Question bank SMO 2023

In a convex quadrilateral $ABCD$, the diagonals intersect at $O$, $M$ and $N$ are points on the segments $OA$ and $OD$ respectively. Suppose $MN$ is parallel to $AD$ and $NC$ is parallel to $AB$. Prove that $\angle ABM=\angle NCD$.

What is the maximum number of integers that can be chosen from $1,2,\ldots,99$ so that the chosen integers can be arranged in a circle with the property that the product of every pair of neighbouring integers is a 3-digit integer.

Define a domino to be a $1\times2$ rectangular block. A $2023\times2023$ square grid is filled with non-overlapping dominoes, leaving a single $1\times1$ gap. John then repeatedly slides dominoes into the gap; each domino is moved at most once. What is the maximum number of times that John could have moved a domino?

(Example: In the $3\times3$ grid shown below, John could move $2$ dominoes: $D$, followed by $A$.)

Two distinct 2-digit prime numbers $p,q$ can be written one after the other in 2 different ways to form two 4-digit numbers. For example, 11 and 13 yield 1113 and 1311. If the two 4-digit numbers formed are both divisible by the average value of $p$ and $q$, find all possible pairs $(p,q)$.

Find all positive integers $k$ such that there exists positive integers $a,b$ such that $a^2+4=(k^2-4)b^2$.

In a scalene triangle $ABC$ with centroid $G$ and circumcircle $\omega$ centred at $O$, the extension of $AG$ meets $\omega$ at $M$, lines $AB$ and $CM$ intersect at $P$, and lines $AC$ and $BM$ intersect at $Q$. Suppose the circumcentre $S$ of the triangle $APQ$ lies on $\omega$ and $A,O,S$ are collinear. Prove that $\angle AGO=90^\circ$.

A grid of cells is tiled with dominoes such that every cell is covered by exactly one domino. A subset $S$ of dominoes is chosen. Is it true that at least one of the following $2$ statements is false?

(1) There are $2022$ more horizontal dominoes than vertical dominoes in $S$.

(2) The cells covered by the dominoes in $S$ can be tiled completely and exactly by the tetrominoes shown below (including rotations & reflections):

Let $n\ge2$ be a positive integer. For any integer $a$, let $Q_a(x)$ denote the polynomial $x^n+ax$. Let $p$ be a prime number and $S_a$ be the set
$$S_a=\{b\mid0\le b\le p-1,\ \exists c\in\mathbb Z,\ Q_a(c)\equiv b\pmod p\}.$$
Show that the expression $\dfrac1{p-1}\sum_{a=1}^{p-1}|S_a|$ is an integer.

Find all functions $f:\mathbb Z\to\mathbb Z$ such that
$$f(x+y)\bigl((f(x)-f(y))^2+f(xy)\bigr)=f(x^3)+f(y^3).$$

Determine all real numbers $x$ between $0$ and $180$ such that it is possible to partition an equilateral triangle into finitely many triangles, each of which has an angle of $x^\circ$.

Let $ABCD$ be a square, $E$ be a point on the side $DC$, $F$ and $G$ be the feet of the altitudes from $B$ to $AE$ and from $A$ to $BE$, respectively. Suppose $DF$ and $CG$ intersect at $H$. Prove that $\angle AHB=90^\circ$.

Find all positive integers $k$ such that there exist positive integers $a,b$ such that $a^2+4=(k^2-4)b^2$.

Let $n$ be a positive integer. There are $n$ islands with $n-1$ bridges connecting them such that one can travel from any island to another. One afternoon, a fire breaks out in one of the islands. Every morning, it spreads to all neighbouring islands. (Two islands are neighbours if they are connected by a bridge.) To control the spread, one bridge is destroyed every night until the fire has nowhere to spread the next day. Let $X$ be the minimum possible number of bridges one has to destroy before the fire stops spreading. Find the maximum possible value of $X$ over all possible configurations of bridges and islands where the fire starts at.

Find all positive integers $m,n$ satisfying $n!+2^{n-1}=2^m$.

Colour a $20000\times20000$ square grid using $2000$ different colours with $1$ colour in each square. Two squares are neighbours if they share a vertex. A path is a sequence of squares so that $2$ successive squares are neighbours. Mark $k$ of the squares. For each unmarked square $x$, there is exactly $1$ marked square $y$ of the same colour so that $x$ and $y$ are connected by a path of squares of the same colour. For any $2$ marked squares of the same colour, any path connecting them must pass through squares of all the colours. Find the maximum value of $k$.

In a certain company, one-third of the employees have a university degree. Half of the employees who have a university degree are men and 40% of the employees who do not have a university degree are women. If 102 of the employees are men, how many of the employees are women?

How many non-congruent triangles with integer side lengths have perimeter 7?

The median and mean of five distinct numbers, $4,7,10,11,N$, are equal. Find the sum of all possible values of $N$.

The following diagram shows two semicircles whose diameters lie on the same line. $AB$ is a chord of the larger semicircle that is tangent to the smaller semicircle at the point $C$ and is parallel to the diameter $DE$ of the larger semicircle. If $|AB|=16$ cm, what is the area of the shaded region in $\text{cm}^2$?

The graph (not drawn to scale) of $y=ax^2+bx+c$ with the maximum point at $x=-1$ is shown below. Which of the following must be true?



(I) $a-b+c\gt 0$; (II) $a+b-c\gt 0$; (III) $abc\gt 0$; (IV) $abc\lt 0$.