Plan ahead for the next school year. Schedule your class today!

G
Topic
First Poster
Last Poster
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
and train with the best! Please note that early bird pricing ends August 19th!
Are you tired of the heat and thinking about Fall? You can plan your Fall schedule now with classes at either AoPS Online, AoPS Academy Virtual Campus, or one of our AoPS Academies around the US.

Our full course list for upcoming classes is below:
All classes start 7:30pm ET/4:30pm PT unless otherwise noted.

Introductory: Grades 5-10

Prealgebra 1 Self-Paced

Prealgebra 1
Wednesday, Jul 16 - Oct 29
Sunday, Aug 17 - Dec 14
Tuesday, Aug 26 - Dec 16
Friday, Sep 5 - Jan 16
Monday, Sep 8 - Jan 12
Tuesday, Sep 16 - Jan 20 (4:30 - 5:45 pm ET/1:30 - 2:45 pm PT)
Sunday, Sep 21 - Jan 25
Thursday, Sep 25 - Jan 29
Wednesday, Oct 22 - Feb 25
Tuesday, Nov 4 - Mar 10
Friday, Dec 12 - Apr 10

Prealgebra 2 Self-Paced

Prealgebra 2
Friday, Jul 25 - Nov 21
Sunday, Aug 17 - Dec 14
Tuesday, Sep 9 - Jan 13
Thursday, Sep 25 - Jan 29
Sunday, Oct 19 - Feb 22
Monday, Oct 27 - Mar 2
Wednesday, Nov 12 - Mar 18

Introduction to Algebra A Self-Paced

Introduction to Algebra A
Tuesday, Jul 15 - Oct 28
Sunday, Aug 17 - Dec 14
Wednesday, Aug 27 - Dec 17
Friday, Sep 5 - Jan 16
Thursday, Sep 11 - Jan 15
Sunday, Sep 28 - Feb 1
Monday, Oct 6 - Feb 9
Tuesday, Oct 21 - Feb 24
Sunday, Nov 9 - Mar 15
Friday, Dec 5 - Apr 3

Introduction to Counting & Probability Self-Paced

Introduction to Counting & Probability
Wednesday, Jul 2 - Sep 17
Sunday, Jul 27 - Oct 19
Monday, Aug 11 - Nov 3
Wednesday, Sep 3 - Nov 19
Sunday, Sep 21 - Dec 14 (1:00 - 2:30 pm ET/10:00 - 11:30 am PT)
Friday, Oct 3 - Jan 16
Sunday, Oct 19 - Jan 25
Tuesday, Nov 4 - Feb 10
Sunday, Dec 7 - Mar 8

Introduction to Number Theory
Tuesday, Jul 15 - Sep 30
Wednesday, Aug 13 - Oct 29
Friday, Sep 12 - Dec 12
Sunday, Oct 26 - Feb 1
Monday, Dec 1 - Mar 2

Introduction to Algebra B Self-Paced

Introduction to Algebra B
Friday, Jul 18 - Nov 14
Thursday, Aug 7 - Nov 20
Monday, Aug 18 - Dec 15
Sunday, Sep 7 - Jan 11
Thursday, Sep 11 - Jan 15
Wednesday, Sep 24 - Jan 28
Sunday, Oct 26 - Mar 1
Tuesday, Nov 4 - Mar 10
Monday, Dec 1 - Mar 30

Introduction to Geometry
Monday, Jul 14 - Jan 19
Wednesday, Aug 13 - Feb 11
Tuesday, Aug 26 - Feb 24
Sunday, Sep 7 - Mar 8
Thursday, Sep 11 - Mar 12
Wednesday, Sep 24 - Mar 25
Sunday, Oct 26 - Apr 26
Monday, Nov 3 - May 4
Friday, Dec 5 - May 29

Paradoxes and Infinity
Mon, Tue, Wed, & Thurs, Jul 14 - Jul 16 (meets every day of the week!)
Sat & Sun, Sep 13 - Sep 14 (1:00 - 4:00 PM PT/4:00 - 7:00 PM ET)

