IMO 1997

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

  1. Problem 1

    In the plane the points with integer coordinates are the vertices of unit squares. The squares are colored alternately black and white (as on a chessboard).

    For any pair of positive integers \(m\) and \(n\), consider a right-angled triangle whose vertices have integer coordinates and whose legs, of lengths \(m\) and \(n\), lie along edges of the squares.

    Let \(S_1\) be the total area of the black part of the triangle and \(S_2\) be the total area of the white part. Let \[ f(m,n)=|S_1-S_2|. \]

    (a) Calculate \(f(m,n)\) for all positive integers \(m\) and \(n\) which are either both even or both odd.

    (b) Prove that \(f(m,n)\leq \frac12\max\{m,n\}\) for all \(m\) and \(n\).

    (c) Show that there is no constant \(C\) such that \(f(m,n)<C\) for all \(m\) and \(n\).

    Solution →

  2. Problem 2

    The angle at \(A\) is the smallest angle of triangle \(ABC\). The points \(B\) and \(C\) divide the circumcircle of the triangle into two arcs. Let \(U\) be an interior point of the arc between \(B\) and \(C\) which does not contain \(A\). The perpendicular bisectors of \(AB\) and \(AC\) meet the line \(AU\) at \(V\) and \(W\), respectively. The lines \(BV\) and \(CW\) meet at \(T\). Show that \[ AU = TB + TC. \]

    Solution →

  3. Problem 3

    Let \(x_1,x_2,\ldots,x_n\) be real numbers satisfying the conditions \[ |x_1+x_2+\cdots+x_n|=1 \] and \[ |x_i|\leq \frac{n+1}{2}\qquad i=1,2,\ldots,n. \] Show that there exists a permutation \(y_1,y_2,\ldots,y_n\) of \(x_1,x_2,\ldots,x_n\) such that \[ |y_1+2y_2+\cdots+ny_n|\leq \frac{n+1}{2}. \]

    Solution →

  4. Problem 4

    An \(n \times n\) matrix whose entries come from the set \(S=\{1,2,\ldots,2n-1\}\) is called a silver matrix if, for each \(i=1,2,\ldots,n\), the \(i\)th row and the \(i\)th column together contain all elements of \(S\). Show that

    (a) there is no silver matrix for \(n=1997\);

    (b) silver matrices exist for infinitely many values of \(n\).

    Solution →

  5. Problem 5

    Find all pairs \((a,b)\) of integers \(a,b\geq 1\) that satisfy the equation \[ a^{b^2}=b^a. \]

    Solution →

  6. Problem 6

    For each positive integer \(n\), let \(f(n)\) denote the number of ways of representing \(n\) as a sum of powers of \(2\) with nonnegative integer exponents. Representations which differ only in the ordering of their summands are considered to be the same. For instance, \(f(4)=4\), because the number \(4\) can be represented in the following four ways:

    \[ 4;\ 2+2;\ 2+1+1;\ 1+1+1+1. \]

    Prove that, for any integer \(n\geq 3\),

    \[ 2^{n^2/4}<f(2^n)<2^{n^2/2}. \]

    Solution →

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