problem stringlengths 21 332 | thinking_trace stringlengths 44 456 | answer stringlengths 1 147 |
|---|---|---|
Is 'aeeeecceeeea' a palindrome? | 'aeeeecceeeea'
Compare first and last, then move inward.
Positions: a=a, e=e, e=e, e=e, e=e, c=c
All match -> palindrome. | Yes |
A car starts at 11 m/s and accelerates at 4 m/s^2 for 8s. Find distance and final velocity. | Using kinematics equations:
v_f = v_0 + a*t = 11 + 4*8 = 43 m/s
d = v_0*t + 0.5*a*t^2 = 11*8 + 0.5*4*8^2
= 88 + 0.5*4*64 = 216.0 m
Check: v_avg = (11+43)/2 = 27.0 m/s, d = v_avg*t = 216.0 m [OK] | Distance = 216.0m, Final velocity = 43 m/s |
How many moles are in 52g of H? (Atomic mass: 1.008 g/mol) | Moles = mass / atomic mass
Moles = 52g / 1.008 g/mol = 51.587302 mol
Rounded: 51.5873 mol | 51.5873 mol |
Is 164 a prime number? | Check 164 for primality.
Test divisors up to sqrt(164) ~= 12
Found: 164 mod 2 = 0 -> divisible by 2
Therefore 164 is not prime. Factors: 2 and 82. | No (divisible by 2) |
Reverse the string 'dahd' manually. | String: 'dahd'
Iterate from end to start:
index 3: 'd'
index 2: 'h'
index 1: 'a'
index 0: 'd'
Result: 'dhad' | dhad |
You have a 4-liter jug and a 6-liter jug. How do you measure exactly 1 liters? | Using two jugs: 4L and 6L, target=1L.
Approach: repeatedly fill smaller jug, pour into larger.
When larger is full, empty it and continue.
Amounts in smaller jug cycle through: 4, 2, 0, 4, 2, 0
The target 1 appears in this cycle.
Solution: repeatedly fill 4L, pour into 6L, empty 6L when full. | Fill 4, pour into 6. Repeat until 6 full, empty it, continue. Eventually 1 remains. |
Binary search for 58 in [1, 3, 11, 18, 58, 61, 61, 79, 90, 91, 98]. How many comparisons? | Array len=11, target=58
Binary search halves the search space each iteration.
log2(11) ~= 3.5, so about 4 comparisons.
It took 4 comparisons to find 58. | 4 comparisons |
Plan execution order with dependencies: ['run tests', 'schedule meeting', 'update database', 'review document']. All tasks take 1 unit. 'deploy patch' depends on 'run tests'. 'prepare slides' depends on 'review document'. 'schedule meeting' depends on 'write email'. Find minimum completion time. | Tasks: run tests, schedule meeting, update database, review document
Building dependency graph...
Level 0 (no deps): tasks with no prerequisites
Level 1: tasks whose deps are in level 0
Level 2: remaining
Minimum completion time = longest chain = 3 units | Critical path length: 3 units |
Binary search for 54 in [7, 9, 15, 16, 31, 31, 39, 54, 59, 80, 82]. How many comparisons? | Array len=11, target=54
Binary search halves the search space each iteration.
log2(11) ~= 3.5, so about 4 comparisons.
It took 4 comparisons to find 54. | 4 comparisons |
DNA strand: CCCCTATCACCTGCG. Give the complementary strand. | DNA base-pairing rules: A-T, C-G
Original: CCCCTATCACCTGCG
C -> G
C -> G
C -> G
C -> G
T -> A
A -> T
T -> A
C -> G
A -> T
C -> G
C -> G
T -> A
G -> C
C -> G
G -> C
Complement: GGGGATAGTGGACGC | GGGGATAGTGGACGC |
Plan execution order with dependencies: ['update database', 'write email', 'review document', 'run tests', 'deploy patch', 'schedule meeting']. All tasks take 1 unit. 'deploy patch' depends on 'run tests'. 'prepare slides' depends on 'review document'. 'schedule meeting' depends on 'write email'. Find minimum completio... | Tasks: update database, write email, review document, run tests, deploy patch, schedule meeting
Building dependency graph...
Level 0 (no deps): tasks with no prerequisites
Level 1: tasks whose deps are in level 0
Level 2: remaining
Minimum completion time = longest chain = 3 units | Critical path length: 3 units |
Binary search for 94 in [3, 9, 14, 31, 33, 45, 46, 81, 85, 94, 96, 99]. How many comparisons? | Array len=12, target=94
Binary search halves the search space each iteration.
log2(12) ~= 3.6, so about 4 comparisons.
