2016/17 AHSMC Part 1

If it is 10:00 AM on a Tuesday, which day would it be $2016$ hours later?

If $x > 0$, $x \neq 1$, and $(\log_2 x)^2 = \log_4 x$, then:

A ring of $10$ grams is $60\%$ gold and $40\%$ silver. A jeweller wants to melt it down, add $2$ grams of silver and add enough gold to make it $70\%$ gold. How many grams of gold should be added?

A quadratic polynomial $f(x) = ax^2 + bx + c$, where $a$, $b$ and $c$ are integers, satisfies $f(2) = 4$ and $f(3) = 9$. The number of such polynomials is:

For any integer $n$, the expression $n^2 + 3n + 2$ cannot assume the value

Two straight lines with nonzero $x$ and $y$-intercepts have the following property: the $x$-intercept of the first line equals the $y$-intercept of the second line, and the $x$-intercept of the second line equals the $y$-intercept of the first line. If the slope of the first line is $m$, then the slope of the second line is

The angles of a triangle when measured in degrees are all prime numbers. The smallest possible size of the largest angle is:

How many three-digit numbers can be written after $523$ to yield a six-digit number which is divisible by each of $7$, $8$ and $9$?

When $x^4 + x^5 + x^{10} + x^{17} + x^{100}$ is divided by $x^2 - 1$ the remainder is

Let $D$ be an arbitrary point on the side $BC$ of the equilateral triangle $ABC$. Points $E$ and $F$ are on $AB$ and $AC$ respectively so that $DE \perp AB$ and $DF \perp AC$ and $E_1$, $F_1$ are points on $BC$ such that $EE_1 \perp BC$ and $FF_1 \perp BC$. If $E_1F_1$ has length $\dfrac{1}{2}$ then the length of $BC$ is

A box contains two red balls, two green balls and two yellow balls. If you randomly remove three balls from the box without replacement, what is the probability that you have removed one of each colour?

Let $f \colon \mathbb{R} \to \mathbb{R}$ be a function such that $x f(x) + (1-x)f(-x) = x^2 + x + 1$ for any real number $x$. The greatest real number $M$ for which $f(x) \ge M$ for all real numbers $x$, is

In a school's math club, the number of different $3$-person committees that could be formed containing two girls and one boy is $2016$ more than the number of different $3$-person committees containing two boys and one girl. The number of girls in the club is:

The product of all real numbers $x$ that are solutions of the equation $$\sqrt[3]{x^2 + x + \sqrt[3]{x^2 + x + 3}} = 3$$ is

The area of the trapezoid $ABCD$ with $AB \parallel CD$, $AD \perp AB$ and $AB = 3\,CD$ is equal to $4$. A circle inside the trapezoid is tangent to all of its sides. The radius of the circle is

A quadrilateral is called convex if its diagonals intersect inside the quadrilateral. A convex quadrilateral has side lengths $3$, $3$, $4$, $4$ not necessarily in this order, and its area is a positive integer. The number of non-congruent convex quadrilaterals having these properties is: