Tagged Questions

Problems from or inspired by mathematics competitions. Questions regarding mathematics competitions.

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Is it possible to permute an unknown binary sequence so that two particular bits are equal? [closed]

A blind mathematician is give a $2015$ bit sequence. The mathematician can take any two bits and switch them (so the bit in position $A$ goes to position $B$ and vice-versa). He knows at what position ...
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How to find Bitwise AND of all numbers for a given range?

How can I find Bitwise AND of all numbers for a given range say from A to B, including both? I found a beautiful answer for finding XOR for such range. http://stackoverflow.com/a/10670524/2046703How ...
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Where can Gaussian Elimination be used?

I have searched for this and came to know about it that it is traditionally used to solve linear equations, finding determinant, rank of matrix, inverse of matrix. There was a problem on codechef: ...
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How prove this the number of ordered $n$-tuples $(\varepsilon_{1},\cdots,\varepsilon_{n})$such this following inequality is $2^{n-100}$

Interesting Question: for any complex numbers $z_{1},z_{2},\cdots,z_{n}$ such $$\begin{cases} |z_{1}|^2+|z_{2}|^2+\cdots+|z_{n}|^2=1\\ |z_{i}|\le\dfrac{1}{10},i=1,2,\cdots,n \end{cases}$$ ...
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Motivation for Putnam (soft question)

This question may be too specific and too vague. But I'm curious about this. How highly are the applicants evaluated in PhD admission if they were ranked above the cutoff of honorable-mention in ...
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Set with distinct subset sums

The problem is as follows : Given a set A with distinct positive integer elements, prove that there always exists another set B consisting of positive integers, s.t., The size of B is less than or ...
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$2x^2+ 3y^2+4z^2 =1$ find the maximum of $4x+3y+2z$

If $2x^2+ 3y^2+4z^2 =1$ find the maximum of $4x+3y+2z$. This is a question from a regional math olympiad and thus there must exist solutions without application of calculus. I have no idea how to ...
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Do two altitudes uniquely determine the third

BdMO 2014 Nationals If the lengths of two altitudes drawn from two vertices of a triangle on their opposite sides are $2014$ and $1$ unit, then what will be the length of the altitude drawn from ...
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When does the Putnam release solutions to this year's exam? Has anyone released their own solutions?

I was just wondering when the Putnam committee releases the solutions to this year's exam or if anyone has posted their own solutions.
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Finding the invariant

There are A white, B black, and C red chips on a table. In one step, you may choose two chips of different colors and replace them by a chip of the third color. If just one chip will remain at the ...
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No 37 question in knight of pi math tournament Dec 15, 2012

The five digit integer ABCDE, where each letter represents a digit, not necessary distinct, is divided by the numbers $2$,$3$,$4$,$5$, and $6$. The remainders are A, B, C, D, and E respectively. What ...
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How many consecutive squares can be subtracted from a number?

Let's say I am given a number N. I want to check how many consecutive squares of integers(starting from 1) can be subtracted from this number. Example- For N=13, I will first subtract 1(=1^2), leaves ...
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Taking a Putnam (General Questions) [duplicate]

I've just discovered an undergrad math competition (William Lowell Putnam Competition) and that my school offers it. The competition looks extraordinarily difficult, but I thought I'd give it a go. ...
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Find if there exist some combination of these digits that will be divisible by 8 or not

Let's say I am given some 100 digits and I have to find whether there can be any combination of these digits such that the number formed will be divisible by 8, how can I do that? I know divisibility ...
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How can I determine the center of this circumference?

I have the following question: if I have an irregular symmetric polygon, how can I determinate the circumference with the least area that contains this polygon? I believe (in case that the polygon ...
How many lines in a three dimensional rectangular coordinate system pass through four distinct points of the form $(i, j, k)$ where $i$, $j$, and $k$ are positive integers not exceeding four? ...