Art of Problem Solving
AoPS Online AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy AoPS Academy
Small live classes for advanced math
and language arts learners in grades 1-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Stay ahead of learning milestones! Enroll in a class over the summer!
No tags match your search
M
G
Topic
First Poster
Last Poster
H
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.

WOOT early bird pricing is in effect, don’t miss out! If you took MathWOOT Level 2 last year, no worries, it is all new problems this year! Our Worldwide Online Olympiad Training program is for high school level competitors. AoPS designed these courses to help our top students get the deep focus they need to succeed in their specific competition goals. Check out the details at this link for all our WOOT programs in math, computer science, chemistry, and physics.

Looking for summer camps in math and language arts? Be sure to check out the video-based summer camps offered at the Virtual Campus that are 2- to 4-weeks in duration. There are middle and high school competition math camps as well as Math Beasts camps that review key topics coupled with fun explorations covering areas such as graph theory (Math Beasts Camp 6), cryptography (Math Beasts Camp 7-8), and topology (Math Beasts Camp 8-9)!

Be sure to mark your calendars for the following events:
[list][*]April 3rd (Webinar), 4pm PT/7:00pm ET, Learning with AoPS: Perspectives from a Parent, Math Camp Instructor, and University Professor
[*]April 8th (Math Jam), 4:30pm PT/7:30pm ET, 2025 MATHCOUNTS State Discussion
April 9th (Webinar), 4:00pm PT/7:00pm ET, Learn about Video-based Summer Camps at the Virtual Campus
[*]April 10th (Math Jam), 4:30pm PT/7:30pm ET, 2025 MathILy and MathILy-Er Math Jam: Multibackwards Numbers
[*]April 22nd (Webinar), 4:00pm PT/7:00pm ET, Competitive Programming at AoPS (USACO).[/list]
Our full course list for upcoming classes is below:
All classes run 7:30pm-8:45pm ET/4:30pm - 5:45pm PT unless otherwise noted.

Introductory: Grades 5-10

Prealgebra 1 Self-Paced

Prealgebra 1
Sunday, Apr 13 - Aug 10
Tuesday, May 13 - Aug 26
Thursday, May 29 - Sep 11
Sunday, Jun 15 - Oct 12
Monday, Jun 30 - Oct 20
Wednesday, Jul 16 - Oct 29

Prealgebra 2 Self-Paced

Prealgebra 2
Sunday, Apr 13 - Aug 10
Wednesday, May 7 - Aug 20
Monday, Jun 2 - Sep 22
Sunday, Jun 29 - Oct 26
Friday, Jul 25 - Nov 21

Introduction to Algebra A Self-Paced

Introduction to Algebra A
Monday, Apr 7 - Jul 28
Sunday, May 11 - Sep 14 (1:00 - 2:30 pm ET/10:00 - 11:30 am PT)
Wednesday, May 14 - Aug 27
Friday, May 30 - Sep 26
Monday, Jun 2 - Sep 22
Sunday, Jun 15 - Oct 12
Thursday, Jun 26 - Oct 9
Tuesday, Jul 15 - Oct 28

Introduction to Counting & Probability Self-Paced

Introduction to Counting & Probability
Wednesday, Apr 16 - Jul 2
Thursday, May 15 - Jul 31
Sunday, Jun 1 - Aug 24
Thursday, Jun 12 - Aug 28
Wednesday, Jul 9 - Sep 24
Sunday, Jul 27 - Oct 19

Introduction to Number Theory
Thursday, Apr 17 - Jul 3
Friday, May 9 - Aug 1
Wednesday, May 21 - Aug 6
Monday, Jun 9 - Aug 25
Sunday, Jun 15 - Sep 14
Tuesday, Jul 15 - Sep 30

Introduction to Algebra B Self-Paced

Introduction to Algebra B
Wednesday, Apr 16 - Jul 30
Tuesday, May 6 - Aug 19
Wednesday, Jun 4 - Sep 17
Sunday, Jun 22 - Oct 19
Friday, Jul 18 - Nov 14

Introduction to Geometry
Wednesday, Apr 23 - Oct 1
Sunday, May 11 - Nov 9
Tuesday, May 20 - Oct 28
Monday, Jun 16 - Dec 8
Friday, Jun 20 - Jan 9
Sunday, Jun 29 - Jan 11
Monday, Jul 14 - Jan 19

