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Product of f(m) multiple of odd integers

buzzychaoz 24

N 2 hours ago by cursed_tangent1434

Source: China Team Selection Test 2016 Test 2 Day 2 Q4

Set positive integer $m=2^k\cdot t$ , where $k$ is a non-negative integer, $t$ is an odd number, and let $f(m)=t^{1-k}$ . Prove that for any positive integer $n$ and for any positive odd number $a\le n$ , $\prod_{m=1}^n f(m)$ is a multiple of $a$ .

24 replies

buzzychaoz

Mar 21, 2016

cursed_tangent1434

2 hours ago

Domain of (a, b) satisfying inequality with fraction

Kunihiko_Chikaya 1

N 2 hours ago by Mathzeus1024

Source: 2014 Kyoto University entrance exam/Science, Problem 4

For real constants $a,\ b$ , define a function $f(x)=\frac{ax+b}{x^2+x+1}.$

Draw the domain of the points $(a,\ b)$ such that the inequality :

$f(x) \leq f(x)^3-2f(x)^2+2$

holds for all real numbers $x$ .

1 reply

Kunihiko_Chikaya

Feb 26, 2014

Mathzeus1024

2 hours ago

Generic Real-valued FE

lucas3617 0

2 hours ago

$f: \mathbb{R} -> \mathbb{R}$ , find all functions where $f(2x+f(2y-x))+f(-x)+f(y)=2f(x)+f(y-2x)+f(2y)$ for all $x$ , $y \in \mathbb{R}$

0 replies

lucas3617

2 hours ago

0 replies

Solve All 6 IMO 2024 Problems (42/42), New Framework Looking for Feedback

Blackhole.LightKing 4

N 2 hours ago by Blackhole.LightKing

Hi everyone,

I’ve been experimenting with a different way of approaching mathematical problem solving — a framework that emphasizes recursive structures and symbolic alignment rather than conventional step-by-step strategies.

Using this method, I recently attempted all six problems from IMO 2024 and was able to arrive at what I believe are valid full-mark solutions across the board (42/42 total score, by standard grading).

However, I don’t come from a formal competition background, so I’m sure there are gaps in clarity, communication, or even logic that I’m not fully aware of.

If anyone here is willing to take a look and provide feedback, I’d appreciate it — especially regarding:

The correctness and completeness of the proofs

Suggestions on how to make the ideas clearer or more elegant

Whether this approach has any broader potential or known parallels

I'm here to learn more and improve the presentation and thinking behind the work.

You can download the Solution here.

https://agi-origin.com/assets/pdf/AGI-Origin_IMO_2024_Solution.pdf

Thanks in advance,
— BlackholeLight0

4 replies

Blackhole.LightKing

Yesterday at 12:14 PM

Blackhole.LightKing

2 hours ago

$KH$, $EM$ and $BC$ are concurrent

yunxiu 44

N 2 hours ago by alexanderchew

Source: 2012 European Girls’ Mathematical Olympiad P7

Let $ABC$ be an acute-angled triangle with circumcircle $\Gamma$ and orthocentre $H$ . Let $K$ be a point of $\Gamma$ on the other side of $BC$ from $A$ . Let $L$ be the reflection of $K$ in the line $AB$ , and let $M$ be the reflection of $K$ in the line $BC$ . Let $E$ be the second point of intersection of $\Gamma$ with the circumcircle of triangle $BLM$ .
Show that the lines $KH$ , $EM$ and $BC$ are concurrent. (The orthocentre of a triangle is the point on all three of its altitudes.)

Luxembourg (Pierre Haas)

44 replies

yunxiu

Apr 13, 2012

alexanderchew

2 hours ago

An fe based off of another trivial problem

benjaminchew13 1

N 2 hours ago by Mathzeus1024

Source: Own

Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that for all real numbers $x$ and $y$ , $f(x+y-f(x))f(f(x+y)-y)=f(xy)$

1 reply

benjaminchew13

5 hours ago

Mathzeus1024

2 hours ago

functional equation interesting

skellyrah 6

N 3 hours ago by skellyrah

find all functions IR->IR such that $xf(x+yf(xy)) + f(f(y)) = f(xf(y))^2 + (x+1)f(x)$

6 replies

skellyrah

Yesterday at 8:32 PM

skellyrah

3 hours ago

4 variables with quadrilateral sides

mihaig 1

N 3 hours ago by Quantum-Phantom

Source: VL

Let $a,b,c,d\geq0$ satisfying
$\frac1{a+1}+\frac1{b+1}+\frac1{c+1}+\frac1{d+1}=2.$ Prove
$4\left(abc+abd+acd+bcd\right)\geq3\left(a+b+c+d\right)+4.$

1 reply

mihaig

Today at 5:11 AM

Quantum-Phantom

3 hours ago

4 var inequality

sealight2107 1

N 3 hours ago by sealight2107

Source: Own

Let $a,b,c,d$ be positive reals such that $a+b+c+d+\frac{1}{abcd} = 18$ . Find the minimum and maximum value of $a,b,c,d$

1 reply

sealight2107

Wednesday at 2:40 PM

sealight2107

3 hours ago

Inspired by SXTX (4)2025 Q712

sqing 1

N 4 hours ago by sqing

Source: Own

Let $a ,b,c>0$ and $(a+b)^2+2(b+c)^2+(c+a)^2=12.$ Prove that $abc(a+b+c) \leq \frac{9}{5}$ Let $a ,b,c>0$ and $2(a+b)^2+ (b+c)^2+2(c+a)^2=12.$ Prove that $abc(a+b+c) \leq \frac{9}{8}$

1 reply

sqing

Yesterday at 11:59 AM

sqing

4 hours ago

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