ITMC 2018 Grade 9 National Preliminary set01

Printed positive marks are preserved per question. The source uses different blank-answer and incorrect-answer rules by section, including an optional bonus penalty; the import does not reproduce that complete official scoring scheme.

Consider the ten real numbers $\sqrt0,\sqrt1,\sqrt2,\sqrt3,\sqrt4,\sqrt5,\sqrt6,\sqrt7,\sqrt8,\sqrt9$. How many are rational?

A linear polynomial has degree 1 and form $ax+b$, where $x$ is a variable and $a,b$ are real constants. Examples: $2x+1,x+20,-x-10,3x-10,\frac12x-1$. Which polynomial cannot be written as a product of two linear polynomials?

Which polynomial equals $(x-20)(x-18)$?

Consider: (1) If $\triangle ABC\sim\triangle DEF$, then $\triangle ABC\cong\triangle DEF$. (2) If $\triangle ABC\cong\triangle DEF$, then $\triangle ABC\sim\triangle DEF$. (3) If $\triangle ABC\sim\triangle DEF$ and $AB=DE$, then $\triangle ABC\cong\triangle DEF$. Which option is true?

A rectangular box has length $a$, width $b$ and height $c$. Which expression equals its outside surface area?

A square pyramid has height 1 and base side $a$. A circular cone has base radius $r$ and height 1. Their volumes are equal. Find $a/r$.

Let $a,b$ be positive integers. (1) A smallest positive common divisor exists. (2) A largest positive common divisor exists. (3) A smallest positive common multiple exists. (4) A largest positive common multiple exists. How many statements are true?

A composite number is a positive integer expressible as a product of two or more smaller positive integers greater than 1. For example, $6=2\times3$ is composite, but 17 is not. (1) Every prime number is not composite. (2) Every number that is not composite must be prime. Which option is true?

A phone normally costs 26900 baht. A store reduces the price by 9000 baht, then gives a further 10% credit-card discount. This final price is paid in 10 equal monthly instalments without interest. (Baht is Thailand’s currency.) What is the monthly payment?

Observe $(a-b)(a+b)=a^2-b^2$, $(a-b)(a^2+ab+b^2)=a^3-b^3$, and $(a-b)(a^3+a^2b+ab^2+b^3)=a^4-b^4$. Find $(a-b)(a^5+a^4b+a^3b^2+a^2b^3+ab^4+b^5)$.

In a common year February has 28 days and the year has 365 days. If 5 August is Sunday, what weekday is 13 October of the same year?

At the 2018 FIFA World Cup,32 teams were divided into 8 groups A–H of 4 teams. In the first round, each pair of teams in a group plays one match. A winner earns 3 points and the loser 0; in a draw each team earns 1. The two teams with the highest points in each group advance to round 2.

Group B results are Morocco–Iran 0–1; Portugal–Spain 3–3; Portugal–Morocco 1–0; Iran–Spain 0–1; Iran–Portugal 1–1; Spain–Morocco 2–2. What is the total number of points in group B?

At the 2018 FIFA World Cup,32 teams were divided into 8 groups A–H of 4 teams. In the first round, each pair of teams in a group plays one match. A winner earns 3 points and the loser 0; in a draw each team earns 1. The two teams with the highest points in each group advance to round 2.

After the first round, which of the following can be the total points of the 4 teams in a group?

At the 2018 FIFA World Cup,32 teams were divided into 8 groups A–H of 4 teams. In the first round, each pair of teams in a group plays one match. A winner earns 3 points and the loser 0; in a draw each team earns 1. The two teams with the highest points in each group advance to round 2.

The group totals are A:18, B:15, C:16, D: $x$, E:16, F:18, G: $y$, H:17. If 9 matches in all groups are draws, find $x+y$.

Let real $x$ satisfy $\sqrt{x+30}+\sqrt{x-30}=20$. Find $\sqrt{x+30}-\sqrt{x-30}$.

The real number $x$ satisfying $x/a+a/x=b$ equals which expression?

$6x^3+5x^2-7x-4$ is divisible by $3x^2+x-4$. Find the sum of the quotient’s coefficients.

The parabola $y=-2(x+3)^2+1$ meets the $x$ -axis at $(a,0),(b,0)$ and the $y$ -axis at $(0,c)$. Find $a+b+c$.

The parabola $y=ax^2+bx+c$ has vertex $(20,18)$ and passes through the origin. Which point is not on it?

A right triangle has hypotenuse $c$, other side lengths $a,b$, and area $A$. Find $a+b$.