Intermediate: Grades 8-12

Intermediate Algebra
Monday, Apr 21 - Oct 13
Sunday, Jun 1 - Nov 23
Tuesday, Jun 10 - Nov 18
Wednesday, Jun 25 - Dec 10
Sunday, Jul 13 - Jan 18
Thursday, Jul 24 - Jan 22

Intermediate Counting & Probability
Wednesday, May 21 - Sep 17
Sunday, Jun 22 - Nov 2

Intermediate Number Theory
Friday, Apr 11 - Jun 27
Sunday, Jun 1 - Aug 24
Wednesday, Jun 18 - Sep 3

Precalculus
Wednesday, Apr 9 - Sep 3
Friday, May 16 - Oct 24
Sunday, Jun 1 - Nov 9
Monday, Jun 30 - Dec 8

Advanced: Grades 9-12

Olympiad Geometry
Tuesday, Jun 10 - Aug 26

Calculus
Tuesday, May 27 - Nov 11
Wednesday, Jun 25 - Dec 17

Group Theory
Thursday, Jun 12 - Sep 11

Contest Preparation: Grades 6-12

MATHCOUNTS/AMC 8 Basics
Wednesday, Apr 16 - Jul 2
Friday, May 23 - Aug 15
Monday, Jun 2 - Aug 18
Thursday, Jun 12 - Aug 28
Sunday, Jun 22 - Sep 21
Tues & Thurs, Jul 8 - Aug 14 (meets twice a week!)

MATHCOUNTS/AMC 8 Advanced
Friday, Apr 11 - Jun 27
Sunday, May 11 - Aug 10
Tuesday, May 27 - Aug 12
Wednesday, Jun 11 - Aug 27
Sunday, Jun 22 - Sep 21
Tues & Thurs, Jul 8 - Aug 14 (meets twice a week!)

AMC 10 Problem Series
Friday, May 9 - Aug 1
Sunday, Jun 1 - Aug 24
Thursday, Jun 12 - Aug 28
Tuesday, Jun 17 - Sep 2
Sunday, Jun 22 - Sep 21 (1:00 - 2:30 pm ET/10:00 - 11:30 am PT)
Monday, Jun 23 - Sep 15
Tues & Thurs, Jul 8 - Aug 14 (meets twice a week!)

AMC 10 Final Fives
Sunday, May 11 - Jun 8
Tuesday, May 27 - Jun 17
Monday, Jun 30 - Jul 21

AMC 12 Problem Series
Tuesday, May 27 - Aug 12
Thursday, Jun 12 - Aug 28
Sunday, Jun 22 - Sep 21
Wednesday, Aug 6 - Oct 22

AMC 12 Final Fives
Sunday, May 18 - Jun 15

F=ma Problem Series
Wednesday, Jun 11 - Aug 27

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, May 22 - Aug 7
Sunday, Jun 15 - Sep 14 (1:00 - 2:30 pm ET/10:00 - 11:30 am PT)
Tuesday, Jun 17 - Sep 2
Monday, Jun 30 - Sep 22

Intermediate Programming with Python
Sunday, Jun 1 - Aug 24
Monday, Jun 30 - Sep 22

USACO Bronze Problem Series
Tuesday, May 13 - Jul 29
Sunday, Jun 22 - Sep 1

Physics

Introduction to Physics
Wednesday, May 21 - Aug 6
Sunday, Jun 15 - Sep 14
Monday, Jun 23 - Sep 15

Physics 1: Mechanics
Thursday, May 22 - Oct 30
Monday, Jun 23 - Dec 15

Relativity
Sat & Sun, Apr 26 - Apr 27 (4:00 - 7:00 pm ET/1:00 - 4:00pm PT)
Mon, Tue, Wed & Thurs, Jun 23 - Jun 26 (meets every day of the week!)
0 replies
jlacosta
Apr 2, 2025
0 replies
J
H
k i Adding contests to the Contest Collections
dcouchman   1
N Apr 5, 2023 by v_Enhance
Want to help AoPS remain a valuable Olympiad resource? Help us add contests to AoPS's Contest Collections.

Find instructions and a list of contests to add here: https://artofproblemsolving.com/community/c40244h1064480_contests_to_add
1 reply
dcouchman
Sep 9, 2019
v_Enhance
Apr 5, 2023
J
H
k i Zero tolerance
ZetaX   49
N May 4, 2019 by NoDealsHere
Source: Use your common sense! (enough is enough)
Some users don't want to learn, some other simply ignore advises.
But please follow the following guideline:


To make it short: ALWAYS USE YOUR COMMON SENSE IF POSTING!
If you don't have common sense, don't post.


