IMO 1968 Shortlist
IMO 1968 Shortlist — 26 problems. 26 problems.
IMO 1968 Shortlist
26 problems · Source: IMO Compendium
Problems
| # | Origin | Problem |
|---|---|---|
| 1 | SWE | Two ships sail on the sea with constant speeds and fixed directions. It is… |
| 2 | ROM | Prove that there exists a unique triangle whose side |
| 3 | POL | Prove that in any tetrahedron there is a vertex such that the lengths of its… |
| 4 | BUL | Let …, …, … be real numbers. Prove that the system of equations |
| 5 | BUL | Let … be the apothem (distance from the center to one of the sides) of a… |
| 6 | HUN | If … … are distinct non-zero real numbers, prove that the equation |
| 7 | HUN | Prove that the product of the radii of three circles exscribed to a given… |
| 8 | ROM | Given an oriented line … and a fixed point … on it, consider all trapezoids …… |
| 9 | ROM | Let … be an arbitrary triangle and … a point inside it. Let … be the distances… |
| 10 | ROM | Consider two segments of length … (…) and a segment of length …. |
| 11 | ROM | Find all solutions … of the equation |
| 12 | POL | If … and … are arbitrary positive real numbers and … an integer, prove that |
| 13 | POL | Given two congruent triangles … and … …, prove that there exists a plane such… |
| 14 | BUL | A line in the plane of a triangle … intersects the sides … and … respectively… |
| 15 | GBR | Let … denote the integer part of …, i.e., the greatest integer not exceeding ….… |
| 16 | GBR | A polynomial … with integer coefficients is said to be divisible by an integer… |
| 17 | GBR | Given a point … and lengths …, prove that there exists an equilateral triangle… |
| 18 | ITA | If an acute-angled triangle ABC is given, construct an equilat- |
| 19 | ITA | We are given a fixed point on the circle of radius …, and going from this point… |
| 20 | CZS | Given … (…) points in space such that every three of them form a triangle with… |
| 21 | CZS | Let a0, a1, . . . , ak (k \geq1) be positive integers. Find all positive |
| 22 | CZS | Find all positive integers … for which …, |
| 23 | CZS | Find all complex numbers m such that polynomial |
| 24 | MON | Find the number of all …-digit numbers for which some fixed digit stands only… |
| 25 | MON | Given k parallel lines and a few points on each of them, find |
| 26 | GDR | Let … be a real number and … a real function defined on all of …, satisfying… |