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 <math>(15,15)</math>. | ||
+ | |||
+ | [[2020 CIME I Problems/Problem 1 | Solution]] | ||
==Problem 2== | ==Problem 2== | ||
At the local Blast Store, there are sufficiently many items with a price of <math>$n.99</math> for each nonnegative integer <math>n</math>. A sales tax of <math>7.5\%</math> is applied on all items. If the total cost of a purchase, after tax, is an integer number of cents, find the minimum possible number of items in the purchase. | At the local Blast Store, there are sufficiently many items with a price of <math>$n.99</math> for each nonnegative integer <math>n</math>. A sales tax of <math>7.5\%</math> 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. | ||
+ | |||
+ | [[2020 CIME I Problems/Problem 2 | Solution]] | ||
==Problem 3== | ==Problem 3== | ||
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teams so that each of his students is on exactly one team. Find the sum of all | teams so that each of his students is on exactly one team. Find the sum of all | ||
possible values of <math>n</math>. | possible values of <math>n</math>. | ||
+ | |||
+ | [[2020 CIME I Problems/Problem 3 | Solution]] | ||
==Problem 4== | ==Problem 4== | ||
− | There exists a unique positive real number <math>x</math> satisfying <cmath>x=\sqrt{x^2+\frac{1}{x^2}} - \sqrt{x^2-\frac{1}{x^2}}</cmath> | + | There exists a unique positive real number <math>x</math> satisfying <cmath>x=\sqrt{x^2+\frac{1}{x^2}} - \sqrt{x^2-\frac{1}{x^2}}.</cmath> Given that <math>x</math> can be written in the form <math>x=2^\frac{m}{n} \cdot 3^\frac{-p}{q}</math> for integers <math>m, n, p, q</math> with <math>\gcd(m, n) = \gcd(p, q) = 1</math>, find <math>m+n+p+q</math>. |
+ | |||
+ | [[2020 CIME I Problems/Problem 4 | Solution]] | ||
+ | |||
+ | ==Problem 5== | ||
+ | Let <math>ABCD</math> be a rectangle with sides <math>AB>BC</math> and let <math>E</math> be the reflection of <math>A</math> over <math>\overline{BD}</math>. If <math>EC=AD</math> and the area of <math>ECBD</math> is <math>144</math>, find the area of <math>ABCD</math>. | ||
+ | |||
+ | ==Problem 6== | ||
+ | Find the number of complex numbers <math>z</math> satisfying <math>|z|=1</math> and <math>z^850+z^350+1=0</math>. | ||
+ | |||
+ | ==Problem 7== | ||
+ | For every positive integer <math>n</math> define <cmath>f(n)=\frac{n}{1 \cdot 3 \cdot 5 \cdot \cdot \cdot (2n+1).</cmath> Suppose that the sum <math>f(1)+f(2)+\cdot \cdot \cdot+f(2020)</math> can be expressed as <math>\frac{p}{q}</math> for relatively prime integers <math>p</math> and <math>q</math>. Find the remainder when <math>p</math> is divided by <math>1000</math>. |
Revision as of 13:38, 30 August 2020
2020 CIME I (Answer Key) | AoPS Contest Collections | ||
Instructions
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1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 |
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
.
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
\[f(n)=\frac{n}{1 \cdot 3 \cdot 5 \cdot \cdot \cdot (2n+1).\] (Error compiling LaTeX. Unknown error_msg)
Suppose that the sum can be expressed as
for relatively prime integers
and
. Find the remainder when
is divided by
.