It took 4 comparisons to find 94. | 4 comparisons |
Compute 10! recursively. | Recursive definition: n! = n * (n-1)!, 0! = 1
10! = 10 * 362880
9! = 9 * 40320
8! = 8 * 5040
7! = 7 * 720
6! = 6 * 120
5! = 5 * 24
4! = 4 * 6
3! = 3 * 2
2! = 2 * 1
1! = 1
Therefore 10! = 3628800 | 3628800 |
Given nums = [12, 18, 11, 6], target = 12. Find two indices that sum to target. | Use hash map for O(n) solution.
Target = 12
i=0, v=12, need=0 -> add to map
i=1, v=18, need=-6 -> add to map
i=2, v=11, need=1 -> add to map
i=3, v=6, need=6 -> add to map
Result: No solution | No solution |
You push a heavy box across a rough floor. What happens? Explain. | Friction opposes motion. The rough surface creates more friction, requiring sustained force to overcome it. | It moves slowly and requires continuous effort. |
Plan execution order with dependencies: ['run tests', 'schedule meeting', 'write email', 'review document']. All tasks take 1 unit. 'deploy patch' depends on 'run tests'. 'prepare slides' depends on 'review document'. 'schedule meeting' depends on 'write email'. Find minimum completion time. | Tasks: run tests, schedule meeting, write email, review document
Building dependency graph...
Level 0 (no deps): tasks with no prerequisites
Level 1: tasks whose deps are in level 0
Level 2: remaining
Minimum completion time = longest chain = 3 units | Critical path length: 3 units |
Solve the quadratic equation: 1x^2 + 10x + 9 = 0 | Solve 1x^2 + 10x + 9 = 0.
Using the quadratic formula: x = [-b +/- sqrt(b^2 - 4ac)] / (2a)
b^2 - 4ac = (10)^2 - 4(1)(9) = 100 - 36 = 64
sqrt(64) = 8
x = [-10 +/- 8] / 2
x1 = [-10 + 8] / 2 = -1 = -9
x2 = [-10 - 8] / 2 = -9 = -1
Verification: plugging -9: 1*-9^2 + 10*-9 + 9 = 0 [OK]
Plugging -1: 1*-1^2 + 10*-1 + 9 = 0 [O... | x = -9, x = -1 |
You have a 3-liter jug and a 5-liter jug. How do you measure exactly 1 liters? | Using two jugs: 3L and 5L, target=1L.
Approach: repeatedly fill smaller jug, pour into larger.
When larger is full, empty it and continue.
Amounts in smaller jug cycle through: 3, 1, 4, 2, 0
The target 1 appears in this cycle.
Solution: repeatedly fill 3L, pour into 5L, empty 5L when full. | Fill 3, pour into 5. Repeat until 5 full, empty it, continue. Eventually 1 remains. |
A projectile launched at 29 m/s at 60 deg. Find range, max height, time of flight. (g=9.8) | v=29 m/s, theta=60 deg, g=9.8 m/s^2
v_x = v*cos(60) = 29*0.500 = 14.5 m/s
v_y = v*sin(60) = 29*0.866 = 25.1 m/s
Time of flight: T = 2*v_y/g = 2*25.1/9.8 = 5.1s
Range: R = v_x*T = 14.5*5.1 = 74.3m
Max height: H = v_y^2/(2g) = (25.1)^2/(2*9.8) = 32.2m | Range=74.3m, Max height=32.2m, Time=5.1s |
A projectile launched at 35 m/s at 60 deg. Find range, max height, time of flight. (g=9.8) | v=35 m/s, theta=60 deg, g=9.8 m/s^2
v_x = v*cos(60) = 35*0.500 = 17.5 m/s
v_y = v*sin(60) = 35*0.866 = 30.3 m/s
Time of flight: T = 2*v_y/g = 2*30.3/9.8 = 6.2s
Range: R = v_x*T = 17.5*6.2 = 108.3m
Max height: H = v_y^2/(2g) = (30.3)^2/(2*9.8) = 46.9m | Range=108.3m, Max height=46.9m, Time=6.2s |
Plan execution order with dependencies: ['write email', 'deploy patch', 'schedule meeting', 'prepare slides', 'update database']. All tasks take 1 unit. 'deploy patch' depends on 'run tests'. 'prepare slides' depends on 'review document'. 'schedule meeting' depends on 'write email'. Find minimum completion time. | Tasks: write email, deploy patch, schedule meeting, prepare slides, update database
Building dependency graph...
