# Questions tagged [contest-math]

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

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### Australian Maths Competition Area Question

PQRS is a square. T and U are midpoints of the sides PS and PQ respectively. TQ, SU and PR intersect at V. This is a question from the 2009 Intermediate Division AMC paper. I was given it in a ...
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### $n$ and $m$ are integers and $m >1$. Given: $2^n −2=m(m+1)$, Prove that $n$ can’t be an even number . [on hold]

$n$ and $m$ are integers and $m >1$ we have : $2^n −2=m(m+1).$ Prove that $n$ can’t be an even number .
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### Finding number of cycles in permutation corresponding to translation in a group [University Olympiad]

here is a question from our university olympiad practice manual which is intended to help us prepare for upcoming inter-university math Olympiad. I think I can see, solution would be similar to Proof ...
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### Prove that $A, P, Q$ are collinear.

Two circles $\omega_1$, $\omega_2$ intersect at $A, B$. An arbitrary line through $B$ meets $\omega_1$, $\omega_2$ at $C, D$ respectively. The points $E, F$ are chosen on $\omega_1$, $\omega_2$ ...
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### Providing divisibility condition given fraction identity

If $x,y,z$ are positive integers satisfying $$\frac{1}{x^2}+\frac{1}{y^2}=\frac{1}{z^2}$$ prove that $20{\,\mid\,}xy$. My work: Expanding, we find $$(xz)^2+(yz)^2=(xy)^2$$ I know the Pythagorean ...
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### Minimising expression with absolute values

Let $x,y,z$ be distinct reals and consider the expression $$L=\frac{(|x|+|y|+|z|)^3}{|(x-y)(x-z)(y-z)|}$$ Find the minimum possible value of $L$ over all $(x,y,z)$. My work Using a calculator, the ...
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### Tips for discrete mathematics in Contests and a example problem.

In the last time I did lot's of preparations for future math contests and I discovered a problem where I don't know how to start with. Here is the problem: In a spa there are 100 showers. In ...
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### Find the number of polynomials $P(x)$ with coefficients $0,1,2,3$ such that $P(2) = n$.

Find the number of polynomials $P(x)$ with coefficients $0,1,2,3$ such that $P(2) = n$. Only what I currently know is that I should consider polynomials with the smallest degree $k$, where $k$ is ...
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### Perfect squares of the form $ab^n+c$ and a Diophantine equation

The motivation for this question comes from the following problem from an international Team selection test of 2007 from Chile: Problem: Let $p$ be a prime number. Find all pairs of positive ...