Found problems: 10
MMPC Part II 1996 - 2019, 2016.1
If a polygon has both an inscribed circle and a circumscribed circle, then define the [i]halo[/i] of that polygon to be the region inside the circumcircle but outside the incircle. In particular, all regular polygons and all triangles have halos.
(a) What is the area of the halo of a square with side length 2?
(b) What is the area of the halo of a 3-4-5 right triangle?
(c) What is the area of the halo of a regular 2016-gon with side length 2?
2016 MMPC, 4
It is a fact that every set of 2016 consecutive integers can be partitioned in two sets with the following four
properties:
(i) The sets have the same number of elements.
(ii) The sums of the elements of the sets are equal.
(iii) The sums of the squares of the elements of the sets are equal.
(iv) The sums of the cubes of the elements of the sets are equal.
Let $S =\{n + 1; n + 2;$ [b]. . .[/b] $; n + k\}$ be a set of $k$ consecutive integers.
(a) Determine the smallest value of $k$ such that property (i) holds for $S$.
(b) Determine the smallest value of $k$ such that properties (i) and (ii) hold for $S$.
(c) Show that properties (i), (ii) and (iii) hold for $S$ when $k = 8$.
(d) Show that properties (i), (ii), (iii) and (iv) hold for $S$ when $k = 16$.
2016 MMPC, 2
Let $s_1,s_2,s_3,s_4,...$ be a sequence (infinite list) of $1$s and $0$s. For example $1,0,1,0,1,0,...$, that is, $s_n=1$ if $n$ is odd and $s_n=0$ if $n$ is even, is such a sequence. Prove that it is possible to delete infinitely many terms in $s_1,s_2,s_3,s_4,...$ so that the resulting sequence is the original sequence. For the given example, one can delete $s_3,s_4,s_7,s_8,s_{11},s_{12},...$
2016 MMPC, 3
This problem is about pairs of consecutive whole numbers satisfying the property that one of the numbers is a perfect square and the other one is the double of a perfect square.
(a) The smallest such pairs are $(0,1)$ and $(8,9)$, Indeed $0=2 \cdot 0^2$ and $1=1^2$; $8=2 \cdot 2^2$ and $9=3^2$. Show that there are infinitely many pairs of the form $(2a^2,b^2)$ where the smaller number is the double of a perfect square satisfying the given property.
(b) Find a pair of integers satisfying the property that is not in the form given in the first part, that is, find a
pair of integers such that the smaller one is a perfect square and the larger one is the double of a perfect
square.
2016 MMPC, 5
Consider four real numbers $x$, $y$, $a$, and $b$, satisfying $x + y = a + b$ and $x^2 + y^2 = a^2 + b^2$. Prove that $x^n + y^n = a^n + b^n$, for all $n \in \mathbb{N}$.
MMPC Part II 1996 - 2019, 2016.3
This problem is about pairs of consecutive whole numbers satisfying the property that one of the numbers is a perfect square and the other one is the double of a perfect square.
(a) The smallest such pairs are $(0,1)$ and $(8,9)$, Indeed $0=2 \cdot 0^2$ and $1=1^2$; $8=2 \cdot 2^2$ and $9=3^2$. Show that there are infinitely many pairs of the form $(2a^2,b^2)$ where the smaller number is the double of a perfect square satisfying the given property.
(b) Find a pair of integers satisfying the property that is not in the form given in the first part, that is, find a
pair of integers such that the smaller one is a perfect square and the larger one is the double of a perfect
square.
MMPC Part II 1996 - 2019, 2016.5
Consider four real numbers $x$, $y$, $a$, and $b$, satisfying $x + y = a + b$ and $x^2 + y^2 = a^2 + b^2$. Prove that $x^n + y^n = a^n + b^n$, for all $n \in \mathbb{N}$.
MMPC Part II 1996 - 2019, 2016.2
Let $s_1,s_2,s_3,s_4,...$ be a sequence (infinite list) of $1$s and $0$s. For example $1,0,1,0,1,0,...$, that is, $s_n=1$ if $n$ is odd and $s_n=0$ if $n$ is even, is such a sequence. Prove that it is possible to delete infinitely many terms in $s_1,s_2,s_3,s_4,...$ so that the resulting sequence is the original sequence. For the given example, one can delete $s_3,s_4,s_7,s_8,s_{11},s_{12},...$
MMPC Part II 1996 - 2019, 2016.4
It is a fact that every set of 2016 consecutive integers can be partitioned in two sets with the following four
properties:
(i) The sets have the same number of elements.
(ii) The sums of the elements of the sets are equal.
(iii) The sums of the squares of the elements of the sets are equal.
(iv) The sums of the cubes of the elements of the sets are equal.
Let $S =\{n + 1; n + 2;$ [b]. . .[/b] $; n + k\}$ be a set of $k$ consecutive integers.
(a) Determine the smallest value of $k$ such that property (i) holds for $S$.
(b) Determine the smallest value of $k$ such that properties (i) and (ii) hold for $S$.
(c) Show that properties (i), (ii) and (iii) hold for $S$ when $k = 8$.
(d) Show that properties (i), (ii), (iii) and (iv) hold for $S$ when $k = 16$.
2016 MMPC, 1
If a polygon has both an inscribed circle and a circumscribed circle, then define the [i]halo[/i] of that polygon to be the region inside the circumcircle but outside the incircle. In particular, all regular polygons and all triangles have halos.
(a) What is the area of the halo of a square with side length 2?
(b) What is the area of the halo of a 3-4-5 right triangle?
(c) What is the area of the halo of a regular 2016-gon with side length 2?