small problem

by sadwinter, Jun 2, 2025, 8:22 AM

Let $0\leq a,b \leq1$ . Prove that
$0\leq(a+2b)^2-4a(4b-a-3ab^2)(2a^2+b^2)\leq9$

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

Circle Midpoint Config

by Fuyuki, Jun 2, 2025, 8:18 AM

In triangle ABC, point D is the midpoint of BC. Let the second intersection of AD and (ABC) be E. Then, F is the intersection of EC and AB. G is the intersection of BE and AC. Prove that BC is parallel to FG.

geometry

A weird problem

by jayme, Jun 2, 2025, 6:52 AM

Dear Mathlinkers,

1. ABC a triangle
2. 0 the circumcircle
3. I the incenter
4. 1 a circle passing througn B and C
5. X, Y the second points of intersection of 1 wrt BI, CI
6. 2 the circumcircle of the triangle XYI
7. M, N the symetrics of B, C wrt XY.

Question : if 2 is tangent to 0 then, 2 is tangent to MN.

Sincerely
Jean-Louis

geometry

interesting geo config (2/3)

by Royal_mhyasd, May 31, 2025, 11:36 PM

Let $\triangle ABC$ be an acute triangle and $H$ its orthocenter. Let $P$ be a point on the parallel through $A$ to $BC$ such that $\angle APH = |\angle ABC-\angle ACB|$ . Define $Q$ and $R$ as points on the parallels through $B$ to $AC$ and through $C$ to $AB$ similarly. If $P,Q,R$ are positioned around the sides of $\triangle ABC$ as in the given configuration, prove that $P,Q,R$ are collinear.

Attachments:

Problem 10

by SlovEcience, May 30, 2025, 7:57 AM

Let $x, y, z$ be positive real numbers satisfying
$xy + yz + zx = 3xyz.$ Prove that
$\sqrt{\frac{x}{3y^2z^2 + xyz}} + \sqrt{\frac{y}{3x^2z^2 + xyz}} + \sqrt{\frac{z}{3x^2y^2 + xyz}} \le \frac{3}{2}.$

inequalities

algebra

greatest volume

by hzbrl, May 8, 2025, 9:56 AM

A large sphere with radius 7 contains three smaller balls each with radius 3 . The three balls are each externally tangent to the other two balls and internally tangent to the large sphere. There are four right circular cones that can be inscribed in the large sphere in such a way that the bases of the cones are tangent to all three balls. Of these four cones, the one with the greatest volume has volume $n \pi$ . Find $n$ .

geometry algebra

Purple Comet

algebra

NT game with products

by Kimchiks926, Nov 12, 2022, 4:50 PM

Ingrid and Erik are playing a game. For a given odd prime $p$ , the numbers $1, 2, 3, ..., p-1$ are written on a blackboard. The players take turns making moves with Ingrid starting. A move consists of one of the players crossing out a number on the board that has not yet been crossed out. If the product of all currently crossed out numbers is $1 \pmod p$ after the move, the player whose move it was receives one point, otherwise, zero points are awarded. The game ends after all numbers have been crossed out.

The player who has received the most points by the end of the game wins. If both players have the same score, the game ends in a draw. For each $p$ , determine which player (if any) has a winning strategy

This post has been edited 2 times. Last edited by Kimchiks926, Nov 12, 2022, 4:51 PM
Reason: typo

modular arithmetic

number theory

set with c+2a>3b

by VicKmath7, Jul 12, 2022, 12:55 PM

Let $n$ be a positive integer. Given is a subset $A$ of $\{0,1,...,5^n\}$ with $4n+2$ elements. Prove that there exist three elements $a<b<c$ from $A$ such that $c+2a>3b$ .

Proposed by Dominik Burek and Tomasz Ciesla, Poland

This post has been edited 3 times. Last edited by VicKmath7, Dec 22, 2022, 4:09 PM

Problem 61: Poland MO Finals 2003, Problem 3

by henderson, Jan 30, 2017, 8:04 PM

$\color{red}\bf{Problem \ 61}$ Find all polynomials $W$ with integer coefficients satisfying the following condition: For every natural number $n, 2^n - 1$ is divisible by $W(n).$

This post has been edited 1 time. Last edited by henderson, Jan 30, 2017, 8:05 PM

number theory

Justin Stevens

Poland MO

Problem 57: Prime factorization

by henderson, Jan 4, 2017, 2:32 PM

${\color{red}\bf{Problem \ 57}}$ Prove that for every nonnegative integer $n,$ the number $7^{7^{n}}+1$ is the product of at least $2n+3$ (not necessarily distinct) primes.
(USAMO 2007)
${\color{red}\bf{Solution}}$ Let's apply induction.

