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k a June Highlights and 2025 AoPS Online Class Information
jlacosta   0
Jun 2, 2025
Congratulations to all the mathletes who competed at National MATHCOUNTS! If you missed the exciting Countdown Round, you can watch the video at this link. Are you interested in training for MATHCOUNTS or AMC 10 contests? How would you like to train for these math competitions in half the time? We have accelerated sections which meet twice per week instead of once starting on July 8th (7:30pm ET). These sections fill quickly so enroll today!

[list][*]MATHCOUNTS/AMC 8 Basics
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Summer camps are starting this month at the Virtual Campus in math and language arts that are 2 - to 4 - weeks in duration. Spaces are still available - don’t miss your chance to have a transformative summer experience. 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)!

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0 replies
jlacosta
Jun 2, 2025
0 replies
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Calculus-flavored sum
NamelyOrange   2
N 31 minutes ago by CyanSwan27
Evaluate $\sum_{n = 0}^{\infty}\frac{1}{(4n)!}$.

(Source: AOPS user @SomeDumbChild)
2 replies
NamelyOrange
Tuesday at 11:15 PM
CyanSwan27
31 minutes ago
J
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Perimeter relationship
Silverfalcon   5
N 4 hours ago by redracer
The perimeter of an equilateral triangle exceeds the perimeter of a square by $1989 \ \text{cm}$. The length of each side of the triangle exceeds the length of each side of the square by $d \ \text{cm}$. The square has perimeter greater than 0. How many positive integers are NOT possible value for $d$?

$\text{(A)} \ 0 \qquad \text{(B)} \ 9 \qquad \text{(C)} \ 221 \qquad \text{(D)} \ 663 \qquad \text{(E)} \ \text{infinitely many}$
5 replies
Silverfalcon
Dec 19, 2005
redracer
4 hours ago
J
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A little problem
Zootieroaz   11
N 4 hours ago by redracer
The set of all numbers x for which \[x+\sqrt{x^{2}+1}-\frac{1}{x+\sqrt{x^{2}+1}}\]is a rational number is the set of all:
$\textbf{(A)}\ \text{ integers } x \qquad \textbf{(B)}\ \text{ rational } x \qquad \textbf{(C)}\ \text{ real } x\qquad \textbf{(D)}\ x \text{ for which } \sqrt{x^2+1} \text{ is rational} \qquad \textbf{(E)}\ x \text{ for which } x+\sqrt{x^2+1} \text{ is rational }$
11 replies
Zootieroaz
Nov 30, 2005
redracer
4 hours ago
J
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cot 10 + tan 5
Silverfalcon   7
N 5 hours ago by martianrunner
$\cot 10 + \tan 5 =$

$\textbf{(A)}\ \csc 5 \qquad \textbf{(B)}\ \csc 10 \qquad \textbf{(C)}\ \sec 5 \qquad \textbf{(D)}\ \sec 10 \qquad \textbf{(E)}\ \sin 15$
7 replies
Silverfalcon
Oct 22, 2005
martianrunner
5 hours ago
J
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Triangle in a circle
agolsme   13
N Yesterday at 9:23 PM by mudkip42
A triangle is inscribed in a circle. The vertices of the triangle divide the circle into three arcs of lengths $3$, $4$, and $5$. What is the area of the triangle?
$\textbf{(A)}\ 6 \qquad \textbf{(B)}\ \frac{18}{\pi^2} \qquad \textbf{(C)}\ \frac{9}{\pi^2}\left(\sqrt{3}-1\right) \qquad \textbf{(D)}\ \frac{9}{\pi^2}\left(\sqrt{3}+1\right) \qquad \textbf{(E)}\ \frac{9}{\pi^2}\left(\sqrt{3}+3\right)$
13 replies
agolsme
Dec 4, 2005
mudkip42
Yesterday at 9:23 PM
J
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Lattice points in a line
Silverfalcon   3
N Yesterday at 9:08 PM by mudkip42
A lattice point is a point in the plane with integer coordinates. How many lattice points are on the line segment whose endpoints are (3,17) and (48,281)? (Include both endpoints of the segment in your count.)

