math-eval/TAL-SCQ5K huggingface dataset:
Problem: What is the missing number in the sequence below?  $1, 3, 7, 15, 31,$~\uline{~~~~~~~~~~}~ 
Options:
aoVal: A
content: $$63$$ 
aoVal: B
content: $$47$$ 
aoVal: C
content: $$57$$ 
aoVal: D
content: $$59$$ 
aoVal: E
content: None of the above 
Answer: A
Analysis: 1, 3, 7, 15, 31, \cdots ..  2. 4. 8.~ 16.~ 32  31+32 = 64 
Knowledge points: Overseas Competition->Knowledge Point->Combinatorics->Patterns of Figures
Source: 其它
Problem: Rose and Kylie are playing a game. Here are the rules：  1. There are $$16$$ 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: A
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.  $$16\div (1+2)=5 \textbackslash{} \text{R} 1$$ Rose should take away the remainder, i.e. one marble, 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: Chris, Molly, Sara, and Lynn each has some coins. Chris gives Molly $20$ coins, Molly gives Sara $25$ coins, Sara gives Lynn $30$ coins, and Lynn gives Chris $30$ coins. Now, which of the following is correct if they each wants to have the same number of coins as in the beginning? 
Options:
aoVal: A
content: Lynn gives Sara $5$ coins, and Molly gives Sara $5$ coins. 
aoVal: B
content: Chris gives Molly $5$ coins, and Chris gives Lynn $10$ coins. 
aoVal: C
content: Chris gives Molly $5$ coins, and Chris gives Sara $5$ coins. 
aoVal: D
content: Chris gives Lynn $10$ coins. 
aoVal: E
content: Chris gives Sara $10$ coins. 
Answer: C
Analysis: Chris gives Molly $20$ coins, and Lynn gives Chris $30$ coins. Now, Chris has $10$ more.  Chris gives Molly $20$ coins, and Molly gives Sara $25$ coins. Now, Molly has $5$ less.  Molly gives Sara $25$ coins, and Sara gives Lynn $30$ coins. Now, Sara has $5$ less.  Sara gives Lynn $30$ coins, and Lynn gives Chris $30$ coins. Now, Lynn has the same as beginning.  Thus, Chris can give Molly $5$ coins, and give Sara $5$ coins. 
Knowledge points: Overseas Competition->Knowledge Point->Combinatorics->Logical Reasoning
Problem: In the following column puzzle, different letters represent different one-digit numbers. Find the three-digit number represented by $$ABC$$.  $$\begin{matrix}\& \& 8 \& 8\& \boxed C \textbackslash\textbackslash{} \&\&5\& \boxed B \&4 \textbackslash\textbackslash{} + \&\&\boxed A\&1 \&7 \textbackslash\textbackslash{} \hline \&2~\&0 \&2 \&1\textbackslash\textbackslash{} \end{matrix}$$ 
Options:
aoVal: A
content: $$625$$ 
aoVal: B
content: $$620$$ 
aoVal: C
content: $$602$$ 
aoVal: D
content: $$260$$ 
aoVal: E
content: $$206$$ 
Answer: B
Analysis: Start from the ones digit, $$4+7=11$$, $$0+1=1$$, so $$C=0$$.  In the tens place, $$8+1+1=10$$, $$0+2=2$$, so $$B=2$$.  In the hundreds place, $$8+5+1=14$$, $$14+6=20$$, so $$A=6$$. 
Knowledge points: Overseas Competition->Knowledge Point->Combinatorics->Number Puzzles
Problem: Tom wrote various words in code in such a way that different digits represent different letters and the same digit represents the same letter. For example, the word $BALL$ was coded as $$3488$$. One of the words below was coded as $$6155491$$. Which one? 
Options:
aoVal: A
content: $$SURGEON$$ 
aoVal: B
content: $$HARBORS$$ 
aoVal: C
content: $SWEATER$ 
aoVal: D
content: $MESSAGE$ 
Answer: D
Analysis: According to the pattern: the same digit represents the same letter, the code $$6155491$$ should represent a word with two same letters at the third and fourth place. 
