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
4 years, 6 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.

- 4 years, 6 months ago

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 $c$.

- 4 years, 6 months ago

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, $f(2014^2)+f(2015^3)+f(2016^5) = 49/3$.

- 4 years, 6 months ago

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.

- 4 years, 6 months ago

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.

- 4 years, 6 months ago

What age group is this aimed at?

- 4 years, 6 months ago

12 years to 15 years.

- 4 years, 6 months ago

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

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

$b < 1$

- 4 years, 6 months ago

Sorry, forgot to state that the roots were distinct.

- 4 years, 6 months ago

I let the roots are $k-d,k,k+d$ such that $d > 0$. This will definitely give you distinct roots.

- 4 years, 6 months ago

Doesn't $b < 1$ come directly from the fact that $b = 1 - d^2$, since $d^2$ can take any value in the interval $(0,\infty)$

- 4 years, 6 months ago

Yeah I can be dumdum with IQ of room temperature at any time. =w=

- 4 years, 6 months ago

could u explain how u got a - b = 2?

- 4 years, 6 months ago

Subtract $3-d^{2} = a$ and $1-d^{2} = b$.

- 4 years, 6 months ago

thanks!!

- 4 years, 6 months ago