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k a July Highlights and 2025 AoPS Online Class Information
jwelsh   0
Jul 1, 2025
We are halfway through summer, so be sure to carve out some time to keep your skills sharp and explore challenging topics at AoPS Online and our AoPS Academies (including the Virtual Campus)!

[list][*]Over 60 summer classes are starting at the Virtual Campus on July 7th - check out the math and language arts options for middle through high school levels.
[*]At AoPS Online, we have accelerated sections where you can complete a course in half the time by meeting twice/week instead of once/week, starting on July 8th:
[list][*]MATHCOUNTS/AMC 8 Basics
[*]MATHCOUNTS/AMC 8 Advanced
[*]AMC Problem Series[/list]
[*]Plus, AoPS Online has a special seminar July 14 - 17 that is outside the standard fare: Paradoxes and Infinity
[*]We are expanding our in-person AoPS Academy locations - are you looking for a strong community of problem solvers, exemplary instruction, and math and language arts options? Look to see if we have a location near you and enroll in summer camps or academic year classes today! New locations include campuses in California, Georgia, New York, Illinois, and Oregon and more coming soon![/list]

MOP (Math Olympiad Summer Program) just ended and the IMO (International Mathematical Olympiad) is right around the corner! This year’s IMO will be held in Australia, July 10th - 20th. Congratulations to all the MOP students for reaching this incredible level and best of luck to all selected to represent their countries at this year’s IMO! Did you know that, in the last 10 years, 59 USA International Math Olympiad team members have medaled and have taken over 360 AoPS Online courses. Take advantage of our Worldwide Online Olympiad Training (WOOT) courses
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0 replies
jwelsh
Jul 1, 2025
0 replies
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Probability that x/y is odd
Stear14   2
N 3 hours ago by Stear14
Part A. $\ $ Let $\ x,y\in U[0,1]$. $\ $ Find the probability that the nearest integer to the ratio $\ x/y\ $ is odd.

Part B. $\ $ Let $\ x,y\in N(0,1)$. $\ $ Find the probability that the nearest integer to the ratio $\ x/y\ $ is odd.

In both cases, give the answers not as infinite series, but in terms of elementary functions and known constants.
2 replies
Stear14
Yesterday at 5:41 AM
Stear14
3 hours ago
J
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Another Interesting ineqaulity
BomboaneMentos316   0
4 hours ago
Prove that for any x,y,z real and positive the following is true:

\begin{align*} \frac{x^3}{y^2 + y z + z^2} \;+\; \frac{y^3}{2 z^2 + y z x} \;+\; \frac{z^3}{x^2 + x y + y^2} \;\ge\; \frac{x + y + z}{3}. \end{align*}
0 replies
BomboaneMentos316
4 hours ago
0 replies
J
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Original Question #8
Siopao_Enjoyer   4
N 4 hours ago by P0tat0b0y
Let $x$, $y$, $z$ be real numbers such that:
\[ \begin{cases} x+y+z&=1\\ x^2+y^2+z^2&=12 \\ x^3+y^3+z^3&=18 \end{cases} \]Find the value of $xyz$.

Answer Confirmation
4 replies
Siopao_Enjoyer
Yesterday at 11:21 PM
P0tat0b0y
4 hours ago
J
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Korea csat problem, so-called “Killer problem”..
darrime627   0
5 hours ago
Find the values of \( a \) and \( b \) such that the function \( f(x) \), which has a second derivative for all \( x \in \mathbb{R} \), satisfies the following conditions:

\[ (\gamma) \quad [f(x)]^5 + [f(x)]^3 + ax + b = \ln \left( x^2 + x + \frac{5}{2} \right) \]
\[ (\delta) \quad f(-3) f(3) < 0, \quad f'(2) > 0 \]
0 replies
darrime627
5 hours ago
0 replies
J
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a cute combinatorics (?) problem
pzzd   6
N 5 hours ago by pzzd
here’s a cute little problem that can be solved with binomial coefficients, but is also related to some very common sequences in mathematics :3

say you have a $2$-inch-wide rectangle of some length $l$, and a bunch of $2$x$1$ dominos. how many different ways can you completely cover the rectangle with dominos? you can place the dominos horizontally or vertically - for example, for a $2$-by-$3$ rectangle, a valid arrangement of dominos is $1$ vertical domino on the left and $2$ horizontal dominos on the right.

hope you find this interesting!
6 replies
pzzd
Yesterday at 2:48 PM
pzzd
5 hours ago
J
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Nice recurrence finding remainder
Kyj9981   1
N 5 hours ago by Kyj9981
Source: PMO22 Areas Part II.2

Let $a_1, a_2, \dots$ be a sequence of integers defined by $a_1 = 3$, $a_2 = 3$, and $a_{n+2} = a_{n+1}a_n - a_{n+1} - a_n + 2$ for all $n \geq 1$. Find the remainder when $a_{2020}$ is divided by $22$.
1 reply
Kyj9981
6 hours ago
Kyj9981
5 hours ago
J
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Weird parity (idk maybe) problem
Ro.Is.Te.   0
5 hours ago
Given the equation:
$\frac{1}{x - y - z} = \frac{1}{y} + \frac{1}{z}$
How many ordered triples $(x,y,z)$ are either prime numbers or the negatives of prime numbers?
0 replies
Ro.Is.Te.
5 hours ago
0 replies
J
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Inequality
Martin.s   5
N 6 hours ago by solyaris


