2022 IIMC keystage3-IWYMIC individual

It is known that the equation $x^2+px+q=0$ has two positive integer roots. If $p+q=16$, then what is the value of $q$?

How many positive integers $n$ are there such that $n^2+n$ has exactly $6$ positive divisors?

Let $a$, $b$ and $c$ be positive real numbers such that the expression $\frac{3a^2+b^2+3c^2}{ab+bc+ac}$ attains its minimum value. If $abc=432$, then find the value of $3a+b+3c$.

In the diagram below, let $P$ be a point on the circumcircle of the equilateral triangle $ABC$ such that $PB=24$ cm and $PC=8$ cm. If $AP$ and $BC$ intersect at $D$, then find the length, in cm, of $PD$.

Let $a$, $b$, $c$ and $d$ be positive integers such that $0\lt a\lt b\lt c\lt d\lt 2022$, $a+d=b+c$ and $bc-ad=2021$. How many ordered quadruples $(a,b,c,d)$ are there?

Let $f(n)$ denote the number of occurrences of the digit $2$ in the following positive integer sequence $1,2,3,4,\ldots,n$. For example: $f(23)=7$, as the digit $2$ occurs once in each of the numbers $2,12,20,21,23$ and twice in the number $22$, so $f(23)=5+2=7$. Find a positive integer $n$ such that $f(n)=n$.

Let $a$ and $b$ be the two distinct roots of $x^2+2018x+1=0$, while $c$ and $d$ be the two distinct roots of $x^2-2022x+1=0$.

Find the value of $(a+c)(a-d)(b+c)(b-d)$.

From the grid shown in the diagram below, in how many ways can we choose three different points, such that the three chosen points form a triangle?

In the diagram below, let $ABCDEF$ be a regular hexagon. Let $G$ be the midpoint of $CD$ and $H$ be the intersection point of $AG$ and $BF$, as shown in the diagram below. If the length of $BF$ is $140$ cm, then find the length, in cm, of $BH$.

One thousand non-zero numbers $x_1,x_2,x_3,x_4,\ldots,x_{1000}$ are written in a cyclic order around a circle. It turns out that each number that is written in an odd position is equal to the sum of its two neighbors and that each number that is written in an even position is equal to the product of its two neighbors. What are the possible values for the sum of these one thousand numbers?

How many $10$-digit positive integers are there such that the product of their digits is $120$ and the sum of their digits is $20$?

In the diagram below, triangle $ABC$ is isosceles with $AB=AC$ and $\angle A=108^\circ$. Let $M$ be a point in the interior of $ABC$ such that $\angle MAB=30^\circ$ and $\angle MBA=12^\circ$. Find the angle measure, in degrees, of $\angle MCB$.

Anna and Boris play a game using a $6\times8$ board that contains one token on each unit square. Anna goes first, and they take alternating turns thereafter. In her turn, Anna takes two tokens on adjacent squares in the same row or column. In his turn, Boris takes only one token. However, if Anna can no longer take two adjacent tokens when her turn comes, Boris will get all the remaining tokens. What is the maximum number of tokens Boris is guaranteed to get?

Let $OA$ and $OB$ be two radii of a circle with centre $O$, such that $OA\perp OB$, as shown in the diagram below. Let $P$ be a point on the circumference and $Q$ be the point on $AP$ such that $AP=4AQ$. If $OA=8$ cm, then what is the minimum possible length, in cm, of $BQ$?

Let $\overline{abcd}$ be a four-digit number, where $a,c\ne0$, such that $\frac{\sqrt{\overline{abcd}}}{\sqrt{\overline{ab}}+\sqrt{\overline{cd}}}$ is a rational number. Find all possible value(s) of $\overline{abcd}$.