math-eval/TAL-SCQ5K huggingface dataset:
Problem: The Gellers went on a trip. They left home at 9:15 in the morning and arrived at the hotel at 8:22 in the evening. How long did the journey take? 
Options:
aoVal: A
content: $$11$$ hours $$22$$ minutes 
aoVal: B
content: $$11$$ hours $$15$$ minutes 
aoVal: C
content: $$11$$ hours $$7$$ minutes 
aoVal: D
content: $$1$$ hours $$7$$ minutes 
aoVal: E
content: $$12$$ hours $$7$$ minutes 
Answer: C
Analysis: $3$h-$15$min + $8$h+$22$min = $11$h$7$min 
Knowledge points: Overseas Competition->Knowledge Point->Combinatorics->Time Problem->Time Calculation
Problem: The first student wrote the number $1$ on the board, the second student wrote the number $2$, and the third one and each of the following students wrote a number that was the quotient of the number written just before the last number and the last number. What did the tenth student write? 
Options:
aoVal: A
content: $\frac1{2^{10}}$ 
aoVal: B
content: $$256$$ 
aoVal: C
content: $\frac1{2^{13}}$ 
aoVal: D
content: $$1024$$ 
aoVal: E
content: $$2^{34}$$ 
Answer: E
Analysis: The ten numbers are: $1(2^{0}), 2^{1}, \frac1{2^{1}}, 2^{2}, \frac1{2^{3}}, 2^{5} \frac1{2^{8}}\cdots$  Thus the tenth exponent is $34$. 
Knowledge points: Overseas Competition->Knowledge Point->Combinatorics->Strategies and Operations->Operational Problem
Source: 其它
Problem: In Think Academy, there are $147$ students. Among these $147$ students, we can guarantee that at least~\uline{~~~~~~~~~~}~students belong to the zodiac sign with the most students. 
Options:
aoVal: A
content: $$12$$ 
aoVal: B
content: $$13$$ 
aoVal: C
content: $$14$$ 
aoVal: D
content: $$15$$ 
aoVal: E
content: $$136$$ 
Answer: B
Analysis: $147\div12=12R3$, $12+1=13$. 
Knowledge points: Overseas Competition->Knowledge Point->Combinatorics->Pigeonhole Principle
Source: 其它
Problem: From $1$ to $23$, how many integers can be chosen at most to ensure that no two of chosen numbers differ by $4$? 
Options:
aoVal: A
content: $$9$$ 
aoVal: B
content: $$10$$ 
aoVal: C
content: $$11$$ 
aoVal: D
content: $$12$$ 
Answer: D
Analysis: $1, 5, 9, 13, 17, 21$  $2, 6, 10, 14, 18, 22$  $3, 7, 11, 15, 19, 23$  $4, 8, 12, 16, 20$ 
Knowledge points: Overseas Competition->Knowledge Point->Combinatorics->Constructing and Proving
Source: 其它
Problem: Rose and Kylie are playing a game. Here are the rules：  1. There are $$14$$ marbles placed in a row.  2. The players take turns removing either $1$ or $2$ marbles each turn.  3. Whoever picks the last marble wins the game.  Rose starts first and is followed by Kylie. To ensure her victory, how many marbles must Rose take away in the first turn? 
Options:
aoVal: A
content: $$1$$ 
aoVal: B
content: $$2$$ 
Answer: B
Analysis: If there are n marbles in total, the first player has a winning strategy for all $$n$$ that is not a multiple of $$3$$. For $$n$$ being a multiple of $$3$$, the second player can always win, regardless of what strategy the first player plays.  $$14\div (1+2)=4 \textbackslash{} \text{R} 2$$ Rose should take away the remainder, i.e. two marbles, to make herself the second player when there is a multiple of $3$ marbles left. 
