Question

14 students have taken part in a math olympiad. Every problem inthat olympiad has been solved by exactly 7 students. Furthermore,the first 13 students have all solved 10 problems each. If the 14thstudent has solved exactly k problems, then what is the sum of allpossible values of k?একটা গণিত অলিম্পিয়াডে 14 জন শিক্ষার্থী অংশ নিয়েছে। অলিম্পিয়াডেরপ্রতিটা সমস্যাই ঠিক 7 জন শিক্ষার্থী সমাধান করেছে। আরও বলে দেওয়াআছে যে, প্রথম 13 জন শিক্ষার্থী ঠিক 10টা করে সমস্যার সমাধান করেছে।14তম শিক্ষার্থী যদি ঠিক kটা সমস্যার সমাধান করে থাকে, তাহলে }-এর

Answer 1

Answer:

Let’s introduce another variable: p, the number of problems.

Since each problem has been solved by 7 students, there are 7*p solves in total.

Since the first 13 students solved 10 problems each, that accounts for 130 solves. With the 14th student we have 130 + k solves in total. It follows that 7p = 130 + k is a necessary condition to have a valid configuration of problems and solves.

Since there are at least 130 solves, p will have to be at least 19(otherwise if p = 18, we’d have 18*7 = 126 solves, which is too few). This also tells us that k must be at least 3, as 19*7 = 133 solves.

What’s the maximum value of p? Even if the 14th student answered every single problem, we’d still need to have 6 other students answer each of those problems as well. In other words, from our bank of 130 solves from the first 13 students, at least 6 must be used per problem. 130 / 6 = 21.66… so there can’t be more than 21 problems(22 problems would need 22*6 = 132 solves that aren’t from the 14th student, which can’t possibly work) so, p <= 21.

Alright, that’s already pretty restrictive. The number of problems can only be 19,20, or 21. We don’t know for sure that these actually give viable configurations, so we’ll investigate further.

Note the earlier equation we had, 7p = 130 + k ==> k = 7p - 130.

Plugging in p = 19,20,21 we get the corresponding values of k = 3,10,17. So far these values of k are plausible, the 14th student isn’t answering more questions than exist, nor less than 0.

Step-by-step explanation:

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