+\frac83a^4b+\frac83ab^4+\frac{22}3a^2b^2c-\frac{14}3a^3b^2-\frac{14}3a^2b^3-\frac{16}3a^2b^2c\right).\]By the $uvw$ method, it is not hard to show that $f(a,b,c)\ge0$ is true.

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NguyenVanHoa29
7 posts
NguyenVanHoa29
#3 h  Apr 29, 2025, 9:51 AM • 1 Y
Y by arqady
I think it is a concave function according to w^3 and the rest is easy checking.
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