maximum

260 Questions
128 Answers

-2

148

1

+835

quick question

In a right triangle, one of the angles is $45^\circ,$ and the hypotenuse is $6$ inches. Find the area of the triangle in square inche

maximum Aug 15, 2023

-2

86

2

+835

need help

Can anyone give a hint here?
Given a positive integer $n,$ prove that there exists a positive integer $m > 1000$ such that $m \equiv 7 \pmod{n}$ and $\gcd(m,n) = 1.$

maximum Aug 15, 2023

-1

157

1

+835

hard number theory congruence

does anybody have any ideas?
The prime $p$ has the property that the congruence $$n^2 \equiv -1\pmod{p}$$ has no solutions $n$. Determine the sum of all such primes $p$.

maximum Aug 14, 2023

-2

76

2

+835

help with squares

help I have question about squares
Let $a$ and $b$ be non-zero integers. If both $a^2$ and $b^2$ have at most three digits, then what is the greatest possible difference between $a^2$ and $b^2?$

maximum Aug 14, 2023

-2

96

2

+835

pythagorean theorem

The hypotenuse of right triangle $ABC$ is $AB$. The area of square $BCDE$ is $144$. If $AB$ and $AC$ are each integers less than $20$, how many possibilities are there for their lengths?

maximum Aug 14, 2023

-2

134

1

+835

functions

Help me plz
For integers $n$, let \[f(n) = \left\{
\begin{array}{cl}
n & \text{ if }n\text{ is odd}, \\
n -1 & \text{ if }n\text{ is even}.
\end{array}<#> \right.\] Find $f(f(f(f(f(4)))))$. read more ..

maximum Aug 14, 2023

-2

116

1

+835

piecewise functions

Let \[f(x) =
\begin{cases}
x^2&\text{if } x\le 3, \\
3ax &\text{if } x>3.
\end{cases}<#> \]<#> Find the value of $a$ if the graph of $y=f(x)$ is continuous (which means the graph can be drawn without lifting your read more ..

maximum Aug 14, 2023

-1

152

3

+835

plz help with digit

I need help with this
Find the number of times the digit $9$ appears in the list of all integers from $1$ to $150$. (The number $ 99 $, for example, is counted twice, because $9$ appears two times in it.)

maximum Aug 14, 2023

-1

87

1

+835

congruence

What is the smallest positive integer $n$ such that $3n \equiv 1356 \pmod{5}?$

maximum Aug 14, 2023

-1

116

0

+835

polynomial roots

Find all real numbers $a$ such that the roots of the polynomial
\[x^3 - 12x^2 + 73x + a\]
form an arithmetic progression and are not all real.

maximum Aug 13, 2023

+835

-1

Do not post AoPS homework.

maximumJan 20, 2025