Very interesting inequality

In a chess tournament with n ≥ 5 players, each player played all other players. One gets a point for a
win, half a point for a draw, and zero points for a loss. At the end of the tournament, each player had
a different number of points. Prove that the second and third ranked players had together more points
than the winner of the tournament.

0 replies

slimshady360

24 minutes ago

0 replies

Mathematics

slimshady360 0

25 minutes ago

Solve this

0 replies

slimshady360

25 minutes ago

0 replies

Olympiad question

slimshady360 0

27 minutes ago

Let a,b,c be positive real numbers such that a + b+c = 3abc. Prove that
a2 +b2 +c2 +3 ≥2(ab+bc+ca)

0 replies

slimshady360

27 minutes ago

0 replies

Infinite sequences.. welp

navi_09220114 2

N 29 minutes ago by ja.

Source: Own. Malaysian IMO TST 2025 P1

Determine all integers $n\ge 2$ such that for any two infinite sequences of positive integers $a_1<a_2< \cdots$ and $b_1, b_2, \cdots$ , such that $a_i\mid a_j$ for all $i<j$ , there always exists a real number $c$ such that $\lfloor{ca_i}\rfloor \equiv b_i \pmod {n}$ for all $i\ge 1$ .

Proposed by Wong Jer Ren & Ivan Chan Kai Chin

2 replies

navi_09220114

Yesterday at 12:52 PM

ja.

29 minutes ago

An important lemma of isogonal conjugate points

buratinogigle 0

37 minutes ago

Source: Own

Let $P$ and $Q$ be two isogonal conjugate with respect to triangle $ABC$ . Let $S$ and $T$ be two points lying on the circle $(PBC)$ such that $PS$ and $PT$ are perpendicular and parallel to bisector of $\angle BAC$ , respectively. Prove that $QS$ and $QT$ bisect two arcs $BC$ containing $A$ and not containing $A$ , respectively, of $(ABC)$ .

0 replies

+1 w

buratinogigle

37 minutes ago

0 replies

Orders and primes

GreekIdiot 0

an hour ago

Find whether there exist prime numbers $p$ such that there exists an integer $a$ satisfying
$i)a^7 \equiv 1 \: mod \: p$ , with $ord_{p}(a)=7$
$ii)a^3+a+1 \equiv 0 \: mod \: p$

0 replies

GreekIdiot

an hour ago

0 replies

Divisibility

RenheMiResembleRice 2

N an hour ago by navier3072

Source: Byer

Prove that for all n ∈ ℕ, 133|( $11^{\left(n+2\right)}+12^{\left(2n+1\right)}$ ).

2 replies

RenheMiResembleRice

Today at 3:07 AM

navier3072

an hour ago

Integer FE

GreekIdiot 2

N an hour ago by GreekIdiot

Let $\mathbb{N}$ denote the set of positive integers
Find all $f: \mathbb{N} \rightarrow \mathbb{N}$ such that for all $a,b \in \mathbb{N}$ it holds that $f(ab+f(b+1))|bf(a+b)f(3b-2+a)$

2 replies

GreekIdiot

Yesterday at 8:53 PM

GreekIdiot

an hour ago

Cyclic ine

m4thbl3nd3r 7

N 2 hours ago by Victoria_Discalceata1

Let $a,b,c>0$ such that $a+b+c=3$ . Prove that $a^3b+b^3c+c^3a+9abc\le 12$

7 replies

m4thbl3nd3r

Yesterday at 3:17 PM

Victoria_Discalceata1

2 hours ago

weird FE on R

frac 6

N 2 hours ago by NicoN9

Source: probably own

Find all functions $f:\mathbb{R} \rightarrow \mathbb{R}$ such that
$f(x+y)^2=xf(x+f(y))+yf(f(y))+f(xy)$ for all $x,y\in \mathbb{R}$ .

6 replies

frac

Jan 4, 2025

NicoN9

2 hours ago

Very interesting inequality

sqing 0

Mar 19, 2025

Source: Own

Let $a,b,c\geq 2 .$ Prove that
$(a-1)(b^2-2)(c^3-3)- \frac{5}{2}abc\geq -10$ $(a-\frac{3}{2})(b^2-2)(c^3-3)- \frac{5}{2}abc\geq -15$ $(a-\frac{3}{2})(b^2-\frac{3}{2})(c^3-3)- \frac{25}{8}abc\geq - \frac{155}{8}$ $(a-\frac{3}{2})(b^2-\frac{3}{2})(c^3-3)- 3abc\geq - \frac{363}{20}$ $(a-\frac{3}{2})(b^2-\frac{3}{2})(c^3-\frac{5}{2})- \frac{55}{16}abc\geq - \frac{341}{16}$