Knowledge points: Overseas Competition->Knowledge Point->Combinatorics->Strategies and Operations->Game Strategy
Problem: The time shown on the electronic clock is $6:08$. How long before the numbers appear again? (adapted from 2007 Math Kangaroo Problem, Level 3 - 4, Question \#16) 
Options:
aoVal: A
content: $$2$$ h $$2$$ min 
aoVal: B
content: $$3$$ h 
aoVal: C
content: $$3$$ h $$2$$ min 
aoVal: D
content: $$2$$ h $$12$$ min 
aoVal: E
content: $$2$$ h 
Answer: A
Analysis: $8:06$=$6:08$+$2$h$2$min 
Knowledge points: Overseas Competition->Knowledge Point->Combinatorics->Time Problem->Time Calculation
Problem: We are given a certain number. We double the number and then subtract $1$ from the result. We repeat this operation $4$ more times. If the end result is $2^{6}+1$, then what is the number that we started with? (Adapted from $2005$ Math Kangaroo Problems, Level $11-12$, Question \#$29$) 
Options:
aoVal: A
content: $$1$$ 
aoVal: B
content: $$2$$ 
aoVal: C
content: $$4$$ 
aoVal: D
content: $$6$$ 
aoVal: E
content: $$3$$ 
Answer: E
Analysis: After repeating the operation $4+1=5$ times, the result is $2^{6}+1$.  After repeating the operation $4$ times, the result is $(2^{6}+1+1)\div2=2^{5}+1$.  After repeating the operation $3$ times, the result is $2^{4}+1$.  $$\cdots $$  After doing the operation for the first time, the result is $2^{2}+1$.  Before we do anything, the number is $2^{1}+1$, which is $3$. 
Knowledge points: Overseas Competition->Knowledge Point->Combinatorics->Strategies and Operations->Operational Problem
Source: 其它
Problem: When number $\overline{OK}$ is divided by $K$, the result will give the same quotient as the divisor and leave the remainder of $O$. The same letter represents the same digit, and different letters represent different digits. Then, what is the product of the two digits, $O$ and $K$? 
Options:
aoVal: A
content: $$18$$ 
aoVal: B
content: $$36$$ 
aoVal: C
content: $$63$$ 
aoVal: D
content: $$72$$ 
aoVal: E
content: $$81$$ 
Answer: D
Analysis: $89\div9=9R8$ 
Knowledge points: Overseas Competition->Knowledge Point->Combinatorics->Number Puzzles
Source: 其它
Problem: How many minutes are there in one week? 
Options:
aoVal: A
content: $$7\times24$$ 
aoVal: B
content: $$7\times 2\times 12$$ 
aoVal: C
content: $$7\times 12 \times 60$$ 
aoVal: D
content: $$7\times 2 \times 12 \times 60$$ 
aoVal: E
content: None of the above 
Answer: D
Analysis: One week $$7$$ days.  $$24$$ hours per day.  $$60$$ minutes per hour. 
Knowledge points: Overseas Competition->Knowledge Point->Combinatorics->Time Problem
Source: 其它
Problem: We left for a summer camp at $4:32$ PM and got to our destination at $6:11$ PM. How long did we travel? (Adapted from 2015 Math Kangaroo Problem, Level 1--2, Question \#24) 
Options:
aoVal: A
content: $1$ hour $39$ minutes 
aoVal: B
content: $2$ hours $39$ minutes 
aoVal: C
content: $2$ hours $21$ minutes 
aoVal: D
content: $1$ hour $21$ minutes 
aoVal: E
content: $39$ minutes 
Answer: A
Analysis: $6:11$ - $4:43$ = $1$ hour $39$ minutes 
Knowledge points: Overseas Competition->Knowledge Point->Combinatorics->Time Problem
Source: 其它
Problem: Apply this operation to $$(50, 20)$$. What are the last two numbers when the operation stops? 
Options:
aoVal: A
content: $(2,2) 
aoVal: B
content: $(3,3)$ 
aoVal: C
content: $(5,5)$ 
aoVal: D
content: $(7,7)$ 
aoVal: E
content: None of the above 
Answer: E
Analysis: E 
Knowledge points: Overseas Competition->Knowledge Point->Combinatorics->Patterns of Figures->Special Changes
Source: 其它
Problem: Amy calculate $47\times86$ wrong. Her teacher says her result which has a difference of $172$ from the correct one. After checking, Amy finds that she writes one of the four digits two multipliers the result as $9$. Which digits does she write incorrectly? 
