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Difference between revisions of "1988 IMO Problems"

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1988 IMO:
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Problems of the 1988 [[IMO]].
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==Day I==
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===Problem 1===
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Consider 2 concentric circle radii <math> R</math> and <math> r</math> (<math> R > r</math>) with centre <math> O.</math> Fix <math> P</math> on the small circle and consider the variable chord <math> PA</math> of the small circle. Points <math> B</math> and <math> C</math> lie on the large circle; <math> B,P,C</math> are collinear and <math> BC</math> is perpendicular to <math> AP.</math>
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i.) For which values of <math> \angle OPA</math> is the sum <math> BC^2 + CA^2 + AB^2</math> extremal?
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ii.) What are the possible positions of the midpoints <math> U</math> of <math> BA</math> and <math> V</math> of <math> AC</math> as <math> \angle OPA</math> varies?
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[[1988 IMO Problems/Problem 1|Solution]]
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===Problem 2===
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Let <math> n</math> be an even positive integer. Let <math> A_1, A_2, \ldots, A_{n + 1}</math> be sets having <math> n</math> elements each such that any two of them have exactly one element in common while every element of their union belongs to at least two of the given sets. For which <math> n</math> can one assign to every element of the union one of the numbers 0 and 1 in such a manner that each of the sets has exactly <math> \frac {n}{2}</math> zeros?
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[[1988 IMO Problems/Problem 2|Solution]]
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===Problem 3===
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A function <math> f</math> defined on the positive integers (and taking positive integers values) is given by:
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<math> \begin{matrix} f(1) = 1, f(3) = 3 \\
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f(2 \cdot n) = f(n) \\
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f(4 \cdot n + 1) = 2 \cdot f(2 \cdot n + 1) - f(n) \\
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f(4 \cdot n + 3) = 3 \cdot f(2 \cdot n + 1) - 2 \cdot f(n), \end{matrix}</math>
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for all positive integers <math> n.</math> Determine with proof the number of positive integers <math> \leq 1988</math> for which <math> f(n) = n.</math>
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[[1988 IMO Problems/Problem 3|Solution]]
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==Day II==
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===Problem 4===
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Show that the solution set of the inequality
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\[ \sum^{70}_{k = 1} \frac {k}{x - k} \geq \frac {5}{4}
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\]
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is a union of disjoint intervals, the sum of whose length is 1988.
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[[1988 IMO Problems/Problem 4|Solution]]
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===Problem 5===
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In a right-angled triangle <math> ABC</math> let <math> AD</math> be the altitude drawn to the hypotenuse and let the straight line joining the incentres of the triangles <math> ABD, ACD</math> intersect the sides <math> AB, AC</math> at the points <math> K,L</math> respectively. If <math> E</math> and <math> E_1</math> dnote the areas of triangles <math> ABC</math> and <math> AKL</math> respectively, show that
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\[ \frac {E}{E_1} \geq 2.
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\]
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[[1988 IMO Problems/Problem 5|Solution]]
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===Problem 6===
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Let <math> a</math> and <math> b</math> be two positive integers such that <math> a \cdot b + 1</math> divides <math> a^{2} + b^{2}</math>. Show that <math> \frac {a^{2} + b^{2}}{a \cdot b + 1}</math> is a perfect square.
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[[1988 IMO Problems/Problem 6|Solution]]
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* [[1988 IMO]]
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* [http://www.artofproblemsolving.com/Forum/resources.php?c=1&cid=16&year=1988 IMO 1988 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=1988|before=[[1987 IMO]]|after=[[1989 IMO]]}}

Revision as of 10:29, 30 January 2021

Problems of the 1988 IMO.

Day I

Problem 1

Consider 2 concentric circle radii $R$ and $r$ ($R > r$) with centre $O.$ Fix $P$ on the small circle and consider the variable chord $PA$ of the small circle. Points $B$ and $C$ lie on the large circle; $B,P,C$ are collinear and $BC$ is perpendicular to $AP.$

i.) For which values of $\angle OPA$ is the sum $BC^2 + CA^2 + AB^2$ extremal?

ii.) What are the possible positions of the midpoints $U$ of $BA$ and $V$ of $AC$ as $\angle OPA$ varies?

Solution

Problem 2

Let $n$ be an even positive integer. Let $A_1, A_2, \ldots, A_{n + 1}$ be sets having $n$ elements each such that any two of them have exactly one element in common while every element of their union belongs to at least two of the given sets. For which $n$ can one assign to every element of the union one of the numbers 0 and 1 in such a manner that each of the sets has exactly $\frac {n}{2}$ zeros?

Solution

Problem 3

A function $f$ defined on the positive integers (and taking positive integers values) is given by:

$\begin{matrix} f(1) = 1, f(3) = 3 \\ f(2 \cdot n) = f(n) \\ f(4 \cdot n + 1) = 2 \cdot f(2 \cdot n + 1) - f(n) \\ f(4 \cdot n + 3) = 3 \cdot f(2 \cdot n + 1) - 2 \cdot f(n), \end{matrix}$

for all positive integers $n.$ Determine with proof the number of positive integers $\leq 1988$ for which $f(n) = n.$

Solution

Day II

Problem 4

Show that the solution set of the inequality \[ \sum^{70}_{k = 1} \frac {k}{x - k} \geq \frac {5}{4} \] is a union of disjoint intervals, the sum of whose length is 1988.

Solution

Problem 5

In a right-angled triangle $ABC$ let $AD$ be the altitude drawn to the hypotenuse and let the straight line joining the incentres of the triangles $ABD, ACD$ intersect the sides $AB, AC$ at the points $K,L$ respectively. If $E$ and $E_1$ dnote the areas of triangles $ABC$ and $AKL$ respectively, show that \[ \frac {E}{E_1} \geq 2. \]

Solution

Problem 6

Let $a$ and $b$ be two positive integers such that $a \cdot b + 1$ divides $a^{2} + b^{2}$. Show that $\frac {a^{2} + b^{2}}{a \cdot b + 1}$ is a perfect square.

Solution

1988 IMO (Problems) • Resources
Preceded by
1987 IMO 		1 • 2 • 3 • 4 • 5 • 6 Followed by
1989 IMO
All IMO Problems and Solutions
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