Knowledge points: Overseas Competition->Knowledge Point->Combinatorics->Logical Reasoning->Reasoning by Conditions->Complex Reasoning
Problem: The area of a square is $16$, its side length is~\uline{~~~~~~~~~~}~. 
Options:
aoVal: A
content: $$2$$ 
aoVal: B
content: $$4$$ 
aoVal: C
content: $$8$$ 
aoVal: D
content: $$16$$ 
Answer: B
Analysis: $A=s^{2}$  $s^{2}=16$  $s=4$ 
Knowledge points: Overseas Competition->Knowledge Point->Combinatorics->Magic Square
Source: 其它
Problem: 2 foxes and 2 rabbits want to cross a river. There is a boat that can hold 2 animals. When the number of fox is less than the number of rabbits, the rabbits fight with the foxes. For 2 foxes and 2 rabbits to cross the river safely, at least how many times does the boat need to cross the river? (count the round trip as twice and every trip need one animal row the boat)~\uline{~~~~~~~~~~}~． 
Options:
aoVal: A
content: $$1$$ 
aoVal: B
content: $$2$$ 
aoVal: C
content: $$3$$ 
aoVal: D
content: $$4$$ 
aoVal: E
content: $$5$$ 
Answer: E
Analysis: Assume the foxes as A, and the rabbits as B. The way they cross the river as following:  $$AA\xrightarrow{BB}$$  $$AA\xleftarrow{B}B$$  $$B\xrightarrow{AA}B$$  $$B\xleftarrow{B}AA$$  $$\xrightarrow{BB}AA$$ 
Knowledge points: Overseas Competition->Knowledge Point->Combinatorics->Fun Problems in Math->Fun Math Problems
Problem: There are two groups of balls. Group $A$ has $585$ balls and Group $B$ has $590$ balls. DQ and Justin want to play a game with these balls. They will take turns to take balls from the two groups. Each person can take $1$ to $6$ balls from a group at a time. The person who takes the last ball in the two groups will win the game. If DQ starts the game first, does she have the winning strategy? 
Options:
aoVal: A
content: Yes, she can take $5$ balls from group $A$ at first. 
aoVal: B
content: Yes, she can take $3$ balls from group $B$ at first. 
aoVal: C
content: Yes, she can take $6$ balls from group $A$ at first. 
aoVal: D
content: Yes, she can take $5$ balls from group $B$ at first. 
aoVal: E
content: No, she doesn\textquotesingle t have winning strategy. 
Answer: D
Analysis: DQ needs to take the $590-585=5$ balls in Group $B$ which will make the two groups have the same amount of balls. Then, no matter how many balls Justin takes in a group, DQ will take as many balls as Justin took before in the other group. 
Knowledge points: Overseas Competition->Knowledge Point->Combinatorics->Strategies and Operations->Game Strategy
Source: 其它
Problem: Jack and Edward want to take turns to count off numbers from $1$ to $124$ in order. Each person can count $1$ to $5$ numbers at a time. The person who counts off number $124$ will win the game. Does Jack have the winning strategy if he counts off numbers first? 
Options:
aoVal: A
content: Yes, he has. 
aoVal: B
content: No, he doesn\textquotesingle t. 
Answer: A
Analysis: $124\div(1+5)=20R4$. So, Jack should count off $1$ to $4$ and there will be $120$ numbers left. Jack will become the second mover and he has the winning strategy. 
Knowledge points: Overseas Competition->Knowledge Point->Combinatorics->Strategies and Operations->Game Strategy
Source: 其它
Problem: The ancient Romans used Roman numerals. We still use them today. Here are some examples: $$\rm I=1$$, $$\rm II=2$$, $$\rm V=5$$, $$\rm IX=9$$, $$\rm X=10$$, $$\rm XI=11$$, $$\rm XX=20$$. This year($2022$) we celebrate Math Kangaroo number $$\rm XX$$. What year was Math Kangaroo number $$\rm XV$$? 
