Inspired by JK1603JK

by sqing, Apr 4, 2025, 3:31 AM

Let $ a,b,c\geq 0 $ and $ab+bc+ca=1.$ Prove that$$\frac{abc-2}{abc-1}\ge \frac{4(a^2b+b^2c+c^2a)}{a^3b+b^3c+c^3a+1} $$$$\frac{abc-1}{abc-2}\ge \frac{(\sqrt 2-1)(a^2b+b^2c+c^2a+1)}{a^3b+b^3c+c^3a+1} $$
inequalities
L

Geometry problem-second time posting

by kjhgyuio, Apr 4, 2025, 3:18 AM

A square is cut into several rectangles, none of which is a square ,so that the sides of each rectangles are parallel to the sides of a square .For each rectangle with sides a,b,a<b compute the ratio a/b Prove that the sum of these ratios is at least 1
geometry
hard
Fractions
L

Inequality from China

by sqing, Apr 3, 2025, 1:11 PM

Let $x\in (0,\frac{\pi}{2}) . $ Prove that $$tanx\ge x^k$$Where $ k=1,2,3,4.$
This post has been edited 1 time. Last edited by sqing, Yesterday at 1:12 PM
inequalities proposed
inequalities
L

pretty well known

by dotscom26, Apr 3, 2025, 2:03 AM

Let $\triangle ABC$ be a scalene triangle such that $\Omega$ is its incircle. $AB$ is tangent to $\Omega$ at $D$. A point $E$ ($E \notin \Omega$) is located on $BC$.

Let $\omega_1$, $\omega_2$, and $\omega_3$ be the incircles of the triangles $BED$, $ADE$, and $AEC$, respectively.

Show that the common tangent to $\omega_1$ and $\omega_3$ is also tangent to $\omega_2$.
geometry
Inversion
L

Olympiad Geometry problem-second time posting

by kjhgyuio, Apr 2, 2025, 1:03 AM

In trapezium ABCD,AD is parallel to BC and points E and F are midpoints of AB and DC respectively. If
Area of AEFD/Area of EBCF =√3 + 1/3-√3 and the area of triangle ABD is √3 .find the area of trapezium ABCD
geometry
SMO
Olympiad Geometry
Olympiad
trapezium
L

A board with crosses that we color

by nAalniaOMliO, Mar 28, 2025, 8:37 PM

In some cells of the table $2025 \times 2025$ crosses are placed. A set of 2025 cells we will call balanced if no two of them are in the same row or column. It is known that any balanced set has at least $k$ crosses.
Find the minimal $k$ for which it is always possible to color crosses in two colors such that any balanced set has crosses of both colors.
This post has been edited 1 time. Last edited by nAalniaOMliO, Mar 29, 2025, 10:41 AM
combinatorics
L

Proving ZA=ZB

by nAalniaOMliO, Mar 28, 2025, 8:36 PM

Point $H$ is the foot of the altitude from $A$ of triangle $ABC$. On the lines $AB$ and $AC$ points $X$ and $Y$ are marked such that the circumcircles of triangles $BXH$ and $CYH$ are tangent, call this circles $w_B$ and $w_C$ respectively. Tangent lines to circles $w_B$ and $w_C$ at $X$ and $Y$ intersect at $Z$.
Prove that $ZA=ZH$.
Vadzim Kamianetski
This post has been edited 1 time. Last edited by nAalniaOMliO, Mar 29, 2025, 6:45 PM
geometry
L

Unsolved NT, 3rd time posting

by GreekIdiot, Mar 26, 2025, 11:40 AM

Solve $5^x-2^y=z^3$ where $x,y,z \in \mathbb Z$
Hint
This post has been edited 2 times. Last edited by GreekIdiot, Mar 26, 2025, 11:41 AM
number theory
modular arithmetic
Exponents
unsolved
L

NMO (Nepal) Problem 4

by khan.academy, Mar 17, 2024, 2:24 PM

Find all integer/s $n$ such that $\displaystyle{\frac{5^n-1}{3}}$ is a prime or a perfect square of an integer.

Proposed by Prajit Adhikari, Nepal
This post has been edited 1 time. Last edited by khan.academy, Mar 17, 2024, 2:39 PM
number theory
L

2019 Nepal National Mathematics Olympiad

by Piinfinity, Oct 13, 2020, 5:56 AM

Problem 31
Let $f:\mathbb{R}\to\mathbb{R}$ be a function such that
$f(f(x))=x^2-x+1$
for all real numbers $x$. Determine $f(0)$.
function
algebra
L

The oldest, shortest words — "yes" and "no" — are those which require the most thought.

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