More specifically:

For new threads:


a) Good, meaningful title:
The title has to say what the problem is about in best way possible.
If that title occured already, it's definitely bad. And contest names aren't good either.
That's in fact a requirement for being able to search old problems.

Examples:
Bad titles:
- "Hard"/"Medium"/"Easy" (if you find it so cool how hard/easy it is, tell it in the post and use a title that tells us the problem)
- "Number Theory" (hey guy, guess why this forum's named that way¿ and is it the only such problem on earth¿)
- "Fibonacci" (there are millions of Fibonacci problems out there, all posted and named the same...)
- "Chinese TST 2003" (does this say anything about the problem¿)
Good titles:
- "On divisors of a³+2b³+4c³-6abc"
- "Number of solutions to x²+y²=6z²"
- "Fibonacci numbers are never squares"


b) Use search function:
Before posting a "new" problem spend at least two, better five, minutes to look if this problem was posted before. If it was, don't repost it. If you have anything important to say on topic, post it in one of the older threads.
If the thread is locked cause of this, use search function.

Update (by Amir Hossein). The best way to search for two keywords in AoPS is to input
[code]+"first keyword" +"second keyword"[/code]
so that any post containing both strings "first word" and "second form".


c) Good problem statement:
Some recent really bad post was:
[quote]$lim_{n\to 1}^{+\infty}\frac{1}{n}-lnn$[/quote]
It contains no question and no answer.
If you do this, too, you are on the best way to get your thread deleted. Write everything clearly, define where your variables come from (and define the "natural" numbers if used). Additionally read your post at least twice before submitting. After you sent it, read it again and use the Edit-Button if necessary to correct errors.


For answers to already existing threads:


d) Of any interest and with content:
Don't post things that are more trivial than completely obvious. For example, if the question is to solve $x^{3}+y^{3}=z^{3}$, do not answer with "$x=y=z=0$ is a solution" only. Either you post any kind of proof or at least something unexpected (like "$x=1337, y=481, z=42$ is the smallest solution). Someone that does not see that $x=y=z=0$ is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
Similar, posting "I have solved this problem" but not posting anything else is not welcome; it even looks that you just want to show off what a genius you are.

e) Well written and checked answers:
Like c) for new threads, check your solutions at least twice for mistakes. And after sending, read it again and use the Edit-Button if necessary to correct errors.



To repeat it: ALWAYS USE YOUR COMMON SENSE IF POSTING!


Everything definitely out of range of common sense will be locked or deleted (exept for new users having less than about 42 posts, they are newbies and need/get some time to learn).

The above rules will be applied from next monday (5. march of 2007).
Feel free to discuss on this here.
49 replies
ZetaX
Feb 27, 2007
NoDealsHere
May 4, 2019
J
H
NC State Math Contest Wake Tech Regional Problems and Solutions
mathnerd_101   5
N 20 minutes ago by mathnerd_101
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
5 replies
mathnerd_101
Today at 11:40 AM
mathnerd_101
20 minutes ago
J
H
thank you
Piwbo   1
N 42 minutes ago by CHESSR1DER
Let $p_n$ be the n-th prime number in increasing order for $n\geq 1$. Prove that there exists a sequence of distinct prime numbers $q_n$ satisfying $q_1+q_2+...+q_n=p_n$ for all $n\geq 1 $
1 reply
Piwbo
Yesterday at 11:22 AM
CHESSR1DER
42 minutes ago
J
H
Difficult lattice point coloring problem
CBMaster   0
43 minutes ago
Source: Own
Is it possible to color all lattice points in plane into 3 colors such that

1. every line passing through lattice points and parallel to x axis has these three colors infinitely many(that is, every color appears infinitely many times in those lines).

2. every line passing through lattice points and not parallel to x axis cannot have three different color lattice points on it.
0 replies
CBMaster
43 minutes ago
0 replies
J
H
Stop Projecting your insecurities
naman12   52
N an hour ago by ihategeo_1969
Source: 2022 USA TST #2
Let $ABC$ be an acute triangle. Let $M$ be the midpoint of side $BC$, and let $E$ and $F$ be the feet of the altitudes from $B$ and $C$, respectively. Suppose that the common external tangents to the circumcircles of triangles $BME$ and $CMF$ intersect at a point $K$, and that $K$ lies on the circumcircle of $ABC$. Prove that line $AK$ is perpendicular to line $BC$.

