Congruences & Divisibility
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Prove that among any five distinct integers
there are always three with sum divisible by three.
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Suppose that \(a\) and \(b\) are integers.
Prove that each of these expressions is never zero:
\( a^2-4b-2 \) and \(a^2-4b-3. \)
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Moscow Mathematical Olympiad (1995–1996)
Prove that in the product \(1!\,2!\,3!\,\dotsb\,100!,\)
one of the factors can be erased
so that the remaining product is a perfect square.
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Prove that the equation \(x^2 + y^2 - 3 = 0\)
has no solutions \((x,y)\) such that \(x\) and \(y\)
are simultaneously rational numbers.
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Putnam & Beyond
Show that the number \(2002^{2002}\) can be written
as the sum of four perfect cubes,
but not as the sum of three perfect cubes.
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Russian Mathematical Olympiad (1999)
Show that each positive integer
can be written as the difference of two positive integers
having the same number of prime factors.
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Prove: among any ten consecutive integers,
at least one is relatively prime to each of the others.
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Determine all positive integers \(n\)
for which there exist positive integers \(a,\) \(b,\) and \(c\)
satisfying \(2a^n+3b^n = 4c^n.\)
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Show that every positive rational number can be written
as a quotient of products of factorials
of (not necessarily distinct) primes.
E.g.
\[ \frac{10}{9} = \frac{2!\cdot 5!}{3!\cdot 3! \cdot 3!}. \]
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Find all ordered pairs \((a,b)\)
or positive integers for which
\[\frac 1 a + \frac 1 b = \frac{3}{2018}\,.\]
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What is the unit’s digit (right-most digit) of this number?
\[ \left\lceil \frac{10^{20000}}{10^{100}+3} \right\rceil \]
(Note that \(\lceil x \rceil\) denotes the ceiling function
which returns the greatest integer less than or equal to \(x.\))
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Putnam & Beyond
Prove that if \(n\) is a positive integer
that is divisible by at least two distinct primes,
then there exists an \(n\)-gon with all angles equal
and with side lengths the numbers \(1,\) \(2,\)…, \(n\)
in some order.