Cheesy's math casino and probability

by pithon_with_an_i, May 14, 2025, 1:33 PM

There are $p$ people are playing a game at Cheesy's math casino, where $p$ is a prime number. Let $n$ be a positive integer. A subset of length $s$ from the set of integers from $1$ to $n$ inclusive is randomly chosen, with an equal probability ( $s \leq n$ and is fixed). The winner of Cheesy's game is person $i$ , if the sum of the chosen numbers are congruent to $i \pmod p$ for $0 \leq i \leq p-1$ .
For each $n$ , find all values of $s$ such that no person will sue Cheesy for creating unfair games (i.e. all the winning outcomes are equally likely).

(Proposed by Jaydon Chieng, Yeoh Teck En)

Remark

combinatorics

probability

Weird n-variable extremum problem

by pithon_with_an_i, May 14, 2025, 1:26 PM

Let $n$ be a positive integer greater or equal to $2$ and let $a_1$ , $a_2$ , ..., $a_n$ be a sequence of non-negative real numbers. Find the maximum value of $3(a_1 + a_2 + \cdots + a_n) - (a_1^2 + a_2^2 + \cdots + a_n^2) - a_1a_2 \cdots a_n$ in terms of $n$ .

(Proposed by Cheng You Seng)

This post has been edited 1 time. Last edited by pithon_with_an_i, an hour ago
Reason: Typo

algebra

inequalities

Three lines meet at one point

by TUAN2k8, May 14, 2025, 1:01 PM

Let $ABC$ be an acute triangle incribed in a circle $\omega$ .Let $M$ be the midpoint of $BC$ .Let $AD,BE$ and $CF$ be altitudes from $A,B$ and $C$ of triangle $ABC$ , respectively, and let them intersect at $H$ .Let $K$ be the intersection point of tangents to the circle $\omega$ at points $B,C$ .Prove that $MH,KD$ and $EF$ are concurrent.

geometry

Triangle

2025 IMO TEAMS

by Oksutok, May 14, 2025, 12:52 PM

Good Luck in Sunshine Coast, Australia

IMO

Problem 7

by SlovEcience, May 14, 2025, 11:03 AM

Consider the sequence $(u_n)$ defined by $u_0 = 5$ and
$u_{n+1} = \frac{1}{2}u_n^2 - 4 \quad \text{for all } n \in \mathbb{N}.$ a) Prove that there exist infinitely many positive integers $n$ such that $u_n > 2020n$ .

b) Compute
$\lim_{n \to \infty} \frac{2u_{n+1}}{u_0u_1\cdots u_n}.$

arithmetic sequence

algebra

polonomials

by Ducksohappi, May 8, 2025, 8:36 AM

$P\in \mathbb{R}[x]$ with even-degree
Prove that there is a non-negative integer k such that
$Q_k(x)=P(x)+P(x+1)+...+P(x+k)$
has no real root

algebra

Partitioning coprime integers to arithmetic sequences

by sevket12, Feb 8, 2025, 12:33 PM

For a positive integer $n$ , let $S_n$ be the set of positive integers that do not exceed $n$ and are coprime to $n$ . Define $f(n)$ as the smallest positive integer that allows $S_n$ to be partitioned into $f(n)$ disjoint subsets, each forming an arithmetic progression.

Prove that there exist infinitely many pairs $(a, b)$ satisfying $a, b > 2025$ , $a \mid b$ , and $f(a) \nmid f(b)$ .

This post has been edited 1 time. Last edited by sevket12, Feb 8, 2025, 12:39 PM

number theory

TST

Combinatorics Problem

by P.J, Dec 28, 2024, 10:16 AM

Calculate the sum of 1 x 1000 + 2 x 999 + ... + 999 x 2 + 1000 x 1

combinatorics

Inequality with a^2 + b^2 + c^2 + abc = 4

by Nguyenhuyen_AG, Oct 1, 2020, 10:00 AM

Let $a,\,b,\,c$ positive real numbers such that $a^2+b^2+c^2+abc=4.$ Prove that
$\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+(k+5)(a+b+c) \geqslant 3(k+6),$ for all $0 \leqslant k \leqslant k_0 = \frac{3\big(\sqrt[3]{2}+\sqrt[3]{4}\big)-7}{2}.$
hide

inequalities

geometric inequality

Coaxal Circles

by fattypiggy123, Mar 13, 2017, 1:07 AM

Let $ABCD$ be a quadrilateral and let $l$ be a line. Let $l$ intersect the lines $AB,CD,BC,DA,AC,BD$ at points $X,X',Y,Y',Z,Z'$ respectively. Given that these six points on $l$ are in the order $X,Y,Z,X',Y',Z'$ , show that the circles with diameter $XX',YY',ZZ'$ are coaxal.