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k a April Highlights and 2025 AoPS Online Class Information
jlacosta   0
Apr 2, 2025
Spring is in full swing and summer is right around the corner, what are your plans? At AoPS Online our schedule has new classes starting now through July, so be sure to keep your skills sharp and be prepared for the Fall school year! Check out the schedule of upcoming classes below.

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[list][*]April 3rd (Webinar), 4pm PT/7:00pm ET, Learning with AoPS: Perspectives from a Parent, Math Camp Instructor, and University Professor
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[*]April 22nd (Webinar), 4:00pm PT/7:00pm ET, Competitive Programming at AoPS (USACO).[/list]
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0 replies
jlacosta
Apr 2, 2025
0 replies
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Factorise (x+1)(x+2)(x+3)(x+4)-3
Idiot_of_the64squares   5
N 42 minutes ago by Idiot_of_the64squares
On expansion the expression becomes:
$ x^4+10x^3+35x^2 +50x+ 21 $
I cannot solve it further
5 replies
Idiot_of_the64squares
4 hours ago
Idiot_of_the64squares
42 minutes ago
J
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NC State Math Contest Wake Tech Regional Problems and Solutions
mathnerd_101   4
N an hour ago by Aaronjudgeisgoat
Problem 1: Determine the area enclosed by the graphs of $$y=|x-2|+|x-4|-2, y=-|x-3|+4.$$ Hint
Solution to P1

Problem 2: Calculate the sum of the real solutions to the equation $x^\frac{3}{2} -9x-16x^\frac{1}{2} +144=0.$
Hint
Solution to P2



Problem 3: List the two transformations needed to convert the graph $\frac{x-1}{x+2}$ to $\frac{3x-6}{x-1}.$
Hint
Solution to P3

Problem 4: Let $a,b$ be positive integers such that $a^2-b^2=20,$ and $a^3-b^3=120.$ Determine the value of $a+\frac{b^2}{a+b}.$
Hint
Solution for P4

Problem 5: Eve and Oscar are playing a game where they roll a fair, six-sided die. If an even number occurs on two consecutive rolls, then Eve wins. If an odd number is immediately followed by an even number, Oscar wins. The die is rolled until one person wins. What is the probability that Oscar wins?
Hint
Solution to P5

Problem 6: In triangle $ABC,$ $M$ is on point $\overline{AB}$ such that $AM = x+32$ and $MB=x+12$ and $N$ is a point on $\overline{AC}$ such that $MN=2x+1$ and $BC=x+22.$ Given that $\overline{MN} || \overline{BC},$ calculate $MN.$
Hint
Solution to P6

Problem 7: Determine the sum of the zeroes of the quadratic of polynomial $Q(x),$ given that $$Q(0)=72, Q(1) = 75, Q(3) = 63.$$
Hint

Solution to Problem 7

Problem 8:
Hint
Solution to P8

Problem 9:
Find the sum of all real solutions to $$(x-4)^{log_8(4x-16)} = 2.$$ Hint
Solution to P9

Problem 10:
Define the function
\[f(x) = \begin{cases} x - 9, & \text{if } x > 100 \\ f(f(x + 10)), & \text{if } x \leq 100 \end{cases}\]
Calculate \( f(25) \).

Hint

Solution to P10

Problem 11:
Let $a,b,x$ be real numbers such that $$log_{a-b} (a+b) = 3^{a+b}, log_{a+b} (a-b) = 125 \cdot 15^{b-a}, a^2-b^2=3^x. $$Find $x.$
Hint

Solution to P11

Problem 12: Points $A,B,C$ are on circle $Q$ such that $AC=2,$ $\angle AQC = 180^{\circ},$ and $\angle QAB = 30^{\circ}.$ Determine the path length from $A$ to $C$ formed by segment $AB$ and arc $BC.$

Hint
Solution to P12

Problem 13: Determine the number of integers $x$ such that the expression $$\frac{\sqrt{522-x}}{\sqrt{x-80}} $$is also an integer.
Hint

Solution to Problem 13

Problem 14: Determine the smallest positive integer $n$ such that $n!$ is a multiple of $2^15.$

Hint
Solution to Problem 14

Problem 15: Suppose $x$ and $y$ are real numbers such that $x^3+y^3=7,$ and $xy(x+y)=-2.$ Calculate $x-y.$
Funnily enough, I guessed this question right in contest.