Level 0 (no deps): tasks with no prerequisites
Level 1: tasks whose deps are in level 0
Level 2: remaining
Minimum completion time = longest chain = 3 units | Critical path length: 3 units |
Compute 2^15 mod 17 using modular exponentiation. | Using binary exponentiation.
15 in binary = 1111
bit 0 (2^0): multiply by 2 -> 2
bit 1 (2^1): multiply by 4 -> 8
bit 2 (2^2): multiply by 16 -> 9
bit 3 (2^3): multiply by 1 -> 9
Result: 2^15 mod 17 = 9 | 9 |
Write pseudocode for FizzBuzz up to 49. | Loop from 1 to 49:
- If i % 15 == 0: print 'FizzBuzz'
- Else if i % 3 == 0: print 'Fizz'
- Else if i % 5 == 0: print 'Buzz'
- Else: print i
Order matters: check 15 first (which is 3*5) to avoid double-matching. | Print 1..49, replacing multiples of 3 with 'Fizz', 5 with 'Buzz', both with 'FizzBuzz' |
Compute P(5, 3) (permutations of 5 items taken 3 at a time). | P(5, 3) = 5! / (5 - 3)!
P(5, 3) = 5! / 2!
= 5 * 4 * 3
= 60 | 60 |
You have a 6-liter jug and a 10-liter jug. How do you measure exactly 6 liters? | Using two jugs: 6L and 10L, target=6L.
Approach: repeatedly fill smaller jug, pour into larger.
When larger is full, empty it and continue.
Amounts in smaller jug cycle through: 6, 2, 8, 4, 0, 6, 2, 8, 4, 0
The target 6 appears in this cycle.
Solution: repeatedly fill 6L, pour into 10L, empty 10L when full. | Fill 6, pour into 10. Repeat until 10 full, empty it, continue. Eventually 6 remains. |
Given nums = [13, 11, 6, 16, 16, 12, 17], target = 23. Find two indices that sum to target. | Use hash map for O(n) solution.
Target = 23
i=0, v=13, need=10 -> add to map
i=1, v=11, need=12 -> add to map
i=2, v=6, need=17 -> add to map
i=3, v=16, need=7 -> add to map
i=4, v=16, need=7 -> add to map
i=5, v=12, need=11 -> found at index 1
i=6, v=17, need=6 -> found at index 2
Result: [1, 5] | [1, 5] |
You have a 6-liter jug and a 7-liter jug. How do you measure exactly 4 liters? | Using two jugs: 6L and 7L, target=4L.
Approach: repeatedly fill smaller jug, pour into larger.
When larger is full, empty it and continue.
Amounts in smaller jug cycle through: 6, 5, 4, 3, 2, 1, 0
The target 4 appears in this cycle.
Solution: repeatedly fill 6L, pour into 7L, empty 7L when full. | Fill 6, pour into 7. Repeat until 7 full, empty it, continue. Eventually 4 remains. |
Is 329 a prime number? | Check 329 for primality.
Test divisors up to sqrt(329) ~= 18
Found: 329 mod 7 = 0 -> divisible by 7
Therefore 329 is not prime. Factors: 7 and 47. | No (divisible by 7) |
Given nums = [11, 4, 3, 11, 10, 10], target = 21. Find two indices that sum to target. | Use hash map for O(n) solution.
