IMO 2023

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

  1. Problem 1

    Determine all composite integers \(n>1\) that satisfy the following property: if \(d_1,d_2,\ldots,d_k\) are all the positive divisors of \(n\) with \(1=d_1<d_2<\cdots<d_k=n\), then \(d_i\) divides \(d_{i+1}+d_{i+2}\) for every \(1\leq i\leq k-2\).

    Solution →

  2. Problem 2

    Let \(ABC\) be an acute-angled triangle with \(AB<AC\). Let \(\Omega\) be the circumcircle of \(ABC\). Let \(S\) be the midpoint of the arc \(CB\) of \(\Omega\) containing \(A\). The perpendicular from \(A\) to \(BC\) meets \(BS\) at \(D\) and meets \(\Omega\) again at \(E\ne A\). The line through \(D\) parallel to \(BC\) meets line \(BE\) at \(L\). Denote the circumcircle of triangle \(BDL\) by \(\omega\). Let \(\omega\) meet \(\Omega\) again at \(P\ne B\). Prove that the line tangent to \(\omega\) at \(P\) meets line \(BS\) on the internal angle bisector of \(\angle BAC\).

    Solution →

  3. Problem 3

    For each integer \(k \geqslant 2\), determine all infinite sequences of positive integers \(a_1,a_2,\ldots\) for which there exists a polynomial \(P\) of the form \(P(x)=x^k+c_{k-1}x^{k-1}+\cdots+c_1x+c_0\), where \(c_0,c_1,\ldots,c_{k-1}\) are non-negative integers, such that \[ P(a_n)=a_{n+1}a_{n+2}\cdots a_{n+k} \] for every integer \(n\geqslant 1\).

    Solution →

  4. Problem 4

    Let \(x_1,x_2,\ldots,x_{2023}\) be pairwise different positive real numbers such that \[ a_n=\sqrt{(x_1+x_2+\cdots+x_n)\left(\frac1{x_1}+\frac1{x_2}+\cdots+\frac1{x_n}\right)} \] is an integer for every \(n=1,2,\ldots,2023\). Prove that \(a_{2023}\ge3034\).

    Solution →

  5. Problem 5

    Let \(n\) be a positive integer. A Japanese triangle consists of \(1+2+\cdots+n\) circles arranged in an equilateral triangular shape such that for each \(i=1,2,\ldots,n\), the \(i^{\mathrm{th}}\) row contains exactly \(i\) circles, exactly one of which is coloured red. A ninja path in a Japanese triangle is a sequence of \(n\) circles obtained by starting in the top row, then repeatedly going from a circle to one of the two circles immediately below it and finishing in the bottom row. Here is an example of a Japanese triangle with \(n=6\), along with a ninja path in that triangle containing two red circles.

    \[ \begin{array}{c} {\color{red}\bigcirc}\\[-5pt] \quad\searrow\\[-5pt] {\color{red}\bigcirc}\quad\bigcirc\\[-5pt] \quad\swarrow\\[-5pt] \bigcirc\quad\bigcirc\quad{\color{red}\bigcirc}\\[-5pt] \quad\searrow\\[-5pt] \bigcirc\quad\bigcirc\quad{\color{red}\bigcirc}\quad\bigcirc\\[-5pt] \quad\swarrow\\[-5pt] \bigcirc\quad\bigcirc\quad\bigcirc\quad{\color{red}\bigcirc}\quad\bigcirc\\[-5pt] \swarrow\quad\\[-5pt] {\color{red}\bigcirc}\quad\bigcirc\quad\bigcirc\quad\bigcirc\quad\bigcirc\quad\bigcirc\\ n=6 \end{array} \]

    In terms of \(n\), find the greatest \(k\) such that in each Japanese triangle there is a ninja path containing at least \(k\) red circles.

    Solution →

  6. Problem 6

    Let \(ABC\) be an equilateral triangle. Let \(A_1,B_1,C_1\) be interior points of \(ABC\) such that \(BA_1=A_1C\), \(CB_1=B_1A\), \(AC_1=C_1B\), and \[ \angle BA_1C+\angle CB_1A+\angle AC_1B=480^\circ. \] Let \(BC_1\) and \(CB_1\) meet at \(A_2\), let \(CA_1\) and \(AC_1\) meet at \(B_2\), and let \(AB_1\) and \(BA_1\) meet at \(C_2\). Prove that if triangle \(A_1B_1C_1\) is scalene, then the three circumcircles of triangles \(AA_1A_2\), \(BB_1B_2\) and \(CC_1C_2\) all pass through two common points.

    (Note: a scalene triangle is one where no two sides have equal length.)

    Solution →

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