Intermediate: Grades 8-12

Intermediate Algebra
Sunday, Jul 13 - Jan 18
Thursday, Jul 24 - Jan 22
Friday, Aug 8 - Feb 20
Tuesday, Aug 26 - Feb 24
Sunday, Sep 28 - Mar 29
Wednesday, Oct 8 - Mar 8
Sunday, Nov 16 - May 17
Thursday, Dec 11 - Jun 4

Intermediate Counting & Probability
Sunday, Sep 28 - Feb 15
Tuesday, Nov 4 - Mar 24

Intermediate Number Theory
Wednesday, Sep 24 - Dec 17

Precalculus
Wednesday, Aug 6 - Jan 21
Tuesday, Sep 9 - Feb 24
Sunday, Sep 21 - Mar 8
Monday, Oct 20 - Apr 6
Sunday, Dec 14 - May 31

Advanced: Grades 9-12

Calculus
Sunday, Sep 7 - Mar 15
Wednesday, Sep 24 - Apr 1
Friday, Nov 14 - May 22

Contest Preparation: Grades 6-12

MATHCOUNTS/AMC 8 Basics
Tues & Thurs, Jul 8 - Aug 14 (meets twice a week!)
Sunday, Aug 17 - Nov 9
Wednesday, Sep 3 - Nov 19
Tuesday, Sep 16 - Dec 9
Sunday, Sep 21 - Dec 14 (1:00 - 2:30 pm ET/10:00 - 11:30 am PT)
Monday, Oct 6 - Jan 12
Thursday, Oct 16 - Jan 22
Tues, Thurs & Sun, Dec 9 - Jan 18 (meets three times a week!)

MATHCOUNTS/AMC 8 Advanced
Tues & Thurs, Jul 8 - Aug 14 (meets twice a week!)
Sunday, Aug 17 - Nov 9
Tuesday, Aug 26 - Nov 11
Thursday, Sep 4 - Nov 20
Friday, Sep 12 - Dec 12
Monday, Sep 15 - Dec 8
Sunday, Oct 5 - Jan 11
Tues, Thurs & Sun, Dec 2 - Jan 11 (meets three times a week!)
Mon, Wed & Fri, Dec 8 - Jan 16 (meets three times a week!)

AMC 10 Problem Series
Tues & Thurs, Jul 8 - Aug 14 (meets twice a week!)
Sunday, Aug 10 - Nov 2
Thursday, Aug 14 - Oct 30
Tuesday, Aug 19 - Nov 4
Mon & Wed, Sep 15 - Oct 22 (meets twice a week!)
Mon, Wed & Fri, Oct 6 - Nov 3 (meets three times a week!)
Tue, Thurs & Sun, Oct 7 - Nov 2 (meets three times a week!)

AMC 10 Final Fives
Friday, Aug 15 - Sep 12
Sunday, Sep 7 - Sep 28
Tuesday, Sep 9 - Sep 30
Monday, Sep 22 - Oct 13
Sunday, Sep 28 - Oct 19 (1:00 - 2:30 pm ET/10:00 - 11:30 am PT)
Wednesday, Oct 8 - Oct 29
Thursday, Oct 9 - Oct 30

AMC 12 Problem Series
Wednesday, Aug 6 - Oct 22
Sunday, Aug 10 - Nov 2
Monday, Aug 18 - Nov 10
Mon & Wed, Sep 15 - Oct 22 (meets twice a week!)
Tues, Thurs & Sun, Oct 7 - Nov 2 (meets three times a week!)

AMC 12 Final Fives
Thursday, Sep 4 - Sep 25
Sunday, Sep 28 - Oct 19
Tuesday, Oct 7 - Oct 28

AIME Problem Series A
Thursday, Oct 23 - Jan 29

AIME Problem Series B
Tuesday, Sep 2 - Nov 18

F=ma Problem Series
Tuesday, Sep 16 - Dec 9
Friday, Oct 17 - Jan 30

WOOT Programs
Visit the pages linked for full schedule details for each of these programs!


