IMO 1972 Shortlist
IMO 1972 Shortlist — 12 problems. 12 problems.
IMO 1972 Shortlist
12 problems · Source: IMO Compendium
Problems
| # | Origin | Problem |
|---|---|---|
| 1 | BUL | Let f and ϕ be real functions defined on the set R satisfying |
| 2 | CZS | We are given 3n points A1, A2, . . . , A3n in the plane, no three |
| 3 | CZS | Let x1, x2, . . . , xn be real numbers satisfying x1+x2+\cdot \cdot \cdot+xn = |
| 4 | GDR | Let n1, n2 be positive integers. Consider in a plane E two dis- |
| 5 | GDR | Prove the following assertion: The four altitudes of a tetrahe- |
| 6 | GDR | Show that for any n ̸\equiv0 (mod 10) there exists a multiple of |
| 7 | GBR | (a) A plane \pi passes through the vertex O of the regular |
| 8 | GBR | Let m and n be nonnegative integers. Prove that m!n!(m+ |
| 9 | NET | Find all solutions in positive real numbers xi (i = |
| 10 | NET | Prove that for each n \geq4 every cyclic quadrilateral can |
| 11 | NET | Consider a sequence of circles K1, K2, K3, K4, . . . of radii |
| 12 | USS | A set of 10 positive integers is given such that the decimal |