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inequalities
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Inspired by IMO 1984
sqing   0
29 minutes ago
Source: Own
Let $ a,b,c\geq 0 $ and $a+b+c=1$. Prove that
$$a^2+b^2+c+ ab +9abc\leq 1$$$$ a^2+b^2+c +ab+10 abc\leq\frac{28}{27}$$$$a^2+b^2+c+ ab +\frac{19}{2}abc\leq\frac{55}{54}$$
0 replies
2 viewing
sqing
29 minutes ago
0 replies
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Integers with determinant \pm 1
anantmudgal09   32
N 37 minutes ago by anudeep
Source: INMO 2021 Problem 1
Suppose $r \ge 2$ is an integer, and let $m_1, n_1, m_2, n_2, \dots, m_r, n_r$ be $2r$ integers such that $$\left|m_in_j-m_jn_i\right|=1$$for any two integers $i$ and $j$ satisfying $1 \le i<j \le r$. Determine the maximum possible value of $r$.

Proposed by B Sury
32 replies
anantmudgal09
Mar 7, 2021
anudeep
37 minutes ago
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Long polynomial factorization
wassupevery1   0
38 minutes ago
Source: 2025 Vietnam IMO TST - Problem 6
For each prime $p$ of the form $4k+3$ with $k \in \mathbb{Z}^+$, consider the polynomial $$Q(x)=px^{2p} - x^{2p-1} + p^2x^{\frac{3p+1}{2}}+2(p^2+1)x^p + p^2 x^{\frac{p-1}{2}} -x + p.$$Determine all ordered pairs of polynomials $f, g$ with integer coefficients such that $Q(x)=f(x)g(x)$.
0 replies
wassupevery1
38 minutes ago
0 replies
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Good set of cells
wassupevery1   0
41 minutes ago
Source: 2025 Vietnam IMO TST - Problem 5
There is an $n \times n$ grid which has rows and columns numbered from $1$ to $n$; the cell at row $i$ and column $j$ is denoted as the cell at $(i, j)$. A subset $A$ of the cells is called good if for any two cells at $(x_1, y), (x_2, y)$, the cells $(u, v)$ satisfying $x_1 < u \leq x_2, v<y$ or $x_1 \leq u < x_2, v>y$ are not in $A$. Determine the minimal number of good sets such that they are pairwise disjoint and every cell of the board belongs to exactly one good set.
0 replies
wassupevery1
41 minutes ago
0 replies
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Find the value
sqing   2
N 44 minutes ago by sqing
Source: Hunan changsha 2025
Let $ a,b,c $ be real numbers such that $ abc\neq 0,2a-b+c= 0 $ and $ a-2b-c=0. $ Find the value of $\frac{a^2+b^2+c^2}{ab+bc+ca}.$
Let $ a,b,c $ be real numbers such that $ abc\neq 0,a+2b+3c= 0 $ and $ 2a+3b+4c=0. $ Find the value of $\frac{ab+bc+ca}{a^2+b^2+c^2}.$
2 replies
sqing
3 hours ago
sqing
44 minutes ago
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Binomial Non-divisibility
wassupevery1   0
an hour ago
Source: 2025 Vietnam IMO TST - Problem 4
Find all positive integers $k$ for which there are infinitely many positive integers $n$ such that $\binom{(2025+k)n}{2025n}$ is not divisible by $kn+1$.
0 replies
wassupevery1
an hour ago
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Inspired by IMO 1984
sqing   2
N an hour ago by sqing
Source: Own
Let $ a,b,c\geq 0 $ and $a+b+c=1$. Prove that
$$a^2+b^2+ ab +24abc\leq\frac{81}{64}$$Equality holds when $a=b=\frac{3}{8},c=\frac{1}{4}.$
$$a^2+b^2+ ab +18abc\leq\frac{343}{324}$$Equality holds when $a=b=\frac{7}{18},c=\frac{2}{9}.$
2 replies
sqing
5 hours ago
sqing
an hour ago
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2025 Caucasus MO Juniors P1
BR1F1SZ   1
N an hour ago by pco
Source: Caucasus MO
Anya and Vanya’s houses are located on the straight road. The distance between their houses is divided by a shop and a school into three equal parts. If Anya and Vanya leave their houses at the same time and walk towards each other, they will meet near the shop. If Anya rides a scooter, then her speed will increase by $150\,\text{m/min}$, and they will meet near the school. Find Vanya’s speed of walking.
1 reply
BR1F1SZ
Today at 12:54 AM
pco
an hour ago
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Probability-hard
Noname23   3
N 2 hours ago by Noname23
problem
3 replies
Noname23
3 hours ago
Noname23
2 hours ago
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Sequence and prime factors
USJL   0
2 hours ago
Source: 2025 Taiwan TST Round 2 Independent Study 1-N
Let $a_0,a_1,\ldots$ be a sequence of positive integers with $a_0=1$, $a_1=2$ and
\[a_n = a_{n-1}^{a_{n-1}a_{n-2}}-1\]for all $n\geq 2$. Show that if $p$ is a prime less than $2^k$ for some positive integer $k$, then there exists $n\leq k+1$ such that $p\mid a_n$.
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USJL
2 hours ago
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inequality
ehuseyinyigit   3
N Mar 23, 2025 by ehuseyinyigit
Source: Nice
For all positive real numbers $a,b$ and $c$, prove
$$\sum_{cyc}{\dfrac{1}{b\left(a^4+a^3c+b^2c^2\right)}}\geq \dfrac{27}{(a+b+c)(a^2+b^2+c^2)^2}$$
3 replies
ehuseyinyigit
Feb 3, 2025
ehuseyinyigit
Mar 23, 2025
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inequality
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Reveal topic tags
inequalities
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G H BBookmark kLocked kLocked NReply
Source: Nice
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ehuseyinyigit
785 posts
ehuseyinyigit
#1 h  Feb 3, 2025, 3:28 PM
Y by
For all positive real numbers $a,b$ and $c$, prove
$$\sum_{cyc}{\dfrac{1}{b\left(a^4+a^3c+b^2c^2\right)}}\geq \dfrac{27}{(a+b+c)(a^2+b^2+c^2)^2}$$
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arqady
30154 posts
arqady
#2 h  Feb 3, 2025, 3:51 PM • 2 Y
Y by ehuseyinyigit, MyLifeMyChoice
ehuseyinyigit wrote:
For all positive real numbers $a,b$ and $c$, prove
$$\sum_{cyc}{\dfrac{1}{b\left(a^4+a^3c+b^2c^2\right)}}\geq \dfrac{27}{(a+b+c)(a^2+b^2+c^2)^2}$$
C-S and Vasc :-D
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ehuseyinyigit
785 posts
ehuseyinyigit
#3 h  Feb 4, 2025, 3:35 PM
Y by
Yes arqady, I was playing around the Vasc's :-D

$$LHS=\sum_{cyc}{\dfrac{1}{b(a^4+a^3c+b^2c^2)}}\geq \dfrac{9}{\sum{b(a^4+a^3c+b^2c^2)}}$$and Vasc's inequality implies
$$LHS\geq \dfrac{9}{\sum{b(a^4+a^3c+b^2c^2)}}=\dfrac{9}{(a+b+c)(a^3b+b^3c+c^3a)}\geq \dfrac{27}{(a+b+c)(a^2+b^2+c^2)}$$as desired.
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ehuseyinyigit
785 posts
ehuseyinyigit
#4 h  Mar 23, 2025, 7:20 PM
Y by
Does Vasc Inequality has a general form, like $n$-variable ?
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