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Difference between revisions of "1977 IMO Problems"
 
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Problems of the 19th [[IMO]] 1977 in Yugoslavia.
 
Problems of the 19th [[IMO]] 1977 in Yugoslavia.
  
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==Problem 1==
+
==Day I==
 +
===Problem 1===
 
In the interior of a square <math>ABCD</math> we construct the equilateral triangles <math>ABK, BCL, CDM, DAN.</math> Prove that the midpoints of the four segments <math>KL, LM, MN, NK</math> and the midpoints of the eight segments <math>AK, BK, BL, CL, CM, DM, DN, AN</math> are the 12 vertices of a regular dodecagon.
 
In the interior of a square <math>ABCD</math> we construct the equilateral triangles <math>ABK, BCL, CDM, DAN.</math> Prove that the midpoints of the four segments <math>KL, LM, MN, NK</math> and the midpoints of the eight segments <math>AK, BK, BL, CL, CM, DM, DN, AN</math> are the 12 vertices of a regular dodecagon.
  
 
[[1977 IMO Problems/Problem 1|Solution]]
 
[[1977 IMO Problems/Problem 1|Solution]]
  
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==Problem 2==
+
===Problem 2===
 
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.
 
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 Problems/Problem 2|Solution]]
 
[[1977 IMO Problems/Problem 2|Solution]]
  
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==Problem 3==
+
===Problem 3===
 
Let <math>n</math> be a given number greater than 2. We consider the set <math>V_n</math> of all the integers of the form <math>1 + kn</math> with <math>k = 1, 2, \ldots</math> A number <math>m</math> from <math>V_n</math> is called indecomposable in <math>V_n</math> if there are not two numbers <math>p</math> and <math>q</math> from <math>V_n</math> so that <math>m = pq.</math> Prove that there exist a number <math>r \in V_n</math> that can be expressed as the product of elements indecomposable in <math>V_n</math> in more than one way. (Expressions which differ only in order of the elements of <math>V_n</math> will be considered the same.)
 
Let <math>n</math> be a given number greater than 2. We consider the set <math>V_n</math> of all the integers of the form <math>1 + kn</math> with <math>k = 1, 2, \ldots</math> A number <math>m</math> from <math>V_n</math> is called indecomposable in <math>V_n</math> if there are not two numbers <math>p</math> and <math>q</math> from <math>V_n</math> so that <math>m = pq.</math> Prove that there exist a number <math>r \in V_n</math> that can be expressed as the product of elements indecomposable in <math>V_n</math> in more than one way. (Expressions which differ only in order of the elements of <math>V_n</math> will be considered the same.)
  
 
[[1977 IMO Problems/Problem 3|Solution]]
 
[[1977 IMO Problems/Problem 3|Solution]]
  
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==Problem 4==
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==Day II==
 +
===Problem 4===
 
Let <math>a,b,A,B</math> be given reals. We consider the function defined by<cmath> f(x) = 1 - a \cdot \cos(x) - b \cdot \sin(x) - A \cdot \cos(2x) - B \cdot \sin(2x). </cmath>Prove that if for any real number <math>x</math> we have <math>f(x) \geq 0</math> then <math>a^2 + b^2 \leq 2</math> and <math>A^2 + B^2 \leq 1.</math>
 
Let <math>a,b,A,B</math> be given reals. We consider the function defined by<cmath> f(x) = 1 - a \cdot \cos(x) - b \cdot \sin(x) - A \cdot \cos(2x) - B \cdot \sin(2x). </cmath>Prove that if for any real number <math>x</math> we have <math>f(x) \geq 0</math> then <math>a^2 + b^2 \leq 2</math> and <math>A^2 + B^2 \leq 1.</math>
  
 
[[1977 IMO Problems/Problem 4|Solution]]
 
[[1977 IMO Problems/Problem 4|Solution]]
  
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==Problem 5==
+
===Problem 5===
 
Let <math>a,b</math> be two natural numbers. When we divide <math>a^2+b^2</math> by <math>a+b</math>, we the the remainder <math>r</math> and the quotient <math>q.</math> Determine all pairs <math>(a, b)</math> for which <math>q^2 + r = 1977.</math>
 
Let <math>a,b</math> be two natural numbers. When we divide <math>a^2+b^2</math> by <math>a+b</math>, we the the remainder <math>r</math> and the quotient <math>q.</math> Determine all pairs <math>(a, b)</math> for which <math>q^2 + r = 1977.</math>
  
 
[[1977 IMO Problems/Problem 5|Solution]]
 
[[1977 IMO Problems/Problem 5|Solution]]
  
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==Problem 6==
+
===Problem 6===
 
Let <math>\mathbb{N}</math> be the set of positive integers. Let <math>f</math> be a function defined on <math>\mathbb{N}</math>, which satisfies the inequality <math>f(n + 1) > f(f(n))</math> for all <math>n \in \mathbb{N}</math>. Prove that for any <math>n</math> we have <math>f(n) = n.</math>
 
Let <math>\mathbb{N}</math> be the set of positive integers. Let <math>f</math> be a function defined on <math>\mathbb{N}</math>, which satisfies the inequality <math>f(n + 1) > f(f(n))</math> for all <math>n \in \mathbb{N}</math>. Prove that for any <math>n</math> we have <math>f(n) = n.</math>
  

Latest revision as of 15:54, 29 January 2021

Problems of the 19th IMO 1977 in Yugoslavia.

Day I

Problem 1

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.

Solution

Problem 2

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.

Solution

Problem 3

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.)

Solution

Day II

Problem 4

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 $A^2 + B^2 \leq 1.$

Solution

Problem 5

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 $q^2 + r = 1977.$

Solution

Problem 6

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 $f(n) = n.$

Solution

1977 IMO (Problems) • Resources
Preceded by
1976 IMO 		1 • 2 • 3 • 4 • 5 • 6 Followed by
1978 IMO
All IMO Problems and Solutions
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