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Algebra Exam Paper

Junior Exam J3

Each problem is worth 7 marks.

Time: 4 hours

No books, notes or calculators permitted

Note: You must answer with proof.

Q1

Let \(a\), \(b\), \(x\), \(y\) and \(z\) be integers such that

\[0 \leq x \leq a \leq y \leq b \leq z\]

\[a + b + x + y + z = 2016\]

(a) Find the maximum value of \(x + y + z\). Find all \(x\), \(y\) and \(z\) which satisfy this.

(b) Find the minimum value of \(x + y + z\). Find all \(x\), \(y\) and \(z\) which satisfy this.

Q2

The polynomial

\[P(x) = x^3 + 3x^2 + ax + b\]

has 3 distinct integer roots which can be arranged to form an arithmetic progression. Find all \(a\) and \(b\) which satisfy this.

Q3

Let \(A\) denote the set containing all non-zero real numbers.

(a) Find all functions \(f:\mathbb {R} \rightarrow \mathbb {R}\) such that \(x f(xy) + f(-y) = x f(x)\).

(b) Find all functions \(f:A \rightarrow \mathbb {R}\) such that \(x f(xy) + f(-y) = x f(x)\).

Q4

Assuming \(a\), \(b\), \(c\) and \(d\) are positive reals:

(a) Find the value of \(a + b + c + d\) if

\[\dfrac {1}{a} + \dfrac {1}{b} + \dfrac {4}{c} + \dfrac {16}{d} = \dfrac {64}{a + b + c + d}\]

(b) Prove that

\[\dfrac {1}{a} + \dfrac {1}{b} + \dfrac {4}{c} + \dfrac {16}{d} \geq \dfrac {64}{a + b + c + d}\]

Q5

\(f:\mathbb {N}^{+} \rightarrow \mathbb {R}\) is a function such that

\[f(p) + f(pn) = f(n)\]

where \(n\) is a number and \(p\) is a prime number. It is also known that

\[f(2^{2014}) + f(3^{2015}) + f(5^{2016}) = 2013\].

Find the value of

\[f(2014^2) + f(2015^3) + f(2016^5)\].

Note by Sharky Kesa
2 years, 7 months ago

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5) Choose arbitrary primes \(p_1\) and \(p_2\). Then, \(f(p_1)+f(p_1p_2) = f(p_2)\) and \(f(p_2)+f(p_1p_2) = f(p_1)\). This gives that \(f(p_1) = f(p_2)\) and \(f(p_1p_2) = 0\). Let \(f(p) = c\) for arbitrary prime \(p\).

Let \(n\) be an arbitrary positive composite number with prime factorization \(p_1p_2\cdots p_k\), where the \(p_i\) are not necessarily unique. Then, we propose that \(f(n) = (2-k)c\). We will show this by induction. From the formula given, \(f(n) = f(p_2\cdots p_k)-f(p_1)\). First, note that if \(k = 2\), \(f(n) = 0\) and if \(k = 3\), \(f(n) = -f(p_1) = -c\). Let \(n_i\) have \(i\) non-unique prime factors. Then, if \(f(n_k) = (2-k)c\), we have from our equation that \(f(n_{k+1}) = (2-k)c-c = (2-(k+1))c\). Our induction is complete.

Note that \(2014 = 2\times 19\times 53\), giving \(f(2014^2) = -4c\); \(2015 = 5\times 13\times 31\), giving \(f(2015^3) = -7c\); and \(2016 = 2^5\cdot 3^2\cdot 7\), giving \(f(2016^5) = -38c\). Thus, \(f(2014^2)+f(2015^3)+f(2016^5) = -49c\).

We have \(f(2^{2014})+f(3^{2015})+f(5^{2016}) = -2012c-2013c-2014c = 2013\), which gives \(c = -1/3\). Thus, \(f(2014^2)+f(2015^3)+f(2016^5) = 49/3\). Michael Lee · 2 years, 7 months ago

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3a) If we let \(y = 0\), we have \(f(0)+x\cdot f(0) = x\cdot f(x)\), or \(f(x) = \frac{f(0)+x\cdot f(0)}{x}\) for \(x \neq 0\). If we go back to our original equation and also let \(x = 0\), we find that \(f(0) = 0\). Thus, \(f(x) = 0\) for all \(x \in \mathbb{R}\).