Kevin Cong
52 replies
naman12
Dec 12, 2022
ihategeo_1969
an hour ago
J
H
Digit sum
Disjeje   3
N an hour ago by jasperE3
Let’s say S(n) is digit sum of n does n exists thatS(n)>S(n^2)?
3 replies
Disjeje
4 hours ago
jasperE3
an hour ago
J
H
R+ Functional Equation
Mathdreams   3
N an hour ago by jasperE3
Source: Nepal TST 2025, Problem 3
Find all functions $f : \mathbb{R}^+ \rightarrow \mathbb{R}^+$ such that \[f(f(x)) + xf(xy) = x + f(y)\]for all positive real numbers $x$ and $y$.

(Andrew Brahms, USA)
3 replies
Mathdreams
5 hours ago
jasperE3
an hour ago
J
H
Factorise (x+1)(x+2)(x+3)(x+4)-3
Idiot_of_the64squares   5
N an hour 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
an hour ago
J
H
TST Junior Romania 2025
ant_   0
an hour ago
Source: ssmr
Consider the isosceles triangle $ABC$, with $\angle BAC > 90^\circ$, and the circle $\omega$ with center $A$ and radius $AC$. Denote by $M$ the midpoint of side $AC$. The line $BM$ intersects the circle $\omega$ for the second time in $D$. Let $E$ be a point on the circle $\omega$ such that $BE \perp AC$ and $DE \cap AC = {N}$. Show that $AN = 2AB$.
0 replies
ant_
an hour ago
0 replies
J
H
Functional equation
Pmshw   16
N an hour ago by jasperE3
Source: Iran 2nd round 2022 P2
Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that for any real value of $x,y$ we have:
$$f(xf(y)+f(x)+y)=xy+f(x)+f(y)$$
16 replies
Pmshw
May 8, 2022
jasperE3
an hour ago
J
H
Line passes through the fix point
moony_   1
N an hour ago by GioOrnikapa
Source: idk
Let $ABC$ be a triangle. $P$ and $Q$ are points, such that $PA = PB$, $QA$ = $QC$ and $\angle{PBC} =\angle{QCB}$ ($P$ - inside $\triangle{ABC}$ and $Q$ - oitside). Proove that line $PQ$ passes through the fix point.
1 reply
moony_
4 hours ago
GioOrnikapa
an hour ago
J
H
Subsets of points lying in disks
oVlad   4
N an hour ago by Andyexists
Source: Romania TST 2023 Day 2 P4
Fix a positive integer $n.{}$ Consider an $n{}$-point set $S{}$ in the plane. An eligible set is a non-empty set of the form $S\cap D,{}$ where $D$ is a closed disk in the plane. In terms of $n,$ determine the smallest possible number of eligible subsets $S{}$ may contain.

Proposed by Cristi Săvescu
4 replies
oVlad
Apr 7, 2024
Andyexists
an hour ago
J
H
hard ...
REGNA   3
N an hour ago by KenWuMath
find function $f:\mathbb{R^+}\rightarrow \mathbb{R^+}$ such that :
$f()\frac{f(x)}{y}=yf(y)f(f(x)) \;\;\;\;\;(x,y> 0)$
3 replies
REGNA
Oct 21, 2022
KenWuMath
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 2 hours 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
2 hours 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
J
H
J
H
Distance vs time swimming problem
smalkaram_3549   1
N Apr 5, 2025 by Lankou
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.
1 reply
smalkaram_3549
Apr 5, 2025
Lankou
Apr 5, 2025
J
Distance vs time swimming problem
G H J
Reveal topic tags
rate problems
L
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.
smalkaram_3549
168 posts
smalkaram_3549
#1 h  Apr 5, 2025, 2:57 AM
Y by
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:
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Lankou
1384 posts
Lankou
#2 h  Apr 5, 2025, 11:54 AM
Y by
$l, s_1, s_2$: length of the pool, speed of the first swimmer, speed of the second swimmer
$\frac{18}{s_1}=\frac{l-18}{s_2}$
$\frac{l+26}{s_1}=\frac{2l-26}{s_2}$
Solve for $l$
Z K Y
N Quick Reply
G
H
=
a