IMO 1967

6 problems with complete solutions · Official problem paper (PDF)

  1. Problem 1

    Let \(ABCD\) be a parallelogram with side lengths \(AB = a, AD = 1\), and with \(\angle BAD = \alpha\). If \(\triangle ABD\) is acute, prove that the four circles of radius \(1\) with centers \(A, B, C, D\) cover the parallelogram if and only if \[ a \le \cos \alpha + \sqrt{3}\sin \alpha. \]

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  2. Problem 2

    Prove that if one and only one edge of a tetrahedron is greater than 1, then its volume is \(\le 1/8\).

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  3. Problem 3

    Let \(k,m,n\) be natural numbers such that \(m+k+1\) is a prime greater than \(n+1\). Let \(c_s=s(s+1)\). Prove that the product \[ (c_{m+1}-c_k)(c_{m+2}-c_k)\cdots(c_{m+n}-c_k) \] is divisible by the product \(c_1c_2\cdots c_n\).

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  4. Problem 4

    Let \(A_0B_0C_0\) and \(A_1B_1C_1\) be any two acute-angled triangles. Consider all triangles \(ABC\) that are similar to \(\Delta A_1B_1C_1\) (so that vertices \(A_1,B_1,C_1\) correspond to vertices \(A,B,C\), respectively) and circumscribed about triangle \(A_0B_0C_0\) (where \(A_0\) lies on \(BC\), \(B_0\) on \(CA\), and \(AC_0\) on \(AB\)). Of all such possible triangles, determine the one with maximum area, and construct it.

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  5. Problem 5

    Consider the sequence \(\{c_n\}\), where \[ \begin{aligned} c_1 &= a_1+a_2+\cdots+a_8\\ c_2 &= a_1^2+a_2^2+\cdots+a_8^2\\ &\quad\cdots\\ c_n &= a_1^n+a_2^n+\cdots+a_8^n\\ &\quad\cdots \end{aligned} \] in which \(a_1,a_2,\cdots,a_8\) are real numbers not all equal to zero. Suppose that an infinite number of terms of the sequence \(\{c_n\}\) are equal to zero. Find all natural numbers \(n\) for which \(c_n=0\).

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  6. Problem 6

    In a sports contest, there were \(m\) medals awarded on \(n\) successive days (\(n > 1\)). On the first day, one medal and \(1/7\) of the remaining \(m - 1\) medals were awarded. On the second day, two medals and \(1/7\) of the now remaining medals were awarded; and so on. On the \(n\)-th and last day, the remaining \(n\) medals were awarded. How many days did the contest last, and how many medals were awarded altogether?

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