inequality

by SunnyEvan, Jun 6, 2025, 6:53 AM

Let $a,b > 0 ,$ such that : $a+b \geq \frac{3(a^4+b^4)}{a^2+b^2+1}\sqrt{\frac{\frac{1}{a}+\frac{1}{b}}{a+b}}.$
Prove that: $\frac{a^2+b^2+2}{a^6b^2+a^2b^6} \geq 2$

This post has been edited 1 time. Last edited by SunnyEvan, an hour ago

inequalities

A beautiful Lemoine point problem

by phonghatemath, Jun 6, 2025, 5:01 AM

Given triangle $ABC$ inscribed in a circle with center $O$ . $P$ is any point not on (O). $AP, BP, CP$ intersect $(O)$ at $A', B', C'$ . Let $L, L'$ be the Lemoine points of triangle $ABC, A'B'C'$ respectively. Prove that $P, L, L'$ are collinear.

geometry

inequality

by SunnyEvan, Jun 5, 2025, 1:51 PM

Let $x,y \geq 0 ,$ such that : $\frac{x^2}{x^3+y}+\frac{y^2}{x+y^3} \geq 1 .$
Prove that : $x^2+y^2-xy \leq x+y$ $(x+\frac{1}{2})^2+(x+\frac{1}{2})^2 \leq \frac{5}{2}$ $(x+1)^2+(y+1)^2 \leq 5$ $(x+2)^2+(y+2)^2 \leq 13$

This post has been edited 2 times. Last edited by SunnyEvan, Yesterday at 2:48 PM

inequalities

k colorings and triangles

by Rijul saini, Jun 4, 2025, 7:11 PM

In the city of Timbuktu, there is an orphanage. It shelters children from the new mysterious disease that causes children to explode. There are m children in the orphanage. To try to cure this disease, a mad scientist named Myla has come up with an innovative cure. She ties every child to every other child using medicinal ropes. Every child is connected to every other child using one of $k$ different ropes. She then performs a experiment that causes $3$ children, each connected to each other with the same type of rope, to be cured. Two experiments are said to be of the same type, if each of the ropes connecting the children has the same medicine imbued in it. She then unties them and lets them go back home.

We let $f(n, k)$ be the minimum m such that Myla can perform at least $n$ experiments of the same type. Prove that:

$i.$ For every $k \in \mathbb N$ there exists a $N_k \in N$ and $a_k, b_k \in \mathbb Z$ such that for all $n > N_k$ , $f(n, k) = a_kn + b_k.$
$ii.$ Find the value of $a_k$ for every $k \in \mathbb N$ .

This post has been edited 1 time. Last edited by Rijul saini, Wednesday at 7:12 PM

graph theory

combinatorics

Write down sum or product of two numbers

by Rijul saini, Jun 4, 2025, 6:56 PM

Suppose Alice's grimoire has the number $1$ written on the first page and $n$ empty pages. Suppose in each of the next $n$ seconds, Alice can flip to the next page, and write down the sum or product of two numbers (possibly the same) which are already written in her grimoire.

Let $F(n)$ be the largest possible number such that for any $k < F(n)$ , Alice can write down the number $k$ on the last page of her grimoire. Prove that there exists a positive integer $N$ such that for all $n>N$ , we have that $n^{0.99n}\leqslant F(n)\leqslant n^{1.01n}.$
Proposed by Rohan Goyal and Pranjal Srivastava

This post has been edited 1 time. Last edited by Rijul saini, Wednesday at 7:26 PM

combinatorics

A scalene triangle and nine point circle

by ariopro1387, May 27, 2025, 10:24 AM

In a scalene triangle $ABC$ , points $Y$ and $X$ lie on $AC$ and $BC$ respectively such that $BC \perp XY$ . Points $Z$ and $T$ are the reflections of $X$ and $Y$ with respect to the midpoints of sides $BC$ and $AC$ , respectively. Point $P$ lies on segment $ZT$ such that the circumcenter of triangle $XZP$ coincides with the circumcenter of triangle $ABC$ .
Prove that the nine-point circle of triangle $ABC$ passes through the midpoint of segment $XP$ .

Iran TST Starter

by M11100111001Y1R, May 27, 2025, 7:36 AM

Let $a_n$ be a sequence of positive real numbers such that for every $n > 2025$ , we have:
$a_n = \max_{1 \leq i \leq 2025} a_{n-i} - \min_{1 \leq i \leq 2025} a_{n-i}$ Prove that there exists a natural number $M$ such that for all $n > M$ , the following holds:
$a_n < \frac{1}{1404}$

Sequence

ISI UGB 2025 P8

by SomeonecoolLovesMaths, May 11, 2025, 11:20 AM

Let $n \geq 2$ and let $a_1 \leq a_2 \leq \cdots \leq a_n$ be positive integers such that $\sum_{i=1}^{n} a_i = \prod_{i=1}^{n} a_i$ . Prove that $\sum_{i=1}^{n} a_i \leq 2n$ and determine when equality holds.

number theory

gcd (a^n+b,b^n+a) is constant

by EthanWYX2009, Jul 16, 2024, 1:17 PM

Determine all pairs $(a,b)$ of positive integers for which there exist positive integers $g$ and $N$ such that
$\gcd (a^n+b,b^n+a)=g$ holds for all integers $n\geqslant N.$ (Note that $\gcd(x, y)$ denotes the greatest common divisor of integers $x$ and $y.$ )

Proposed by Valentio Iverson, Indonesia

This post has been edited 5 times. Last edited by EthanWYX2009, Jul 19, 2024, 5:26 AM

number theory

IMO

Bushy and Jumpy and the unhappy walnut reordering

by popcorn1, Jul 20, 2021, 8:39 PM

Two squirrels, Bushy and Jumpy, have collected 2021 walnuts for the winter. Jumpy numbers the walnuts from 1 through 2021, and digs 2021 little holes in a circular pattern in the ground around their favourite tree. The next morning Jumpy notices that Bushy had placed one walnut into each hole, but had paid no attention to the numbering. Unhappy, Jumpy decides to reorder the walnuts by performing a sequence of 2021 moves. In the $k$ -th move, Jumpy swaps the positions of the two walnuts adjacent to walnut $k$ .

Prove that there exists a value of $k$ such that, on the $k$ -th move, Jumpy swaps some walnuts $a$ and $b$ such that $a<k<b$ .