For \( n = 2, 3, \dots \), the following inequalities hold:

\[ -\frac{1}{3} \leq \frac{\sin(n\theta)}{n \sin \theta} \leq \frac{\sqrt{6}}{9} \quad \text{for } \frac{\pi}{n} \leq \theta \leq \pi - \frac{\pi}{n}, \]
and

\[ -\frac{1}{3} \leq \frac{\sin(n\theta)}{n \sin \theta} \leq \frac{1}{5} \quad \text{for } \frac{\pi}{n} \leq \theta \leq \frac{\pi}{2}. \]
5 replies
Martin.s
Jun 23, 2025
solyaris
6 hours ago
J
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Sum of recurrence
Kyj9981   1
N Today at 12:46 PM by Kyj9981
source: Sipnayan SHS Elims 2018/V1

Let $s_0=6$, $s_1=6$, and $s_n=2s_{n-1}+8s_{n-2}$ for $n \geq 2$. Define
\[A_n=\sum_{i=0}^n s_{i}\]Find $A_{2018}$. Express your answer in the form $a^b+c^d$, where $a$, $b$, $c$, and $d$ are positive integers.
1 reply
Kyj9981
Today at 12:34 PM
Kyj9981
Today at 12:46 PM
J
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Quadratic surface and its tangent plane
RainbowNeos   0
Today at 12:18 PM
Given a n*n symmetric real matrix $A$ with full rank. Suppose that $A$ has at least two positive eigenvalues and at least one negative eigenvalue. Show that for all $x\in\mathbb{R}^{n*1}$ such that $x^T A x=1$, there exists $y\neq x$ such that $y^T A x = y^T A y = 1$.
0 replies
RainbowNeos
Today at 12:18 PM
0 replies
J
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positive derivative at local max
tobiSALT   1
N Today at 12:15 PM by Mathzeus1024
Source: CIMA Math Olympiad 2023 P3
Let $f : [0, 1] \to \mathbb{R}$ be a function with continuous derivative such that $f(0) = 0$ and $f(1) = 1$. Show that there exists a real number $t$ such that $f'(t) > 0$ and $f(t) > f(s)$ for all $s$ such that $0 \le s < t$.
1 reply
tobiSALT
Nov 18, 2024
Mathzeus1024
Today at 12:15 PM
J
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[PMO27 Areas] I.13 are you sure
BinariouslyRandom   4
N Today at 11:19 AM by Kyj9981
The sequence of real numbers $x_1, x_2, \dots$, satisfies the recurrence relation
\[ \frac{x_{n+1}}{x_n} = \frac{(x_{n+1})^2 + 27}{x_n^2 + 27} \]for all positive integers $n$. Suppose that $x_{20} = x_{25} = 3$. Let $M$ be the maximum value of
\[ \sum_{n=1}^{2025} x_n. \]What is $M \pmod{1000}$?
4 replies
BinariouslyRandom
Jan 25, 2025
Kyj9981
Today at 11:19 AM
J
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Challenge: Make as many positive integers from 2 zeros
Biglion   25
N Today at 11:05 AM by littleduckysteve
How many positive integers can you make from at most 2 zeros, any math operation and cocatination?
New Rule: The successor function can only be used at most 3 times per number
Starting from 0, 0=0
25 replies
Biglion
Jul 2, 2025
littleduckysteve
Today at 11:05 AM
J
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[PMO26 Qualifying II.12] Equality
kae_3   5
N Today at 10:18 AM by fruitmonster97
The real numbers $x,y$ are such that $x\neq y$ and \[\frac{x}{26-x^2}=\frac{y}{26-y^2}=\frac{xy}{26-(xy)^2}.\]What is $x^2+y^2$?

$\text{(a) }626\qquad\text{(b) }650\qquad\text{(c) }677\qquad\text{(d) }729$

Answer Confirmation
5 replies
kae_3
Feb 21, 2025
fruitmonster97
Today at 10:18 AM
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J
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Investigating functions
mikejoe   1
N May 14, 2025 by Mathzeus1024
Source: Edwards and Penney
Investigate the function $f(x) = (x-2) \sqrt{x+1}$
Also determine its domain and range.
1 reply
mikejoe
Nov 2, 2012
Mathzeus1024
May 14, 2025
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Investigating functions
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Source: Edwards and Penney
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mikejoe
8 posts
mikejoe
#1 h  Nov 2, 2012, 8:02 PM • 2 Y
Y by Adventure10, Mango247
Investigate the function $f(x) = (x-2) \sqrt{x+1}$
Also determine its domain and range.
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Mathzeus1024
1058 posts
Mathzeus1024
#2 h  May 14, 2025, 1:08 PM
Y by
The domain is clearly $x \in [-1, \infty)$ due to the function's radicand. As for the range, taking the derivative of $f$ equal to zero yields the critical value:

$f'(x) = \sqrt{x+1}+\frac{x-2}{2\sqrt{x+1}}=0 \Rightarrow 2(x+1) = 2-x \Rightarrow x = 0$;

and second derivative check of $f$ against this critical value yields:

$f''(x) = \frac{1}{2\sqrt{x+1}} + \frac{x+4}{4(x+1)^{3/2}} \Rightarrow f''(0) = \frac{3}{2} > 0$ (a global minimum).

Thus, the range of $f$ is $[-2,\infty)$.
This post has been edited 1 time. Last edited by Mathzeus1024, May 14, 2025, 1:11 PM
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