Options:
aoVal: A
content: $$4$$ 
aoVal: B
content: $$7$$ 
aoVal: C
content: $$8$$ 
aoVal: D
content: $$6$$ 
aoVal: E
content: It\textquotesingle s impossible to determine. 
Answer: B
Analysis: The difference between the wrong answer and the correct one is $172$. $172\div86=2$, $172\div47=3R31$.That means Amy writes neither $ 8$ nor $6$ wrong. Then, compare $49\times86$ and $97\times86$, so we can get the correct answer. $49\times86-47\times86=172.$ 
Knowledge points: Overseas Competition->Knowledge Point->Combinatorics->Number Puzzles
Source: 其它
Problem: Lucy, Maria, and Anna have a meeting at $$12:30$$. Lucy\textquotesingle s walk takes $$10$$ minutes, Maria\textquotesingle s walk takes a quarter of an hour, and Anna\textquotesingle s walk takes $$30$$ minutes. At what time must the person who needs the longest time to get to the meeting leave her house? (Adapted from 2006 Math Kangaroo Problem, Level 1-2, Question \#9) 
Options:
aoVal: A
content: $$12:00$$ 
aoVal: B
content: $$12:10$$ 
aoVal: C
content: $$12:15$$ 
aoVal: D
content: $$12:20$$ 
aoVal: E
content: $$11:50$$ 
Answer: A
Analysis: $12:30$ - $30$ min = $12:00$ 
Knowledge points: Overseas Competition->Knowledge Point->Combinatorics->Time Problem->Time Calculation
Source: 其它
Problem: From Monday to Wednesday Mark always lies. For the rest of the week he tells the truth. One day, Mark said to Mary:  $$1$$) "Yesterday I lied." and  $$2$$) "Starting the day after tomorrow, I will be lying for two consecutive days."  On what day of the week did Mark talk to Mary? 
Options:
aoVal: A
content: Monday 
aoVal: B
content: Tuesday 
aoVal: C
content: Wednesday 
aoVal: D
content: Thurday 
aoVal: E
content: Friday 
Answer: A
Analysis: It was on Monday, so the answer is A. 
Knowledge points: Overseas Competition->Knowledge Point->Combinatorics->Logical Reasoning->Reasoning by Conditions->Complex Reasoning
Problem: Which of the following month has $30$ days? (adapted from $$2011$$ Math kangaroo Problems, Level $$1-2$$, Question \#$$5$$) 
Options:
aoVal: A
content: $February$ 
aoVal: B
content: $$June$$ 
aoVal: C
content: $$July$$ 
aoVal: D
content: $$October$$ 
aoVal: E
content: $$December$$ 
Answer: B
Analysis: June has $30$ days. 
Knowledge points: Overseas Competition->Knowledge Point->Combinatorics->Time Problem->Reading the Clock
Problem: A certain play has two $$45$$-minute parts and an intermission between them. The play started at $$10:50$$ and ended at $$12:40$$. How many minutes long was the intermission? 
Options:
aoVal: A
content: $$10$$ 
aoVal: B
content: $$15$$ 
aoVal: C
content: $$20$$ 
aoVal: D
content: $$25$$ 
Answer: C
Analysis: From $$10:50$$ to $$12:40$$ = $$1$$ h $$50$$ min, $$1$$ h $$50$$ min = $$110$$ min, two $$45$$-minute parts = $$90$$ min, and $$110 - 90 = 20$$ min. 
Knowledge points: Overseas Competition->Knowledge Point->Combinatorics->Time Problem
Source: 其它
Problem: The distance between the school and the park is $$2400$$ $\text{m}$. Lily and Candy depart from the school towards the park at the same time, and return immediately after arriving at the park. Lily travels $$47$$ $\text{m/min}$, and Candy travels $$53$$ $\text{m/min}$. They will meet each other for the first time after~\uline{~~~~~~~~~~}~minutes. 
