Beyond Solutions

[Soln] 2013 AIME I Problem 2

Posted on 5 Jun 2016 by kjytay

2013 AIME I 2. Find the number of five-digit positive integers, $n$ , that satisfy the following conditions:

(a) the number $n$ is divisible by 5,
(b) the first and last digits of $n$ are equal, and
(c) the sum of the digits of $n$ is divisible by 5.

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Posted in Grade 12, USA | Tagged 2013, AIME, Combinatorics | Leave a comment

2013 AIME I Problem 2

Posted on 5 Jun 2016 by kjytay

2013 AIME I 2. Find the number of five-digit positive integers, $n$ , that satisfy the following conditions:

(a) the number $n$ is divisible by 5,
(b) the first and last digits of $n$ are equal, and
(c) the sum of the digits of $n$ is divisible by 5.

Posted in Grade 12, USA | Tagged 2013, AIME, Combinatorics | Leave a comment

2013 AIME I 1. The AIME Triathlon consists of a half-mile swim, a 30-mile bicycle, and an eight-mile run. Tom swims, bicycles, and runs at constant rates. He runs five times as fast as he swims, and he bicycles twice as fast as he runs. Tom completes the AIME Triathlon in four and a quarter hours. How many minutes does he spend bicycling?

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Posted in Grade 12, USA | Tagged 2013, AIME, Algebra | Leave a comment

2013 AIME I Problem 1

Posted on 4 Jun 2016 by kjytay

Posted in Grade 12, USA | Tagged 2013, AIME, Algebra | Leave a comment

[Soln] 2016 Pan-African Math Olympiad Problem 3

Posted on 3 Jun 2016 by kjytay

2016 PAMO 3. For any positive integer $n$ , we define the integer $P(n)$ as

$P(n) = n(n+1)(2n+1)(3n+1) \dots (16n + 1).$

Find the greatest common divisor of the integers $P(1), P(2), P(3), \dots, P(2016)$ .

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Posted in Grade 12, Intl/Regional | Tagged 2016, Number Theory, PAMO | Leave a comment

[Hints] 2016 Pan-African Math Olympiad Problem 3

Posted on 3 Jun 2016 by kjytay

Click for hints for 2016 Pan-African Math Olympiad Problem 3

Posted in Grade 12, Intl/Regional | Tagged 2016, Number Theory, PAMO | Leave a comment

2016 Pan-African Math Olympiad Problem 3

Posted on 2 Jun 2016 by kjytay

2016 PAMO 3. For any positive integer $n$ , we define the integer $P(n)$ as

$P(n) = n(n+1)(2n+1)(3n+1) \dots (16n + 1).$

Find the greatest common divisor of the integers $P(1), P(2), P(3), \dots, P(2016)$ .

Posted in Grade 12, Intl/Regional | Tagged 2016, Number Theory, PAMO | Leave a comment

[Soln] 2015 IMC Day 1 Problem 2

Posted on 1 Jun 2016 by kjytay

2015 IMC 1.2. For a positive integer $n$ , let $f(n)$ be the number obtained by writing $n$ in binary and replacing every $0$ with $1$ and vice versa. For example, $n = 23$ is $10111$ in binary, so $f(n)$ is $1000$ in binary, therefore $f(23) = 8$ . Prove that