2022 IIMC keystage3-IWYMIC team

A palindromic number is a number that is the same when we read it forward and backward. How many $11$-digit palindromic numbers are divisible by $101$?

The function $f(x)$ is defined for every real number $x$ and satisfies the following condition:
$$f(x)+2f(y)=3f\left(\frac{x+2y}{3}\right)$$
for every two real numbers $x$ and $y$. If $f(2)=1$ and $f(5)=7$, then find $f(2022)$.

An Indonesian high-tech company has invented an airbag that can be dropped from tall buildings without bursting. An engineer will be conducting the test at the Autograph Tower Thamrin Nine, currently the tallest building in Indonesia, which has $75$ floors. His goal is to find the highest floor of this building from which the airbag won't burst when dropped. The engineer only made two identical prototypes of this airbag for testing, which can be dropped from various floors of the building. If an airbag doesn't burst after being dropped, it may be reused without suffering any loss of quality. But if both airbags burst before the engineer has determined the highest floor, then the test is considered to be a failure. What is the least number of times the engineer must drop the airbags in order to determine the highest floor?

In the diagram below, points $A_1$, $A_2$, $A_3$, $A_4$, $A_5$ lie on the circumference of a unit circle and
$$\angle A_5A_2A_4=\angle A_1A_3A_5=\angle A_2A_4A_1=\angle A_3A_5A_2=30^\circ.$$
Let $A_2A_4$ and $A_3A_5$, $A_3A_5$ and $A_4A_1$, $A_4A_1$ and $A_5A_2$, $A_5A_2$ and $A_1A_3$ intersect at points $B_1$, $B_2$, $B_3$, $B_4$, $B_5$, respectively. Find the area, in square units, of the pentagon $B_1B_2B_3B_4B_5$.



**Source note.** The printed sentence lists four pairs of chords but five intersection labels. The supplied diagram identifies the remaining point as $B_5=A_1A_3\cap A_2A_4$; the original sentence and diagram are preserved above.

If $ax+by=7$, $ax^2+by^2=49$, $ax^3+by^3=133$ and $ax^4+by^4=406$, then find the numerical value of $2022(x+y)+7(a+b)+2xy$.

How many different arrangements of $1,2,3,4,5,6$ and $7$ in a row are there such that the sum of any $2$ consecutive numbers is a prime?

Let $PQRS$ be a trapezoid such that $PS\parallel QR$ and $T$ is the intersection point of $PR$ and $QS$, as shown in the diagram below. Let $I$, $M$ and $C$ be the midpoints of $ST$, $PQ$ and $RT$, respectively. If $\angle STR=120^\circ$, $\angle SQR=\angle SPR$ and the area of triangle $IMC$ is equal to $1024\sqrt3$ square centimeters, then find the length, in centimeters, of $PQ$.

In the diagram below, $ABC$ is an equilateral triangle, where $M$ and $N$ are points on $AC$ and $BM$, respectively, such that $\angle ABM=15^\circ$ and $\angle BAN=30^\circ$. Let $P$ be the intersection point of the line $AN$ with the circumcircle of triangle $ABM$. Prove that $AN=NP$.

Determine how many ordered pairs of integer numbers $(a,b)$ satisfy the following conditions:

- $1\le a,b\le100$ and $a\ne b$.
- $a$ is even.
- $(a-1)\mid(b-1)$, $a\mid b$ and $(a+1)\mid(b+1)$.

Andy constructed the smallest subset $S$ of positive integers that satisfies the following properties:

- If number $t$ is in $S$, then the numbers $6t$ and $6t+1$ are also in $S$.
- The number $1$ is in $S$, but the numbers $2,3,4$ and $5$ are not in $S$.

Then, Andy calculates the sum of two distinct numbers in $S$, where the two numbers are both less than $2022$. How many different possible sums can Andy obtain?