Options:
aoVal: A
content: $$24$$ 
aoVal: B
content: $$48$$ 
aoVal: C
content: $$72$$ 
aoVal: D
content: $$96$$ 
Answer: B
Analysis: $2400\times2\div(53+47)=48$min 
Knowledge points: Overseas Competition->Knowledge Point->Combinatorics
Source: 其它
Problem: If $a,b,c,d,e,f,g,h,j$ represents different digits from $1$ to $9$ such that $a+\frac{b}{c}+\frac{d}{e}\times f-\left(g+\frac{h}{j} \right) = N$, where $N$ is a whole number. What is the value of $\frac{d}{e}\times f$ to maximize $N$? 
Options:
aoVal: A
content: $$36$$ 
aoVal: B
content: $$56$$ 
aoVal: C
content: $$63$$ 
aoVal: D
content: $$72$$ 
aoVal: E
content: $$90$$ 
Answer: D
Analysis: D 
Knowledge points: Overseas Competition->Knowledge Point->Combinatorics->Combinatorics Involving Extreme Values->Forming a Maximum/Minimum Multi-Digit Numbers with Fixed Sums
Source: 其它
Problem: True or False:  A number can\textquotesingle t be equal to its opposite. 
Options:
aoVal: A
content: True 
aoVal: B
content: False 
Answer: B
Analysis: $0$\textquotesingle s opposite number is $0$ 
Knowledge points: Overseas Competition->Knowledge Point->Combinatorics->Mathematical Thoughts->Absolute Value
Source: 其它
Problem: There are two groups of flowers. Group $A$ has $15$ flowers and Group $B$ has $13$ flowers. Cathy and Ivy want to play a game with these flowers. They will take turns to take flowers from the two groups. Each person can take any number of flowers from a group at a time. The person who takes the last flower in two groups will win the game. If Ivy starts the game first, who has the winning strategy? 
Options:
aoVal: A
content: Ivy has the winning strategy. 
aoVal: B
content: Cathy has the winning strategy. 
Answer: A
Analysis: Two groups of flowers are not equal. Thus, Ivy needs to take away 2 flowers from Group A and two groups will have the same amount of flowers. Then, Ivy will become the second mover and she has the winning strategy. 
Knowledge points: Overseas Competition->Knowledge Point->Combinatorics->Strategies and Operations->Game Strategy
Source: 其它
Problem: Bella got up at 8:10. It took 10 minutes to wash and 30 minutes to eat breakfast. At this moment, she began to read. After reading, she went out. She went out at 9:45. How long did she read? 
Options:
aoVal: A
content: $$10$$ minutes 
aoVal: B
content: $$15$$ minutes 
aoVal: C
content: $$20$$ minutes 
aoVal: D
content: $$25$$ minutes 
aoVal: E
content: $$30$$ minutes 
Answer: D
Analysis: $$8:10$$+$$10$$min+$$30$$min+reading=$$9:45$$  reading=$$25$$min 
Knowledge points: Overseas Competition->Knowledge Point->Combinatorics->Time Problem->Time Calculation
Problem: London time is 7 hours behind of Hong Kong time. Mike is planning to have a flight to Hong Kong from London at $$1$$am of $$12$$th June. If the flight takes 13 hours. When will Mike arrive Hong Kong?(In Hong Kong time) 
Options:
aoVal: A
content: $$1$$am $$12$$th June 
aoVal: B
content: $$9$$pm $$12$$th June 
aoVal: C
content: $$8$$am $$12$$th June 
aoVal: D
content: $$7$$am $$12$$th June 
Answer: B
Analysis: Because the flight takes 13 hours, Mike will arrive Hong Kong at $$2$$pm of $$12$$th June(London time). As london time is 7 hours behind Hong Kong time, it will be~$2+7=9$pm(Hong Kong time) 
Knowledge points: Overseas Competition->Knowledge Point->Combinatorics->Time Problem
Source: 其它
Problem: Grace was going to meet her friends at the bus station. Right now, it is $8:40$. Grace arrived at the bus station half an hour ago. The trip from her home to the station was $1$ hour long. What time did Grace depart toward the bus station? 
