- This is Problem 4b from the 2017 Euclid Contest.
In an arithmetic sequence with \(5\) terms, the sum of the squares of the first \(3\) terms equals the sum of the squares of the last \(2\) terms. If the first term is \(5\), determine all possible values of the fifth term.
- This is Problem 4b from the 1998 Euclid Contest.
The \(n\)th term of an arithmetic sequence is given by \(t_n=555-7n\). If \(S_n = t_1+t_2+\cdots + t_n\), determine the smallest value of \(n\) for which \(S_n \lt 0\).
- This is Problem 5a from the 2003 Euclid Contest.
In the series of odd numbers \(1+3+5-7-9-11+13+15+17-19-21-23 \cdots\) the signs alternate every three terms, as shown. What is the sum of the first \(300\) terms of the series?
- This is Problem 5b from the 2010 Euclid Contest.
A geometric sequence has \(20\) terms. The sum of its first two terms is \(40\). The sum of its first three terms is \(76\). The sum of its first four terms is \(130\). Determine how many of the terms in the sequence are integers.
- This is Problem 6b from the 2016 Euclid Contest.
Determine all possible values for the area of a right-angled triangle with one side length equal to \(60\) and with the property that its side lengths form an arithmetic sequence.
- This is Problem 6b from the 2014 Euclid Contest.
The geometric sequence with \(n\) terms \(t_1, ~t_2, \ldots, ~t_{n-1}, ~t_n\)has \(t_1t_n=3\). Also, the product of all \(n\) terms equals \(59~049\) (that is, \(t_1t_2t_3\cdot\cdot\cdot t_{n-1}t_n=59~049\)). Determine the value of \(n\).
- This is Problem 6b from the 2008 Euclid Contest.
The numbers \(a\), \(b\), and \(c\), in that order, form a three term arithmetic sequence and \(a+b+c=60\).
The numbers \(a-2\), \(b\), and \(c+3\), in that order, form a three term geometric sequence.
Determine all possible values of \(a\), \(b\), and \(c\).
- This is Problem 7b from the 2006 Euclid Contest.
Suppose that \(a\), \(b\), and \(c\) are three consecutive terms in an arithmetic sequence. Prove that \(a^2-bc\), \(b^2-ac\), and \(c^2-ab\) are also three consecutive terms in an arithmetic sequence.
- This is Problem 8 from the 2011 Euclid Contest.
The Sieve of Sundaram uses the following infinite table of positive integers:
\(\begin{array} {ccccc} 4 & 7 & 10 & 13 & \cdots \\ 7 & 12 & 17 & 22 & \cdots \\ 10 & 17 & 24 & 31 & \cdots \\ 13 & 22 & 31 & 40 & \cdots \\ \vdots & \vdots & \vdots & \vdots & \ddots \end{array}\)
The numbers in each row in the table form an arithmetic sequence. The numbers in each column in the table form an arithmetic sequence. The first four entries in each of the first four rows and columns are shown.
- Determine the number in the \(50\)th row and \(40\)th column.
- Determine a formula for the number in the \(R\)th row and \(C\)th column.
- Prove that if \(N\) is an entry in the table, then \(2N+1\) is composite.