IMO 1966 Longlist
IMO 1966 Longlist — 57 problems.
IMO 1966 Longlist
57 problems · Source: IMO Compendium
| Problem | Origin | Statement |
|---|---|---|
| BUL3 | BUL | A regular triangular prism has height h and a base of side length |
| BUL12 | BUL | Find digits x, y, z such that the equality |
| BUL21 | BUL | Prove that the volume V and the lateral area S of a right circular |
| BUL22 | BUL | Assume that two parallelograms P, P ′ of equal areas have sides |
| BUL23 | BUL | Three faces of a tetrahedron are right triangles, while the fourth |
| BUL32 | BUL | The sides a, b, c of a triangle ABC form an arithmetic progression; |
| BUL33 | BUL | Two circles touch each other from inside, and an equilateral |
| BUL34 | BUL | Determine all pairs of positive integers (x, y) satisfying the equa- |
| CZS1 | CZS | We are given n > 3 points in the plane, no three of which lie on |
| CZS11 | CZS | Does there exist an integer z that can be written in two different |
| CZS16 | CZS | We are given a circle K with center S and radius 1 and a square |
| CZS26 | CZS | (a) Prove that (a1 +a2 +\cdot \cdot \cdot+ak)2 \leqk(a2 |
| CZS28 | CZS | Let there be given a circle with center S and radius 1 in the plane, |
| CZS40 | CZS | For a positive real number p, find all real solutions to the equation |
| CZS41 | CZS | If A1A2 . . . An is a regular n-gon (n \geq3), how many different |
| CZS42 | CZS | Let a1, a2, . . . , an (n \geq2) be a sequence of integers. Show that |
| CZS43 | CZS | Five points in a plane are given, no three of which are collinear. |
| GDR2 | GDR | Given n positive real numbers a1, a2, . . . , an such that a1a2 \cdot \cdot… |
| GDR10 | GDR | How many real solutions are there to the equation x = |
| GDR25 | GDR | Show that tan 7◦30′ = |
| GDR27 | GDR | We are given a circle K and a point P lying on a line g. Construct |
| HUN18 | HUN | Solve the equation |
| HUN19 | HUN | Construct a triangle given the three exradii. |
| HUN20 | HUN | We are given three equal rectangles with the same center in |
| POL4 | POL | Five points in the plane are given, no three of which are collinear. |
| POL14 | POL | Compute the largest number of regions into which one can divide |
| POL15 | POL | Points A, B, C, D lie on a circle such that AB is a diameter and |
| POL24 | POL | There are n \geq2 people in a room. Prove that there exist two |
| POL35 | POL | If a, b, c, d are integers such that ad is odd and bc is even, prove |
| POL36 | POL | Let ABCD be a cyclic quadrilateral. Show that the centroids of |
| POL37 | POL | Prove that the perpendiculars drawn from the midpoints of the |
| ROM9 | ROM | Find x such that |
| ROM17 | ROM | Suppose ABCD and A′B′C′D′ are two parallelograms arbi- |
| ROM29 | ROM | (a) Find the number of ways 500 can be represented as a sum of |
| ROM30 | ROM | If n is a natural number, prove that |
| ROM31 | ROM | Solve the equation |x2 −1| + |x2 −4| = mx as a function of the |
| ROM38 | ROM | Two concentric circles have radii R and r respectively. Determine |
| ROM39 | ROM | In a plane, a circle with center O and radius R and two points |
| ROM47 | ROM | Find the number of lines dividing a given triangle into two parts |
| USS5 | USS | Prove the inequality |
| USS6 | USS | A convex planar polygon M with perimeter l and area S is given. |
| USS7 | USS | For which arrangements of two infinite circular cylinders does |
| USS8 | USS | We are given a bag of sugar, a two-pan balance, and a weight of |
| USS48 | USS | Find all positive numbers p for which the equation x2+px+3p = 0 |
| USS49 | USS | Two mirror walls are placed to form an angle of measure \alpha. There |
| USS50 | USS | Given a quadrangle of sides a, b, c, d and area S, show that S \leq |
| USS51 | USS | In a school, n children numbered 1 to n are initially arranged in |
| USS52 | USS | A figure of area 1 is cut out from a sheet of paper and divided |
| USS53 | USS | Prove that in every convex hexagon of area S one can draw |
| USS54 | USS | Find the last two digits of a sum of eighth powers of 100 |
| USS55 | USS | Given the vertex A and the centroid M of a triangle ABC, |
| USS56 | USS | Let ABCD be a tetrahedron such that AB \perpCD, |
| USS57 | USS | Is it possible to choose a set of 100 (or 200) points on the |
| YUG13 | YUG | Let a1, a2, . . . , an be positive real numbers. Prove the inequality |
| YUG44 | YUG | What is the greatest number of balls of radius 1/2 that can be |
| YUG45 | YUG | An alphabet consists of n letters. What is the maximal length |
| YUG46 | YUG | Let |