$\boxed{\color{blue}{1}}$ The base case is $n=0,$ which is $8=2^3,$ so it has $2n+3$ prime factors.

$\boxed{\color{blue}{2}}$ Now, assume that $7^{7^{n}}+1$ is the product of $2n+3$ primes.

$\boxed{\color{blue}{3}}$ We wish to prove that $7^{7^{n+1}}+1$ is the product of $2(n+1)+3=2n+5$ primes.
Let $x = 7^{7^{n}}.$ $\quad$ $x+1$ is the product of $2n+3$ primes, so we want to show that $x^7+1$ is the product of $2n+5$ primes.

Note that $x^7+1 = (x+1)(x^6-x^5+x^4-x^3+x^2-x+1).$ By the inductive hypothesis, we know that $x+1$ is the product of $2n+3$ primes, so it suffices to show that $x^6-x^5+x^4-x^3+x^2-x+1$ is composite.

Since $x^6-x^5+x^4-x^3+x^2-x+1=(x+1)^6 - 7x(x^2+x+1)^2$ and $x=7^{7^{n}}$ , $x^6-x^5+x^4-x^3+x^2-x+1$ is the difference of two squares, therefore is composite.

Thus, $7^{7^{n+1}}+1$ is the product of at least $(2n+3)+2=2n+5$ primes, and we are done. $\square$

This post has been edited 4 times. Last edited by henderson, Jan 12, 2017, 11:11 AM

number theory

prime numbers

USAMO

Problem 56: Brazil MO 2013, Day 1, Problem 2

by henderson, Dec 16, 2016, 3:48 PM

${\color{red}\bf{Problem \ 56}}$ Arnaldo and Bernaldo play the following game: given a fixed finite set of positive integers $A$ known by both players, Arnaldo picks a number $a \in A$ but doesn't tell it to anyone. Bernaldo thens pick an arbitrary positive integer $b$ (not necessarily in $A$ ). Then Arnaldo tells the number of divisors of $ab$ . Show that Bernaldo can choose $b$ in a way that he can find out the number $a$ chosen by Arnaldo.
(Brazil MO 2013)
${\color{red}\bf{Solution}}$ Let us say the primes that divide at least one element from $A$ are $p_0,p_1,\ldots,p_k$ . An element $a\in A$ can be represented then as $a=\prod_{j=0}^k p_j^{\alpha_j}$ , with $\alpha_j \geq 0$ . When $b=\prod_{j=0}^k p_j^{\beta_j}$ , the number of divisors of $ab$ is $\tau(ab) = \prod_{j=0}^k (1+ \alpha_j + \beta_j)$ . Let us plug in $\beta_j = x^{2^j}$ ; then $P(x) = \prod_{j=0}^k (1+\alpha_j + x^{2^j})$ is a polynomial in $x$ of degree $2^{k+1} - 1$ , where the coefficient of $x^{2^{k+1} - 2^j - 1}$ is precisely $1+\alpha_j$ . In fact, if we take $n > \prod_{j=0}^k (1+\alpha_j)$ , then $P(n)$ is the writing in basis $n$ of some (huge) integer, and all "digits" can be determined, namely also the values $\alpha_j$ . So all is left to do is to take $n > \prod_{j=0}^k (1+a_j)$ , where $a_j = \max_{A} \alpha_j$ , and $\beta_j = n^{2^j}$ .

This post has been edited 3 times. Last edited by henderson, Jan 4, 2017, 2:42 PM

number theory

Combinatorial Number Theory

combinatorics

IMO ShortList 2001, combinatorics problem 4

by orl, Sep 30, 2004, 5:26 PM

A set of three nonnegative integers $\{x,y,z\}$ with $x < y < z$ is called historic if $\{z-y,y-x\} = \{1776,2001\}$ . Show that the set of all nonnegative integers can be written as the union of pairwise disjoint historic sets.

Attachments:

This post has been edited 2 times. Last edited by orl, Oct 25, 2004, 12:12 AM

combinatorics

Combinatorial Number Theory

IMO ShortList 2003, combinatorics problem 4

by darij grinberg, May 17, 2004, 4:19 PM

Let $x_1,\ldots, x_n$ and $y_1,\ldots, y_n$ be real numbers. Let $A = (a_{ij})_{1\leq i,j\leq n}$ be the matrix with entries $a_{ij} = \begin{cases}1,&\text{if }x_i + y_j\geq 0;\\0,&\text{if }x_i + y_j < 0.\end{cases}$ Suppose that $B$ is an $n\times n$ matrix with entries $0$ , $1$ such that the sum of the elements in each row and each column of $B$ is equal to the corresponding sum for the matrix $A$ . Prove that $A=B$ .