IMO 1967 Longlist

IMO 1967 Longlist — 59 problems.

59 items

IMO 1967 Longlist

59 problems · Source: IMO Compendium

Problem	Origin	Statement
BUL1	BUL	Prove that all numbers in the sequence
BUL2	BUL	Prove that 1
BUL3	BUL	Prove the trigonometric inequality cos x < 1 −x2
BUL4	BUL	Suppose medians ma and mb of a triangle are orthogonal.
BUL5	BUL	Solve the system
BUL6	BUL	Solve the system
CZS7	CZS	Find all real solutions of the system of equations
CZS8	CZS	ABCD is a parallelogram; AB = a, AD = 1, \alpha is the size
CZS9	CZS	The circle k and its diameter AB are given. Find the locus of
CZS10	CZS	The square ABCD is to be decomposed into n triangles
CZS11	CZS	Let n be a positive integer. Find the maximal number of non-
CZS12	CZS	Given a segment AB of the length 1, deﬁne the set M of points
GBR17	GBR	Let k, m, and n be positive integers such that m+k + 1 is
GBR18	GBR	If x is a positive rational number, show that x can be uniquely
GBR19	GBR	The n points P1, P2, . . . , Pn are placed inside or on the bound-
GDR13	GDR	Find whether among all quadrilaterals whose interiors lie inside
GDR14	GDR	Which fraction p/q, where p, q are positive integers less than
GDR15	GDR	Suppose tan \alpha = p/q, where p and q are integers and q ̸= 0.
GDR16	GDR	Prove the following statement: If r1 and r2 are real numbers
HUN20	HUN	In space, n points (n \geq3) are given. Every pair of points
HUN21	HUN	Without using any tables, ﬁnd the exact value of the product
HUN22	HUN	The distance between the centers of the circles k1 and k2 with
HUN23	HUN	Prove that for an arbitrary pair of vectors f and g in the
HUN24	HUN	Father has left to his children several identical gold coins.
HUN25	HUN	Three disks of diameter d are touching a sphere at their centers.
ITA26	ITA	Let ABCD be a regular tetrahedron. To an arbitrary point
ITA27	ITA	Which regular polygons can be obtained (and how) by cutting
ITA28	ITA	Find values of the parameter u for which the expression
ITA29	ITA	The triangles A0B0C0 and A′B′C′ have all their angles
MON30	MON	Given m+n numbers ai (i = 1, 2, . . . , m), bj (j = 1, 2, . . ., n),
MON31	MON	An urn contains balls of k diﬀerent colors; there are ni balls
MON32	MON	Determine the volume of the body obtained by cutting the
MON33	MON	In what case does the system
MON34	MON	The faces of a convex polyhedron are six squares and eight
MON35	MON	Prove the identity
POL36	POL	Prove that the center of the sphere circumscribed around a
POL37	POL	Prove that for arbitrary positive numbers the following in-
POL38	POL	Does there exist an integer such that its cube is equal to
POL39	POL	Show that the triangle whose angles satisfy the equality
POL40	POL	Exactly one side of a tetrahedron is of length greater than
POL41	POL	A line l is drawn through the intersection point H of the
ROM42	ROM	Decompose into real factors the expression 1 −sin5 x−cos5 x.
ROM43	ROM	The equation
ROM44	ROM	Suppose p and q are two diﬀerent positive integers and x is a
ROM45	ROM	(a) Solve the equation
ROM46	ROM	If x, y, z are real numbers satisfying the relations x+y+z = 1
ROM47	ROM	Prove the inequality
SWE48	SWE	Determine all positive roots of the equation xx = 1/
SWE49	SWE	Let n and k be positive integers such that 1 \leqn \leqN + 1,
SWE50	SWE	The function ϕ(x, y, z), deﬁned for all triples (x, y, z) of real
SWE51	SWE	A subset S of the set of integers 0, . . . , 99 is said to have
SWE52	SWE	In the plane a point O and a sequence of points P1, P2, P3, . . .
SWE53	SWE	In making Euclidean constructions in geometry it is permit-
USS54	USS	Is it possible to put 100 (or 200) points on a wooden cube such
USS55	USS	Find all x for which for all n,
USS56	USS	In a group of interpreters each one speaks one or several foreign
USS57	USS	Consider the sequence (cn):
USS58	USS	A linear binomial l(z) = Az + B with complex coeﬃcients A
USS59	USS	On the circle with center O and radius 1 the point A0 is