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Difference between revisions of "1979 IMO Problems"

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Problems of the 21st [[IMO]] 1979 in the United Kingdom.
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==Day I==
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===Problem 1===
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If <math>p</math> and <math>q</math> are natural numbers so that<cmath> \frac{p}{q}=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+ \ldots -\frac{1}{1318}+\frac{1}{1319}, </cmath>prove that <math>p</math> is divisible by <math>1979</math>.
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[[1979 IMO Problems/Problem 1|Solution]]
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===Problem 2===
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We consider a prism which has the upper and inferior basis the pentagons: <math>A_{1}A_{2}A_{3}A_{4}A_{5}</math> and <math>B_{1}B_{2}B_{3}B_{4}B_{5}</math>. Each of the sides of the two pentagons and the segments <math>A_{i}B_{j}</math> with <math>i,j=1,\ldots</math>,5 is colored in red or blue. In every triangle which has all sides colored there exists one red side and one blue side. Prove that all the 10 sides of the two basis are colored in the same color.
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[[1979 IMO Problems/Problem 2|Solution]]
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===Problem 3===
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Two circles in a plane intersect. <math>A</math> is one of the points of intersection. Starting simultaneously from <math>A</math> two points move with constant speed, each travelling along its own circle in the same sense. The two points return to <math>A</math> simultaneously after one revolution. Prove that there is a fixed point <math>P</math> in the plane such that the two points are always equidistant from <math>P.</math>
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[[1979 IMO Problems/Problem 3|Solution]]
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==Day II==
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===Problem 4===
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We consider a point <math>P</math> in a plane <math>p</math> and a point <math>Q \not\in p</math>. Determine all the points <math>R</math> from <math>p</math> for which<cmath> \frac{QP+PR}{QR}  </cmath>is maximum.
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[[1979 IMO Problems/Problem 4|Solution]]
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===Problem 5===
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Determine all real numbers a for which there exists positive reals <math>x_{1}, \ldots, x_{5}</math> which satisfy the relations <math> \sum_{k=1}^{5} kx_{k}=a,</math> <math> \sum_{k=1}^{5} k^{3}x_{k}=a^{2},</math> <math> \sum_{k=1}^{5} k^{5}x_{k}=a^{3}.</math>
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[[1979 IMO Problems/Problem 5|Solution]]
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===Problem 6===
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Let <math>A</math> and <math>E</math> be opposite vertices of an octagon. A frog starts at vertex <math>A.</math> From any vertex except <math>E</math> it jumps to one of the two adjacent vertices. When it reaches <math>E</math> it stops. Let <math>a_n</math> be the number of distinct paths of exactly <math>n</math> jumps ending at <math>E</math>. Prove that:<cmath> a_{2n-1}=0, \quad a_{2n}={(2+\sqrt2)^{n-1} - (2-\sqrt2)^{n-1} \over\sqrt2}. </cmath>
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[[1979 IMO Problems/Problem 6|Solution]]
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* [[1979 IMO]]
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* [http://www.artofproblemsolving.com/Forum/resources.php?c=1&cid=16&year=1979 IMO 1979 Problems on the Resources page]
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* [[IMO Problems and Solutions, with authors]]
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* [[Mathematics competition resources]] {{IMO box|year=1979|before=[[1978 IMO]]|after=[[1980 IMO]]}}

Latest revision as of 10:07, 14 June 2024

Problems of the 21st IMO 1979 in the United Kingdom.

Day I

Problem 1

If $p$ and $q$ are natural numbers so that\[\frac{p}{q}=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+ \ldots -\frac{1}{1318}+\frac{1}{1319},\]prove that $p$ is divisible by $1979$.

Solution

Problem 2

We consider a prism which has the upper and inferior basis the pentagons: $A_{1}A_{2}A_{3}A_{4}A_{5}$ and $B_{1}B_{2}B_{3}B_{4}B_{5}$. Each of the sides of the two pentagons and the segments $A_{i}B_{j}$ with $i,j=1,\ldots$,5 is colored in red or blue. In every triangle which has all sides colored there exists one red side and one blue side. Prove that all the 10 sides of the two basis are colored in the same color.

Solution

Problem 3

Two circles in a plane intersect. $A$ is one of the points of intersection. Starting simultaneously from $A$ two points move with constant speed, each travelling along its own circle in the same sense. The two points return to $A$ simultaneously after one revolution. Prove that there is a fixed point $P$ in the plane such that the two points are always equidistant from $P.$

Solution

Day II

Problem 4

We consider a point $P$ in a plane $p$ and a point $Q \not\in p$. Determine all the points $R$ from $p$ for which\[\frac{QP+PR}{QR}\]is maximum.

Solution

Problem 5

Determine all real numbers a for which there exists positive reals $x_{1}, \ldots, x_{5}$ which satisfy the relations $\sum_{k=1}^{5} kx_{k}=a,$ $\sum_{k=1}^{5} k^{3}x_{k}=a^{2},$ $\sum_{k=1}^{5} k^{5}x_{k}=a^{3}.$

Solution

Problem 6

Let $A$ and $E$ be opposite vertices of an octagon. A frog starts at vertex $A.$ From any vertex except $E$ it jumps to one of the two adjacent vertices. When it reaches $E$ it stops. Let $a_n$ be the number of distinct paths of exactly $n$ jumps ending at $E$. Prove that:\[a_{2n-1}=0, \quad a_{2n}={(2+\sqrt2)^{n-1} - (2-\sqrt2)^{n-1} \over\sqrt2}.\]

Solution

1979 IMO (Problems) • Resources
Preceded by
1978 IMO 		1 • 2 • 3 • 4 • 5 • 6 Followed by
1980 IMO
All IMO Problems and Solutions
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