# Tagged Questions

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

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### Roots are the reciprocal of $f(x)$

I don't understand if $f(x)$ has roots, $r_1, r_2$ for example and $g(x)$ has roots $\frac{1}{r_1}, \frac{1}{r_2}$ Then how is $g(x) = x^2f(\frac{1}{x})$ What does $$f(\frac{1}{x})$$ have to do ...
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### Cognitive processes involved solving IMO level problems [closed]

I am currently 16 years old and, though I'm obviously not as good as most of the people on this site, I have always been considerably better than most of my classmates in mathematics. This, of course, ...
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### Prove this Complicated Inequality

Let $a$, $b$, $c$ be positive real numbers such that $a^2 + b^2 + c^2 + (a + b + c)^2 \le 4$. Prove that $$\frac{ab + 1}{(a + b)^2} + \frac{bc + 1}{(b + c)^2} + \frac{ca + 1}{(c + a)^2} \ge 3.$$ ...
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### What is wrong with this proof of a number theory competition problem?

Let $a$ and $b$ be positive integers. Suppose $a^n+n| b^n+n$ for any positive integer $n$, prove that $a=b$. My trial: Clearly $b\geq a$, write $b=a+d$, we must show that $d=0$. Now by assumption and ...
313 views

### Increasing sequence of divisors of a quadratic trinomial

This question is from a Russian contest, the 2011 Tuymaada Olympiad. It's the fourth question on day two for the problems at grade level 2. Let $P(n)$ be a quadratic trinomial with integer ...
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### Picking K counters out of K buckets containing NK counters, N of each different colour, up to N in each

This is a generalisation of a question that recently came up while solving a TopCoder problem. Suppose we have N blue counters, N red counters, N white counters, and so forth, K colours in total. We ...
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### My answer to this combi problem doesn't match the answer in the book (Problem-Solving Strategies)

[Problems 31 and 32 from Arthur Engel's Problem-Solving Strategies.] Let $n$ children be seated in a line. How many ways can they change their places if they may only move by one place at most? ...
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### Evaluate powers in fraction

This is abstracted from 2007 British Mathematical Olympiad Question 1.I wish to practice mathematics olympiad question for the upcoming Singapore Mathematics Olympiad Secondary 2 (Grade 8). Find ...
5k views

### How to solve 3 variable in 2 equation?

This paper is abstracted from 2007 British Mathematics Olympiad Round 1 Question 2. I am currently practicing grade 8 (Singapore Secondary 2) for the upcoming Singapore Mathematics Olympiad(SMO). ...
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### Accurate Formula and One Old-Exam Questions?! [closed]

We get stuck in a problem on old-exam. \begin{equation*} A=\sqrt{x+ \frac {2}{x}} -\sqrt{x- \frac {2}{x}}~\text{and}~x>>1. \end{equation*} For calculating $A$ which of the following option ...
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### How long will it take for one of them or both of them?

One knight can storm a castle in 15 days. He and his partner can do it in 10 days. How long does it take the partner to storm the same castle alone? Pipe A can fill a pool in 5 hours, while pipe B ...
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### Finding the number of real roots of an unusual(!) equation [closed]

How many real roots does the below equation have? \begin{equation*} \frac{x^{2000}}{2001}+2\sqrt{3}x^2-2\sqrt{5}x+\sqrt{3}=0 \end{equation*} A) 0 B) 11 C) 12 D) 1 E) None of these I could not come ...
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### MOSP $2002$ Combinatorics Problem

I only want a hint(I already have the solution near me, but the book doesn't give a hint (MOSP) Assume that each of the $30$ MOPpers has exactly one favorite chess variant and exactly one ...
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### Use Lagrange Interpolation polynomial to find this $\sum_{cyc}\frac{x^3}{(x^2-y^2)(x^2-z^2)}$

Let $x,y,z$ be the solutions of the equation $t^3-t^2+2t-3=0$. Find the sum $$\dfrac{x^3}{(x^2-y^2)(x^2-z^2)}+\dfrac{y^3}{(y^2-x^2)(y^2-z^2)}+\dfrac{z^3}{(z^2-x^2)(z^2-y^2)}$$ How can I use the ...
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### Logic problems : references

I'm looking for problems from mathematical contests about logic (similar to the problem PMWC Problem T5).
I am getting crazy with this one. Suppose $a_n=(1^2+2^2+3^2+\ldots+n^2)^n$ and $b_n=n^n(n!)^2$. Show that $a_n>b_n$ for all $n$. They suggest to use the AM-GM inequality.