Difference between revisions of "2020 CIME I Problems"
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==Problem 1== | ==Problem 1== | ||
− | A knight begins on the point <math>(0,0)</math> in the coordinate plane. From any point <math>(x,y)</math> the knight moves to either <math>(x+2,y+1)</math> or <math>(x+1,y+2)</math>. Find the number of ways the knight can reach <math>(15,15)</math>. | + | A knight begins on the point <math>(0,0)</math> in the coordinate plane. From any point <math>(x,y)</math> the knight moves to either <math>(x+2,y+1)</math> or <math>(x+1,y+2)</math>. Find the number of ways the knight can reach the point <math>(15,15)</math>. |
[[2020 CIME I Problems/Problem 1 | Solution]] | [[2020 CIME I Problems/Problem 1 | Solution]] | ||
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==Problem 8== | ==Problem 8== | ||
− | A person has been declared the first to inhabit a certain planet on day <math>N=0</math>. For each positive integer <math>N | + | A person has been declared the first to inhabit a certain planet on day <math>N=0</math>. For each positive integer <math>N>0</math>, if there is a positive number of people on the planet, then either one of the following three occurs, each with probability <math>\frac{1}{3}</math>: |
:(i) the population stays the same; | :(i) the population stays the same; | ||
Line 63: | Line 63: | ||
==Problem 12== | ==Problem 12== | ||
− | Define a sequence <math>a_0, a_1, a_2, ...</math> by <cmath>a_i = \underbrace{1 \ | + | Define a sequence <math>a_0, a_1, a_2, ...</math> by <cmath>a_i=\underbrace{1\ldots1}_{2^{i}\text{ 1's}}\underbrace{0\ldots0}_{(2^i-1)\text{ 0's}}1_2,</cmath> where <math>a_i</math> is expressed in binary. Let <math>S</math> be the sum of the digits when <math>a_0 a_1 a_2 \cdots a_{10}</math> is expressed in binary. Find the remainder when <math>S</math> is divided by <math>1000</math>. |
+ | |||
+ | [[2020 CIME I Problems/Problem 12 | Solution]] | ||
+ | |||
+ | ==Problem 13== | ||
+ | Chris writes on a piece of paper the positive integers from <math>1</math> to <math>8</math> in that order. Then, he randomly writes either <math>+</math> or <math>\times</math> between every two adjacent numbers, each with equal probability. The expected value of the expression he writes can be expressed as <math>\frac{p}{q}</math> for relatively prime positive integers <math>p</math> and <math>q</math>. Find the remainder when <math>p+q</math> is divided by <math>1000</math>. | ||
+ | |||
+ | [[2020 CIME I Problems/Problem 13 | Solution]] | ||
+ | |||
+ | ==Problem 14== | ||
+ | Let ABC be a triangle with sides <math>AB = 5, BC = 7, CA = 8</math>. Denote by <math>O</math> and <math>I</math> the circumcenter and incenter of <math>\triangle ABC</math>, respectively. The incircle of <math>\triangle ABC</math> touches <math>\overline{BC}</math> at <math>D</math>, and line <math>OD</math> intersects the circumcircle of <math>\triangle AID</math> again at <math>K</math>. Then the length of <math>DK</math> can be expressed in the form <math>\frac{m \sqrt n}{p}</math>, where <math>m, n, p</math> are positive integers, <math>m</math> and <math>p</math> are relatively prime, and <math>n</math> is not divisible by the square of any prime. Find <math>m+n+p</math>. | ||
+ | |||
+ | [[2020 CIME I Problems/Problem 14 | Solution]] | ||
+ | |||
+ | ==Problem 15== | ||
+ | Find the number of integer sequences <math>a_1, a_2, \ldots, a_6</math> such that | ||
+ | :(1) <math>0 \le a_1 < 6</math> and <math>12 \le a_6 < 18</math>, | ||
+ | :(2) <math>1 \le a_{k+1}-a_k < 6</math> for all <math>1 \le k < 6</math>, and | ||
+ | :(3) there do not exist <math>1 \le i < j \le 6</math> such that <math>a_j-a_i</math> is divisible by <math>6</math>. | ||
+ | |||
+ | [[2020 CIME I Problems/Problem 15 | Solution]] | ||
+ | |||
+ | {{CIME box|year=2020|n=I|before=[[2019 CIME II Problems]]|after=[[2020 CIME II Problems]]}} |
Latest revision as of 18:00, 31 August 2020
2020 CIME I (Answer Key) | AoPS Contest Collections | ||
Instructions
| ||
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 |
Contents
Problem 1
A knight begins on the point in the coordinate plane. From any point
the knight moves to either
or
. Find the number of ways the knight can reach the point
.
