2024 InIMC keystage3-IWYMIC individual

Let $N=\dfrac{1^4+2022^4+2023^4}{1^2+2022^2+2023^2}$. What is the remainder when $N$ is divided by 2024?

Let $a$ and $b$ be positive real numbers such that $\dfrac{a}{a+b^2}+\dfrac{b}{b+a^2}=\dfrac78$ and $ab=2$. What is the value of $a^9+b^9$?

Let $a$ and $b$ be positive integers such that $(a+1)(a+2)(a+3)\cdots(a+b)=2527k$, where $k$ is a positive integer. What is the smallest possible value of $a+b$?

In the diagram below, $ABC$ is an isosceles triangle with $AB=AC$ and $\angle BAC=80^\circ$. The points $D$ and $K$ are on $BC$ and $AC$, respectively, such that $\angle CAD=\angle ABK=30^\circ$. What is the measure, in degrees, of $\angle BKD$?

Let $\alpha$ and $\beta$ be the two roots of the equation $x^2-x-1=0$. What is the value of $\alpha^9+13\alpha^8\beta^9+\beta^8$?

In the diagram below, $ABC$ is a triangle, where the angle bisectors of $\angle ABC$ and $\angle ACB$ intersect at point $P$. Let points $D$ and $E$ lie on sides $AB$ and $AC$, respectively, such that $DE$ passes through point $P$ and $\angle AED=\angle ABP$. If the areas of triangles $BDP$, $CEP$ and $BPC$ are equal to 18 cm², 36 cm² and 57 cm², respectively, then what is the area, in cm², of $ABC$?

Use three different digits $A$, $B$ and $C$ exactly once to create 6 three-digit numbers. Among these numbers, it is known that:

- Only one number is divisible by 16.
- Excluding the above number, only one number is divisible by 4.
- Excluding the above two numbers, only two numbers are divisible by 2.
- One number is a prime number.
- One number is a square of a prime number.

What is the largest three-digit number that can be formed by using digits $A$, $B$ and $C$ exactly once?

In the diagram below, there are 14 points on a circle. In how many ways can we draw 7 non-intersecting chords between these points? (Note that the chords cannot have common ends.)

My father’s vault is protected with a code that can only be unlocked by entering a three-digit combination password, with each digit being either 1, 2 or 3.



The vault will only be opened if the buttons corresponding to the three digits of the password are pressed in succession. For example, anyone can press 1, 2, 3, 1, 2 and the vault will unlock if the password happens to be 312.

My mother doesn’t know the password of the vault, but she wants to open it. What is the minimum number of buttons that she must press to be sure that the vault will be opened? Note that the vault does not have an [ENTER] button.

For any positive real number $x$, we define $[x]$ to be the greatest integer that does not exceed $x$ and $\{x\}$ to be the fractional part of $x$ such that $\{x\}=x-[x]$. Let $a$, $b$ and $c$ be positive real numbers that satisfy the following system of equations:
$$\begin{cases}a+2[b]+2\{c\}=4.6,\\\left[a\right]+\{b\}-c=1,\\2\{a\}+2b+[c]=4.5.\end{cases}$$
What is the largest possible value of $a+b+c$?

In an infinite grid, each unit square is coloured either white or grey. If squares $X$ and $Y$ are located in the same row while squares $Y$ and $Z$ are located in the same column such that squares $X$ and $Z$ are white and square $Y$ is grey, then the ordered triple $(X,Y,Z)$ is called an “IMC triangle”. For example, as shown in the grid below, $(X,Y,Z)$ is an IMC triangle.



What is the maximum possible number of “IMC triangles” that can be found on a $15\times15$ grid?

Each square of an infinite grid has been filled with exactly one digit. Elsa places an O-tetromino on this grid covering four squares along the grid lines:



The minimum number of different digits used to fill the infinite grid so that the four squares will have different digits, regardless the position of the O-tetromino, is 4. The following grid is an example of such filling.



When Elsa places a L-tetromino on this grid covering four squares along the grid lines, what is the minimum number of different digits that must be used for the infinite grid so that the four squares covered by the L-tetromino will always have different digits? (Elsa may flip or rotate the L-tetromino)

Consider the following finite sequence of numbers: $\sqrt1,\sqrt2,\sqrt3,\ldots,\sqrt{2023},\sqrt{2024}$. Now, at each turn, select two numbers, say $x$ and $y$, from the sequence which are to be removed and then replace them in the sequence by a single number:
$$z=\sqrt{\frac{1}{\frac1{x^2}+\frac1{y^2}+\frac1{x^2y^2}}}.$$
After doing this 2023 times, a single number is left. What is this number?



Section B instruction: Show your detailed solution.

In the diagram below, $A_1A_2A_3$, $B_1B_2B_3$ and $C_1C_2C_3$ are three triangles such that $A_1A_2A_3$ is drawn inside $B_1B_2B_3$ and $B_1B_2B_3$ is drawn inside $C_1C_2C_3$, where $B_i$ is the midpoint of $A_iC_i$ for all $1\le i\le3$.



If the perimeters of $A_1A_2A_3$ and $C_1C_2C_3$ are $a$ cm and $c$ cm, respectively, what is the maximum possible value of the perimeter, in cm, of $B_1B_2B_3$?



Section B instruction: Show your detailed solution.

Anne and Bob will play a number game, where they use one piece of paper with only the number 2024 written on it. Starting with Anne, each player takes turns writing down a positive integer that is at least one-third of the previous number but less than the previous number. For the first step, Anne should write a number less than 2024 but no less than $\dfrac{2024}{3}$. The player who writes down the number 1 wins the game. Determine, with complete proof, which player has a winning strategy? What is the strategy?



Section B instruction: Show your detailed solution.