Target = 21
i=0, v=11, need=10 -> add to map
i=1, v=4, need=17 -> add to map
i=2, v=3, need=18 -> add to map
i=3, v=11, need=10 -> add to map
i=4, v=10, need=11 -> found at index 3
i=5, v=10, need=11 -> found at index 3
Result: [3, 4] | [3, 4] |
A projectile launched at 20 m/s at 60 deg. Find range, max height, time of flight. (g=9.8) | v=20 m/s, theta=60 deg, g=9.8 m/s^2
v_x = v*cos(60) = 20*0.500 = 10.0 m/s
v_y = v*sin(60) = 20*0.866 = 17.3 m/s
Time of flight: T = 2*v_y/g = 2*17.3/9.8 = 3.5s
Range: R = v_x*T = 10.0*3.5 = 35.3m
Max height: H = v_y^2/(2g) = (17.3)^2/(2*9.8) = 15.3m | Range=35.3m, Max height=15.3m, Time=3.5s |
How many moles are in 49g of N? (Atomic mass: 14.01 g/mol) | Moles = mass / atomic mass
Moles = 49g / 14.01 g/mol = 3.497502 mol
Rounded: 3.4975 mol | 3.4975 mol |
A projectile launched at 38 m/s at 30 deg. Find range, max height, time of flight. (g=9.8) | v=38 m/s, theta=30 deg, g=9.8 m/s^2
v_x = v*cos(30) = 38*0.866 = 32.9 m/s
v_y = v*sin(30) = 38*0.500 = 19.0 m/s
Time of flight: T = 2*v_y/g = 2*19.0/9.8 = 3.9s
Range: R = v_x*T = 32.9*3.9 = 127.6m
Max height: H = v_y^2/(2g) = (19.0)^2/(2*9.8) = 18.4m | Range=127.6m, Max height=18.4m, Time=3.9s |
A car starts at 11 m/s and accelerates at 10 m/s^2 for 8s. Find distance and final velocity. | Using kinematics equations:
v_f = v_0 + a*t = 11 + 10*8 = 91 m/s
d = v_0*t + 0.5*a*t^2 = 11*8 + 0.5*10*8^2
= 88 + 0.5*10*64 = 408.0 m
Check: v_avg = (11+91)/2 = 51.0 m/s, d = v_avg*t = 408.0 m [OK] | Distance = 408.0m, Final velocity = 91 m/s |
Plan execution order with dependencies: ['deploy patch', 'write email', 'run tests', 'review document', 'schedule meeting']. All tasks take 1 unit. 'deploy patch' depends on 'run tests'. 'prepare slides' depends on 'review document'. 'schedule meeting' depends on 'write email'. Find minimum completion time. | Tasks: deploy patch, write email, run tests, review document, schedule meeting
Building dependency graph...
Level 0 (no deps): tasks with no prerequisites
Level 1: tasks whose deps are in level 0
Level 2: remaining
Minimum completion time = longest chain = 3 units | Critical path length: 3 units |
Plan execution order with dependencies: ['run tests', 'deploy patch', 'schedule meeting', 'write email', 'review document', 'update database']. All tasks take 1 unit. 'deploy patch' depends on 'run tests'. 'prepare slides' depends on 'review document'. 'schedule meeting' depends on 'write email'. Find minimum completio... | Tasks: run tests, deploy patch, schedule meeting, write email, review document, update database
Building dependency graph...
Level 0 (no deps): tasks with no prerequisites
Level 1: tasks whose deps are in level 0
Level 2: remaining
Minimum completion time = longest chain = 3 units | Critical path length: 3 units |
Solve the quadratic equation: 1x^2 + 15x + 54 = 0 | Solve 1x^2 + 15x + 54 = 0.
Using the quadratic formula: x = [-b +/- sqrt(b^2 - 4ac)] / (2a)
b^2 - 4ac = (15)^2 - 4(1)(54) = 225 - 216 = 9
sqrt(9) = 3
x = [-15 +/- 3] / 2
x1 = [-15 + 3] / 2 = -6 = -9
x2 = [-15 - 3] / 2 = -9 = -6
Verification: plugging -9: 1*-9^2 + 15*-9 + 54 = 0 [OK]
Plugging -6: 1*-6^2 + 15*-6 + 54 = 0... | x = -9, x = -6 |
Data points (x,y): [(1, 36), (2, 40), (3, 44), (4, 43), (5, 50), (6, 50), (7, 55)]. Estimate the linear trend (slope). | n = 7 points
mean_x = 4.00, mean_y = 45.43
Using least squares: slope = sum((x-mx)(y-my)) / sum((x-mx)^2)
Numerator = 83.00, Denominator = 28.00
Estimated slope = 2.9643 ~= 2.96
Interpretation: y increases by ~3.0 per unit x. | Slope ~= 2.96 |
Compute 7^10 mod 21 using modular exponentiation. | Using binary exponentiation.
10 in binary = 1010
bit 1 (2^1): multiply by 7 -> 7
bit 3 (2^3): multiply by 7 -> 7
Result: 7^10 mod 21 = 7 | 7 |
You have a 3-liter jug and a 5-liter jug. How do you measure exactly 3 liters? | Using two jugs: 3L and 5L, target=3L.
Approach: repeatedly fill smaller jug, pour into larger.
When larger is full, empty it and continue.
Amounts in smaller jug cycle through: 3, 1, 4, 2, 0
The target 3 appears in this cycle.
