Inspired by Kosovo 2010

by sqing, May 9, 2025, 3:56 AM

Let $a,b>0 , a+b\leq k$ . Prove that
$\left(1+\frac{1}{a(b+1)}\right)\left(1+\frac{1}{b(a+1)}\right)\geq\left(1+\frac{4}{k(k+2)}\right)^2$ $\left(1+\frac {a}{b(a+1)}\right)\left(1+\frac {b}{a(b+1)}\right) \geq\left(1+\frac{2}{k+2}\right)^2$ Let $a,b>0 , a+b\leq 2$ . Prove that
$\left(1+\frac{1}{a(b+1)}\right)\left(1+\frac{1}{b(a+1)}\right)\geq \frac{9}{4}$ $\left(1+\frac {a}{b(a+1)}\right)\left(1+\frac {b}{a(b+1)}\right) \geq \frac{9}{4}$

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

inequalities proposed

algebra

inequalities

Geometry Parallel Proof Problem

by CatalanThinker, May 9, 2025, 3:33 AM

Let M be the midpoint of the side BC of triangle ABC. The bisector of the exterior angle of point A intersects the side BC in D. Let the circumcircle of triangle ADM intersect the lines AB and AC in E and F respectively. If the midpoint of EF is N, prove that MN || AD.
I have done some constructions, but still did not quite get to the answer, see diagram attached below

Attachments:

geometry

Interesting inequalities

by sqing, May 9, 2025, 3:02 AM

Let $a,b >0$ and $a^2-ab+b^2\leq 1$ . Prove that
$a^4 +b^4+\frac{a }{b +1}+ \frac{b }{a +1} \leq 3$ $a^3 +b^3+\frac{a^2}{b^2+1}+ \frac{b^2}{a^2+1} \leq 3$ $a^4 +b^4-\frac{a}{b+1}-\frac{b}{a+1} \leq 1$ $a^4+b^4 -\frac{a^2}{b^2+1}- \frac{b^2}{a^2+1}\leq 1$ $a^3+b^3 -\frac{a^3}{b^3+1}- \frac{b^3}{a^3+1}\leq 1$

This post has been edited 1 time. Last edited by sqing, 3 hours ago

inequalities proposed

inequalities

Is this FE is solvable?

by ItzsleepyXD, May 9, 2025, 1:45 AM

Find all function $f : \mathbb{R} \to \mathbb{R}$ such that for all $x,y \in \mathbb{R}$ . $f(x+f(y))+f(x+y)=2x+f(y)+f(f(y))$ . Original

function

algebra

sus

Equilateral triangle formed by circle and Fermat point

by Mimii08, May 8, 2025, 10:36 PM

Hi! I found this interesting geometry problem and I would really appreciate help with the proof.

Let ABC be an acute triangle, and let T be the Fermat (Torricelli) point of triangle ABC. Let A1, B1, and C1 be the feet of the perpendiculars from T to the sides BC, AC, and AB, respectively. Let ω be the circle passing through points A1, B1, and C1. Let A2, B2, and C2 be the second points where ω intersects the sides BC, AC, and AB, respectively (different from A1, B1, C1).

Prove that triangle A2B2C2 is equilateral.

Hard combi

by EeEApO, May 8, 2025, 6:08 PM

In a quiz competition, there are a total of $100$ questions, each with $4$ answer choices. A participant who answers all questions correctly will receive a gift. To ensure that at least one member of my family answers all questions correctly, how many family members need to take the quiz?

Now, suppose my spouse and I move into a new home. Every year, we have twins. Starting at the age of $16$ , each of our twin children also begins to have twins every year. If this pattern continues, how many years will it take for my family to grow large enough to have the required number of members to guarantee winning the quiz gift?

combinatorics

Find all functions $f$ is strictly increasing : $\mathbb{R^+}$ $\rightarrow$

by guramuta, May 8, 2025, 1:45 PM

Find all functions $f$ is strictly increasing : $\mathbb{R^+}$ $\rightarrow$ $\mathbb{R^+}$ such that:
i) $f(2x)$ $\geq$ $2f(x)$
ii) $f(f(x)f(y)+x) = f(xf(y)) + f(x)$

Gheorghe Țițeica 2025 Grade 9 P2

by AndreiVila, Mar 28, 2025, 9:09 PM

Let $a,b,c$ be three positive real numbers with $ab+bc+ca=4$ . Find the minimum value of the expression $E(a,b,c)=\frac{a^2+b^2}{ab}+\frac{b^2+c^2}{bc}+\frac{c^2+a^2}{ca}-(a-b)^2.$