Olympiad problems

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Found problems: 2

1973 AMC 12/AHSME, 24

Tags: systems of equations

The check for a luncheon of 3 sandwiches, 7 cups of coffee and one piece of pie came to $ \$3.15$. The check for a luncheon consisting of 4 sandwiches, 10 cups of coffee and one piece of pie came to $ \$4.20$ at the same place. The cost of a luncheon consisting of one sandwich, one cup of coffee, and one piece of pie at the same place will come to $ \textbf{(A)}\ \$1.70 \qquad \textbf{(B)}\ \$1.65 \qquad \textbf{(C)}\ \$1.20 \qquad \textbf{(D)}\ \$1.05 \qquad \textbf{(E)}\ \$0.95$

2020 Puerto Rico Team Selection Test, 2

Tags: number theory , systems of equations , Digits , Diophantine Equations , diophantine

The cost of $1000$ grams of chocolate is $x$ dollars and the cost of $1000$ grams of potatoes is $y$ dollars, the numbers $x$ and $y$ are positive integers and have not more than $2$ digits. Mother said to Maria to buy $200$ grams of chocolate and $1000$ grams of potatoes that cost exactly $N$ dollars. Maria got confused and bought $1000$ grams of chocolate and $200$ grams of potatoes that cost exactly $M$ dollars ($M >N$). It turned out that the numbers $M$ and $N$ have no more than two digits and are formed of the same digits but in a different order. Find $x$ and $y$.