MathWOOT Level 1
MathWOOT Level 2
ChemWOOT
CodeWOOT
PhysicsWOOT


Programming

Introduction to Programming with Python
Thursday, Aug 14 - Oct 30
Sunday, Sep 7 - Nov 23
Tuesday, Dec 2 - Mar 3

Intermediate Programming with Python
Friday, Oct 3 - Jan 16

USACO Bronze Problem Series
Wednesday, Sep 3 - Dec 3
Thursday, Oct 30 - Feb 5
Tuesday, Dec 2 - Mar 3

Physics

Introduction to Physics
Tuesday, Sep 2 - Nov 18
Sunday, Oct 5 - Jan 11
Wednesday, Dec 10 - Mar 11

Physics 1: Mechanics
Sunday, Sep 21 - Mar 22
Sunday, Oct 26 - Apr 26
0 replies
jwelsh
Jul 1, 2025
0 replies
ISI UGB 2025 P1
SomeonecoolLovesMaths   10
N 2 hours ago by Cats_on_a_computer
Source: ISI UGB 2025 P1
Suppose $f \colon \mathbb{R} \longrightarrow \mathbb{R}$ is differentiable and $| f'(x)| < \frac{1}{2}$ for all $x \in \mathbb{R}$. Show that for some $x_0 \in \mathbb{R}$, $f \left( x_0 \right) = x_0$.
10 replies
SomeonecoolLovesMaths
May 11, 2025
Cats_on_a_computer
2 hours ago
2017 Mock AIME II #10: Special Property Returns
DeathLlama9   6
N Today at 1:24 AM by Tetra_scheme
Find the number of integers $k$ with $2 \le k \le 5000$ satisfying the following property: for any prime $p$ dividing $k$, $p + 2$ divides $k$ as well.

Proposed by CantonMathGuy
6 replies
DeathLlama9
May 1, 2017
Tetra_scheme
Today at 1:24 AM
2024 Putnam A1
KevinYang2.71   23
N Today at 12:05 AM by megahertz13
Determine all positive integers $n$ for which there exists positive integers $a$, $b$, and $c$ satisfying
\[
2a^n+3b^n=4c^n.
\]
23 replies
KevinYang2.71
Dec 10, 2024
megahertz13
Today at 12:05 AM
A^2+B^2=AB+BA
mathisreal   0
Today at 12:03 AM
Source: OIMU 2023 #4
Determine all pairs of real matrices $(A,B)$ of size $2\times 2$ with $A\neq B$ such that
\[A^2+B^2=AB+BA=2I\]
0 replies
mathisreal
Today at 12:03 AM
0 replies
Matrices satisfy three conditions
ThE-dArK-lOrD   4
N Yesterday at 7:53 PM by Adustat
Source: IMC 2019 Day 1 P5
Determine whether there exist an odd positive integer $n$ and $n\times n$ matrices $A$ and $B$ with integer entries, that satisfy the following conditions:
[list=1]
[*]$\det (B)=1$;[/*]
[*]$AB=BA$;[/*]
[*]$A^4+4A^2B^2+16B^4=2019I$.[/*]
[/list]
(Here $I$ denotes the $n\times n$ identity matrix.)