Solution: repeatedly fill 3L, pour into 5L, empty 5L when full. | Fill 3, pour into 5. Repeat until 5 full, empty it, continue. Eventually 3 remains. |
Data points (x,y): [(1, 33), (2, 39), (3, 46), (4, 54), (5, 63), (6, 71), (7, 77)]. Estimate the linear trend (slope). | n = 7 points
mean_x = 4.00, mean_y = 54.71
Using least squares: slope = sum((x-mx)(y-my)) / sum((x-mx)^2)
Numerator = 213.00, Denominator = 28.00
Estimated slope = 7.6071 ~= 7.61
Interpretation: y increases by ~7.6 per unit x. | Slope ~= 7.61 |
A projectile launched at 45 m/s at 30 deg. Find range, max height, time of flight. (g=9.8) | v=45 m/s, theta=30 deg, g=9.8 m/s^2
v_x = v*cos(30) = 45*0.866 = 39.0 m/s
v_y = v*sin(30) = 45*0.500 = 22.5 m/s
Time of flight: T = 2*v_y/g = 2*22.5/9.8 = 4.6s
Range: R = v_x*T = 39.0*4.6 = 178.9m
Max height: H = v_y^2/(2g) = (22.5)^2/(2*9.8) = 25.8m | Range=178.9m, Max height=25.8m, Time=4.6s |
Find GCD(45, 45) using Euclidean algorithm. | Euclidean algorithm: gcd(a,b) = gcd(b, a mod b)
gcd(45, 45)
Final: gcd = 45
Check: 45/45 = 1, 45/45 = 1 -> coprime. | 45 |
Is 'bddebaabeddb' a palindrome? | 'bddebaabeddb'
Compare first and last, then move inward.
Positions: b=b, d=d, d=d, e=e, b=b, a=a
All match -> palindrome. | Yes |
Find the mean and median of: [92, 57, 12, 88, 41, 86] | Dataset: [92, 57, 12, 88, 41, 86]
Sum = 376, Count = 6
Mean = 376 / 6 = 62.6667 ~= 62.67
Sorted: [12, 41, 57, 86, 88, 92]
Median: n=6, even, position 3 -> 71.5 | Mean = 62.67, Median = 71.5 |
Solve the quadratic equation: 3x^2 + 3x + -168 = 0 | Solve 3x^2 + 3x + -168 = 0.
Using the quadratic formula: x = [-b +/- sqrt(b^2 - 4ac)] / (2a)
b^2 - 4ac = (3)^2 - 4(3)(-168) = 9 - -2016 = 2025
sqrt(2025) = 45
x = [-3 +/- 45] / 6
x1 = [-3 + 45] / 6 = 7 = -8
x2 = [-3 - 45] / 6 = -8 = 7
Verification: plugging -8: 3*-8^2 + 3*-8 + -168 = 0 [OK]
Plugging 7: 3*7^2 + 3*7 + -1... | x = -8, x = 7 |
Reverse the string 'ihjghf' manually. | String: 'ihjghf'
Iterate from end to start:
index 5: 'f'
index 4: 'h'
index 3: 'g'
index 2: 'j'
index 1: 'h'
index 0: 'i'
Result: 'fhgjhi' | fhgjhi |
Data points (x,y): [(1, 23), (2, 25), (3, 29), (4, 28), (5, 32), (6, 36), (7, 38)]. Estimate the linear trend (slope). | n = 7 points
mean_x = 4.00, mean_y = 30.14
Using least squares: slope = sum((x-mx)(y-my)) / sum((x-mx)^2)
Numerator = 70.00, Denominator = 28.00
Estimated slope = 2.5000 ~= 2.50
Interpretation: y increases by ~2.5 per unit x. | Slope ~= 2.50 |
Solve the quadratic equation: 3x^2 + -15x + 12 = 0 | Solve 3x^2 + -15x + 12 = 0.
Using the quadratic formula: x = [-b +/- sqrt(b^2 - 4ac)] / (2a)
b^2 - 4ac = (-15)^2 - 4(3)(12) = 225 - 144 = 81
sqrt(81) = 9
x = [15 +/- 9] / 6
x1 = [15 + 9] / 6 = 4 = 1
x2 = [15 - 9] / 6 = 1 = 4
Verification: plugging 1: 3*1^2 + -15*1 + 12 = 0 [OK]
Plugging 4: 3*4^2 + -15*4 + 12 = 0 [OK] | x = 1, x = 4 |
Solve for x: 3x + 20 = 65 | I need to solve 3x + 20 = 65 for x.
First, subtract 20 from both sides: 3x = 65 - 20 = 45
Now divide both sides by 3: x = 45 / 3 = 15
Check: 3*15 + 20 = 45 + 20 = 65. Correct. | 15 |
Plan execution order with dependencies: ['write email', 'prepare slides', 'review document', 'update database']. All tasks take 1 unit. 'deploy patch' depends on 'run tests'. 'prepare slides' depends on 'review document'. 'schedule meeting' depends on 'write email'. Find minimum completion time. | Tasks: write email, prepare slides, review document, update database
Building dependency graph...
