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Calculator
SaadFlash1001   3
N 35 minutes ago 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
2 hours ago
Ravensrule8
35 minutes ago
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[Own Problem] Alice and Bob gets arrested
lrnnz   3
N 35 minutes ago 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
an hour ago
Ravensrule8
35 minutes ago
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Problem of the Day
Silverfalcon   164
N 42 minutes ago 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
42 minutes ago
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cot 10 + tan 5
Silverfalcon   6
N an hour ago by mudkip42
$\cot 10 + \tan 5 =$

$\textbf{(A)}\ \csc 5 \qquad \textbf{(B)}\ \csc 10 \qquad \textbf{(C)}\ \sec 5 \qquad \textbf{(D)}\ \sec 10 \qquad \textbf{(E)}\ \sin 15$
6 replies
Silverfalcon
Oct 22, 2005
mudkip42
an hour ago
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Area of the figure
Silverfalcon   3
N an hour ago 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
an hour ago
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Traffic issue!
Silverfalcon   4
N an hour ago 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
an hour ago
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[Sipnayan's POTD] GCD and LCM
lrnnz   2
N 2 hours ago 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
2 hours ago
lrnnz
2 hours ago
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[PMO 22] III.1 GCD and LCM
lrnnz   1
N 2 hours ago 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
2 hours ago
lrnnz
2 hours ago
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Factoring basically..
Silverfalcon   4
N 2 hours ago 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
2 hours ago
J
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Median, altitude, and the angle chasing
Silverfalcon   2
N 2 hours ago by mudkip42
In $\triangle ABC, \angle A = 100^\circ, \angle B = 50^\circ, \angle C = 30^\circ, \overline{AH}$ is an altitude, and $\overline{BM}$ is a median. Then $\angle MHC =$

IMAGE

$\text{(A)} \ 15^\circ \qquad \text{(B)} \ 22.5^\circ \qquad \text{(C)} \ 30^\circ \qquad \text{(D)} \ 40^\circ \qquad \text{(E)} \ 45^\circ$
2 replies
Silverfalcon
Dec 18, 2005
mudkip42
2 hours ago
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Probability
CatsMeow12   10
N Jul 3, 2024 by CatsMeow12
Two pairs of distinct points are chosen from the $3$-by-$3$ grid below. What is the probability that the lines formed by both pairs are perpendicular?

IMAGE

I would greatly appreciate solutions to this so I can check my work. Thanks! :-D
10 replies
CatsMeow12
Jul 3, 2024
CatsMeow12
Jul 3, 2024
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Counting and Probability
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CatsMeow12
2254 posts
CatsMeow12
#1 h  Jul 3, 2024, 12:57 AM
Y by
Two pairs of distinct points are chosen from the $3$-by-$3$ grid below. What is the probability that the lines formed by both pairs are perpendicular?

[asy]unitsize(32); for (int x=0; x<3; ++x) { for (int y=0; y<3; ++y) { dot((x,y)); } } [/asy]

I would greatly appreciate solutions to this so I can check my work. Thanks! :-D
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ethanzhang1001
1060 posts
ethanzhang1001
#2 h  Jul 3, 2024, 1:05 AM
Y by
$\frac{3\cdot 3}{\binom{9}{4}}=\frac{1}{14}$

(oops wrong i spent like 2 seconds before typing this)
This post has been edited 1 time. Last edited by ethanzhang1001, Jul 3, 2024, 1:13 AM
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CatsMeow12
2254 posts
CatsMeow12
#3 h  Jul 3, 2024, 1:06 AM
Y by
ethanzhang1001 wrote:
$\frac{3\cdot 3}{\binom{9}{4}}=\frac{1}{14}$

Wait... how did you get this?
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Elephant200
1472 posts
Elephant200
#4 h  Jul 3, 2024, 1:07 AM
Y by
Are you sure there's only $9$ successful outcomes? I see a lot more.

Also, 4 points don't uniquely determine two pairs of segments.
This post has been edited 1 time. Last edited by Elephant200, Jul 3, 2024, 1:08 AM
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cknk
89 posts
cknk
#5 h  Jul 3, 2024, 1:41 AM
Y by
I, uh... there must be a better way to do this... how would we know that we had all the successful outcomes?
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miyukina
1219 posts
miyukina
#6 h  Jul 3, 2024, 2:39 AM • 1 Y
Y by CatsMeow12
not sure
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CatsMeow12
2254 posts
CatsMeow12
#7 h  Jul 3, 2024, 3:17 AM • 1 Y
Y by miyukina
miyukina wrote:
not sure

I got 4 fewer possibilities... but your solution was almost identical to mine
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miyukina
1219 posts
miyukina
#8 h  Jul 3, 2024, 4:16 AM
Y by
CatsMeow12 wrote:
miyukina wrote:
not sure

I got 4 fewer possibilities... but your solution was almost identical to mine

If we both get 8 pairs with slopes of ±1/2 and ±2 , where do you think we differ at?
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CatsMeow12
2254 posts
CatsMeow12
#9 h  Jul 3, 2024, 4:18 AM
Y by
miyukina wrote:
CatsMeow12 wrote:
miyukina wrote:
not sure

I got 4 fewer possibilities... but your solution was almost identical to mine

If we both get 8 pairs with slopes of ±1/2 and ±2 , where do you think we differ at?

In the middle term of the numerator, the second term is 1 x 5... I only got one possibility there. You might be correct though, what does the 1 x 5 term represent?
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miyukina
1219 posts
miyukina
#10 h  Jul 3, 2024, 4:25 AM
Y by
CatsMeow12 wrote:
miyukina wrote:
CatsMeow12 wrote:

I got 4 fewer possibilities... but your solution was almost identical to mine

If we both get 8 pairs with slopes of ±1/2 and ±2 , where do you think we differ at?

In the middle term of the numerator, the second term is 1 x 5... I only got one possibility there. You might be correct though, what does the 1 x 5 term represent?

The longest diagonal (0, 0) to (2, 2) paired with
1) the other longest diagonal
2) (0, 1) to (1, 0)
3) (0, 2) to (1, 1)
4) (1, 1) to (2, 0)
5) (1, 2) to (2, 1)
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CatsMeow12
2254 posts
CatsMeow12
#11 h  Jul 3, 2024, 7:35 PM
Y by
miyukina wrote:
The longest diagonal (0, 0) to (2, 2) paired with
1) the other longest diagonal
2) (0, 1) to (1, 0)
3) (0, 2) to (1, 1)
4) (1, 1) to (2, 0)
5) (1, 2) to (2, 1)

Cases (2) and (5) should be already covered by the 4 x 3 term, but cases (3) and (4) weren't in my solution... these two cases can also be applied by picking the other diagonal, so your answer is actually correct :-D
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