International Mathematical Olympiad National Selection Test 
Malaysia 2020 Round 1 Primary
Time: 2.5 hours [hide=Rules]
$\bullet$ For each problem you have to submit the answer only. The answer to each problem is a non-negative integer.
$\bullet$ No mark is deducted for a wrong answer.
$\bullet$ The maximum number of points is (1 + 2 + 3 + 4) x 5 = 50 points.[/hide]


[b]Part A[/b] (1 point each)

p1. Annie asks his brother four questions, "What is $20$ plus $20$? What is $20$ minus $20$? What is $20$ times $20$? What is $20$ divided by $20$?". His brother adds the answers to these four questions, and then takes the (positive) square root of the result. What number does he get?

p2. A broken watch moves slower than a regular watch. In every $7$ hours, the broken watch lags behind a regular watch by $10$ minutes. In one week, how many hours does the broken watch lags behind a regular watch?

p3. Given a square $ABCD$. A point $P$ is chosen outside the square so that triangle $BCP$ is equilateral. Find $\angle APC$, in degrees.

p4. Hussein throws 4 dice simultaneously, and then adds the number of dots facing up on all $4$ dice. How many possible sums can Hussein get?

Note: For example, he can get sum $14$, by throwing $4$, $6$, $3$, and $ 1$. Assume these are regular dice, with $1$ to $6$ dots on the faces.

p5. Mrs. Sheila says, "I have $5$ children. They were born one by one every $3$ years. The age of my oldest child is $7$ times the age of my youngest child." What is the age of her third child?


[b]Part B [/b](2 points each)


p6. The number $N$ is the smallest positive integer with the sum of its digits equal to $2020$. What is the first (leftmost) digit of $N$?


p7. At a food stall, the price of $16$ banana fritters is $k$ RM , and the price of $k$ banana fritters is $ 1$ RM . What is the price of one banana fritter, in sen?

Note: $1$ RM is equal to $100$ sen.


p8. Given a trapezium $ABCD$ with $AD \parallel$ to $BC$, and $\angle A = \angle B = 90^o$. It is known that the area of the trapezium is 3 times the area of $\vartriangle ABD$. Find$$\frac{area \,\, of \,\, \vartriangle ABC}{area \,\, of \,\, \vartriangle ABD}.$$

p9. Each $\vartriangle$ symbol in the expression below can be substituted either with $+$ or $-$:$$\vartriangle 1 \vartriangle 2 \vartriangle 3 \vartriangle 4.$$How many possible values are there for the resulting arithmetic expression?

Note: One possible value is $-2$, which equals $-1 - 2 - 3 + 4$.


p10. How many $3$-digit numbers have its sum of digits equal to $4$?


[b]Part C[/b] (3 points each)


p11. Find the value of$$+1 + 2 + 3 - 4 - 5 - 6 + 7 + 8 + 9 - 10 - 11 - 12 +... - 2020$$where the sign alternates between $+$ and $-$ after every three numbers.


p12. If Natalie cuts a round pizza with $4$ straight cuts, what is the maximum number of pieces that she can get?

Note: Assume that all the cuts are vertical (perpendicular to the surface of the pizza). She cannot move the pizza pieces until she finishes cutting.


p13. Given a square with area $ A$. A circle lies inside the square, such that the circle touches all sides of the square. Another square with area $ B$ lies inside the circle, such that all its vertices lie on the circle. Find the value of $A/B$.


p14. This sequence lists the perfect squares in increasing order:$$0, 1, 4, 9, 16, ... ,a, 10^8, b, ...$$Determine the value of $b - a$.


p15. Determine the last digit of $5^5 + 6^6 + 7^7 + 8^8 + 9^9$


[b]Part D[/b] (4 points each)


p16. Find the sum of all integers between $-\sqrt{1442}$ and $\sqrt{2020}$.


p17. Three brothers own a painting company called Tiga Abdul Enterprise. They are hired to paint a building.
Wahab says, "I can paint this building in $3$ months if I work alone". Wahib says, "I can paint this building in $2$ months if I work alone". Wahub says, "I can paint this building in $k$ months if I work alone". If they work together, they can finish painting the building in $1$ month only. What is $k$?


p18. Given a rectangle $ABCD$ with a point $P$ inside it. It is known that $PA = 17$, $PB = 15$, and $PC = 6$. What is the length of $PD$?


p19. What is the smallest positive multiple of $225$ that can be written using digits $0$ and $ 1$ only?


p20. Given positive integers $a, b$, and $c$ with $a + b + c = 20$. Determine the number of possible integer values for $\frac{a + b}{c}$.


PS. Problems 6-20 were also used in [url=https://artofproblemsolving.com/community/c4h2675966p23194287]Juniors [/url]as 1-15. Problems 11-20 were also used in Seniors 1-10.
### Problem 6
The number \( N \) is the smallest positive integer with the sum of its digits equal to \( 2020 \). What is the first (leftmost) digit of \( N \)?

1. To minimize \( N \), we need to maximize the leftmost digit.
2. The largest possible digit is \( 9 \).
3. We can use as many \( 9 \)'s as possible to get close to \( 2020 \).
4. \( 2020 \div 9 \approx 224.44 \), so we can use \( 224 \) nines.
5. The sum of \( 224 \) nines is \( 224 \times 9 = 2016 \).
6. We need \( 4 \) more to reach \( 2020 \), so the number \( N \) is \( 999\ldots 9994 \) (with \( 224 \) nines followed by a \( 4 \)).

Conclusion: The first (leftmost) digit of \( N \) is \( \boxed{9} \).