3b) Since the value of \(f(0)\) is undefined, we take our previous solution of \(f(x) = \frac{f(0)+x\cdot f(0)}{x}\) and substitute \(c\) for \(f(0)\) to get \(f(x) = c\frac{x+1}{x}\). In 3a), we must consider that this \(f\) does not map \(0\) to a single value in \(\mathbb{R}\) except in the case that \(f(0) = 0\), but as \(0 \notin A\), we avoid that problem here. Michael Lee · 2 years, 7 months ago

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2) Let \(P(x) = P(x+c) = P(x+2c) = 0\). Then, expand to find that \(\frac{P(x+2c)-P(x)}{2} = 3cx^2+(6c^2+6c)x+(4c^3+6c^2+ac) = 0\) and \(P(x+c)-P(x) = 3cx^2+(3c^2+6c)x+(c^3+3c^2+ac) = 0\). Then, \([P(x+c)-P(x)]-\left[\frac{P(x+2c)-P(x)}{2}\right] = 3c^2x+(3c^3+3c^2) = 0\), which gives \(x = -1-c\). Thus, the three roots in arithmetic sequence are \(x = -1-c\), \(x+c = -1\), and \(x+2c = -1+c\). From Vieta's formulas, this gives \(a = (-1-c)(-1)+(-1-c)(-1+c)+(-1)(-1+c) = 3-c^2\) and \(b = -(-1-c)(-1)(-1+c) = 1-c^2\) for integer \(c\). Michael Lee · 2 years, 7 months ago

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4a) does not have a unique answer. The equality is satisfied by any \(a\), \(b\), \(c\), and \(d\) that satisfy \(a = b = d/4\) and \(c = d/2\) as long as \(d \neq 0\), which gives \(a+b+c+d = 2d\), which can obviously take on any positive value within the constraints of the problem. Michael Lee · 2 years, 7 months ago

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Question 4: Using Cauchy-Schwarz, the inequality is proven trivially.

Using the equality case for C-S, this implies that a^2 = b^2 = (c^2)/4 = (d^2)/16 implying that a = b = c/2 = d/4. Performing the substitutions gives a + b + c + d = 8. John Ashley Capellan · 2 years, 7 months ago

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What age group is this aimed at? Curtis Clement · 2 years, 7 months ago

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@Curtis Clement 12 years to 15 years. Sharky Kesa · 2 years, 7 months ago

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Q2

Let the roots are \(k-d,k,k+d\) such that \(d > 0\).

By Vieta's formula, we get \((k-d)+(k)+(k+d) = -3\), which gives \(k = -1\).

Therefore, the roots are \((-1-d, -1, -1+d\).

Also, we get \((-1-d)(-1)(-1+d) = -b\) and \((-1-d)(-1) + (-1)(-1+d) + (-1+d)(-1-d) = a\).

\(1-d^{2} = b\) and \(1+d+1-d+1-d^{2} = a\).

\(1-d^{2} = b\) and \(3-d^{2} = a\).

Therefore, \((a,b) = (3-d^{2},1-d^{2})\) for all positive integers \(d\).


My life is complicated. This is the original solution if I didn't state \(d > 0\).

Therefore, \(a-b = 2\).

Substitute back to equation we get

\(x^{3}+3x^{2}+(b+2)x+b\).

We know that \(-1\) is the root of equation. We can factor \((x+1)\) out of the polynomial.

\((x+1)(x^{2}+2x+b)\).

If \(x^{2}+2x+b = 0\) has 2 other solutions, we get

\(\Delta = 2^{2} - 4\times 1\times b > 0\).

\(b < 1\) Samuraiwarm Tsunayoshi · 2 years, 7 months ago

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@Samuraiwarm Tsunayoshi Sorry, forgot to state that the roots were distinct. Sharky Kesa · 2 years, 7 months ago

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@Sharky Kesa I let the roots are \(k-d,k,k+d\) such that \(d > 0\). This will definitely give you distinct roots. Samuraiwarm Tsunayoshi · 2 years, 7 months ago

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@Samuraiwarm Tsunayoshi could u explain how u got a - b = 2? Nihar Mahajan · 2 years, 7 months ago

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@Nihar Mahajan Subtract \(3-d^{2} = a\) and \(1-d^{2} = b\). Samuraiwarm Tsunayoshi · 2 years, 7 months ago

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@Samuraiwarm Tsunayoshi thanks!! Nihar Mahajan · 2 years, 7 months ago

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@Samuraiwarm Tsunayoshi Doesn't \( b < 1\) come directly from the fact that \( b = 1 - d^2 \), since \( d^2 \) can take any value in the interval \( (0,\infty) \) Siddhartha Srivastava · 2 years, 7 months ago

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@Siddhartha Srivastava Yeah I can be dumdum with IQ of room temperature at any time. =w= Samuraiwarm Tsunayoshi · 2 years, 7 months ago

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