IMO 1974 Longlist
IMO 1974 Longlist — 40 problems.
IMO 1974 Longlist
40 problems · Source: IMO Compendium
| Problem | Origin | Statement |
|---|---|---|
| BUL2 | BUL | Let {un} be the Fibonacci sequence, i.e., u0 = 0, u1 = 1, |
| BUL3 | BUL | Let ABCD be an arbitrary quadrilateral. Let squares ABB1A2, |
| BUL4 | BUL | Let Ka, Kb, Kc with centers Oa, Ob, Oc be the excircles of a |
| BUL5 | BUL | A straight cone is given inside a rectangular parallelepiped |
| CUB6 | CUB | Prove that the product of two natural numbers with their sum |
| CUB7 | CUB | Let P be a prime number and n a natural number. Prove that |
| CZS9 | CZS | Solve the following system of linear equations with unknown |
| CZS10 | CZS | A regular octagon P is given whose incircle k has diameter 1. |
| CZS11 | CZS | Given a line p and a triangle \trianglein the plane, construct an |
| CZS12 | CZS | A circle K with radius r, a point D on K, and a convex |
| FIN13 | FIN | Prove that 2147 −1 is divisible by 343. |
| FIN14 | FIN | Let n and k be natural numbers and a1, a2, . . . , an positive real |
| GBR16 | GBR | A pack of 2n cards contains n different pairs of cards. Each |
| GBR17 | GBR | Show that there exists a set S of 15 distinct circles on the |
| GBR19 | GBR | (Alternative to GBR 2) Prove that there exists, for n \geq4, a |
| NET20 | NET | For which natural numbers n do there exist n natural numbers |
| NET21 | NET | Let M be a nonempty subset of Z+ such that for every element |
| POL25 | POL | Let f : R \toR be of the form f(x) = x + \epsilon sin x, where |
| POL26 | POL | Let g(k) be the number of partitions of a k-element set M, i.e., |
| ROM27 | ROM | Let C1 and C2 be circles in the same plane, P1 and P2 arbitrary |
| ROM28 | ROM | Let M be a finite set and P = {M1, M2, . . . , Mk} a partition |
| ROM29 | ROM | Let A, B, C, D be points in space. If for every point M on the |
| ROM31 | ROM | Let y\alpha = n |
| SWE32 | SWE | Let a1, a2, . . . , an be n real numbers such that 0 < a \leqak \leqb |
| SWE33 | SWE | Let a be a real number such that 0 < a < 1, and let n be a |
| SWE35 | SWE | If p and q are distinct prime numbers, then there are integers |
| SWE36 | SWE | Consider infinite diagrams |
| USA37 | USA | Let a, b, and c denote the three sides of a billiard table in the |
| USA38 | USA | Consider the binomial coefficients |
| USA39 | USA | Let n be a positive integer, n \geq2, and consider the polynomial |
| USA41 | USA | Through the circumcenter O of an arbitrary acute-angled trian- |
| USS43 | USS | An (n2 +n+1)\times(n2 +n+1) matrix of zeros and ones is given. |
| USS44 | USS | We are given n mass points of equal mass in space. We define |
| USS46 | USS | Outside an arbitrary triangle ABC, triangles ADB and BCE |
| VIE47 | VIE | Given two points A, B outside of a given plane P, find the |
| VIE48 | VIE | Let a be a number different from zero. For all integers n define |
| VIE49 | VIE | Determine an equation of third degree with integral coefficients |
| YUG50 | YUG | Let m and n be natural numbers with m > n. Prove that |
| YUG51 | YUG | There are n points on a flat piece of paper, any two of them |
| YUG52 | YUG | A fox stands in the center of the field which has the form of an |