Olympiad problems

1977 IMO Longlists, 20

Tags: function , trigonometry , Trigonometric inequality , Inequality , IMO , IMO 1977

Let $a,b,A,B$ be given reals. We consider the function defined by \[ f(x) = 1 - a \cdot \cos(x) - b \cdot \sin(x) - A \cdot \cos(2x) - B \cdot \sin(2x). \] Prove that if for any real number $x$ we have $f(x) \geq 0$ then $a^2 + b^2 \leq 2$ and $A^2 + B^2 \leq 1.$

1977 IMO Shortlist, 10

Tags: number theory , prime numbers , prime factorization , Dirichlet s Theorem , IMO , IMO 1977

Let $n$ be a given number greater than 2. We consider the set $V_n$ of all the integers of the form $1 + kn$ with $k = 1, 2, \ldots$ A number $m$ from $V_n$ is called indecomposable in $V_n$ if there are not two numbers $p$ and $q$ from $V_n$ so that $m = pq.$ Prove that there exist a number $r \in V_n$ that can be expressed as the product of elements indecomposable in $V_n$ in more than one way. (Expressions which differ only in order of the elements of $V_n$ will be considered the same.)

1977 IMO, 1

Tags: function , trigonometry , Trigonometric inequality , Inequality , IMO , IMO 1977

Let $a,b,A,B$ be given reals. We consider the function defined by \[ f(x) = 1 - a \cdot \cos(x) - b \cdot \sin(x) - A \cdot \cos(2x) - B \cdot \sin(2x). \] Prove that if for any real number $x$ we have $f(x) \geq 0$ then $a^2 + b^2 \leq 2$ and $A^2 + B^2 \leq 1.$

1977 IMO, 3

Tags: number theory , prime numbers , prime factorization , Dirichlet s Theorem , IMO , IMO 1977

Let $n$ be a given number greater than 2. We consider the set $V_n$ of all the integers of the form $1 + kn$ with $k = 1, 2, \ldots$ A number $m$ from $V_n$ is called indecomposable in $V_n$ if there are not two numbers $p$ and $q$ from $V_n$ so that $m = pq.$ Prove that there exist a number $r \in V_n$ that can be expressed as the product of elements indecomposable in $V_n$ in more than one way. (Expressions which differ only in order of the elements of $V_n$ will be considered the same.)

1977 IMO Shortlist, 3

Tags: number theory , Additive Number Theory , remainder , Divisibility , IMO , IMO 1977

Let $a,b$ be two natural numbers. When we divide $a^2+b^2$ by $a+b$, we the the remainder $r$ and the quotient $q.$ Determine all pairs $(a, b)$ for which $q^2 + r = 1977.$

1977 IMO Shortlist, 15

Tags: linear algebra , algebra , Sequence , maximization , Extremal combinatorics , IMO , IMO 1977

In a finite sequence of real numbers the sum of any seven successive terms is negative and the sum of any eleven successive terms is positive. Determine the maximum number of terms in the sequence.

1977 IMO Longlists, 27

Tags: number theory , prime numbers , prime factorization , Dirichlet s Theorem , IMO , IMO 1977

Let $n$ be a given number greater than 2. We consider the set $V_n$ of all the integers of the form $1 + kn$ with $k = 1, 2, \ldots$ A number $m$ from $V_n$ is called indecomposable in $V_n$ if there are not two numbers $p$ and $q$ from $V_n$ so that $m = pq.$ Prove that there exist a number $r \in V_n$ that can be expressed as the product of elements indecomposable in $V_n$ in more than one way. (Expressions which differ only in order of the elements of $V_n$ will be considered the same.)