Proposed by Orif Ibrogimov, ETH Zurich and National University of Uzbekistan
4 replies
ThE-dArK-lOrD
Jul 31, 2019
Adustat
Yesterday at 7:53 PM
Fractions Equal Integers
joml88   6
N Yesterday at 6:42 PM by OWOW
For how many ordered pairs of positive integers $(x,y),$ with $y<x\le 100,$ are both $\frac xy$ and $\frac{x+1}{y+1}$ integers?
6 replies
joml88
Dec 22, 2005
OWOW
Yesterday at 6:42 PM
Aime and Math prize for girls
erniuda   0
Yesterday at 3:48 PM
Hi, I know aime is harder than mp4g
If I got 5 in Aime. What score should I get in MP4G?
If I get 14 in MP4G, what score should I get in Aime?
0 replies
erniuda
Yesterday at 3:48 PM
0 replies
January 2025 Mock AIME #9 - expected value convergence
fidgetboss_4000   3
N Yesterday at 10:11 AM by zhaohm
$9$ bullets are placed in a row, with the bullet at position $1$ being a fake bullet. Chisato is bored and wants to amuse herself, so each second, she randomly chooses two of the $9$ bullets and switches their positions in the row. Let $E_n$ be the expected position in the row of the fake bullet after $n$ seconds have elapsed. What is the least value of $n$ such that $E_n \geq 4.999999$?
3 replies
fidgetboss_4000
Feb 10, 2025
zhaohm
Yesterday at 10:11 AM
January 2025 Mock AIME #11 - bipartitions
fidgetboss_4000   2
N Yesterday at 9:59 AM by zhaohm
Call two integers $a$ and $b$ $S$-worthy if there exists an integer $m \in S$ such that $am - b$ is divisible by $83.$ How many subsets $S \subset {1, 2, 3, ..., 82}$ are there such that the integers ${1, 2, 3, ..., 82}$ can be partitioned into two disjoint sets $A$ and $B$ such neither $A$ nor $B$ contains a pair of distinct integers that is $S$-worthy? Express your answer modulo $1000.$
2 replies
fidgetboss_4000
Feb 10, 2025
zhaohm
Yesterday at 9:59 AM
Expected value of OH^2 (OTIS Mock AIME 2024 #9)
v_Enhance   22
N Jul 3, 2025 by Kempu33334
Let $\omega$ be a circle with center $O$ and radius $12$. Points $A$, $B$, and $C$ are chosen uniformly at random on the circumference of $\omega$. Let $H$ denote the orthocenter of $\triangle ABC$. Compute the expected value of $OH^2$.

Jiahe Liu
22 replies
v_Enhance
Jan 16, 2024
Kempu33334
Jul 3, 2025
Intermediate Algebra: How to Study
HamstPan38825   45
N Jul 1, 2025 by AbhayAttarde01
I saw some users making comments about how to study from Introduction to Geometry, so I thought I'd put something down for Intermediate Algebra as well.
Chapters 1, 2, and 4 are extremely important review; if you think you have completely mastered the topic, these chapters are skippable but if you have not completed Introduction to Algebra, they don't take too long and are very good review.
Chapter 3 is not very important: reference Precalculus for deeper complex number studies.
Chapter 5, as stated, is mostly unimportant but quite interesting. Do this if you have time, (though 5.3 is the only really important section), but it's not vitally important.
Chapter 6 is one of those not negligible but not very important chapters: skim through this chapter lightly, since it really only gives quite basic information.
Chapters 7 and 8 are very important: Personally, Chapter 8 is the chapter more directed towards contests and Chapter 7 is a standard Algebra II chapter; the concepts are simple, but be sure to attempt all the challenge problems.
Chapter 9 is interesting but not very useful. Some of the concepts are interesting, but this chapter is not that important.
Chapters 10 and 11 are the most important chapters up to this point: Chapter 10 is arguably the most important chapter in the book unless you are preparing for contests. Chapter 11 is very general, a great tool for polishing up your bashing skills.
Chapter 12 is the hardest chapter in the book (excluding 17-20), and was my personal favorite chapter. Inequalities are important when training for olympiads, but if you're only aiming for AMC/AIME, it's not as important as Chapters 10, 11, etc. However, if you have time and enough persistence, the chapter is very useful (and in my opinion, one of the best-written chapters in the book.)
Chapters 13-16 all address a type of function: chapter 14 is the shortest chapter in the book, and the least important of the four; chapters 13 and 16 are the most important, with logarithmic identities and custom piecewise functions being a huge topic, especially in the AIME. Chapter 15 is a standard-oriented chapter, but some of the challenge problems are quite interesting.
Chapter 17 is the supplement to Chapter 10: if you're preparing for the AIME, Chapter 17 is just about as important as Chapter 10; the recursion section (17.1) is especially important.
Chapter 18 is the supplement to Chapter 12: the topics discussed here will likely not be used until the USAMO, so this chapter is probably the least important of the last four chapters. It is the second-hardest chapter in the entire book.
Chapter 19 is an interesting chapter; it is the only one of the last four chapters to bring up new material. Functional Equations are very high-level concepts: they frequently appear in the final AIME problems and various olympiads. Study this chapter, but you're not likely to completely understand it until you master the other chapters (and the AMC/AIME.)
Chapter 20 is the "monster" of all the chapters; this chapter is the most important and hardest of all the chapters when preparing for contests. Symmetry is probably most important here, though the problems given are insufficient for you to master symmetry. Attempt all problems in this chapter.