Level 0 (no deps): tasks with no prerequisites
Level 1: tasks whose deps are in level 0
Level 2: remaining
Minimum completion time = longest chain = 3 units | Critical path length: 3 units |
Reverse the string 'hiajgeg' manually. | String: 'hiajgeg'
Iterate from end to start:
index 6: 'g'
index 5: 'e'
index 4: 'g'
index 3: 'j'
index 2: 'a'
index 1: 'i'
index 0: 'h'
Result: 'gegjaih' | gegjaih |
How many moles are in 75g of O? (Atomic mass: 16.0 g/mol) | Moles = mass / atomic mass
Moles = 75g / 16.0 g/mol = 4.687500 mol
Rounded: 4.6875 mol | 4.6875 mol |
Given nums = [6, 5, 14, 17, 2, 4, 17, 5], target = 16. Find two indices that sum to target. | Use hash map for O(n) solution.
Target = 16
i=0, v=6, need=10 -> add to map
i=1, v=5, need=11 -> add to map
i=2, v=14, need=2 -> add to map
i=3, v=17, need=-1 -> add to map
i=4, v=2, need=14 -> found at index 2
i=5, v=4, need=12 -> add to map
i=6, v=17, need=-1 -> add to map
i=7, v=5, need=11 -> add to ... | [2, 4] |
DNA strand: TCTGGGTC. Give the complementary strand. | DNA base-pairing rules: A-T, C-G
Original: TCTGGGTC
T -> A
C -> G
T -> A
G -> C
G -> C
G -> C
T -> A
C -> G
Complement: AGACCCAG | AGACCCAG |
Write pseudocode for FizzBuzz up to 85. | Loop from 1 to 85:
- If i % 15 == 0: print 'FizzBuzz'
- Else if i % 3 == 0: print 'Fizz'
- Else if i % 5 == 0: print 'Buzz'
- Else: print i
Order matters: check 15 first (which is 3*5) to avoid double-matching. | Print 1..85, replacing multiples of 3 with 'Fizz', 5 with 'Buzz', both with 'FizzBuzz' |
DNA strand: ATCCAGACCC. Give the complementary strand. | DNA base-pairing rules: A-T, C-G
Original: ATCCAGACCC
A -> T
T -> A
C -> G
C -> G
A -> T
G -> C
A -> T
C -> G
C -> G
C -> G
Complement: TAGGTCTGGG | TAGGTCTGGG |
A projectile launched at 46 m/s at 30 deg. Find range, max height, time of flight. (g=9.8) | v=46 m/s, theta=30 deg, g=9.8 m/s^2
v_x = v*cos(30) = 46*0.866 = 39.8 m/s
v_y = v*sin(30) = 46*0.500 = 23.0 m/s
Time of flight: T = 2*v_y/g = 2*23.0/9.8 = 4.7s
Range: R = v_x*T = 39.8*4.7 = 187.0m
Max height: H = v_y^2/(2g) = (23.0)^2/(2*9.8) = 27.0m | Range=187.0m, Max height=27.0m, Time=4.7s |
Solve for x: 1x + 3 = 4 | I need to solve 1x + 3 = 4 for x.
First, subtract 3 from both sides: 1x = 4 - 3 = 1
Now divide both sides by 1: x = 1 / 1 = 1
Check: 1*1 + 3 = 1 + 3 = 4. Correct. | 1 |
Solve the quadratic equation: 2x^2 + -36x + 160 = 0 | Solve 2x^2 + -36x + 160 = 0.
Using the quadratic formula: x = [-b +/- sqrt(b^2 - 4ac)] / (2a)
b^2 - 4ac = (-36)^2 - 4(2)(160) = 1296 - 1280 = 16
sqrt(16) = 4
x = [36 +/- 4] / 4
x1 = [36 + 4] / 4 = 10 = 10
x2 = [36 - 4] / 4 = 8 = 8
Verification: plugging 10: 2*10^2 + -36*10 + 160 = 0 [OK]
Plugging 8: 2*8^2 + -36*8 + 160... | x = 10, x = 8 |
Reverse the string 'cfgaifgdc' manually. | String: 'cfgaifgdc'
Iterate from end to start:
index 8: 'c'
index 7: 'd'
index 6: 'g'
index 5: 'f'
index 4: 'i'
index 3: 'a'
index 2: 'g'
index 1: 'f'
index 0: 'c'
Result: 'cdgfiagfc' | cdgfiagfc |
Find the 7th term of the arithmetic sequence: 1, 3, 5, 7, ... | The sequence starts at 1 with common difference 2.
