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**B1.** Let \( n \) be an odd positive number greater than 1. We have the condition that \( n \mid 2023^n-1 \).

(i) Find the two smallest values of \( n \).

(ii) Prove that there are infinitely many such \( n \mathrm{~s}\).

**B2. **Solve for \( f: \mathbb{Z}^{+} \rightarrow \mathbb{Z}^{+} \).

(i) \( f(m+n)=f(m)+f(n)+m n(m+n) \)

(ii) Show that \(f \) is of the form \( \sum_{i=0}^d c_i n^i \). Find necessary and sufficient conditions on \( d, c_0, c_1, \ldots, c_d\)

**B3.** The polynomial

\( p(x)=x^4+a x^3+b x^2+c x+d \) has exactly one real number \( r \) such that \( p(r)=0 \).

(i) Show that if \( a, b, c, d \) are rational, \( r \) is also rational.

(ii) Show that if \( a, b, c, d \) are integers, \( r \) is also an integer.

Hint: Consider the roots of \( p^{\prime}(x) \).

**B4.** In a class, there are \( n \) 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 \( 1 \leq i \leq n \), let \( b_i \) denote the number of students in front of the \( i \) th student who are taller than the \( i \) th student. For example, consider the sequence \( [4,1,2,6,7,5]\), with 4 being the front 5 being the end. Then \( b_1=0, b_2=1, b_3=1, b_4=0, b_5=0, b_6=2 \). Define the badness of an ordering as the \( \max _{i \in[n]} b_i \). The example sequence has badness of 2. Let \( f_k(n) \) denote the number of orderings of \( [n]\) that have badness \( k \). Find \( f_k(n) \).

Hint: Consider \( g_k(n) \) as the number of orderings of \( n \) with badness less than or equal to \( k \).

**B5.** In whatever follows \( f \) denotes a differentiable function from \( R \) to \( R \). \( f \circ f\) denotes the composition of \( f(x) \).

(i) If \( f(f(x))=f(x) \forall x \in R \) then for all \( x, f^{\prime}(x)= \) or \( f^{\prime}(f(x))= \) . Fill in the blanks and justify.

(ii) Assume that the range of \( f\) is of the form \( (-\infty,+\infty),[a, \infty),(-\infty, b],[a, b]\). Show that if \( f \circ f=f \), then the range of \( f \) is \( \mathbf{R}\). (Hint: Consider a maximal element in the range of \( f \) ). (iii) If \( g \) satisfies \( g \circ g \circ g=g \), then \( g \) is onto. Prove that \( g \) is either strictly increasing or strictly decreasing. Furthermore show that if \( g\) is strictly increasing, then \( g \) is unique.

**B6. **For each integer \( a_0>1 \), define the sequence \( a_0, a_1, a_2, \ldots\), by

$$

a_{n+1}= \begin{cases}\sqrt{a_n} & \text { if } \sqrt{a_n} \text { is an integer } \ a_n+3 & \text { otherwise }\end{cases}

$$

for each \( n \geq 0 \). Let \( S \) denote the set of distinct numbers in the sequence \( a_0, a_1, a_2, \ldots \) Find all values of \( a_0 \) for which the set \( S \) is finite.

If you can't prove the above, you may solve these for partial credit.

(i) Show that there is no \( a_0 \) for which \( |S|=1 \) or 2 .

(ii) Prove that \( |S|=3 \) only if \( a_0=3 \).

(iii) For an given \( k \geq 3 \), there exists an \( a_0 \) such that \( |S|=k\).

(iv) Find one \( a_0 \) such that \( |S|=\infty \)

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