Final Note: Don't skim over sections and only do problems you think are sufficient. Do every single problem presented in the book, and all Review and Challenge Problems. 1,100 problems still might not be sufficient for you to master all the topics, and don't be intimidated by problems sourced from olympiads, USAMO, and IMO (personally, the last few challenge problems from Chapter 12 were way easier than their source: 12.77 was extremely easy, 12.80 was literally an easier version of 12.79).

Edit:
Most important chapters for AMC 10/12: Chapters 1, 2, 4, 8, 10, 13, 20 (maybe)
Most important chapters for AIME: Chapter 2, 8, 10, 12, 13, 16, 17, 20
Most important chapters for olympiads: Chapters 2, 8, 10, 11, 12, 13, 16, 17, 18, 19, 20

~HamstPan38825 (this took so long-)
45 replies
HamstPan38825
Nov 10, 2020
AbhayAttarde01
Jul 1, 2025
Problem 21 from AMC12B 2024
Zusen   2
N Jun 29, 2025 by Zusen
The measures of the smallest angles of three different right triangles sum to $90^\circ$. All three triangles have side lengths that are primitive Pythagorean triples. Two of them are $3-4-5$ and $5-12-13$. What is the perimeter of the third triangle?

$\textbf{(A) } 40 \qquad\textbf{(B) } 126 \qquad\textbf{(C) } 154 \qquad\textbf{(D) } 176 \qquad\textbf{(E) } 208$
2 replies
Zusen
Jun 29, 2025
Zusen
Jun 29, 2025
AMC10 2000 Problem #6/AMC12 2000 Problem #4
Avy11   4
N Jun 21, 2025 by Avy11
Here is the link to this problem and the solution https://artofproblemsolving.com/wiki/index.php/2000_AMC_12_Problems/Problem_4

It is not clear to me what is the question asking. If I lay down all the unit digits they keep repeating. Both the solutions stop at 6 not explaining why others are wrong.

1,1,2,3,5,8,3,1,4,5,9,4,3,7,0,7,7,4,1,5,6,1,7,8,5,3,8,1,9,0,9,9,8,7,5,2,7,9,6,5,1,1,6,7,3,0,3,3,6,9,...

