2024 InIMC keystage2-EMIC team

The save rate of a goalkeeper is the percentage given by the number of shots on his target that he has saved divided by the total number of shots on his target. At a given time during a football match, a certain goalkeeper’s save rate was 75%. After failing to save the next three shots on his target, his save rate went down to 60%. What is the minimum number of shots on target that he needs to save in order to raise his save rate back to at least 75%?

The following number:
$$123456789101112131415\cdots444424444344444$$
is made up by writing all the integers from 1 to 44444 consecutively in increasing order from left to right.

How many times does the string “2024” appear in the given number?



Paper instruction: Give a full solution.

How many moments in a day (from 00:00 to 23:59) will the hour and the minute hands of an analog clock (12 hours system) form an angle of $15k$ degrees, where $k$ is an integer so that $0\le k<24$? (The diagram shown is one moment at 04:30, having an angle of $15^\circ\times3=45^\circ$)

In the diagram below, $ABC$ is a triangle, where point $M$ lies on the side $AB$, point $N$ lies on the side $BC$ and $MN\parallel AC$. It is also known that $P$ is the midpoint of $BN$, and that $BN:PC=2:5$. If the area of the triangle $MBP$ is 1 cm², what is the area, in cm², of triangle $ABC$?





Paper instruction: Give a full solution.

The column 1 to column 7 of the table below will continue downward, from left to right and are being filled with the positive integers starting from “1”, written in increasing order and skipping all multiples of 5 or 7.



If 2024 is located in row $x$ and column $y$, what is the value of $x+y$?

In the diagram below, triangle $ABC$ is a right triangle at $A$ with $AC>AB$. Let point $M$ be on ray $BA$ such that $BM=AC$. Let point $N$ be outside triangle $ABC$ and on the ray $AC$ such that $CN=AM$. If ray $MC$ intersects $BN$ at $P$, what is the measure, in degrees, of $\angle MPB$?





Paper instruction: Give a full solution.

There is a regular hexagon-shaped kiosk in the middle of a garden. To get to the kiosk, there’s a spiral path that leads right up to one of the sides of the hexagon. To create the path, the builder used an interesting technique. He tied a rope counterclockwise around the kiosk and used a pencil attached to the end of the rope to draw the outer edge of the path starting from one vertex of the hexagon. After that, he untangled the rope clockwise to draw the inside border. The same process was repeated, starting from the next vertex of the hexagon, as shown in the diagram below. What is the total area (shaded area in the diagram not including the kiosk), in m², of the path to the kiosk if the kiosk measures 6 meters on each side? (Take $\pi=\frac{22}{7}$)

A special calculator has three commands (A, B and C) that can be used to modify the number that is displayed on the calculator.

The three commands are as follows:

- Command A: Insert 2 as the rightmost digit of the number.
- Command B: Replace the number with the sum of all its digits.
- Command C: Replace each digit $d$ of the number with the digit $10-d$, except that 0 is replaced with 0. (For example, 1085 would become 9025.)

For example, this is how one can arrive at 82 from 20 in 3 steps



If the calculator initially displays the number 26, show the step by step process on how to get the calculator to display 99 by using exactly six commands.



Paper instruction: Give a full solution.

Ann and Bob are playing a game where they take turns moving a counter on a grid. Ann goes first. The goal is to move the counter to the bottom-left corner.

In a single move, a player must perform exactly one of the following two actions:

- Move the counter one or more cells to the left.
- Move the counter one or more cells down.

However, the counter **CANNOT** be moved onto or across a shaded cell.

Ann has a “winning cell” if, when the game starts with the counter on that cell and Ann and Bob both play optimally, Ann has a winning strategy. Otherwise, the cell is known as a “losing cell”. For example, in the diagram below, cell $L$ is a losing cell for Ann since her only option is to move the counter down 1 cell, after which Bob can win by moving the counter 3 cells to the left. Cell $W$ is a winning cell for Ann because Ann can move the counter 1 cell to the left, causing Bob to move the counter down 1 cell. Ann can then win by moving the counter 3 cells to the left.



In the $8\times8$ grid below, how many winning cells does Ann have?

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 grid on the right is an example of such filling.



When Elsa places a T-tetromino on this grid covering four squares along the grid lines, what is the minimum number $n$ of different digits that must be used for the infinite grid so that the four squares covered by the tetromino will always have different digits?

The T-tetromino can be flipped or rotated.



(Note: You must explain why using $n-1$ different digits is insufficient and give a filling example to show that using $n$ digits is sufficient.)



Paper instruction: Give a full solution.