Hint
Solution to Problem 15

Problem 16: A sequence of points $p_i = (x_i, y_i)$ will follow the rules such that
\[ p_1 = (0,0), \quad p_{i+1} = (x_i + 1, y_i) \text{ or } (x_i, y_i + 1), \quad p_{10} = (4,5). \]How many sequences $\{p_i\}_{i=1}^{10}$ are possible such that $p_1$ is the only point with equal coordinates?

Hint
Solution to P16

Problem 18: (Also stolen from akliu's blog post)
Calculate

$$\sum_{k=0}^{11} (\sqrt{2} \sin(\frac{\pi}{4}(1+2k)))^k$$
Hint
Solution to Problem 18

Problem 19: Determine the constant term in the expansion of $(x^3+\frac{1}{x^2})^{10}.$

Hint
Solution to P19

Problem 20:

In a magical pond there are two species of talking fish: trout, whose statements are always true, and \emph{flounder}, whose statements are always false. Six fish -- Alpha, Beta, Gamma, Delta, Epsilon, and Zeta -- live together in the pond. They make the following statements:
Alpha says, "Delta is the same kind of fish as I am.''
Beta says, "Epsilon and Zeta are different from each other.''
Gamma says, "Alpha is a flounder or Beta is a trout.''
Delta says, "The negation of Gamma's statement is true.''
Epsilon says, "I am a trout.''
Zeta says, "Beta is a flounder.''

How many of these fish are trout?

Hint
Solution to P20
SHORT ANSWER QUESTIONS:
1. Five people randomly choose a positive integer less than or equal to $10.$ The probability that at least two people choose the same number can be written as $\frac{m}{n}.$ Find $m+n.$

Hint
Solution to S1

2. Define a function $F(n)$ on the positive integers using the rule that for $n=1,$ $F(n)=0.$ For all prime $n$, $F(n) = 1,$ and for all other $n,$ $F(xy)=xF(y) + yF(x).$ Find the smallest possible value of $n$ such that $F(n) = 2n.$

Hint
Solution to S2

3. How many integers $n \le 2025$ can be written as the sum of two distinct, non-negative integer powers of $3?$
Huge shoutout to OTIS for teaching me how to solve problems like this.

Hint

Solution to S3

4. Let $S$ be the set of positive integers of $x$ such that $x^2-5y^2=1$ for some other positive integer $y.$ Find the only three-digit value of $x$ in $S.$
Hint
Solution to S4

5. Let $N$ be a positive integer and let $M$ be the integer that is formed by removing the first three digits from $N.$ Find the value of $N$ with least value such that $N = 2025M.$
Hint

Solution to S5
4 replies
mathnerd_101
Today at 11:40 AM
Aaronjudgeisgoat
an hour ago
J
H
Find all the natural numbers a and b such that: if c and d are real that
bo_ngu_toan   4
N an hour ago by imnotgoodatmathsorry
Find all the natural numbers a and b such that: if c and d are real that x²+ax+1=c and x²+bx+1=d have roots then x²+(a+b)x +1 = cd have root
4 replies
bo_ngu_toan
May 14, 2023
imnotgoodatmathsorry
an hour ago
J
H
pls help me
Leo1608   4
N 2 hours ago by imnotgoodatmathsorry
If a, b, c are integers and a-b is divisible by c then ab is divisible by $c^2$. Prove that a is divisible by c and b is divisible by c
4 replies
Leo1608
Aug 21, 2024
imnotgoodatmathsorry
2 hours ago
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Inequality
lbh_qys   1
N Apr 1, 2025 by lbh_qys
Let \(a,b,c \geq 0\) and \(5(a^2+b^2+c^2)-3(a+b+c)+2abc\leq 8\). Prove that \(a^3+b^3+c^3+2abc\leq 5\).
1 reply
lbh_qys
Mar 28, 2025
lbh_qys
Apr 1, 2025
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Inequality
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lbh_qys
#1 h  Mar 28, 2025, 9:28 AM
Y by
Let \(a,b,c \geq 0\) and \(5(a^2+b^2+c^2)-3(a+b+c)+2abc\leq 8\). Prove that \(a^3+b^3+c^3+2abc\leq 5\).
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lbh_qys
507 posts
lbh_qys
#2 h  Apr 1, 2025, 9:52 AM
Y by
Let \(a,b,c \geq 0\) and \(2(a^2+b^2+c^2)+(ab+bc+ca) \leq 3(a+b+c)\). Prove that

\[a^3+b^3+c^3+2abc\leq 5\]
Let \(a,b,c \geq 0\) and \(7(a^2+b^2+c^2)+8(ab+bc+ca) \geq 15(a+b+c)\). Prove that

\[a^3+b^3+c^3+2abc\geq 5\]
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