# Tagged Questions

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

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### Set Theory problem with unique numbers

Let $A_0$ be the set {$1, 2, 3, 4$}. Let $A_{i+1}$ be the set of all possible sums which can be obtained by adding two numbers in $A_i$ , where the two numbers do not have to be different. How many ...
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### If $abc=1$ then $\sum_{cyc}^{}{\frac{1}{b(a+b)}}\ge \frac{3}{2}$

If $abc=1$ for positive $a,b,c$, then $\sum_{cyc}^{}{\dfrac{1}{b(a+b)}}\ge \dfrac{3}{2}$ I have tried the following,in decreasing order of success: 1)AM-GM:$a+b+c\ge 3$ and $ab+bc+ca\ge 3$ ...
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### Two Place Position and Model Question

! i get trouble in one multiple choice question in logic course: any one could help me with some description ? if we have Two-place position predicate, like : 1) all models of $\varphi$ is ...
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### First Order Logic and Some Validity Checking

I'm sorry for put an image insted of typing it... infact this is an 2012-exam on Logic. i found the solution of this quiz that wrote by one TA. he wrote just the second line is not valid logically in ...
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### In how many ways can we distribute 6 identical pears?

In how many ways can we distribute 6 identical pears between 3 children so that each child receives at least one pear? I am not too sure. I thought, 6 ways to distribute to first, 5 ways to second, ...
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### Does performance in math competitions accurately reflect natural aptitude in mathematics? [closed]

Many great and respected mathematicians have won accolades in math (ex: IMO), does that necessarily mean that these competitions reflect one's potential to be a great mathematician?
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### Using two coins to select a person fairly.

Good evening, I would like to know if the solution to this problem, I know it can be solved because it is from a Hungarian Olympiad. The problem is as follows: You need to fairly select a person ...
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### Given $A \subseteq \mathbf{Z}$ and $x\in \mathbf{Z}$, we say that $x$ is $A$-mirrored if and only if $−x\in A$. We also define…

Sorry if this question seems kind of long but I am confused for part C. My proof for part C that $M_a$ is closed under addition is as follows: The set $M_a$ is closed. Let $x$ be in $M_a$ and ...
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### Number of paths in 3D coordinates

A cute problem which is an extension of a well-known counting problem: Find the number of paths of length $12$ from $(0,0,0)$ to $(4,4,4)$ passing through adjacent lattice points (for two ajacent ...
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### Functional equation defined over non-negative real numbers

I'm new to this forum and I don't know how to write mathematical symbols. I have the following functional equation: $f$ defined on $[0, +\infty)$ with values in $[0, +\infty)$ $f$ is bijective and ...
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### Prove that $\left( \frac{M+z_2+\dots+z_{2n}}{2n} \right)^2\ge\left( \frac{x_1+\dots+x_n}{n} \right)\left(\frac{y_1+\dots+y_n}{n} \right).$

Let $n$ be a positive integer and let $(x_1,\ldots,x_n)$, $(y_1,\ldots,y_n)$ be two sequences of positive real numbers. Suppose $(z_2,\ldots,z_{2n})$ is a sequence of positive real numbers such that ...
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### Sequence of non-collinear integer points.

This is a question from a British Olympiad, I've completed the first 3 but this one had me rather stumped. Given two points $P$ and $Q$ with integer coordinates, we say that $P$ sees $Q$ if the ...
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### Arranging identical balls in a circle

In how many ways can 4 identical red balls and two identical white balls be arranged in a circle? This is an elementary problem, but many tries have not yet yielded results. I tried by taking the ...
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### 2003 Putnam A-1 Help needed about sequences

Okay so for $n=1$ there is only one way. For $n=2$ you have, $1+1, 2 + 0$ for $n=3$ you have: $1+1+1, 1+ 2, 3 + 0$ three ways. So $P(n): n$ ways, we must prove the $P(n+1): n + 1$ statement is ...
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### How to use totient function here?

I have asked this before, but I had no idea how to use Totient, now I do here is the questions: How many positive integers $< 2013$ cannot be divided by $2, 3, 5$ ?? An advice given was find ...
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### Find all points on the line 9x-21y=6

For this equation we are suppose to use the Euclidean Algorithm. But I run into a problem For the GCD (9,-21)= i tried 9=(-21)(0)+9 -21=9(3)+6 9=6(1)+3 6=3(2) +0 which gives a gcd of 3 and the ...
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### Arc length contest! Minimize the arc length of $f(x)$ when given three conditions.

Contest: Give an example of a continuous function $f$ that satisfies three conditions: $f(x) \geq 0$ on the interval $0\leq x\leq 1$; $f(0)=0$ and $f(1)=0$; the area bounded by the graph of $f$ and ...