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1. We know that 7a=b, 8b=c, and 9c=d. If d=2016, what is a?

 

2. For some numbers \(p\) and \(q\), the solution of the system of equations

 \(\begin{align*} x+3y &= 5 , \\ 3x+py &= 7 \end{align*}\)
is \((x,y)=(-1,q)\). What is the value of \(p\)?

 

3. If \(\frac{x}{2} + \frac{y}{3} = 1\) and \(\frac{x}{4} + \frac{2y}{3} = -1 \), what is the value of the sum of \(x\) and \(y\)?

 

 

4. Let a, b, c, and d be positive integers. We know that

 \(\begin{align*} a+b&=kc,\\ b+c&=kd,\\ c+d&=ka,\\ d+a&=kb. \end{align*}\)
Find the maximum of \(\dfrac ab + \dfrac bc + \dfrac cd + \dfrac da.\)

 Dec 10, 2022
 #1
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1. a = 20

 

2. Solving the system, we get p = -4.

 Dec 10, 2022
 #2
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3. x + y = 5/3.

 

4. The maximum of a/b + b/c + c/d + d/a is 15/7.

 Dec 11, 2022

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