Library / /333 of 490

, 2024

Algebra Difficulty 2.6 Junior Find the answer Canada

Each of the variables $a$ ,
$b$ , $c$ , $d$ , and $e$ represents a positive integer with the
properties that $\begin{align*} b+d &> a + d\\ c+e &> b+e\\ b+d&= c\\ a+c &= b+e\end{align*}$ Which of the variables has the
greatest value?

Pick one

Solution

Since $\text{b + d gt; a + d}$ , then
$\text{b gt; a}$ . This means that $a$ does not have the greatest value.

Since $\text{c + e gt; b + e}$ , then $\text{c gt; b}$ . This means that $b$ does not have the greatest value.

Since $b + d = c$ and each of $b$ , $c$ , $d$
is positive, then $\text{d lt; c}$ , which
means that $d$ does not have the
greatest value.

Consider the last equation $a + c = b + e$ along with the fact that $\text{a lt; b lt; c}$ .

From this, we see that $e = c + (a-b)$ .

Since $\text{a lt; b}$ , then $a-b$ is negative and so $\text{e lt; c}$ .

This means that $c$ has the greatest
value.

Want a route through all this instead of an archive? The track puts 2,604 problems in a working order, from Junior Challenge level to the IMO shortlist.

Source: CEMC, University of Waterloo, licensed CC-BY-NC-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.