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MT #2: Algebra

Timings: 17:00-20:00 Hrs    Date: 18 Feb 2021

Part A: Short-answer type questions

Submission file: Write answers to all the ten questions on a single sheet of paper. Email a picture of your answer sheet. Name the file as PartA.jpg.

For this part, answers must be written without any explanation.

  1. Let \(p_1(x),p_2(x),\ldots,p_m(x)\) be \(m\) distinct polynomials such that:
    \[ p_1(x) = p_2(x) = \ldots = p_m(x) \mbox{ if } x<0 \] What is the largest possible value of \(m\)? For example, if it is possible for \(m\) to be 4, then the polynomials might look like this:

  2. Find a polynomial \(q(x)\) with integer coefficients with \(\sqrt{3}+i\) as a root. In case there are multiple candidates, pick a polynomial with the least degree.

  3. (i) Find the remainder when \(f(x)=6x^{16}+4x^{22}+5x^{12}+x^{2}\) is divided by \((x^{2}+1)\). [2 marks]

    (ii) Find the remainder when \(x f(x)\) is divided by \((x^{2}+1)\). [2 marks]

  4. Find two quadratic polynomials \( p(x) \) and \( g(x) \) that satisfy the following conditions:
    1. Both \(p(x)\) and \(g(x)\) have two distinct real roots.
    2. The sum \( p(x)+g(x) \) has no real root.

  5. A grazing field has \(10\) kgs of grass. Every Sunday a herd of cows eats \(x\) kgs of grass. Over a week's time the grass grows by \( 10\% \). What is the maximum value of \(x\) that will allow the cows to feed indefinitely without running out of grass?

  6. Consider the simultaneous equations in variables \(x\) and \(y\), where \(k\) is a constant: \begin{align*} 2x + y &= kx + 4 \\ x + 2y &= ky + 6k \end{align*} For what values of \(k\) does the system not have a solution?

  7. Notation. \( { }^nC_{k} \) represents the binomial coefficient.
    Consider the set of prime numbers less than 100 (listed below). Pick two numbers \(n\) and \(k\) with \(n > k \) from this set such that \( {}^nC_{k} \) is maximized.

  8. We have numbers \( x_1,\ldots,x_{51} \) such that each \(x_i\) is either \(+1\) or \(-1\). What is the minimum value of following sum? \[ \left\lvert \sum_{ 1\leq i < j \leq 51 } x_ix_j \right\rvert \]

    Problem source: PRMO 2019.

  9. Notation. \( [n] \) denotes the set of numbers \( \{1,2,\ldots,n\} \). Assume \(n>100\) for this problem.

    A function \(f:[n]\rightarrow \mathbb{R} \) is defined as follows:
    \[ f(x) = \begin{cases} 0 &\mbox{if } x = 1 \\ 1 & \mbox{if } x = n \\ \frac{1}{2} (f(x-1) + f(x+1)) & \mbox{if } 1 < x < n \end{cases} \] What is the value of \( f(3) \) in terms of \(n\)?

  10. Let \( f(x) = 37 x^{4}-37 x^{3}-x^{2}+9 x-2 \). Let the four roots of \( f(x) \) be \( r_{1}, r_{2}, r_{3}\) and \( r_{4} \). Find the value of \[ \left(r_{1}+r_{2}+r_{3}\right)\left(r_{1}+r_{2}+r_{4}\right)\left(r_{1}+r_{3}+r_{4}\right)\left(r_{2}+r_{3}+r_{4}\right) \] Problem source: Stanford Math Tournament.

Part B: Subjective questions

Submission files: Each question in this part must be answered on a page of its own. Name the files as B1.jpg, B2.jpg, etc. In case you have multiple files for the same question, say B4, name the corresponding files as B4-1.jpg, B4-2.jpg, etc.

Clearly explain your entire reasoning. No credit will be given without reasoning. Partial solutions may get partial credit.

B1. Prove the following inequality for \( a, b>0 \): \[ \large \sqrt[2021]{a b^{2020}} \leq \frac{ a+2020b }{2021} \]

Problem source: Problem solving strategies. Prob. 4, Inequalities.

B2. Consider the polynomial \(f(x) = a_{n}x^n +a_{n-1}x^{n-1} + \cdots + a_{1}x+a_{0}\), where each coefficient \(a_{i}\) is either \(0\) or \(1\). If \( f(2) = 14\), find the polynomial \(f(x)\).

B3. Compute the smallest value \(p\) such that, for all \(q> p\), the polynomial \( x^3 - 7x^2 +qx +16 = 0 \) has exactly one real root.

Problem source: Stanford Math Tournament.

B4. Let \(f(x)=a_{0} x^{n}+\cdots+a_{n}\) be a non-constant polynomial with real coefficients satisfying \[ f(x) f\left(x^{2}\right)=f\left(x^{3}+x\right) \] for all real numbers \(x\). Show that \(f\) has no real root.

B5. Show that \(p(x)\) does not have any real roots where: \[ p(x)=x^{2 n}-2 x^{2 n-1}+3 x^{2 n-2}-4 x^{2 n-3}+\cdots-2 n x+(2 n+1) \]
Problem source: Problem solving strategies. Prob. 5, Polynomials.

B6. (i) Simplify \[ \sum_{j=0}^{n} \sum_{i=j}^{n}{ }^{n} C_{i}{ }^{i} C_{j} \]

(ii) Calculate the value of the expression (i) when \( n=5 \).

Problem sources for B4 and B6: Madhava Mathematics Competition 2019.