0
votes
1answer
47 views

time and distance

Dexter and Prexter are competing with each other in a friendly community competition in a pool of 50m length and the race is for 1000m. Dexter crosses 50m in 2 min and Prexter in 3 min 15 sec. Each ...
0
votes
2answers
39 views

Is my solution of this time and distance problem correct or wrong?

P and Q start running in opposite directions (towards each other) on a circular track starting at diametrically opposite points. They first meet after P has run for 75m and then they next meet after Q ...
1
vote
0answers
41 views

Prove that eventually Hannah and the other swimmer will settle into a pattern where they pass each other (Please refer to the context in my question)

From the 2014 Mathcamp quiz: Hannah is about to get into a swimming pool in which every lane already has one swimmer in it. Hannah wants to choose a lane in which she would have to encounter the other ...
3
votes
2answers
338 views

Putnam $\bf 2001$ problem A$\bf 1$ (Binary operation)

Let $*$ be a binary operation acting on a non-empty set $S$, such that $$(a*b)*a=b,$$ for all $a,b\in S$. Prove that $$a*(b*a)=b,$$ for all $a,b \in S$.
0
votes
1answer
65 views

Proving using squeeze principle

This problem sounds very confusing. Please help me solve this problem.
1
vote
1answer
57 views

Prove the derivative

Let $f(x) = (x^2-1)^{\frac{1}{2}}, x>1$. How do I prove that the $n$th derivative of $f(x) > 0$ for odd $n$, and the $n$th derivative of $f(x) < 0$ for even $n$?
1
vote
0answers
22 views

Convex quadrilateral

In a convex quadrilateral (the two diagonals are interior to the quadrilateral) prove that the sum lengths of the diagonals is less than the perimeter but great than one-half the perimeter.
2
votes
1answer
103 views

Fifteen pennies lie on the table in the shape of a triangle

Fifteen pennies lie on the table in the shape of a triangle, with five pennies on each side. For some reason, the pennies are painted either black or white. Prove that there exist three pennies of ...
0
votes
1answer
94 views

What is the meaning of $(x^2+y^2)^n$? Is this an already known geometric object?

We all know that $x^2+y^2=r^2$ is a circle. What does $(x^2+y^2)^2$ signify? In general, what is $(x^2+y^2)^n$?
-1
votes
1answer
69 views

Please help me solve this problem [closed]

Every month, a girl gets an allowance. Assume one year ago she had no money, and so saved each month's allowance over the past year. Then, she spends $\frac{1}{2}$ of her money on clothes; then ...
7
votes
1answer
175 views

A functional relation which is satisfied by $\cos x$ and $\sin x$

Assume that the functions $f,g : \mathbb R\to \mathbb R$ satisfy the relations \begin{align} \left\{ \begin{array}{ll} f(x+y) &=& f(x)f(y)-g(x)g(y), \\ g(x+y) &=& f(x)g(y)+f(y)g(x), ...
1
vote
3answers
659 views

Find the four digit number?

Find a four digit number which is an exact square such that the first two digits are the same and also its last two digits are also the same.
4
votes
1answer
193 views

No primes in this sequence

Here's a fun little problem: Prove that the sequence $$10001, 100010001, 1000100010001, \cdots$$ contains no prime numbers.
1
vote
2answers
50 views

Recursive formular and closed-form questions

Follow the question the $f(n)=4n-1$ and $F(n)=\sum_{k=0}^nf(k)$. And it ask you to write the recursive of $F(n)$. But I only know the recursive of $f(n)$ is $$f(n)=\begin{cases} -1,&\text{if ...
0
votes
1answer
65 views

Conditions for convergence of a geometric series [duplicate]

This question concerns the infinite geometric series formula. It turns out there is a nice formula for the sum of an infinite geometric series. Consider the infinite geometric series ...
0
votes
0answers
21 views

Number Of Triangles of All Sizes in an Equilateral Triangle [duplicate]

https://mail.google.com/mail/u/0/?ui=2&ik=4622e6803e&view=att&th=1422d3806080ed0d&attid=0.1&disp=emb&realattid=ii_1422415e014f71c5&zw&atsh=1 Consider an ...
5
votes
1answer
186 views

Tricky Puzzle!! Please help.

