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# Contest math

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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