# Tagged Questions

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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $a,b \in S$.
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### 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 $n$?
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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.
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### 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 ...
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### 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 $(x^2+y^2)^n$?
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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 ...
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### 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), ...
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### 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.
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### No primes in this sequence

Here's a fun little problem: Prove that the sequence $$10001, 100010001, 1000100010001, \cdots$$ contains no prime numbers.
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### 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 ...
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### 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 ...
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### 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 ...
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### 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")? ...
### 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 ...
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 ...