Options:
aoVal: A
content: $7:20$ 
aoVal: B
content: $7:10$ 
aoVal: C
content: $7:50$ 
aoVal: D
content: $8:10$ 
Answer: B
Analysis: Grace arrived at the bus station at $8:10$, and the trip was $1$ hour long, so she departed at $7:10$. 
Knowledge points: Overseas Competition->Knowledge Point->Combinatorics->Time Problem
Source: 其它
Problem: In the calculation shown below, different letters represent different digits.  $AA\times AB=5467$  What is the sum of $A+B$? 
Options:
aoVal: A
content: $$4$$ 
aoVal: B
content: $$5$$ 
aoVal: C
content: $$6$$ 
aoVal: D
content: $$7$$ 
aoVal: E
content: $$8$$ 
Answer: E
Analysis: $5467=77\times71$ 
Knowledge points: Overseas Competition->Knowledge Point->Combinatorics->Number Puzzles
Source: 其它
Problem: Which of the following number is the opposite number of $-2022$ 
Options:
aoVal: A
content: $$-2022$$ 
aoVal: B
content: $$2022$$ 
aoVal: C
content: $\frac{1}{2022}$ 
aoVal: D
content: $-\frac{1}{2022}$ 
Answer: B
Analysis: Opposite number is the number with opposite sign 
Knowledge points: Overseas Competition->Knowledge Point->Combinatorics->Mathematical Thoughts->Absolute Value
Source: 其它
Problem: In a speed skating competition 10 racers reached in final. Tom overtook 3 racers more than those who overtook him. Which place did Tom end up in? 
Options:
aoVal: A
content: $$1$$ 
aoVal: B
content: $$3$$ 
aoVal: C
content: $$4$$ 
aoVal: D
content: $$6$$ 
aoVal: E
content: $$7$$ 
Answer: C
Analysis: 9-3= 6/2=3. Tom ended up in 4th place. 
Knowledge points: Overseas Competition->Knowledge Point->Combinatorics->Logical Reasoning
Source: 其它
Problem: A certain play has three $30$-minute parts and two intermissions among them. The play started at $8:30$ AM and ended at $10:15$ AM. How long did the intermissions last for in total? 
Options:
aoVal: A
content: $$10 min$$ 
aoVal: B
content: $$15 min$$ 
aoVal: C
content: $$20 min$$ 
aoVal: D
content: $$25 min$$ 
Answer: B
Analysis: From $8:30$ to $10:15$ = $1$ hr $45$ min, $1$ hr $45$ min = $105$ min, three $30$-minute parts = $90$ min, $105 - 90$ = $15$ min. 
Knowledge points: Overseas Competition->Knowledge Point->Combinatorics->Time Problem
Source: 其它
Problem: Calculate the following.  $(220+130-310) \div (99-89) \times (2 \times 3) =$ 
Options:
aoVal: A
content: $$21$$ 
aoVal: B
content: $$24$$ 
aoVal: C
content: $$23$$ 
aoVal: D
content: $$22$$ 
Answer: B
Analysis: $$NA.$$ 
Knowledge points: Overseas Competition->Knowledge Point->Combinatorics->Fun Problems in Math
Source: 其它
Problem: When calculating $54\times 96$, Judy fails to write the correct column multiplication. She writes one of the four digits as $7$ and gets a result which has a difference of $1920$ from the correct one. Which digit does she write incorrectly? 
Options:
aoVal: A
content: $$5$$ 
aoVal: B
content: $$4$$ 
aoVal: C
content: $$9$$ 
aoVal: D
content: $$6$$ 
aoVal: E
content: It\textquotesingle s impossible to determine. 
Answer: A
Analysis: The difference between the wrong answer and the correct one is $1920$, whose ones digit is $0$. That means Judy writes neither $4$ nor $6$ wrong. Then, compare $20\times54$ and $20\times96$, so we can get the correct answer. $74\times96-54\times96=1920.$ 
Knowledge points: Overseas Competition->Knowledge Point->Combinatorics->Number Puzzles
Source: 其它
Problem: When calculating $63\times72$ , Judy fails to write the correct column multiplication. She writes one of the four digits as 9 and gets a result which has a difference of 432 from the correct one. Which digit does she write incorrectly? 
