Find the smallest of sum of elements

by hlminh, Apr 24, 2025, 3:37 AM

Let $S=\{1,2,...,2014\}$ and $X=\{a_1,a_2,...,a_{30}\}$ is a subset of $S$ such that if $a,b\in X,a+b\leq 2014$ then $a+b\in X.$ Find the smallest of $\dfrac{a_1+a_2+\cdots+a_{30}}{30}.$

combinatorics

Cute diophantine

by TestX01, Apr 24, 2025, 2:36 AM

Find all sequences of four consecutive integers such that twice their product is perfect square minus nine.

number theory

Pells equations

Complicated FE

by XAN4, Apr 23, 2025, 11:53 AM

Find all solutions for the functional equation $f(xyz)+\sum_{cyc}f(\frac{yz}x)=f(x)\cdot f(y)\cdot f(z)$ , in which $f$ : $\mathbb R^+\rightarrow\mathbb R^+$
Note: the solution is actually quite obvious - $f(x)=x^n+\frac1{x^n}$ , but the proof is important.
Note 2: it is likely that the result can be generalized into a more advanced questions, potentially involving more bash.

functional equation

algebra

Stronger inequality than an old result

by KhuongTrang, Aug 1, 2024, 3:13 PM

Problem. Find the best constant $k$ satisfying $(ab+bc+ca)\left[\frac{1}{(a+b)^{2}}+\frac{1}{(b+c)^{2}}+\frac{1}{(c+a)^{2}}\right]\ge \frac{9}{4}+k\cdot\frac{a(a-b)(a-c)+b(b-a)(b-c)+c(c-a)(c-b)}{(a+b+c)^{3}}$ holds for all $a,b,c\ge 0: ab+bc+ca>0.$

This post has been edited 2 times. Last edited by KhuongTrang, Aug 2, 2024, 6:43 AM

inequalities

Something nice

by KhuongTrang, Nov 1, 2023, 12:56 PM

Problem. Given $a,b,c$ be non-negative real numbers such that $ab+bc+ca=1.$ Prove that

$\sqrt{a+1}+\sqrt{b+1}+\sqrt{c+1}\le 1+2\sqrt{a+b+c+abc}.$

This post has been edited 2 times. Last edited by KhuongTrang, Nov 19, 2023, 11:59 PM

inequalities

\frac{1}{9}+\frac{1}{\sqrt{3}}\geq a^2+\sqrt{a+ b^2} \geq \frac{1}{4}

by sqing, Oct 3, 2023, 8:11 AM

Let $a,b\geq 0$ and $3a+4b =1 .$ Prove that
$\frac{2}{3}\geq a +\sqrt{a^2+ 4b^2}\geq \frac{6}{13}$ $\frac{1}{9}+\frac{1}{\sqrt{3}}\geq a^2+\sqrt{a+ b^2} \geq \frac{1}{4}$ $2\geq a+\sqrt{a^2+16b} \geq \frac{2}{3}\geq a+\sqrt{a^2+16b^3} \geq \frac{2(725-8\sqrt{259})}{729}$

This post has been edited 2 times. Last edited by sqing, Oct 3, 2023, 8:55 AM

inequalities proposed

algebra

inequalities

Inequality

Easy IMO 2023 NT

by 799786, Jul 8, 2023, 4:53 AM

Determine all composite integers $n>1$ that satisfy the following property: if $d_1$ , $d_2$ , $\ldots$ , $d_k$ are all the positive divisors of $n$ with $1 = d_1 < d_2 < \cdots < d_k = n$ , then $d_i$ divides $d_{i+1} + d_{i+2}$ for every $1 \leq i \leq k - 2$ .

This post has been edited 7 times. Last edited by v_Enhance, Sep 18, 2023, 12:33 AM
Reason: minor latex pet peeve

hard binomial sum

by PRMOisTheHardestExam, Mar 6, 2023, 5:35 PM

Find
$\frac{\displaystyle\sum_{k=0}^r \binom nk \binom{n-2k}{r-k}}{\displaystyle\sum_{k=r}^n \binom nk \binom{2k}{2r}\left(\frac{3}{4}\right)^{n-k}\left(\frac{1}{2}\right)^{2k-2r}}$ \
where $n \ge 2r$ .
Options: 1/2, 1, 2, none.

algebra

x_1x_2...x_(n+1)-1 is divisible by an odd prime

by ABCDE, Jul 7, 2016, 9:45 PM

Let $m$ and $n$ be positive integers such that $m>n$ . Define $x_k=\frac{m+k}{n+k}$ for $k=1,2,\ldots,n+1$ . Prove that if all the numbers $x_1,x_2,\ldots,x_{n+1}$ are integers, then $x_1x_2\ldots x_{n+1}-1$ is divisible by an odd prime.

This post has been edited 1 time. Last edited by ABCDE, Jul 7, 2016, 9:47 PM

IMO Shortlist

number theory

Divisibility

IMO 2012/5 Mockup

by v_Enhance, Jul 30, 2013, 5:19 AM

Let $ABC$ be a scalene triangle with $\angle BCA = 90^{\circ}$ , and let $D$ be the foot of the altitude from $C$ . Let $X$ be a point in the interior of the segment $CD$ . Let $K$ be the point on the segment $AX$ such that $BK = BC$ . Similarly, let $L$ be the point on the segment $BX$ such that $AL = AC$ . The circumcircle of triangle $DKL$ intersects segment $AB$ at a second point $T$ (other than $D$ ). Prove that $\angle ACT = \angle BCT$ .

This post has been edited 1 time. Last edited by v_Enhance, May 7, 2015, 1:33 AM
Reason: 90\dg should be 90^{\circ}

Submit

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math_exploration

by hlminh, Apr 24, 2025, 3:37 AM

by TestX01, Apr 24, 2025, 2:36 AM

by XAN4, Apr 23, 2025, 11:53 AM

by KhuongTrang, Aug 1, 2024, 3:13 PM

by KhuongTrang, Nov 1, 2023, 12:56 PM

by sqing, Oct 3, 2023, 8:11 AM

by 799786, Jul 8, 2023, 4:53 AM

by PRMOisTheHardestExam, Mar 6, 2023, 5:35 PM

by ABCDE, Jul 7, 2016, 9:45 PM

by v_Enhance, Jul 30, 2013, 5:19 AM