Very interesting inequality

The question is:
Let $x_1$ , $x_2$ , $\dots$ , $x_n$ be $n$ integers. If $k>n$ is an integer, prove that the only solution to
$x_1^2 + x_2^2 + \dots + x_n^2 = kx_1x_2\dots x_n$ is is $x_1 = x_2 = \dots = x_n = 0$ .

0 replies

Eagle116

an hour ago

0 replies

Geometry with parallel lines.

falantrng 32

N 2 hours ago by endless_abyss

Source: RMM 2018,D1 P1

Let $ABCD$ be a cyclic quadrilateral an let $P$ be a point on the side $AB.$ The diagonals $AC$ meets the segments $DP$ at $Q.$ The line through $P$ parallel to $CD$ mmets the extension of the side $CB$ beyond $B$ at $K.$ The line through $Q$ parallel to $BD$ meets the extension of the side $CB$ beyond $B$ at $L.$ Prove that the circumcircles of the triangles $BKP$ and $CLQ$ are tangent .

32 replies

falantrng

Feb 24, 2018

endless_abyss

2 hours ago

sum divides n-th moment

navi_09220114 1

N 2 hours ago by ja.

Source: Own. Malaysian IMO TST 2025 P9

Given four distinct positive integers $a<b<c<d$ such that $\gcd(a,b,c,d)=1$ , find the maximum possible number of integers $1\le n\le 2025$ such that $a+b+c+d\mid a^n+b^n+c^n+d^n$
Proposed by Ivan Chan Kai Chin

1 reply

navi_09220114

Yesterday at 1:07 PM

ja.

2 hours ago

Nice problem

hanzo.ei 0

2 hours ago

Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that
$f(xy) = f(x)f(y) \;-\; f(x + y) \;+\; 1, \quad \forall x, y \in \mathbb{R}.$

0 replies

hanzo.ei

2 hours ago

0 replies

Find all functions

Jackson0423 0

2 hours ago

Find all functions F:R->R such that
1/(F(F(x))-F(x))=F(x)
I know x+1/x works..

0 replies

Jackson0423

2 hours ago

0 replies

2x+1 is a perfect square but the following x+1 integers are not.

Sumgato 7

N 2 hours ago by Davut1102

Source: Spain Mathematical Olympiad 2018 P1

Find all positive integers $x$ such that $2x+1$ is a perfect square but none of the integers $2x+2, 2x+3, \ldots, 3x+2$ are perfect squares.

7 replies

Sumgato

Mar 17, 2018

Davut1102

2 hours ago

Prove that P1(x), P2(x) ,... Pn(x) = k has no root

truongphatt2668 2

N 2 hours ago by truongphatt2668

Let $n \in \mathbb{N}^*$ and $P_1(x),P_2(x), \ldots P_n(x) \in \mathbb{Z}[x]$ such that $\mathrm{deg} P_i = 2, \forall i = \overline{1,n}$ . Prove that exists many $k \in \mathbb{N}$ such that every equation: $P_i(x) = k, \forall i = \overline{1,n}$ has no real roots

2 replies

truongphatt2668

Today at 2:26 AM

truongphatt2668

2 hours ago

Geo: incircle, escircle, isotomic conjugate

XAN4 1

N 2 hours ago by deraxenrovalo

Source: Own

For $\triangle{ABC}$ , Its incircle $\odot I$ and $A-$ escircle $\odot I_A$ are tangent to $BC$ at $D$ and $E$ respectively. $AI$ intersects line $BC$ at $J$ . Line $AD$ intersects $\odot I$ at $F$ , and line $AE$ intersects $\odot I_A$ at $G$ . Line $FG$ intersects $BC$ at $H$ . Prove that $BJ=CH$ .

1 reply

XAN4

Mar 19, 2025

deraxenrovalo

2 hours ago

special sets

ChubbyTomato426 0

2 hours ago

Let $n$ be a positive integer. A subset $\{a, b, c, d\} \subseteq \{1, 2, . . . , 4n\}$ with four distinct elements is special if there exists a rearrangement $(x, y, z, w)$ of $(a, b, c, d)$ such that $xy -zw = 1$ . Prove that the set $\{1, 2, . . . , 4n \}$ cannot be partitioned into $n$ special disjoint sets.

0 replies

ChubbyTomato426

2 hours ago

0 replies

2, 4, 5-Nim

cjquines0 2

N 2 hours ago by Mathdreams

Source: Philippines MO 2016/4

Two players, $A$ (first player) and $B$ , take alternate turns in playing a game using 2016 chips as follows: the player whose turn it is, must remove $s$ chips from the remaining pile of chips, where $s \in \{ 2,4,5 \}$ . No one can skip a turn. The player who at some point is unable to make a move (cannot remove chips from the pile) loses the game. Who among the two players can force a win on this game?

2 replies

cjquines0

Jan 21, 2017

Mathdreams

2 hours ago

Very interesting inequality

sqing 0

Mar 19, 2025

Source: Own

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