Options:
aoVal: A
content: $$2015$$ 
aoVal: B
content: $$2016$$ 
aoVal: C
content: $$2017$$ 
aoVal: D
content: $$2018$$ 
aoVal: E
content: $$2019$$ 
Answer: C
Analysis: According to the pattern, $$\rm XX$$ is $$20$$, and $$\rm XV$$ is $$15$$.  So, Math Kangaroo number $$15$$ was $$5$$ less than real year, which was $$2017$$. 
Knowledge points: Overseas Competition->Knowledge Point->Combinatorics->Logical Reasoning->Reasoning by Conditions->Complex Reasoning
Problem: If May $6$ falls on a Friday, what day of the week will it be in $25$ days? (Adapted from 2015 Math Kangaroo Problem, Level 3-4, Question \#8) 
Options:
aoVal: A
content: Tuesday 
aoVal: B
content: Wednesday 
aoVal: C
content: Thursday 
aoVal: D
content: Friday 
aoVal: E
content: Sunday 
Answer: A
Analysis: $25\div7=3R4$ 
Knowledge points: Overseas Competition->Knowledge Point->Combinatorics->Time Problem
Source: 其它
Problem: If January $2$ falls on a Saturday, what day of the week will it be in $28$ days? 
Options:
aoVal: A
content: Tuesday 
aoVal: B
content: Wednesday 
aoVal: C
content: Thursday 
aoVal: D
content: Saturday 
aoVal: E
content: Sunday 
Answer: D
Analysis: $28\div7=4$ 
Knowledge points: Overseas Competition->Knowledge Point->Combinatorics->Time Problem
Source: 其它
Problem: James starts classes at $9$ A.M. He has $2$ classes, each class $$45$$ minutes long. After each class, he will take a $$15$$-minute break. When will he finish all $$2$$ classes? 
Options:
aoVal: A
content: $10:20$ A.M. 
aoVal: B
content: $11:00$ A.M. 
aoVal: C
content: $11:10$ A.M. 
aoVal: D
content: $11:20$ A.M. 
aoVal: E
content: $11:30$ A.M. 
Answer: B
Analysis: $$45+15+45+15=120$$ minutes = $2$ hours  So, he will finish at $$11:00$$ A.M. 
Knowledge points: Overseas Competition->Knowledge Point->Combinatorics->Time Problem
Source: 其它
Problem: There are $13$ ping-pong players who will be divided into three teams, and they are going to have a men\textquotesingle s singles. The rule is that the players in the same team will not play with each other, and each of them only plays one game with each player in other teams. What is the maximum number of games happened in total? 
Options:
aoVal: A
content: $$23$$ 
aoVal: B
content: $$30$$ 
aoVal: C
content: $$45$$ 
aoVal: D
content: $$56$$ 
aoVal: E
content: $$72$$ 
Answer: D
Analysis: Dividing the players into three teams as equally as possible can make the maximum number of games. So there are $4$, $4$, and $5$ players in the three teams and they will have $4\times4+4\times5+4\times5=56$ games in total. 
Knowledge points: Overseas Competition->Knowledge Point->Combinatorics->Logical Reasoning->Sports Competition
Problem: We left for a summer camp at $4:30$ PM and got to our destination at $6:40$ PM. How long did we travel?~ 
Options:
aoVal: A
content: $1$ hour $40$ minutes 
aoVal: B
content: $2$ hours $10$ minutes 
aoVal: C
content: $2$ hours $20$ minutes 
aoVal: D
content: $1$ hour $50$ minutes 
aoVal: E
content: $50$ minutes 
Answer: B
Analysis: $6:40$ - $4:30$ = $2$ hours $10$ minutes 
Knowledge points: Overseas Competition->Knowledge Point->Combinatorics->Time Problem
Source: 其它
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: Alvin always tells the truth on Sundays, Tuesdays and Thursdays. He lies on every other days.  One day he said, "I told the truth yesterday."  On which day of the week did he make this statement? 
Options:
aoVal: A
content: Sunday 
aoVal: B
content: Wednesday 
aoVal: C
content: Thursday 
aoVal: D
content: Friday 
aoVal: E
content: Saturday 
Answer: E
Analysis: If "I told the truth yesterday" is a true statement -\/-\textgreater{} mean today I am telling the truth. But, the day that Alvin tell the truth are not consecutive day.  So, if "I told the truth yesterday" is not a true statement -\/-\textgreater{} mean yesterday I lie and today I am also lying.  Alvin lies on Monday, Wednesday, Friday and Saturday. Friday and Saturday is consecutive day he lies, so it must be Saturday. 
