Clocks and Calendars

At 8 o'clock, the hour hand is at 240° (8 × 30°) and the minute hand is at 0°. The angle between them is 240°. However, we take the smaller angle, which is 360° - 240° = 120°. The minute hand moves 360° in 60 minutes (6° per minute), and the hour hand moves 30° in 60 minutes (0.5° per minute). At exactly 8:00, the angle is 120°.

At 5:25, hour hand is at 5 + 25/60 = 5.4167 hours from 12. Position = 5.4167 × 30 = 162.5°. Minute hand at 25 is at 25 × 6 = 150° from 12. Difference = 162.5 - 150 = 12.5°. Formula |30H - 5.5M| gives |150 - 137.5| = 12.5°.

At 4:00, the hour hand is at 120° from 12 and minute hand at 0°. The relative speed is 5.5°/min. For the first right angle (90°), we solve |120 - 5.5t| = 90. This gives 5.5t = 30, so t = 60/11 minutes past 4 o'clock.

From noon (12:00) to 5:10 PM is 5 hours 10 minutes = 310 minutes. The hour hand moves at 0.5 degrees per minute (360 degrees ÷ 720 minutes for 12 hours). Angle turned = 310 × 0.5 = 155 degrees.

Clock hands are in a straight line 11 times every 12 hours (once every hour except at 11, which overlaps with 12). This includes both overlaps (0 minutes apart) and oppositions (180 degrees apart). In 84 hours, which is 84/12 = 7 complete 12-hour cycles, the hands align 7 × 11 = 77 times. Additionally, since 84 starts from 00:00 (midnight), we count the starting position as the 1st occurrence, giving us 77 + 77 = 154 total straight line positions.

At 8:20, hour hand is at 250° (8×30 + 20×0.5), minute hand at 120° (20×6). Angle between them = 130°. To reach 90° separation, minute hand must gain 40° relative to hour hand. Relative speed = 5.5°/min. Time needed = 40/5.5 = 7 4/11 minutes. So 8:20 + 7 4/11 = 8:27 3/11. Option C is correct.

From January 5 to December 12, 2016 (a leap year with 366 days), we count 342 days. When we divide 342 by 7, we get 48 weeks and 6 days remainder. Since January 5, 2016 is a Saturday, adding 6 days gives us Friday. The calculation: January (26 remaining) + February (29) + March through November (275) + December (12) = 342 days. 342 mod 7 = 6. Saturday + 6 = Friday.

At 9:27, the hour hand is at 9 + 27/60 = 9.45, which is 9.45×30 = 283.5° from 12 o'clock. The minute hand is at 27×6 = 162° from 12 o'clock. The angle between them is 283.5−162 = 121.5°.

At 8:40, hour hand is at 8 + 40/60 = 8.67, minute hand is at 8. Angle from hour hand to 12 is 360 - 8.67*30 = 100 degrees. Angle from minute hand to 12 is 8*30 = 240 degrees. Difference = 240 - 100 = 140 degrees, not 20. Wait, I need to recalculate. Actually formula is |30H - 5.5M| = |30*8 - 5.5*40| = |240 - 220| = 20 degrees. This gives the smaller angle between hands, which is 20 degrees.

A century year is a leap year only if divisible by 400. From 1600 to 2400, only 1600, 2000, and 2400 are divisible by 400. So only 3 leap years among these 9 century years. The rule differs from regular years (divisible by 4).

For calendar repetition, calculate total offset days. 3009 is not a leap year (3009 not divisible by 4, not century year divisible by 400). Check each option: 3015 is 6 years later with 1 leap year (3012). Total offset = 6×365 + 1 = 2191 days = 312 weeks + 7 days = 312 weeks + 1 day. Wait, that's not matching. Let me recalculate: From 3009 to 3015: years 3010, 3011, 3012 (leap), 3013, 3014, 3015. That's 6 years with 1 leap year (3012). Days = 6×365 + 1 = 2191. 2191 mod 7 = 2191 - 312×7 = 2191 - 2184 = 7, which is 0 mod 7. Actually wait, 2191 ÷ 7 = 313 exactly with remainder 0. Let me verify: 313 × 7 = 2191. Yes, so offset is 0 days. This means calendars repeat. Check 3009 starting day: 3009 = 3008 + 1. 3008 is divisible by 400? No. Let me use formula. Actually, simpler: total days from Jan 1, 3009 to Jan 1, 3015 = 2191. 2191 mod 7 = 0. So same calendar. The answer A is correct.

Calendars repeat when the number of days between years is a multiple of 7 and leap year status matches. From 2017 to 2023 is 6 years with 1 leap year (2020), totaling 2191 days (2191/7 = 313 weeks exactly). 2017 and 2023 are both non-leap years starting on Sunday, so their calendars are identical. 2024 is a leap year, 2027 is 7 years later, and 2028 is a leap year, so their calendars differ.

2016 is a leap year starting on Friday. A leap year calendar repeats after 28 years when the sum of odd days is exactly divisible by 7 (7 × 4 = 28). 2016 + 28 = 2044, which is also a leap year starting on Friday, so the calendar repeats. Option B (2021) is only 5 years later, option C (2056) is 40 years later, and option D (2040) is 24 years later - none of these produce the same calendar.

From 1900-2001, we count leap years. 1900 is NOT a leap year (divisible by 100 but not 400). The leap years in this range are: 1904, 1908, ..., 1996, 2000. This gives us 25 leap years. Note that 2000 IS a leap year because it's divisible by 400.

In 12 hours, the hour and minute hands form a straight line (0° or 180°) 22 times: at each hour (12 times) and when they overlap in the opposite direction during 5-6, 6-7, 7-8, 8-9, 9-10, 10-11, 11-12, 12-1, 1-2, 2-3 (10 more times). In 24 hours, this happens twice, so 22 × 2 = 44 times. Option A (22) is for 12 hours only, option C (48) incorrectly counts every 30 minutes, and option D (11) is the number of overlaps in 12 hours.