I stumbled upon a puzzle I can't crack. It goes like this: In a certain Code language: 7321=6 5342=3 8645=15 Then 9312=? The Answer is 9. But I can't seem to find the logic behind it??
0
votes
1answer
62 views

Partition of circumference into $3k$ arcs

The following problem is from 1982 Russian Mathematical Olympiad. If you go to this link, and scroll down to the section Russian Math Olympiad, then this is Problem 333 in that text-file. Let $k$ ...
9
votes
5answers
415 views

“If $1/a + 1/b = 1 /c$ where $a, b, c$ are positive integers with no common factor, $(a + b)$ is the square of an integer”

If $1/a + 1/b = 1 /c$ where $a, b, c$ are positive integers with no common factor, $(a + b)$ is the square of an integer. I found this question in RMO 1992 paper ! Can anyone help me to prove ...
1
vote
2answers
375 views

A truth teller and liar puzzle of Ramanujan mathematical olympiad 2013

On an island each person always tells the truth or each person always tells a lie. Three people say $A$ , $B$ and $C$ have a conversation. $A$ says that $B$ is lying , $B$ says that $C$ is lying and ...
-4
votes
3answers
196 views

In how many ways can $16 be divided among 4 people? [closed]

In how many ways can $16 be divided among 4 people, assuming that each person has to get something and there are 5 cent coins and up
2
votes
1answer
127 views

Where can I find Putnam competition questions and solutions online?

Math people: Until recently, at least, there existed at least one Web page containing complete Putnam competition problems and solutions from the past twenty years or so. In retrospect, I see that I ...
0
votes
1answer
97 views

Is it appropriate to use conjectures in contest?

Is it appropriate to use conjectures in math contest? I've been to fair amount of math contest and as far as I know the judges want a solid proof for every step take to make to solve the problem. But ...
12
votes
5answers
817 views

Simplify : $( \sqrt 5 + \sqrt6 + \sqrt7)(− \sqrt5 + \sqrt6 + \sqrt7)(\sqrt5 − \sqrt6 + \sqrt7)(\sqrt5 + \sqrt6 − \sqrt7) $

The question is to simplify $( \sqrt 5 + \sqrt6 + \sqrt7)(− \sqrt5 + \sqrt6 + \sqrt7)(\sqrt5 − \sqrt6 + \sqrt7)(\sqrt5 + \sqrt6 − \sqrt7)$ without using a calculator . My friend has given me ...
2
votes
1answer
424 views

Throw a die three times, and get maximum number of different sums.

The IBM Ponder This problem for July 2013 throws an 8 sided die 3 times, and can get 120 possible different positive integer sums. If all the faces have positive integer sides, what is the lowest ...
0
votes
4answers
341 views

How to verify method used to solve integral was actually the fastest?

Is there any way to verify if the method I chose to integrate (by hand) was indeed fastest, or if there exists some better technique? Can a computer tell me or show me what the fastest method was, ...
1
vote
1answer
122 views

A follow-up question on an arithmetic function satisfying a certain inequality

In the MSE question here, I asked whether the inequality: $$\frac{a}{b} + \frac{b}{a} < \frac{f(a)}{f(b)} + \frac{f(b)}{f(a)}$$ would imply $a < b$ (where $f(x) \in \mathbb{N}$ is a function ...
1
vote
1answer
118 views

A quick question on general mathematics

I have the following question that I am currently unable to satisfactorily answer myself. My question is: Does the inequality $$\frac{a}{b} + \frac{b}{a} < \frac{f(a)}{f(b)} + ...
3
votes
3answers
732 views

Cyclic sums — How do you use them?

Can someone give me an example of how cyclic sums are used? I don't really understand how they're used in problem-solving. For example, $$\sum_{a,b,c}a^2$$ Any help would be appreciated, and I'm not ...
15
votes
5answers
919 views

Is high school contest math useful after high school?

I've been prepping for a lot of high school math competitions this year, and I was just wondering if all the math I learn would actually mean something in college. There is a chance that all of it ...
1
vote
2answers
93 views

combinations problem about apples and pears

Carlo has six apples and six pears: how many ways he can set in a row 6 fruits so that there should never be a pear between two apples? Thanks in advance to everyone who will help me resolving this ...
1
vote
1answer
362 views

Multiplication Table with a frame and picture of equal sum

Is there an $n \times n$ multiplication table such that if you form a border of width $k$ ("the frame") and sum its elements, the total will equal the sum of the remaining elements ("the picture")? ...
2
votes
3answers
669 views

How many unit hexagonal tiles can be placed inside a larger hexagon of sides $a,b,c,a,b,c$?

How many unit hexagonal tiles can be placed inside a larger hexagon of sides $a,b,c,a,b,c$? I originally came across this puzzle on the codeforces website. My first question: what is the mathematics ...
4
votes
2answers
376 views

Counting ordered triples of non-negative integers not greater than 100

Can we find the number of ordered triples $(x,y,z)$ of non-negative integers satisfying (i) $x \leq y \leq z$ (ii) $x + y + z \leq 100$? Source:Regional Mathematics Olympiad India (2003) Thank you.I ...