nth term formula: a_n = a_1 + (n-1)d
a_7 = 1 + (7 - 1)*2
a_7 = 1 + 6*2 = 1 + 12 = 13
Verification: sequence is [1, 3, 5, 7, 9, 11, 13] | 13 |
Data points (x,y): [(1, 42), (2, 47), (3, 50), (4, 56), (5, 63), (6, 65), (7, 71)]. Estimate the linear trend (slope). | n = 7 points
mean_x = 4.00, mean_y = 56.29
Using least squares: slope = sum((x-mx)(y-my)) / sum((x-mx)^2)
Numerator = 136.00, Denominator = 28.00
Estimated slope = 4.8571 ~= 4.86
Interpretation: y increases by ~4.9 per unit x. | Slope ~= 4.86 |
How many moles are in 45g of Cl? (Atomic mass: 35.45 g/mol) | Moles = mass / atomic mass
Moles = 45g / 35.45 g/mol = 1.269394 mol
Rounded: 1.2694 mol | 1.2694 mol |
Given nums = [13, 5, 12, 17, 18, 4], target = 17. Find two indices that sum to target. | Use hash map for O(n) solution.
Target = 17
i=0, v=13, need=4 -> add to map
i=1, v=5, need=12 -> add to map
i=2, v=12, need=5 -> found at index 1
i=3, v=17, need=0 -> add to map
i=4, v=18, need=-1 -> add to map
i=5, v=4, need=13 -> found at index 0
Result: [1, 2] | [1, 2] |
Given nums = [4, 9, 15, 8, 5, 4, 2], target = 23. Find two indices that sum to target. | Use hash map for O(n) solution.
Target = 23
i=0, v=4, need=19 -> add to map
i=1, v=9, need=14 -> add to map
i=2, v=15, need=8 -> add to map
i=3, v=8, need=15 -> found at index 2
i=4, v=5, need=18 -> add to map
i=5, v=4, need=19 -> add to map
i=6, v=2, need=21 -> add to map
Result: [2, 3] | [2, 3] |
Solve for x: 10x + 14 = 54 | I need to solve 10x + 14 = 54 for x.
First, subtract 14 from both sides: 10x = 54 - 14 = 40
Now divide both sides by 10: x = 40 / 10 = 4
Check: 10*4 + 14 = 40 + 14 = 54. Correct. | 4 |
Plan execution order with dependencies: ['run tests', 'prepare slides', 'update database', 'deploy patch']. All tasks take 1 unit. 'deploy patch' depends on 'run tests'. 'prepare slides' depends on 'review document'. 'schedule meeting' depends on 'write email'. Find minimum completion time. | Tasks: run tests, prepare slides, update database, deploy patch
Building dependency graph...
Level 0 (no deps): tasks with no prerequisites
Level 1: tasks whose deps are in level 0
Level 2: remaining
Minimum completion time = longest chain = 3 units | Critical path length: 3 units |
39-year-old parent has a 17-year-old child. In how many years will the parent be exactly twice as old as the child? | Let x = number of years. Then parent age = 39+x, child age = 17+x.
Equation: 39+x = 2(17+x)
39+x = 34+2x
39-34 = x
5 = x
Check: in 5 years, parent=44, child=22. Ratio=2.0 ~= 2 [OK] | 5 |
DNA strand: GTGTATTACATCG. Give the complementary strand. | DNA base-pairing rules: A-T, C-G
Original: GTGTATTACATCG
G -> C
T -> A
G -> C
T -> A
A -> T
T -> A
T -> A
A -> T
C -> G
A -> T
T -> A
C -> G
G -> C
Complement: CACATAATGTAGC | CACATAATGTAGC |
Binary search for 2 in [2, 6, 14, 18, 34, 37, 47, 47, 52, 52, 56, 69, 94, 98]. How many comparisons? | Array len=14, target=2
Binary search halves the search space each iteration.
log2(14) ~= 3.8, so about 3 comparisons.
It took 3 comparisons to find 2. | 3 comparisons |
Find the mean and median of: [72, 25, 47, 71, 37, 10, 50, 65, 58] | Dataset: [72, 25, 47, 71, 37, 10, 50, 65, 58]
Sum = 435, Count = 9
Mean = 435 / 9 = 48.3333 ~= 48.33
Sorted: [10, 25, 37, 47, 50, 58, 65, 71, 72]
Median: n=9, odd, position 5 -> 50 | Mean = 48.33, Median = 50 |
Given nums = [20, 20, 4, 5, 4, 13], target = 8. Find two indices that sum to target. | Use hash map for O(n) solution.