Knowledge points: Overseas Competition->Knowledge Point->Combinatorics->Logical Reasoning
Problem: There are two groups of balls. Group $A$ has $50$ balls and Group $B$ has $40$ balls. Candy and Nini want to play a game with these balls. They will take turns to take balls from the two groups. Each person can take any number of balls from a group at a time, but they should take at least one ball at a time. The person who takes the last ball in the two groups will win the game. If Nini starts the game first, how many balls should she take to guarantee her success? ． 
Options:
aoVal: A
content: $10$ 
aoVal: B
content: $8$ 
aoVal: C
content: $6$ 
aoVal: D
content: $4$ 
aoVal: E
content: $2$ 
Answer: A
Analysis: Nini needs to take $$50-40=10$$ balls in Group A which will make the two groups have the same amount of balls. Then, no matter how many balls Candy takes in a group, Nini will take as many balls as Candy took before in the other group. 
Knowledge points: Overseas Competition->Knowledge Point->Combinatorics->Strategies and Operations->Game Strategy
Source: 其它
Problem: There are $7$ water glasses, all facing up. You need to turn exactly $2$ glasses over in each time. Is it possible to turn all $7$ of them upside down after several moves? 
Options:
aoVal: A
content: Yes, it is possible. 
aoVal: B
content: No, it is impossible. 
aoVal: C
content: I have no idea. 
Answer: B
Analysis: You have to flip a cup an odd number of times to turn it over.  To make all the $7$ cups face down, you have to make an odd number of flips.  Each time you flip exactly $2$ cups, the total number of flips is an even number of times.  Hence it is impossible. 
Knowledge points: Overseas Competition->Knowledge Point->Combinatorics->Strategies and Operations->Operational Problem->Turning Mugs over
Problem: There are $$17$$ balls in a bag. Each ball has a number from $$1$$ to $$17$$ on it. We randomly pick a ball from the bag. What is the smallest number of balls we have to pick in order to be sure that we have at least one pair of balls with a difference equal to $$3$$? (adapted from $$2005$$ Math Kangaroo Problem, Level $$9-10$$, Question \#$$15$$) 
Options:
aoVal: A
content: $$7$$ 
aoVal: B
content: $$8$$ 
aoVal: C
content: $$10$$ 
aoVal: D
content: $$11$$ 
aoVal: E
content: $$16$$ 
Answer: C
Analysis: We can create three drawers:  $(1, 4, 7, 10, 13, 16)$.  $(2, 5, 8, 11, 14, 17)$.  $(3, 6, 9, 12, 15)$.  At least $3\times3+1=10$ numbers should be chosen. 
Knowledge points: Overseas Competition->Knowledge Point->Combinatorics->Pigeonhole Principle->Worst Case in Pigeonhole Principle Problems
Problem: Peter\textquotesingle s father has $$4$$ sons and Peter has $$5$$ brother(s) and sister(s). How many daughter(s) does Peter\textquotesingle s father have? 
Options:
aoVal: A
content: $$1$$ 
aoVal: B
content: $$2$$ 
aoVal: C
content: $$3$$ 
aoVal: D
content: $$4$$ 
aoVal: E
content: $$0$$ 
Answer: B
Analysis: $$5+1-4=2$$ 
Knowledge points: Overseas Competition->Knowledge Point->Combinatorics->Logical Reasoning
Problem: John and Olivia exchanged sweets. First John gave Olivia as many sweets as Olivia had. Then Olivia gave John as many sweets as John had after the first exchange. After these two exchanges, each had $$4$$ sweets. How many sweets did John have at the beginning? 
Options:
aoVal: A
content: $$6$$ 
aoVal: B
content: $$5$$ 
aoVal: C
content: $$4$$ 
aoVal: D
content: $$3$$ 
aoVal: E
content: $$2$$ 
Answer: B
Analysis: NA 
Knowledge points: Overseas Competition->Knowledge Point->Combinatorics->Strategies and Operations->Planning->Simple Time Planning Problems->Working Simultaneously
Problem: SASMO 2016 P2 Q2  How many hours are there in two weeks? 