Options:
aoVal: A
content: $$2$$ 
aoVal: B
content: $$3$$ 
aoVal: C
content: $$6$$ 
aoVal: D
content: $$7$$ 
aoVal: E
content: It\textquotesingle s impossible to determine. 
Answer: B
Analysis: The difference between the wrong answer and the correct one is $432$, whose ones digit is $2$. That means Judy writes neither $2$ nor $3$ wrong.  Then, compare $7\times63$ and $6\times72$, so we can get the correct answer.  $69\times72-63\times72=432.$ 
Knowledge points: Overseas Competition->Knowledge Point->Combinatorics->Number Puzzles
Source: 其它
Problem: Four friends are competing in a math competition. They are Andy, Bob, Cindy and Daisy.  The organiser of the competition told you that:  ($$1$$) Andy performed better than Bob.  ($$2$$) Cindy\textquotesingle s score is higher than Andy\textquotesingle s score.  ($$3$$) Daisy\textquotesingle s score is lower than two of her friends.  Arrange their scores in descending order. 
Options:
aoVal: A
content: Andy, Bob, Cindy, Daisy 
aoVal: B
content: Andy, Cindy, Daisy, Bob 
aoVal: C
content: Cindy, Andy, Daisy, Bob 
aoVal: D
content: Cindy, Andy, Bob, Daisy 
aoVal: E
content: None of the above 
Answer: C
Analysis: The score from highest to lower:  First -\/-\textgreater{} Cindy  Second -\/-\textgreater{} Andy  Thrid -\/-\textgreater{} Daisy  Fourth -\/-\textgreater{} Bob 
Knowledge points: Overseas Competition->Knowledge Point->Combinatorics->Logical Reasoning
Problem: There are $5$ boxes on the table. From left to right, each of them has $8,$ $4,$ $2,$ $1,$ and $5$ balls of the same size, respectively. Every time, Judy can take one ball each from the other four boxes, and then put them into the box with the smallest number of balls. She follows the rules and operates $2023$ times. Now how many balls are there in the first box counting from the left? 
Options:
aoVal: A
content: $$2$$ 
aoVal: B
content: $$3$$ 
aoVal: C
content: $$4$$ 
aoVal: D
content: $$5$$ 
aoVal: E
content: $$6$$ 
Answer: D
Analysis: Without operation: $$8$$, $$4$$, $$2$$, $$1$$, $$5$$  After the first operation: $$7$$, $$3$$, $$1$$, $$5$$, $$4$$  After the second operation: $$6$$, $$2$$, $$5$$, $$4$$, $$3$$  After the third operation: $$5$$, $$6$$, $$4$$, $$3$$, $$2$$  After the fourth operation: $$4$$, $$5$$, $$3$$, $$2$$, $$6$$  After the fifth operation: $$3$$, $$4$$, $$2$$, $$6$$, $$5$$  After the sixth operation: $$2$$, $$3$$, $$6$$, $$5$$, $$4$$  After the seventh operation: $$6$$, $$2$$, $$5$$, $$4$$, $$3$$  We can find that $6-2-5-4-3$ is repeating starting from the second operation.  $$（2023-1）\div5$$R$$2$$  Thus, after the $2023$\textsuperscript{rd}~operation, the result will be the same as the third one, which means there are $5$ balls in the first box. 
Knowledge points: Overseas Competition->Knowledge Point->Combinatorics->Strategies and Operations->Operational Problem
Problem: Lily walks to school every day. She leaves home at $7:15$ and arrives at the school at $7:50$. How many minutes does she spend? 
Options:
aoVal: A
content: $$15$$ 
aoVal: B
content: $$30$$ 
aoVal: C
content: $$35$$ 
aoVal: D
content: $$50$$ 
Answer: C
Analysis: $$7:50-7:15=35$$ min 
Knowledge points: Overseas Competition->Knowledge Point->Combinatorics->Time Problem
Problem: Which of the following number is the opposite number of $12$? 