$\textbf{(A)}\ 2 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 16 \qquad \textbf{(E)}\ 46$
3 replies
Silverfalcon
Oct 22, 2005
mudkip42
Yesterday at 9:08 PM
J
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Calculator
SaadFlash1001   3
N Yesterday at 8:56 PM by Ravensrule8
Hello, I am currently working on the into to algebra book by aops. I am a little bit confused if I should use a calculator or not. On some exercises, it specifics not to use a calculator, so I obviously don't. However, what should I do about all the other problems, review problems, exercise, etc? Am I allowed to use a calculator on those problems?
3 replies
SaadFlash1001
Yesterday at 7:36 PM
Ravensrule8
Yesterday at 8:56 PM
J
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[Own Problem] Alice and Bob gets arrested
lrnnz   3
N Yesterday at 8:55 PM by Ravensrule8
Alice and Bob were caught robbing the International Bank Internationally.
Their sentence is set to a + b years, where
gcd(a, b) = 12, and lcm(a, b) = 360,

The judge let them decide the values (a, b). Help Alice and Bob find the minimum possible value of a + b to lessen their sentence!

Answer: Click to reveal hidden text
3 replies
lrnnz
Yesterday at 8:16 PM
Ravensrule8
Yesterday at 8:55 PM
J
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Problem of the Day
Silverfalcon   164
N Yesterday at 8:49 PM by mudkip42
Hi guys,

I was just reading over old posts that I made last year ( :P ) and saw how much the level of Getting Started became harder. To encourage more people from posting, I decided to start a Problem of the Day. This is how I'll conduct this:

1. In each post (not including this one since it has rules, etc) everyday, I'll post the problem. I may post another thread after it to give hints though.
2. Level of problem.. This is VERY important. All problems in this thread will be all AHSME or problems similar to this level. No AIME. Some AHSME problems, however, that involve tough insight or skills will not be posted. The chosen problems will be usually ones that everyone can solve after working. Calculators are allowed when you solve problems but it is NOT necessary.
3. Response.. All you have to do is simply solve the problem and post the solution. There is no credit given or taken away if you get the problem wrong. This isn't like other threads where the number of problems you get right or not matters. As for posting, post your solutions here in this thread. Do NOT PM me. Also, here are some more restrictions when posting solutions:

A. No single answer post. It doesn't matter if you put hide and say "Answer is ###..." If you don't put explanation, it simply means you cheated off from some other people. I've seen several posts that went like "I know the answer" and simply post the letter. What is the purpose of even posting then? Huh?
B. Do NOT go back to the previous problem(s). This causes too much confusion.
C. You're FREE to give hints and post different idea, way or answer in some cases in problems. If you see someone did wrong or you don't understand what they did, post here. That's what this thread is for.

4. Main purpose.. This is for anyone who visits this forum to enjoy math. I rememeber when I first came into this forum, I was poor at math compared to other people. But I kindly got help from many people such as JBL, joml88, tokenadult, and many other people that would take too much time to type. Perhaps without them, I wouldn't be even a moderator in this forum now. This site clearly made me to enjoy math more and more and I'd like to do the same thing. That's about the rule.. Have fun problem solving!

Next post will contain the Day 1 Problem. You can post the solutions until I post one. :D
164 replies
Silverfalcon
Oct 31, 2005
mudkip42
Yesterday at 8:49 PM
J
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Area of the figure
Silverfalcon   3
N Yesterday at 8:41 PM by mudkip42
Two strips of width 1 overlap at an angle of $\alpha$ as shown. The area of the overlap (shown shaded) is

IMAGE

$\text{(A)} \ \sin \alpha \qquad \text{(B)} \ \frac{1}{\sin \alpha} \qquad \text{(C)} \ \frac{1}{1 - \cos \alpha} \qquad \text{(D)} \ \frac{1}{\sin^2 \alpha} \qquad \text{(E)} \ \frac{1}{(1 - \cos \alpha)^2}$
3 replies
Silverfalcon
Dec 18, 2005
mudkip42
Yesterday at 8:41 PM
J
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Traffic issue!
Silverfalcon   4
N Yesterday at 8:40 PM by mudkip42
The traffic on a certain east-west highway moves at a constant speed of 60 miles per hour in both directions. An eastbound driver passes 20 west-bound vehicles in a five-minute interval. Assume vehicles in the westbound lane are equally spaced. Which of the following is closest to the number of westbound vehicles present in a 100-mile section of highway?