1977 IMO Shortlist, 7

Tags: function , trigonometry , Trigonometric inequality , Inequality , IMO , IMO 1977

Let $a,b,A,B$ be given reals. We consider the function defined by \[ f(x) = 1 - a \cdot \cos(x) - b \cdot \sin(x) - A \cdot \cos(2x) - B \cdot \sin(2x). \] Prove that if for any real number $x$ we have $f(x) \geq 0$ then $a^2 + b^2 \leq 2$ and $A^2 + B^2 \leq 1.$

1977 IMO, 2

Tags: linear algebra , algebra , Sequence , maximization , Extremal combinatorics , IMO , IMO 1977

In a finite sequence of real numbers the sum of any seven successive terms is negative and the sum of any eleven successive terms is positive. Determine the maximum number of terms in the sequence.

1977 IMO Shortlist, 12

Tags: complex numbers , geometry , square , polygon , IMO , IMO 1977

In the interior of a square $ABCD$ we construct the equilateral triangles $ABK, BCL, CDM, DAN.$ Prove that the midpoints of the four segments $KL, LM, MN, NK$ and the midpoints of the eight segments $AK, BK, BL, CL, CM, DM, DN, AN$ are the 12 vertices of a regular dodecagon.

1977 IMO Shortlist, 1

Tags: algebra , Functional inequality , functional equation , IMO 1977 , IMO Shortlist , IMO , Hi

Find all functions $f : \mathbb{N}\rightarrow \mathbb{N}$ satisfying following condition: \[f(n+1)>f(f(n)), \quad \forall n \in \mathbb{N}.\]

1977 IMO Longlists, 57

Tags: linear algebra , algebra , Sequence , maximization , Extremal combinatorics , IMO , IMO 1977

In a finite sequence of real numbers the sum of any seven successive terms is negative and the sum of any eleven successive terms is positive. Determine the maximum number of terms in the sequence.

1977 IMO Longlists, 2

Tags: algebra , Functional inequality , functional equation , IMO 1977 , IMO Shortlist , IMO , Hi

Find all functions $f : \mathbb{N}\rightarrow \mathbb{N}$ satisfying following condition: \[f(n+1)>f(f(n)), \quad \forall n \in \mathbb{N}.\]

1977 IMO, 3

Tags: function , algebra , Functional inequality , functional equation , IMO , IMO 1977

Let $\mathbb{N}$ be the set of positive integers. Let $f$ be a function defined on $\mathbb{N}$, which satisfies the inequality $f(n + 1) > f(f(n))$ for all $n \in \mathbb{N}$. Prove that for any $n$ we have $f(n) = n.$

1977 IMO Longlists, 10

Tags: number theory , Additive Number Theory , remainder , Divisibility , IMO , IMO 1977

Let $a,b$ be two natural numbers. When we divide $a^2+b^2$ by $a+b$, we the the remainder $r$ and the quotient $q.$ Determine all pairs $(a, b)$ for which $q^2 + r = 1977.$

1977 IMO, 2

Tags: number theory , Additive Number Theory , remainder , Divisibility , IMO , IMO 1977

Let $a,b$ be two natural numbers. When we divide $a^2+b^2$ by $a+b$, we the the remainder $r$ and the quotient $q.$ Determine all pairs $(a, b)$ for which $q^2 + r = 1977.$

1977 IMO, 1

Tags: complex numbers , geometry , square , polygon , IMO , IMO 1977

In the interior of a square $ABCD$ we construct the equilateral triangles $ABK, BCL, CDM, DAN.$ Prove that the midpoints of the four segments $KL, LM, MN, NK$ and the midpoints of the eight segments $AK, BK, BL, CL, CM, DM, DN, AN$ are the 12 vertices of a regular dodecagon.

1977 IMO Longlists, 29

Tags: complex numbers , geometry , square , polygon , IMO , IMO 1977

In the interior of a square $ABCD$ we construct the equilateral triangles $ABK, BCL, CDM, DAN.$ Prove that the midpoints of the four segments $KL, LM, MN, NK$ and the midpoints of the eight segments $AK, BK, BL, CL, CM, DM, DN, AN$ are the 12 vertices of a regular dodecagon.