Options:
aoVal: A
content: $$12$$ 
aoVal: B
content: $$-12$$ 
aoVal: C
content: $\frac{1}{12}$ 
aoVal: D
content: $-\frac{1}{12}$ 
Answer: B
Analysis: Opposite number is the number with opposite sign 
Knowledge points: Overseas Competition->Knowledge Point->Combinatorics->Mathematical Thoughts->Absolute Value
Source: 其它
Problem: There is a cube whose six faces are marked with $3$, $4$, $5$, $6$, $7$, and $8$. If the sum of every two numbers that are on the opposite faces are identical, the number on the opposite face of $4$ is~\uline{~~~~~~~~~~}~ 
Options:
aoVal: A
content: $$3$$ 
aoVal: B
content: $$5$$ 
aoVal: C
content: $$6$$ 
aoVal: D
content: $$7$$ 
aoVal: E
content: $$8$$ 
Answer: D
Analysis: $3+8=4+7=5+6=11$ 
Knowledge points: Overseas Competition->Knowledge Point->Combinatorics->Fun Problems in Math->Fun Math Problems->Dice
Source: 其它
Problem: Math workshops start at $5:00$ PM. Today Allan was $15$ minutes late to the workshop. What time did Allan come? 
Options:
aoVal: A
content: $5:00$ PM 
aoVal: B
content: $5:05$ PM 
aoVal: C
content: $5:15$ PM 
aoVal: D
content: $5:20$ PM 
Answer: C
Analysis: $15$ minutes past $5:00$ PM is $5:15$ PM. 
Knowledge points: Overseas Competition->Knowledge Point->Combinatorics->Time Problem
Source: 其它
Problem: insert pic  Seven children are standing in a circle. No two boys are standing next to each other. No three girls are standing next to each other. Which of these is true for the number of girls standing in the circle? 
Options:
aoVal: A
content: only 3 is possible 
aoVal: B
content: 3 and 4 are possible 
aoVal: C
content: only 4 is possible 
aoVal: D
content: 4 and 5 are possible 
aoVal: E
content: only 5 is possible 
Answer: C
Analysis: "No two boys stand next to each other" meaning there are a maximum of 3 boys only. Hence, there are a minimum of 4 girls in the circle. 
Knowledge points: Overseas Competition->Knowledge Point->Combinatorics->Logical Reasoning
Source: 其它
Problem: Thirty apples are distributed among 4 children and each of them can get at least one apple. For the kid who gets the most apples, what\textquotesingle s the least possible number of apples he or she can get? 
Options:
aoVal: A
content: $7$ 
aoVal: B
content: $8$ 
aoVal: C
content: $9$ 
aoVal: D
content: $10$ 
aoVal: E
content: $11$ 
Answer: B
Analysis: $$30\div 4=7 \textbackslash{} \text{R}2$$, thus for the kid who gets the most apples, he or she can get $$7+1=8$$ apples at least. 
Knowledge points: Overseas Competition->Knowledge Point->Combinatorics->Pigeonhole Principle->Simple Pigeonhole Principle Problems
Source: 其它
Problem: Ivy, Vivian, and Candy are playing the truth or lie game. The rule is: the person who picks the truth card can only tell the truth, and the person who picks the lie card can only tell a lie.  Ivy said: "Vivian and Candy lied."  Vivian said: "I didn\textquotesingle t lie."  Candy said: "Vivian lied."  How many of them told the truth? How many of them told a lie?~\hspace{0pt}~． 
Options:
aoVal: A
content: $2$; $1$ 
aoVal: B
content: $0$; $3$ 
aoVal: C
content: $1$; $2$ 
aoVal: D
content: $3$; $0$ 
Answer: C
Analysis: Vivian and Candy told contradictory information, so one of them told the truth and the other one told the lie. Therefore, Ivy has definitely told the lie. So two people told the lie, and one people told the truth. 
Knowledge points: Overseas Competition->Knowledge Point->Combinatorics->Logical Reasoning