$\text{(A)} \ 100 \qquad \text{(B)} \ 120 \qquad \text{(C)} \ 200 \qquad \text{(D)} \ 240 \qquad \text{(E)} \ 400$
4 replies
Silverfalcon
Dec 18, 2005
mudkip42
Yesterday at 8:40 PM
J
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[Sipnayan's POTD] GCD and LCM
lrnnz   2
N Yesterday at 7:59 PM by lrnnz
What is the product of the LCM and the GCD of 4, 8, and 28?

Answer: Click to reveal hidden text
2 replies
lrnnz
Yesterday at 7:53 PM
lrnnz
Yesterday at 7:59 PM
J
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[PMO 22] III.1 GCD and LCM
lrnnz   1
N Yesterday at 7:49 PM by lrnnz
Compute the sum of all possible distinct values of m + n if m and n are positive integers such that
lcm(m, n) + gcd(m, n) = 2(m + n) + 11.

Answer: Click to reveal hidden text
1 reply
lrnnz
Yesterday at 7:37 PM
lrnnz
Yesterday at 7:49 PM
J
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Factoring basically..
Silverfalcon   4
N Yesterday at 7:45 PM by BAM10
For how many integers $n$ between 1 and 100 does $x^2+x-n$ factor into the product of two linear factors with integer coefficients?

$\text{(A)} \ 0 \qquad \text{(B)} \ 1 \qquad \text{(C)} \ 2 \qquad \text{(D)} \ 9 \qquad \text{(E)} \ 10$
4 replies
Silverfalcon
Dec 18, 2005
BAM10
Yesterday at 7:45 PM
J
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J
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b+c <=a/sin(A/2)
lgx57   4
N May 16, 2025 by cosinesine
Prove that: In $\triangle ABC$,$b+c \le \dfrac{a}{\sin \frac{A}{2}}$
4 replies
lgx57
May 16, 2025
cosinesine
May 16, 2025
J
b+c <=a/sin(A/2)
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Reveal topic tags
trigonometry
algebra
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lgx57
60 posts
lgx57
#1 h  May 16, 2025, 1:11 PM
Y by
Prove that: In $\triangle ABC$,$b+c \le \dfrac{a}{\sin \frac{A}{2}}$
Z K Y
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sqing
42883 posts
sqing
#2 h  May 16, 2025, 2:20 PM
Y by
lgx57 wrote:
Prove that: In $\triangle ABC$,$b+c \le \dfrac{a}{\sin \frac{A}{2}}$
$$\sin B+\sin C=2\cos \frac{A}{2}\cos \frac{B-C}{2}\le 2\cos \frac{A}{2} $$
Z K Y
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bogpt
4 posts
bogpt
#3 h  May 16, 2025, 4:00 PM
Y by
b+c = a/sinA * (sinB + sinC)

sinB + sinC <= sinA/sin(A/2)

2sin((B+C)/2)cos((B-C)/2) <= 2sin(A/2)cos(A/2)/sin(A/2)
cos((B-C)/2) <= 1
Z K Y
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Jaydenmii
1 post
Jaydenmii
#4 h  May 16, 2025, 4:06 PM
Y by
.............................................................................................
Z K Y
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cosinesine
62 posts
cosinesine
#5 h  May 16, 2025, 4:27 PM
Y by
By Law of Sines and Jensens,
\[ \frac{a}{b + c} \ge \frac{ \sin A }{\sin B + \sin C} \ge \frac{ \sin A }{2 \sin \frac{B + C}{2} } = \frac{ \sin A }{2\cos \frac{A}{2} } = \frac{ 2 \sin \frac{A}{2} \cos \frac{A}{2} }{2 \cos \frac{A}{2} } = \sin \frac{A}{2} \]
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