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Prove that every convex polyhedron
has at least two faces with the same number of edges.
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There are \(2009\) committees, each with \(45\) members,
any two of which have exactly one person in common.
Show that there is one person who is a member of all the committees.
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Consider an equilateral triangle having side-lengths one inch,
and any set of five points on the interior of that triangle.
Show that two of those points must be within one-half inch of each other.
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Consider the set \(S = \{1,2,\dotsc, 2n\}\).
Prove that any \((n+1)\)-element subset of \(S\)
contains two members that are relatively prime.
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Consider the set \(S = \{1,2,\dotsc, 2n\}\).
Prove that any \((n+1)\)-element subset of \(S\)
contains two members such that one is a multiple of the other.
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St. Petersburg City Math Olympiad 2001 (P&B)
In each square of a \(10\times 10\) checkerboard,
a positive integer from the set \(\{2,3,\dotsc,11\}\) is written
in such a way that any two numbers that appear
in adjacent or diagonally adjacent squares of the board
don’t share a common factor greater than \(1\).
Prove that some number appears at least \(17\) times.
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Let \(G\) be a graph with \(n\) vertices and \(k\) edges,
such that the edges are labelled \(1,\dotsc, k\) in some fashion.
Prove that there must exist a path in \(G\) (with repeated vertices allowed)
of length at least \(2k/n\) for which the labels of edges
in the path occur in increasing order.
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Prove that among any seven real numbers \(x_1, x_2,\dotsc,x_7\),
there must be two, \(x_i\) and \(x_j\), such that
\[ 0 \;\leq\; \frac{x_i-x_j}{1+x_ix_j} \;\leq\; \frac{1}{\sqrt{3}} \]
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Given a positive integer \(n,\) what is the largest \(k\)
such that the numbers \(1,2,\dotsc,n\) can be put into \(k\) boxes
so that the sum of the numbers in each box is the same?
[When \(n=8,\) the example \(\{1,2,3,6\},\{4,8\},\{5,7\}\)
shows that the largest \(k\) is at least \(3.\)]
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Let \(a_j,\) \(b_j,\) \(c_j\) be integers for \(1 \leq j \leq N.\)
Assume for each \(j,\) at least one of \(a_j,\) \(b_j,\) \(c_j\) is odd.
Show that there exist integers \(r,\) \(s,\) \(t\) such that
\(ra_j +sb_j +tc_j\) is odd for at least \(4N/7\) values of \(j,\) \(1 \leq j \leq N.\)
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Prove that for every set \(X = \{x_1,x_2,\dotsc,x_n\}\)
of \(n\) real numbers, there exists a non-empty subset \(S\) of \(X\)
and an integer \(m\) such that
\[ \left|m + \sum_{s \in S}s\right| \;\;\leq\;\;\frac{1}{n+1}\,. \]
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There are \(2010\) boxes labeled \(B_1, B_2, \dotsc, B_{2010}\).
For some positive integer \(n\), a total of \(2010n\) balls
have been distributed among the boxes in some way.
You may redistribute the balls by a sequence of moves of this form:
choose an \(i\) and move exactly \(i\) balls
from box \(B_i\) to another box of your choice.
For which values of \(n\) is it possible,
through a sequence of these moves,
to reach the distribution with exactly \(n\) balls in each box
regardless of the initial distribution of balls?