This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 18

1988 IMO Shortlist, 13
Tags: geometry , incenter , ratio , right triangle , area of a triangle , IMO , IMO 1988
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 \[ \frac {E}{E_1} \geq 2. \]

1988 IMO Longlists, 48
Tags: trigonometry , geometry , perimeter , IMO , IMO 1988
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.$ [b]i.)[/b] For which values of $ \angle OPA$ is the sum $ BC^2 \plus{} CA^2 \plus{} AB^2$ extremal? [b]ii.)[/b] What are the possible positions of the midpoints $ U$ of $ BA$ and $ V$ of $ AC$ as $ \angle OPA$ varies?

1988 IMO, 2
Tags: geometry , incenter , ratio , right triangle , area of a triangle , IMO , IMO 1988
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 \[ \frac {E}{E_1} \geq 2. \]

1988 IMO Longlists, 77
Tags: algebra , functional equation , binary representation , function , IMO , IMO 1988 , david monk
A function $ f$ defined on the positive integers (and taking positive integers values) is given by: $ \begin{matrix} f(1) \equal{} 1, f(3) \equal{} 3 \\ f(2 \cdot n) \equal{} f(n) \\ f(4 \cdot n \plus{} 1) \equal{} 2 \cdot f(2 \cdot n \plus{} 1) \minus{} f(n) \\ f(4 \cdot n \plus{} 3) \equal{} 3 \cdot f(2 \cdot n \plus{} 1) \minus{} 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) \equal{} n.$

1988 IMO, 2
Tags: linear algebra , combinatorics , Set systems , Coloring , IMO , IMO 1988
Let $ n$ be an even positive integer. Let $ A_1, A_2, \ldots, A_{n \plus{} 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?

1988 IMO Longlists, 23
Tags: geometry , incenter , ratio , right triangle , area of a triangle , IMO , IMO 1988
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 \[ \frac {E}{E_1} \geq 2. \]

1988 IMO Longlists, 14
Tags: number theory , Divisibility , Diophantine equation , Perfect Square , Vieta Jumping , IMO , IMO 1988
Let $ a$ and $ b$ be two positive integers such that $ a \cdot b \plus{} 1$ divides $ a^{2} \plus{} b^{2}$. Show that $ \frac {a^{2} \plus{} b^{2}}{a \cdot b \plus{} 1}$ is a perfect square.

1988 IMO, 1
Tags: trigonometry , geometry , perimeter , IMO , IMO 1988
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.$ [b]i.)[/b] For which values of $ \angle OPA$ is the sum $ BC^2 \plus{} CA^2 \plus{} AB^2$ extremal? [b]ii.)[/b] What are the possible positions of the midpoints $ U$ of $ BA$ and $ V$ of $ AC$ as $ \angle OPA$ varies?

1988 IMO Shortlist, 26
Tags: algebra , functional equation , binary representation , function , IMO , IMO 1988 , david monk
A function $ f$ defined on the positive integers (and taking positive integers values) is given by: $ \begin{matrix} f(1) \equal{} 1, f(3) \equal{} 3 \\ f(2 \cdot n) \equal{} f(n) \\ f(4 \cdot n \plus{} 1) \equal{} 2 \cdot f(2 \cdot n \plus{} 1) \minus{} f(n) \\ f(4 \cdot n \plus{} 3) \equal{} 3 \cdot f(2 \cdot n \plus{} 1) \minus{} 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) \equal{} n.$

1988 IMO Longlists, 42
Tags: algebra , polynomial , Vieta , Inequality , roots , IMO , IMO 1988
Show that the solution set of the inequality \[ \sum^{70}_{k \equal{} 1} \frac {k}{x \minus{} k} \geq \frac {5}{4} \] is a union of disjoint intervals, the sum of whose length is 1988.

1988 IMO Shortlist, 5
Tags: linear algebra , combinatorics , Set systems , Coloring , IMO , IMO 1988
Let $ n$ be an even positive integer. Let $ A_1, A_2, \ldots, A_{n \plus{} 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?

1988 IMO, 3
Tags: algebra , functional equation , binary representation , function , IMO , IMO 1988 , david monk
A function $ f$ defined on the positive integers (and taking positive integers values) is given by: $ \begin{matrix} f(1) \equal{} 1, f(3) \equal{} 3 \\ f(2 \cdot n) \equal{} f(n) \\ f(4 \cdot n \plus{} 1) \equal{} 2 \cdot f(2 \cdot n \plus{} 1) \minus{} f(n) \\ f(4 \cdot n \plus{} 3) \equal{} 3 \cdot f(2 \cdot n \plus{} 1) \minus{} 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) \equal{} n.$

1988 IMO Shortlist, 9
Tags: number theory , Divisibility , Diophantine equation , Perfect Square , Vieta Jumping , IMO , IMO 1988
Let $ a$ and $ b$ be two positive integers such that $ a \cdot b \plus{} 1$ divides $ a^{2} \plus{} b^{2}$. Show that $ \frac {a^{2} \plus{} b^{2}}{a \cdot b \plus{} 1}$ is a perfect square.

1988 IMO Shortlist, 18
Tags: trigonometry , geometry , perimeter , IMO , IMO 1988
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.$ [b]i.)[/b] For which values of $ \angle OPA$ is the sum $ BC^2 \plus{} CA^2 \plus{} AB^2$ extremal? [b]ii.)[/b] What are the possible positions of the midpoints $ U$ of $ BA$ and $ V$ of $ AC$ as $ \angle OPA$ varies?

1988 IMO Shortlist, 16
Tags: algebra , polynomial , Vieta , Inequality , roots , IMO , IMO 1988
Show that the solution set of the inequality \[ \sum^{70}_{k \equal{} 1} \frac {k}{x \minus{} k} \geq \frac {5}{4} \] is a union of disjoint intervals, the sum of whose length is 1988.

1988 IMO, 1
Tags: algebra , polynomial , Vieta , Inequality , roots , IMO , IMO 1988
Show that the solution set of the inequality \[ \sum^{70}_{k \equal{} 1} \frac {k}{x \minus{} k} \geq \frac {5}{4} \] is a union of disjoint intervals, the sum of whose length is 1988.

1988 IMO Longlists, 7
Tags: linear algebra , combinatorics , Set systems , Coloring , IMO , IMO 1988
Let $ n$ be an even positive integer. Let $ A_1, A_2, \ldots, A_{n \plus{} 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?

1988 IMO, 3
Tags: number theory , Divisibility , Diophantine equation , Perfect Square , Vieta Jumping , IMO , IMO 1988
Let $ a$ and $ b$ be two positive integers such that $ a \cdot b \plus{} 1$ divides $ a^{2} \plus{} b^{2}$. Show that $ \frac {a^{2} \plus{} b^{2}}{a \cdot b \plus{} 1}$ is a perfect square.