Options:
aoVal: A
content: 7 x 12 
aoVal: B
content: 7 x 2 x 12 
aoVal: C
content: 2 x 7 x 2 x 12 
aoVal: D
content: (7+7) x 12 
aoVal: E
content: None of the above 
Answer: C
Analysis: Conversion of unit. 
Knowledge points: Overseas Competition->Knowledge Point->Combinatorics->Time Problem
Source: 其它
Problem: Alice began reading a 400-page book at 8 am and had read 50 pages by 9:30. If she continues to read at the same rate, when can she finish reading this book? 
Options:
aoVal: A
content: 6 pm 
aoVal: B
content: 7 pm 
aoVal: C
content: 8 pm 
aoVal: D
content: 9 pm 
aoVal: E
content: 10 pm 
Answer: C
Analysis: $$300\div20-300\div30=5$$. 
Knowledge points: Overseas Competition->Knowledge Point->Combinatorics->Time Problem
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 $$40$$ minutes. At what time must the person who needs the longest time to get to the meeting leave her house? (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: E
Analysis: $12:30$ - $40$ min = $11:50$ 
Knowledge points: Overseas Competition->Knowledge Point->Combinatorics->Time Problem->Time Calculation
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 commercials 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: There are two containers, one with a capacity of 6 liters and the other with a capacity of 5 liters, and using them to get 1 liter of water from a bucket requires at leasttimes of operations. 
Options:
aoVal: A
content: $$2$$ 
aoVal: B
content: $$3$$ 
aoVal: C
content: $$4$$ 
aoVal: D
content: $$5$$ 
Answer: A
Analysis: Fill the 6-liter container first,  and then pour the water from the 6-liter container into the 5-liter container.  After the 5-liter container is filled, there is 1 liter of water left in the 6-liter container.  It takes two operations.  So the answer is $$A$$ 
Knowledge points: Overseas Competition->Knowledge Point->Combinatorics->Strategies and Operations->Operational Problem->Pouring Water Problems
Problem: A store opens at 9 a.m. and closes at 5p.m. each day. It is closed from 11.55 a.m. to 12.55 p.m. for a lunch break. How many hours does the store stay open each day? 
Options:
aoVal: A
content: $8$ hours and $5$ minutes 
aoVal: B
content: $$7$$ hours and $30$ minutes 
aoVal: C
content: $8$ hours 
aoVal: D
content: $7$ hours 
aoVal: E
content: $6$ hours 
Answer: D
Analysis: NA 
Knowledge points: Overseas Competition->Knowledge Point->Combinatorics->Time Problem
Source: 其它
Problem: There are two piles of matches. Each pile has $$6$$ matches. Cindy and Doris take turns to pick up matches from either pile. There is no limit to how many matches they can pick up, but they must pick up at least one match each turn. The person who picks up the last match will be the winner. If Cindy starts picking up matches first,~\uline{~~~~~~~~~~}~will definitely be the winner (has a winning strategy). 
Options:
aoVal: A
content: Cindy 
aoVal: B
content: Doris 
Answer: B
Analysis: Doris will be the winner, since she can simply mirror the number of matches Cindy picks up every turn. 
Knowledge points: Overseas Competition->Knowledge Point->Combinatorics->Strategies and Operations->Game Strategy
Problem: Taylor has $2$ brothers and $3$ sisters. How many brothers and sisters does his sister Lucy have? 
Options:
aoVal: A
content: 3 brothers and 4 sisters 
aoVal: B
content: 2 brothers and 3 sisters 
aoVal: C
content: 3 brothers and 3 sisters 
aoVal: D
content: 2 brothers and 4 sisters 
aoVal: E
content: 3 brothers and 2 sisters 
Answer: E
Analysis: NA 
Knowledge points: Overseas Competition->Knowledge Point->Combinatorics->Logical Reasoning
Source: 其它
Problem: Among $42$ people, at least how many people were born in the same month with the most births? 
