High School Math inequalities

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Inequality on non-nagative numbers

TUAN2k8 0

20 minutes ago

Source: My book

Let $a,b,c$ be non-nagative real numbers such that $a+b+c=3$ .
Prove that $ab+bc+ca-abc \leq \frac{9}{4}$ .

0 replies

1 viewing

TUAN2k8

20 minutes ago

0 replies

J

H

Inequalities

sqing 12

N 20 minutes ago by sqing

Let $a,b,c >2$ and $ab+bc+ca \leq 75.$ Show that
$\frac{1}{a-2}+\frac{1}{b-2}+\frac{1}{c-2}\geq 1$ Let $a,b,c >2$ and $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\geq \frac{6}{7}.$ Show that
$\frac{1}{a-2}+\frac{1}{b-2}+\frac{1}{c-2}\geq 2$

12 replies

sqing

May 13, 2025

sqing

20 minutes ago

J

H

Incircle in an isoscoles triangle

Sadigly 2

N 35 minutes ago by Sadigly

Source: own

Let $ABC$ be an isosceles triangle with $AB=AC$ , and let $I$ be its incenter. Incircle touches sides $BC,CA,AB$ at $D,E,F$ , respectively. Foot of altitudes from $E,F$ to $BC$ are $X,Y$ , respectively. Rays $XI,YI$ intersect $(ABC)$ at $P,Q$ , respectively. Prove that $(PQD)$ touches incircle at $D$ .

2 replies

Sadigly

Friday at 9:21 PM

Sadigly

35 minutes ago

J

H

Prove that the triangle is isosceles.

TUAN2k8 7

N an hour ago by TUAN2k8

Source: My book

Given acute triangle $ABC$ with two altitudes $CF$ and $BE$ .Let $D$ be the point on the line $CF$ such that $DB \perp BC$ .The lines $AD$ and $EF$ intersect at point $X$ , and $Y$ is the point on segment $BX$ such that $CY \perp BY$ .Suppose that $CF$ bisects $BE$ .Prove that triangle $ACY$ is isosceles.

7 replies

1 viewing

TUAN2k8

May 16, 2025

TUAN2k8

an hour ago

J

H

Locus of Mobile points on Circle and Square

Kunihiko_Chikaya 1

N an hour ago by Mathzeus1024

Source: 2012 Hitotsubashi University entrance exam, problem 4

In the $xyz$ -plane given points $P,\ Q$ on the planes $z=2,\ z=1$ respectively. Let $R$ be the intersection point of the line $PQ$ and the $xy$ -plane.

(1) Let $P(0,\ 0,\ 2)$ . When the point $Q$ moves on the perimeter of the circle with center $(0,\ 0,\ 1)$ , radius 1 on the plane $z=1$ ,
find the equation of the locus of the point $R$ .

(2) Take 4 points $A(1,\ 1,\ 1) , B(1,-1,\ 1), C(-1,-1,\ 1)$ and $D(-1,\ 1,\ 1)$ on the plane $z=2$ . When the point $P$ moves on the perimeter of the circle with center $(0,\ 0,\ 2)$ , radius 1 on the plane $z=2$ and the point $Q$ moves on the perimeter of the square $ABCD$ , draw the domain swept by the point $R$ on the $xy$ -plane, then find the area.

1 reply

Kunihiko_Chikaya

Feb 28, 2012

Mathzeus1024

an hour ago

J

H

Circle is tangent to circumcircle and incircle

ABCDE 73

N 2 hours ago by AR17296174

Source: 2016 ELMO Problem 6

Elmo is now learning olympiad geometry. In triangle $ABC$ with $AB\neq AC$ , let its incircle be tangent to sides $BC$ , $CA$ , and $AB$ at $D$ , $E$ , and $F$ , respectively. The internal angle bisector of $\angle BAC$ intersects lines $DE$ and $DF$ at $X$ and $Y$ , respectively. Let $S$ and $T$ be distinct points on side $BC$ such that $\angle XSY=\angle XTY=90^\circ$ . Finally, let $\gamma$ be the circumcircle of $\triangle AST$ .

(a) Help Elmo show that $\gamma$ is tangent to the circumcircle of $\triangle ABC$ .

(b) Help Elmo show that $\gamma$ is tangent to the incircle of $\triangle ABC$ .

James Lin

73 replies

ABCDE

Jun 24, 2016

AR17296174

2 hours ago

J

H

Mathematical Olympiad Finals 2013

parkjungmin 0

2 hours ago

Mathematical Olympiad Finals 2013

0 replies

parkjungmin

2 hours ago

0 replies

J

H

n^k + mn^l + 1 divides n^(k+1) - 1

cjquines0 37

N 2 hours ago by alexanderhamilton124

Source: 2016 IMO Shortlist N4

Let $n, m, k$ and $l$ be positive integers with $n \neq 1$ such that $n^k + mn^l + 1$ divides $n^{k+l} - 1$ . Prove that
[list]
[*] $m = 1$ and $l = 2k$ ; or
[*] $l|k$ and $m = \frac{n^{k-l}-1}{n^l-1}$ .
[/list]

37 replies

cjquines0

Jul 19, 2017

alexanderhamilton124

2 hours ago

J

H

A very beautiful geo problem

TheMathBob 4

N 2 hours ago by ravengsd

Source: Polish MO Finals P2 2023

Given an acute triangle $ABC$ with their incenter $I$ . Point $X$ lies on $BC$ on the same side as $B$ wrt $AI$ . Point $Y$ lies on the shorter arc $AB$ of the circumcircle $ABC$ . It is given that $\angle AIX = \angle XYA = 120^\circ.$ Prove that $YI$ is the angle bisector of $XYA$ .

4 replies

TheMathBob

Mar 29, 2023

ravengsd

2 hours ago

J

H

Difficult combinatorics problem

shactal 0

2 hours ago

Can someone help me with this problem? Let $n\in \mathbb N^*$ . We call a distribution the act of distributing the integers from $1$
to $n^2$ represented by tokens to players $A_1$ to $A_n$ so that they all have the same number of tokens in their urns.
We say that $A_i$ beats $A_j$ when, when $A_i$ and $A_j$ each draw a token from their urn, $A_i$ has a strictly greater chance of drawing a larger number than $A_j$ . We then denote $A_i>A_j$ . A distribution is said to be chicken-fox-viper when $A_1>A_2>\ldots>A_n>A_1$ What is $R(n)$
, the number of chicken-fox-viper distributions?

0 replies