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# CMI Entrance 2023 Question Paper and Solutions

Please use the following form to contribute problems of CMI BSc Math Entrance 2023. We will work on the solutions.

Also come back to this page to see the updates on Problems and Solutions of CMI BSc Math Entrance 2023.

B1. Let be an odd positive number greater than 1. We have the condition that .
(i) Find the two smallest values of .
(ii) Prove that there are infinitely many such .

B2. Solve for .
(i)
(ii) Show that is of the form . Find necessary and sufficient conditions on

B3. The polynomial
has exactly one real number such that .
(i) Show that if are rational, is also rational.
(ii) Show that if are integers, is also an integer.
Hint: Consider the roots of .

B4. In a class there are students with unequal heights.
(i) Find the number of orderings of the students such that the shortest person is not at the front and the tallest person is not at the end.
(ii) For , let denote the number of students in front of the th student who are taller than the th student. For example, consider the sequence , with 4 being the front 5 being the end. Then . Define the badness of an ordering as the . The example sequence has badness of 2. Let denote the number of orderings of that have badness . Find .

Hint: Consider as the number of orderings of with badness less than or equal to .

B5. In whatever follows denotes a differentiable function from to . denotes the composition of .
(i) If then for all or . Fill in the blanks and justify.
(ii) Assume that the range of is of the form . Show that if , then the range of is . (Hint: Consider a maximal element in the range of ). (iii) If satisfies , then is onto. Prove that is either strictly increasing or strictly decreasing. Furthermore show that if is strictly increasing, then is unique.

B6. For each integer , define the sequence , by

for each . Let denote the set of distinct numbers in the sequence Find all values of for which the set is finite.

If you can't prove the above, you may solve these for partial credit.
(i) Show that there is no for which or 2 .
(ii) Prove that only if .
(iii) For an given , there exists an such that .
(iv) Find one such that

Please use the following form to contribute problems of CMI BSc Math Entrance 2023. We will work on the solutions.

Also come back to this page to see the updates on Problems and Solutions of CMI BSc Math Entrance 2023.

B1. Let be an odd positive number greater than 1. We have the condition that .
(i) Find the two smallest values of .
(ii) Prove that there are infinitely many such .

B2. Solve for .
(i)
(ii) Show that is of the form . Find necessary and sufficient conditions on

B3. The polynomial
has exactly one real number such that .
(i) Show that if are rational, is also rational.
(ii) Show that if are integers, is also an integer.
Hint: Consider the roots of .

B4. In a class there are students with unequal heights.
(i) Find the number of orderings of the students such that the shortest person is not at the front and the tallest person is not at the end.
(ii) For , let denote the number of students in front of the th student who are taller than the th student. For example, consider the sequence , with 4 being the front 5 being the end. Then . Define the badness of an ordering as the . The example sequence has badness of 2. Let denote the number of orderings of that have badness . Find .

Hint: Consider as the number of orderings of with badness less than or equal to .

B5. In whatever follows denotes a differentiable function from to . denotes the composition of .
(i) If then for all or . Fill in the blanks and justify.
(ii) Assume that the range of is of the form . Show that if , then the range of is . (Hint: Consider a maximal element in the range of ). (iii) If satisfies , then is onto. Prove that is either strictly increasing or strictly decreasing. Furthermore show that if is strictly increasing, then is unique.

B6. For each integer , define the sequence , by

for each . Let denote the set of distinct numbers in the sequence Find all values of for which the set is finite.

If you can't prove the above, you may solve these for partial credit.
(i) Show that there is no for which or 2 .
(ii) Prove that only if .
(iii) For an given , there exists an such that .
(iv) Find one such that

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