Please help me understand.
4 replies
Avy11
Jun 20, 2025
Avy11
Jun 21, 2025
Hard states problem (Chicago Mock AMC 12 problem 20)
xHypotenuse   0
Jun 14, 2025
Kat and Pat each has two fair coins. Both players flip their coins simultaneously. The
person who flips more heads wins immediately and the game ends. If both players flip
the same number of heads, then they both flip again simultaneously. However, Kat only
flips the coins that landed tails the previous turn (the ones that landed heads stay as
heads), while Pat flips both coins every time regardless of the outcomes of the previous
turn. This process continues until a winner is determined. If the probability that Kat
wins is m/n, where m and n are relatively prime positive integers, what is the value of
m + n?
0 replies
xHypotenuse
Jun 14, 2025
0 replies
Approximate the integral
ILOVEMYFAMILY   2
N Jun 5, 2025 by ILOVEMYFAMILY
Approximate the integral
\[
I = \int_0^1 \frac{(2^x + 2)\, dx}{1 + x^4}
\]using the trapezoidal rule with accuracy $10^{-2}$.
2 replies
ILOVEMYFAMILY
Jun 5, 2025
ILOVEMYFAMILY
Jun 5, 2025
Approximate the integral
G H J
G H BBookmark kLocked kLocked NReply
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
ILOVEMYFAMILY
661 posts
#1
Y by
Approximate the integral
\[
I = \int_0^1 \frac{(2^x + 2)\, dx}{1 + x^4}
\]using the trapezoidal rule with accuracy $10^{-2}$.
This post has been edited 1 time. Last edited by ILOVEMYFAMILY, Jun 5, 2025, 9:38 AM
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Aiden-1089
338 posts
#2
Y by
Screw approximations, I'm doing the integral.
$I = \int_0^1 \frac{2x + 2}{1 + x^4} dx = \int_0^1 \frac{2x}{1+(x^2)^2} dx + \int_0^1 \frac{2}{1+x^4} dx = \arctan x^2 \bigg\rvert_0^1 + \int_0^1 \frac{1+\frac{1}{x^2}}{x^2+\frac{1}{x^2}} dx - \int_0^1 \frac{1-\frac{1}{x^2}}{x^2+\frac{1}{x^2}} dx$
$=\frac{\pi}{4} + \int_0^1 \frac{d(x-\frac{1}{x})}{(x-\frac{1}{x})^2+2} - \int_0^1 \frac{d(x+\frac{1}{x})}{(x+\frac{1}{x})^2-2} = \frac{\pi}{4} + \frac{1}{\sqrt{2}} \arctan \frac{x-\frac{1}{x}}{\sqrt{2}} \bigg\rvert_0^1 - \frac{1}{2\sqrt{2}} \ln \big\lvert \frac{x+\frac{1}{x}-\sqrt{2}}{x+\frac{1}{x}+\sqrt{2}} \big\rvert \bigg\rvert_0^1$
$=\frac{\pi}{4} + \frac{1}{\sqrt{2}} (0+\frac{\pi}{2}) - \frac{1}{2\sqrt{2}} (\ln (3-2\sqrt{2}) -0) = \boxed{\frac{\sqrt{2}+1}{4}\pi + \frac{1}{\sqrt{2}} \ln (1+\sqrt{2})}$
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
ILOVEMYFAMILY
661 posts
#3
Y by
Aiden-1089 wrote:
Screw approximations, I'm doing the integral.
$I = \int_0^1 \frac{2x + 2}{1 + x^4} dx = \int_0^1 \frac{2x}{1+(x^2)^2} dx + \int_0^1 \frac{2}{1+x^4} dx = \arctan x^2 \bigg\rvert_0^1 + \int_0^1 \frac{1+\frac{1}{x^2}}{x^2+\frac{1}{x^2}} dx - \int_0^1 \frac{1-\frac{1}{x^2}}{x^2+\frac{1}{x^2}} dx$
$=\frac{\pi}{4} + \int_0^1 \frac{d(x-\frac{1}{x})}{(x-\frac{1}{x})^2+2} - \int_0^1 \frac{d(x+\frac{1}{x})}{(x+\frac{1}{x})^2-2} = \frac{\pi}{4} + \frac{1}{\sqrt{2}} \arctan \frac{x-\frac{1}{x}}{\sqrt{2}} \bigg\rvert_0^1 - \frac{1}{2\sqrt{2}} \ln \big\lvert \frac{x+\frac{1}{x}-\sqrt{2}}{x+\frac{1}{x}+\sqrt{2}} \big\rvert \bigg\rvert_0^1$
$=\frac{\pi}{4} + \frac{1}{\sqrt{2}} (0+\frac{\pi}{2}) - \frac{1}{2\sqrt{2}} (\ln (3-2\sqrt{2}) -0) = \boxed{\frac{\sqrt{2}+1}{4}\pi + \frac{1}{\sqrt{2}} \ln (1+\sqrt{2})}$

Sorry I posted the wrong exercise. I have edited it now
Z K Y
N Quick Reply
G
H
=
a