Apply periodic reasoning to calendar cycles, repeated groupings, coin patterns, swimmers, sliding sums, repeating decimals, aligned marks, paired symbol cycles, Fibonacci remainders, growing figures, alternating transfers, and zigzag arrays. Every problem is written in full here; this lesson is not needed.
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Questions 1–10 · 6 points each · 60 points
Use the simplified Gregorian-calendar rule in this problem: every fourth year is a leap year, a common year has 365 days, and a leap year has 366 days. The year 2012 was a leap year, and New Year's Day in 2012 was Sunday.
What is the next year whose New Year's Day is also Sunday?
On the eve of Teachers' Day, 40 Young Pioneers make large red paper flowers for their teachers. The children receive different numbers of sheets: one child receives 7 sheets, one receives 8 sheets, and so on through 46 sheets.
Each flower must use either 3 sheets or 4 sheets. Children cannot exchange sheets. Each child must use all of their own paper and make as many flowers as possible. Altogether, how many of the flowers use 4 sheets?
Xiaoming arranges saved coins repeatedly in this order:
What is the denomination of the 111th coin? What is the total value of the first 111 coins? Use 100 fen = 1 yuan.
Two swimmers move back and forth in a 30-metre pool. Swimmer A travels at 1 metre per second and Swimmer B at 0.6 metre per second. They start at the same time from opposite ends. Ignore the time needed to turn around.
From the start through exactly 15 minutes, how many times do they meet? Count an encounter whenever they occupy the same position, including one catching up from behind.
An 11-digit number has first digit 9 and last digit 8. The sum of every three adjacent digits is 24.
The question mark is in the fifth digit position.
What digit replaces the question mark? What is the complete 11-digit number?
Write 3/7 as a repeating decimal. What digit is in the 2,014th place after the decimal point?
A wooden stick is 100 centimetres long. Starting from the left end, place a mark every 6 centimetres. Starting from the right end, also place a mark every 5 centimetres. Saw the stick at every marked point. Coincident marks give just one cut. The two ends are boundaries of the stick; they do not create extra pieces. Include pieces next to either end when counting their lengths.
How many of the resulting pieces are 1 centimetre long?
Each top-row character and bottom-row letter form one ordered pair. The top row repeats the five symbols 我、们、爱、数、学, and the bottom row repeats A, B, C, D, E, F, G.
The first pair is (我, A), the second is (们, B), and so on. What is the 100th pair?
Consider the sequence:
Each term after the first two is the sum of the previous two. What is the units digit of the 2,014th term?
The first “little house” uses 5 counters, the second uses 11 counters, and the third uses 17 counters. Continue the construction shown below: at Stage n, each of the six line segments (five outline sides and the horizontal eaves line) has n equal intervals, with a counter at each interval endpoint. Count a counter shared by segments only once. The diagrams show the first three stages.
How many counters are needed in the single Stage 30 house?
Questions 11–13 · 20 points each · 60 points
There are two cups, A and B. Initially Cup A contains 1 kilogram of water and Cup B is empty.
Continue in the same way: directions alternate, and the denominator increases by 1 each time. After 2,011 pours, how many grams of water remain in Cup A? Use 1 kilogram = 1,000 grams.
This saved response used the previous unit. Enter your answer in grams above. Your earlier working is retained.
The sequence is:
The first and second terms are both 1. Starting with the third term, each term is the sum of the two terms before it. What remainder is obtained when the 2,012th term is divided by 3?
Positive integers are arranged in the zigzag table below. The columns are numbered 1 through 8.
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
|---|---|---|---|---|---|---|---|
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
| 15 | 14 | 13 | 12 | 11 | 10 | 9 | |
| 16 | 17 | 18 | 19 | 20 | 21 | 22 | |
| 29 | 28 | 27 | 26 | 25 | 24 | 23 | |
| 30 | 31 | 32 | 33 | 34 | 35 | 36 |
The first row has eight numbers. Every later row has seven. Reverse direction at each end without repeating the endpoint column; continue as shown.
In which column does the number 2,015 appear?