Max value

by Hip1zzzil, May 17, 2025, 3:55 AM

Three distinct nonzero real numbers $x,y,z$ satisfy:

(i) $2x+2y+2z=3$
(ii) $\frac{1}{xz}+\frac{x-y}{y-z}=\frac{1}{yz}+\frac{y-z}{z-x}=\frac{1}{xy}+\frac{z-x}{x-y}$
Find the maximum value of $18x+12y+6z$ .

algebra

Pythagoras...

by Hip1zzzil, May 17, 2025, 3:41 AM

Find the sum of all $k$ s such that:
There exists two odd positive integers $a,b$ such that ${k}^{2}={a}^{2b}+{(2b)}^{4}.$

This post has been edited 1 time. Last edited by Hip1zzzil, 2 hours ago
Reason: E

number theory

Guangxi High School Mathematics Competition 2025 Q12

by sqing, May 17, 2025, 2:58 AM

Let $a,b,c>0$ . Prove that
$abc\geq \frac {a+b+c}{\frac {1}{a^2}+\frac {1}{b^2}+\frac {1}{c^2} }\geq(a+b-c)(b+c-a)(c+a-b)$

This post has been edited 1 time. Last edited by sqing, 2 hours ago

inequalities proposed

algebra

Prove that the triangle is isosceles.

by TUAN2k8, May 16, 2025, 9:55 AM

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.

geometry

Triangle

Inequality on APMO P5

by Jalil_Huseynov, May 17, 2022, 6:50 PM

Let $a,b,c,d$ be real numbers such that $a^2+b^2+c^2+d^2=1$ . Determine the minimum value of $(a-b)(b-c)(c-d)(d-a)$ and determine all values of $(a,b,c,d)$ such that the minimum value is achived.

inequalities

APMO

APMO 2022

Hard Function

by johnlp1234, Jul 8, 2020, 8:21 AM

Find all function $f:\mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ such that:
$f(x^3+f(y))=y+(f(x))^3$

function

Hard Function

by johnlp1234, Jul 7, 2020, 3:40 PM

f:R+--->R+:
f(x^3+f(y))=y+(f(x))^3

function

2018 Hong Kong TST2 problem 4

by YanYau, Oct 21, 2017, 11:54 AM

In triangle $ABC$ with incentre $I$ , let $M_A,M_B$ and $M_C$ by the midpoints of $BC, CA$ and $AB$ respectively, and $H_A,H_B$ and $H_C$ be the feet of the altitudes from $A,B$ and $C$ to the respective sides. Denote by $\ell_b$ the line being tangent tot he circumcircle of triangle $ABC$ and passing through $B$ , and denote by $\ell_b'$ the reflection of $\ell_b$ in $BI$ . Let $P_B$ by the intersection of $M_AM_C$ and $\ell_b$ , and let $Q_B$ be the intersection of $H_AH_C$ and $\ell_b'$ . Defined $\ell_c,\ell_c',P_C,Q_C$ analogously. If $R$ is the intersection of $P_BQ_B$ and $P_CQ_C$ , prove that $RB=RC$ .

geometry

incenter

circumcircle

APMO 2016: one-way flights between cities

by shinichiman, May 16, 2016, 8:52 PM

The country Dreamland consists of $2016$ cities. The airline Starways wants to establish some one-way flights between pairs of cities in such a way that each city has exactly one flight out of it. Find the smallest positive integer $k$ such that no matter how Starways establishes its flights, the cities can always be partitioned into $k$ groups so that from any city it is not possible to reach another city in the same group by using at most $28$ flights.

Warut Suksompong, Thailand

This post has been edited 1 time. Last edited by MellowMelon, May 18, 2017, 3:31 AM
Reason: add proposer

combinatorics

APMO

Circles intersecting each other

by rkm0959, Mar 22, 2015, 5:25 AM

There are $2015$ distinct circles in a plane, with radius $1$ .
Prove that you can select $27$ circles, which form a set $C$ , which satisfy the following.

For two arbitrary circles in $C$ , they intersect with each other or
For two arbitrary circles in $C$ , they don't intersect with each other.

This post has been edited 1 time. Last edited by rkm0959, Jun 6, 2015, 2:14 PM

combinatorics

combinatorial geometry

extremal principle

Submit

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math_exploration