Options:
aoVal: A
content: $$1$$ 
aoVal: B
content: $$2$$ 
aoVal: C
content: $$3$$ 
aoVal: D
content: $$4$$ 
Answer: D
Analysis: There are $12$ months. Thus, $42 \div 12 = 3R6$, $3+1 = 4$ 
Knowledge points: Overseas Competition->Knowledge Point->Combinatorics->Pigeonhole Principle
Source: 其它
Problem: Teacher wrote $$10$$ non-zero natural numbers in order on the blackboard, where the $$1$$st number is $$16$$, and the sum of any $$3$$ adjacent numbers is $$100$$. The $$8$$th biggest number is~\uline{~~~~~~~~~~}~. 
Options:
aoVal: A
content: $$81$$ 
aoVal: B
content: $$82$$ 
aoVal: C
content: $$83$$ 
aoVal: D
content: $$84$$ 
Answer: C
Analysis: Let first 10 numbers be $${{a}\_{1}}$$、$${{a}\_{2}}$$、$${{a}\_{3}}$$、$${{a}\_{4}}$$、$$\ldots \ldots $$、$${{a}\_{10}}$$.  $${{a}\_{1}}+{{a}\_{2}}+{{a}\_{3}}={{a}\_{2}}+{{a}\_{3}}+{{a}\_{4}}$$，$${{a}\_{1}}={{a}\_{4}}$$.  Therefore, $${{a}\_{1}}={{a}\_{4}}={{a}\_{7}}={{a}\_{10}}=16$$，$${{a}\_{8}}=100-{{a}\_{10}}-{{a}\_{9}}=100-16-{{a}\_{9}}=84-{{a}\_{9}}$$.  The maximum value of $${{a}\_{8}}$$ is $$83$$．$$For$$ example：$$16$$、$$83$$、$$1$$、$$16$$、$$83$$、$$1$$、$$16$$、$$83$$、$$1$$、$$16$$． 
Knowledge points: Overseas Competition->Knowledge Point->Combinatorics->Fun Problems in Math
Source: 其它
Problem: There are $$25$$ matches on the table. John and James take turns to remove $$1$$ to $$3$$ matches each time. The person who removes the last match will be the winner. If both of them were to use the best method and John removes first, then~\uline{~~~~~~~~~~}~will win.~ 
Options:
aoVal: A
content: John 
aoVal: B
content: James 
Answer: A
Analysis: $$25\div 4=6\ldots 1$$ John removes $$1$$ match and $$24$$ is a multiple of $$4$$. So, the first player will win the game. 
Knowledge points: Overseas Competition->Knowledge Point->Combinatorics->Strategies and Operations->Game Strategy
Problem: Light Bulb A lights up every $$15$$ minutes while Light Bulb B lights up every $$20$$ minutes. Both light bulbs lit up at the same time at $$8.30$$ a.m.  By noon, how many more times would both light bulbs have lit up at the same time? 
Options:
aoVal: A
content: $$2$$ 
aoVal: B
content: $$3$$ 
aoVal: C
content: $$4$$ 
aoVal: D
content: $$5$$ 
Answer: B
Analysis: The LCM of $$15$$ and $$20$$ is $$60$$.  Both light bulbs will light up at the same time every $$60$$ minutes, at $$9.30$$ a.m., $$10.30$$ a.m. and $$11.30$$ a.m. (total of $$3$$ times). 
Knowledge points: Overseas Competition->Knowledge Point->Combinatorics->Time Problem
Source: 其它
Problem: Adam built fewer sandcastles than Martin but more than Susan. Lucy built more sandcastles than Adam and more than Martin. Dana built more sancastles than Martin but fewer than Lucy.  Who of them built the most sandcastles? 
Options:
aoVal: A
content: Martin 
aoVal: B
content: Adam 
aoVal: C
content: Susan 
aoVal: D
content: Dana 
aoVal: E
content: Lucy 
Answer: E
Analysis: L \textgreater{} D\textgreater{} M \textgreater{} A \textgreater{} S 
Knowledge points: Overseas Competition->Knowledge Point->Combinatorics->Logical Reasoning
Source: 其它