Problem 2
At the local Blast Store, there are sufficiently many items with a price of for each nonnegative integer
. A sales tax of
is applied on all items. If the total cost of a purchase, after tax, is an integer number of cents, find the minimum possible number of items in the purchase.
Problem 3
In a math competition, all teams must consist of between and
members, inclusive. Mr. Beluhov has
students and he realizes that he cannot form teams so that each of his students is on exactly one team. Find the sum of all possible values of
.
Problem 4
There exists a unique positive real number satisfying
Given that
can be written in the form
for integers
with
, find
.
Problem 5
Let be a rectangle with sides
and let
be the reflection of
over
. If
and the area of
is
, find the area of
.
Problem 6
Find the number of complex numbers satisfying
and
.
Problem 7
For every positive integer , define
Suppose that the sum
can be expressed as
for relatively prime integers
and
. Find the remainder when
is divided by
.
Problem 8
A person has been declared the first to inhabit a certain planet on day . For each positive integer
, if there is a positive number of people on the planet, then either one of the following three occurs, each with probability
:
- (i) the population stays the same;
- (ii) the population increases by
; or
- (iii) the population decreases by
. (If there are no greater than
people on the planet, the population drops to zero, and the process terminates.)
The probability that at some point there are exactly people on the planet can be written as
, where
and
are positive integers such that
isn't divisible by
. Find the remainder when
is divided by
.
Problem 9
Let be a cyclic quadrilateral with
. Let
be the point on
such that
. Then
can be expressed in the form
, where
and
are relatively prime positive integers. Find
.
Problem 10
Let be the divisors of a positive integer
. Let
be the sum of all positive integers
satisfying
Find the remainder when
is divided by
.
Problem 11
An of a triangle is a circle tangent to one of the sides of the triangle and the extensions of the other two sides. Let
be a triangle with
and let
denote the radii of the excircles opposite to
, respectively. If
and
, then
can be expressed in the form
, where
and
are positive integers and
isn't divisible by the square of any prime. Find
.
Problem 12
Define a sequence by
where
is expressed in binary. Let
be the sum of the digits when
is expressed in binary. Find the remainder when
is divided by
.
Problem 13
Chris writes on a piece of paper the positive integers from to
in that order. Then, he randomly writes either
or
between every two adjacent numbers, each with equal probability. The expected value of the expression he writes can be expressed as
for relatively prime positive integers
and
. Find the remainder when
is divided by
.
Problem 14
Let ABC be a triangle with sides . Denote by
and
the circumcenter and incenter of
, respectively. The incircle of
touches
at
, and line
intersects the circumcircle of
again at
. Then the length of
can be expressed in the form
, where
are positive integers,
and
are relatively prime, and
is not divisible by the square of any prime. Find
.
Problem 15
Find the number of integer sequences such that
- (1)
and
,
- (2)
for all
, and
- (3) there do not exist
such that
is divisible by
.
2020 CIME I (Problems • Answer Key • Resources) | ||
Preceded by 2019 CIME II Problems |
Followed by 2020 CIME II Problems | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All CIME Problems and Solutions |