Olympiad

by sasu1ke, Apr 5, 2025, 11:52 PM

https://preview.redd.it/9qhdk855w1te1.jpeg?width=1080&crop=smart&auto=webp&s=6f407c03de2b3ae93ac444929b20d75a80f2259b

How to judge a number is prime or not?

by mingzhehu, Apr 5, 2025, 2:45 PM

A=（10X1+1）(10X+1），X1，X∈N+
B=（10 X1+3）（10X+7），X∈N，X1∈N
C=(10 X1+9)(10X+9), X∈N，X1∈N
D=(10 X1+1)(10X+3), X1∈N+，X∈N
E=（10 X1+7）(10X+9），X∈N，X1∈N
F=（10 X1+1）（10X+7），X1∈N+，X∈N
G=（10 X1+3）（10X+9），X∈N，X1∈N
H=（10 X1+1）10X+9），X1∈N+，X∈N
I=（10 X1+3）（10X+3），X1∈N，X∈N
J=( 10X1+7)(10X+7),X∈N，X1∈N

For any natural number P∈{P=10N+1，n∈N}，make P=A or B or C
If P can make the roots of function group(ABC) without any root group completely made up of integer, P will be a prime
For any natural number P∈{P=10N+3，n∈N}，make P=D or E
If P can make the roots of function group(DE) without any root group completely made up
of integer, P will be a prime
For any natural number P∈{P=10N+7，n∈N}，make P=F or G
If P can make the roots of function group(FG) without any root group completely made up
of integer, P will be a prime
For any natural number P∈{P=10N+9，n∈N}，make P=H or I or J
If P can make the roots of function group(GIJ) without any root group completely made up
of integer, P will be a prime

inequality

by revol_ufiaw, Apr 5, 2025, 2:05 PM

Prove that that for any real $x \ge 0$ and natural number $n$ ,
$x^n (n+1)^{n+1} \le n^n (x+1)^{n+1}.$

inequalities

Easy Inequalities

algebra

Inequalities

by sqing, Apr 5, 2025, 1:10 PM

Let $a,b$ be real numbers such that $a^2+b^2+a^3 +b^3=4 .$ Prove that
$a+b \leq 2$ Let $a,b$ be real numbers such that $a+b + a^2+b^2+a^3 +b^3=6 .$ Prove that
$a+b \leq 2$

This post has been edited 1 time. Last edited by sqing, Yesterday at 1:19 PM

inequalities

Distance vs time swimming problem

by smalkaram_3549, Apr 5, 2025, 2:57 AM

How should I approach a problem where we deal with velocities becoming negative and stuff. I know that they both travel 3 Lengths of the pool before meeting a second time.

Attachments:

rate problems

.problem.

by Cobedangiu, Apr 4, 2025, 6:20 AM

Find the integer coefficients after expanding Newton's binomial:
$(\frac{3}{2}-\frac{2}{3}x^2)^n (n \in Z)$

This post has been edited 1 time. Last edited by Cobedangiu, Apr 4, 2025, 6:20 AM

algebra

Inequalities

by sqing, Apr 4, 2025, 3:52 AM

Let $a,b,c> 0$ and $\frac{a}{a^2+ab+c}+\frac{b}{b^2+bc+a}+\frac{c}{c^2+ca+b} \geq 1$ . Prove that
$a+b+c\leq 3$

inequalities

Any nice way to do this?

by NamelyOrange, Apr 2, 2025, 1:11 PM

Source: Taichung P.S.1 math program tryouts

How many ordered pairs $(a,b,c)\in\mathbb{N}^3$ are there such that $c=ab$ and $1\le a\le b\le c\le60$ ?

number theory

that statement is true

by pennypc123456789, Mar 23, 2025, 9:27 AM

we have $a^3+b^3 = 2$ and $3(a^4+b^4)+2a^4b^4 \le 8$ , then we can deduce $a^2+b^2$ \le 2 $ ?

This post has been edited 1 time. Last edited by pennypc123456789, Mar 23, 2025, 10:10 AM

inqualities

What is an isogonal conjugate and why is it useful?

by EaZ_Shadow, Dec 28, 2024, 6:08 PM

What is an isogonal conjugate and why is it useful? People use them in Olympiad geometry proofs but I don’t understand why and what is the purpose, as it complicates me because of me not understanding it.

geometry