Grade what is the units digit of the simplest form of ? Answer: Solution

Answer: 7 candles
Solution:
Assume that he left n candles at each chapel. Before crossing the 3^rd bridge he has (even) candles, which doubled to n, the exact number of candles he left at the 3^rd chapel . Since he left n candles in the 2^nd chapel, he has ( candles after crossing the 2^nd bridge. Before crossing the 2^nd bridge, he has = candles. Before leaving n candles in the 1^st chapel, he has (n + ) or candles, which is the number of candles that he has after crossing the 1^st bridge. Thus, at first he has = . The least value of can be determined if n = 8, so that = 7 candles.
Hence, Juan has 7 candles at first.

22. 
    In a Math test, Ianne scored twice as many as Steph and 30 points more than Audrey did. James scored 50 points more than Steph did. Which of the following statements is precisely true?
    

1. 
   Ianne has the highest score.
   
2. 
   Steph got the lowest score.
   
3. 
   James got the highest score.
   
4. 
   Audrey’s score is higher than Steph.
   
5. 
   Ianne and James together scored higher than Steph and Audrey.
   

Answer: E
Solution: Make a tabulation of their possible scores.

Ianne	Steph	Audrey	James
30	15	0	60
40	20	10	70
100	50	70	100
120	60	90	110
140	70	110	120

Notice that in the 1^st row, statements A, B and D are false but C & E are true. In the fourth row, statement C is false, and the rest are true. Statement E is consistently true in each row.

23. 
    A date is called a lucky date if the sum of the digits when it is written in the form MMDDYY is 13. The date January 9, 2021 when written as 010921 is a lucky date because the sum of the digits is 13 and February 8, 2021, when written as 020821 is a lucky date as well because the sum of the digits is 13. However, March 8, 2021 is not a lucky date because the sum of the digits is 14. How many lucky dates are there in 2021?
    

Ans. 36

Solution:

010921	020821	030721	040621	050521	060421
011821	021721	031621	041521	051421	061321
012721	022621	032521	042421	052321	062221
070321	073021	082021	100921	110821	120721
071221	080221	090121	101821	111721	121621
072121	081121	091021	102721	112621	122521

24. 
    In the following number continuous, the numbers are written several times based on the value of the number. What is the 2021^st digit?
    

122333444455555666666777777788888888. . . 9898989999… 99

Answer: 6
Solution:
The last digit or number in a set of the same number is the same as the sum of all numbers below it. The set of same number closest to 2021 is 63.b
S = = 2016,
2021 – 2016 = 5 digits more are needed after the last 63 in the set.

1223334444555556666667777777. . . 6363…636364646…
Hence, the 2021^st digit is 6.

25. 
    In Alpha camp, campers call each other’s name by number codes. For example, DAN is called 341, ALDEN is called 47361, and BRENT is named 58619. How do the campers name LANDER?
    

Answer: 741368
Solution:
DAN = 341 D = 3, A = 4, N = 1, L = 7, E = 6, R = 8, B = 5 and T = 9
ALDEN = 47361
BRENT = 58619
LANDER = 741368

26. 
    How many ICE-IM ON TOP (International Mathematical Online Training Olympiad Program] students in a class must there be to ensure that 6 students get the same rank (A, B, C, D, E, F)?