Olympiad problems

1978 IMO Longlists, 16

Tags: rearrangement inequality , algebra , Inequality , Summation , IMO , IMO 1978

Let $f$ be an injective function from ${1,2,3,\ldots}$ in itself. Prove that for any $n$ we have: $\sum_{k=1}^{n} f(k)k^{-2} \geq \sum_{k=1}^{n} k^{-1}.$

1978 IMO, 3

Tags: algebra , functional equation , Sequence , partition , IMO , IMO 1978

Let $0<f(1)<f(2)<f(3)<\ldots$ a sequence with all its terms positive$.$ The $n-th$ positive integer which doesn't belong to the sequence is $f(f(n))+1.$ Find $f(240).$

1978 IMO Shortlist, 10

Tags: graph theory , Ramsey Theory , combinatorics , Extremal Graph Theory , Extremal combinatorics , IMO , IMO 1978

An international society has its members from six different countries. The list of members contain $1978$ names, numbered $1, 2, \dots, 1978$. Prove that there is at least one member whose number is the sum of the numbers of two members from his own country, or twice as large as the number of one member from his own country.

1978 IMO Longlists, 41

Tags: geometry , inradius , circumcircle , incenter , Triangle , IMO , IMO 1978

In a triangle $ABC$ we have $AB = AC.$ A circle which is internally tangent with the circumscribed circle of the triangle is also tangent to the sides $AB, AC$ in the points $P,$ respectively $Q.$ Prove that the midpoint of $PQ$ is the center of the inscribed circle of the triangle $ABC.$

1978 IMO, 1

Tags: geometry , inradius , circumcircle , incenter , Triangle , IMO , IMO 1978

In a triangle $ABC$ we have $AB = AC.$ A circle which is internally tangent with the circumscribed circle of the triangle is also tangent to the sides $AB, AC$ in the points $P,$ respectively $Q.$ Prove that the midpoint of $PQ$ is the center of the inscribed circle of the triangle $ABC.$

1978 IMO, 1

Tags: modular arithmetic , number theory , Digits , decimal representation , IMO , IMO 1978

Let $ m$ and $ n$ be positive integers such that $ 1 \le m < n$. In their decimal representations, the last three digits of $ 1978^m$ are equal, respectively, to the last three digits of $ 1978^n$. Find $ m$ and $ n$ such that $ m \plus{} n$ has its least value.

1978 IMO, 3

Tags: graph theory , Ramsey Theory , combinatorics , Extremal Graph Theory , Extremal combinatorics , IMO , IMO 1978

An international society has its members from six different countries. The list of members contain $1978$ names, numbered $1, 2, \dots, 1978$. Prove that there is at least one member whose number is the sum of the numbers of two members from his own country, or twice as large as the number of one member from his own country.

1978 IMO Shortlist, 13

Tags: geometry , 3D geometry , sphere , Locus problems , Locus , IMO , IMO 1978

We consider a fixed point $P$ in the interior of a fixed sphere$.$ We construct three segments $PA, PB,PC$, perpendicular two by two$,$ with the vertexes $A, B, C$ on the sphere$.$ We consider the vertex $Q$ which is opposite to $P$ in the parallelepiped (with right angles) with $PA, PB, PC$ as edges$.$ Find the locus of the point $Q$ when $A, B, C$ take all the positions compatible with our problem.

1978 IMO Shortlist, 9

Tags: algebra , functional equation , Sequence , partition , IMO , IMO 1978

Let $0<f(1)<f(2)<f(3)<\ldots$ a sequence with all its terms positive$.$ The $n-th$ positive integer which doesn't belong to the sequence is $f(f(n))+1.$ Find $f(240).$

1978 IMO Shortlist, 12

Tags: geometry , inradius , circumcircle , incenter , Triangle , IMO , IMO 1978

In a triangle $ABC$ we have $AB = AC.$ A circle which is internally tangent with the circumscribed circle of the triangle is also tangent to the sides $AB, AC$ in the points $P,$ respectively $Q.$ Prove that the midpoint of $PQ$ is the center of the inscribed circle of the triangle $ABC.$

1978 IMO Shortlist, 6

Tags: rearrangement inequality , algebra , Inequality , Summation , IMO , IMO 1978

Let $f$ be an injective function from ${1,2,3,\ldots}$ in itself. Prove that for any $n$ we have: $\sum_{k=1}^{n} f(k)k^{-2} \geq \sum_{k=1}^{n} k^{-1}.$

1978 IMO Longlists, 24

Tags: algebra , functional equation , Sequence , partition , IMO , IMO 1978

Let $0<f(1)<f(2)<f(3)<\ldots$ a sequence with all its terms positive$.$ The $n-th$ positive integer which doesn't belong to the sequence is $f(f(n))+1.$ Find $f(240).$

1978 IMO Shortlist, 3

Tags: modular arithmetic , number theory , Digits , decimal representation , IMO , IMO 1978

Let $ m$ and $ n$ be positive integers such that $ 1 \le m < n$. In their decimal representations, the last three digits of $ 1978^m$ are equal, respectively, to the last three digits of $ 1978^n$. Find $ m$ and $ n$ such that $ m \plus{} n$ has its least value.

1978 IMO, 2

Tags: geometry , 3D geometry , sphere , Locus problems , Locus , IMO , IMO 1978

We consider a fixed point $P$ in the interior of a fixed sphere$.$ We construct three segments $PA, PB,PC$, perpendicular two by two$,$ with the vertexes $A, B, C$ on the sphere$.$ We consider the vertex $Q$ which is opposite to $P$ in the parallelepiped (with right angles) with $PA, PB, PC$ as edges$.$ Find the locus of the point $Q$ when $A, B, C$ take all the positions compatible with our problem.

1978 IMO Longlists, 7

Tags: modular arithmetic , number theory , Digits , decimal representation , IMO , IMO 1978

Let $ m$ and $ n$ be positive integers such that $ 1 \le m < n$. In their decimal representations, the last three digits of $ 1978^m$ are equal, respectively, to the last three digits of $ 1978^n$. Find $ m$ and $ n$ such that $ m \plus{} n$ has its least value.

1978 IMO Longlists, 30

Tags: graph theory , Ramsey Theory , combinatorics , Extremal Graph Theory , Extremal combinatorics , IMO , IMO 1978

An international society has its members from six different countries. The list of members contain $1978$ names, numbered $1, 2, \dots, 1978$. Prove that there is at least one member whose number is the sum of the numbers of two members from his own country, or twice as large as the number of one member from his own country.

1978 IMO, 2

Tags: rearrangement inequality , algebra , Inequality , Summation , IMO , IMO 1978

Let $f$ be an injective function from ${1,2,3,\ldots}$ in itself. Prove that for any $n$ we have: $\sum_{k=1}^{n} f(k)k^{-2} \geq \sum_{k=1}^{n} k^{-1}.$

1978 IMO Longlists, 46

Tags: geometry , 3D geometry , sphere , Locus problems , Locus , IMO , IMO 1978

We consider a fixed point $P$ in the interior of a fixed sphere$.$ We construct three segments $PA, PB,PC$, perpendicular two by two$,$ with the vertexes $A, B, C$ on the sphere$.$ We consider the vertex $Q$ which is opposite to $P$ in the parallelepiped (with right angles) with $PA, PB, PC$ as edges$.$ Find the locus of the point $Q$ when $A, B, C$ take all the positions compatible with our problem.