Target = 8
i=0, v=20, need=-12 -> add to map
i=1, v=20, need=-12 -> add to map
i=2, v=4, need=4 -> add to map
i=3, v=5, need=3 -> add to map
i=4, v=4, need=4 -> found at index 2
i=5, v=13, need=-5 -> add to map
Result: [2, 4] | [2, 4] |
Data points (x,y): [(1, 38), (2, 44), (3, 53), (4, 59), (5, 64), (6, 71), (7, 73)]. Estimate the linear trend (slope). | n = 7 points
mean_x = 4.00, mean_y = 57.43
Using least squares: slope = sum((x-mx)(y-my)) / sum((x-mx)^2)
Numerator = 170.00, Denominator = 28.00
Estimated slope = 6.0714 ~= 6.07
Interpretation: y increases by ~6.1 per unit x. | Slope ~= 6.07 |
Solve for x: 3x + 11 = 50 | I need to solve 3x + 11 = 50 for x.
First, subtract 11 from both sides: 3x = 50 - 11 = 39
Now divide both sides by 3: x = 39 / 3 = 13
Check: 3*13 + 11 = 39 + 11 = 50. Correct. | 13 |
Find GCD(72, 78) using Euclidean algorithm. | Euclidean algorithm: gcd(a,b) = gcd(b, a mod b)
gcd(72, 78)
gcd(78, 72)
gcd(72, 6)
Final: gcd = 6
Check: 72/6 = 12, 78/6 = 13 -> coprime. | 6 |
|A| = 10, |B| = 5, |A intersect B| = 2. Find |A union B|. | By the principle of inclusion-exclusion:
|A union B| = |A| + |B| - |A intersect B|
|A union B| = 10 + 5 - 2 = 13 | 13 |
40-year-old parent has a 14-year-old child. In how many years will the parent be exactly twice as old as the child? | Let x = number of years. Then parent age = 40+x, child age = 14+x.
Equation: 40+x = 2(14+x)
40+x = 28+2x
40-28 = x
12 = x
Check: in 12 years, parent=52, child=26. Ratio=2.0 ~= 2 [OK] | 12 |
Solve the quadratic equation: 2x^2 + 8x + -10 = 0 | Solve 2x^2 + 8x + -10 = 0.
Using the quadratic formula: x = [-b +/- sqrt(b^2 - 4ac)] / (2a)
b^2 - 4ac = (8)^2 - 4(2)(-10) = 64 - -80 = 144
sqrt(144) = 12
x = [-8 +/- 12] / 4
x1 = [-8 + 12] / 4 = 1 = 1
x2 = [-8 - 12] / 4 = -5 = -5
Verification: plugging 1: 2*1^2 + 8*1 + -10 = 0 [OK]
Plugging -5: 2*-5^2 + 8*-5 + -10 = 0 ... | x = 1, x = -5 |
Solve the quadratic equation: 3x^2 + -24x + -27 = 0 | Solve 3x^2 + -24x + -27 = 0.
Using the quadratic formula: x = [-b +/- sqrt(b^2 - 4ac)] / (2a)
b^2 - 4ac = (-24)^2 - 4(3)(-27) = 576 - -324 = 900
sqrt(900) = 30
x = [24 +/- 30] / 6
x1 = [24 + 30] / 6 = 9 = 9
x2 = [24 - 30] / 6 = -1 = -1
Verification: plugging 9: 3*9^2 + -24*9 + -27 = 0 [OK]
Plugging -1: 3*-1^2 + -24*-1 ... | x = 9, x = -1 |
Find GCD(55, 76) using Euclidean algorithm. | Euclidean algorithm: gcd(a,b) = gcd(b, a mod b)
gcd(55, 76)
gcd(76, 55)
gcd(55, 21)
gcd(21, 13)
gcd(13, 8)
gcd(8, 5)
gcd(5, 3)
gcd(3, 2)
gcd(2, 1)
Final: gcd = 1
Check: 55/1 = 55, 76/1 = 76 -> coprime. | 1 |
Data points (x,y): [(1, 52), (2, 59), (3, 68), (4, 74), (5, 83), (6, 88), (7, 98)]. Estimate the linear trend (slope). | n = 7 points
mean_x = 4.00, mean_y = 74.57
Using least squares: slope = sum((x-mx)(y-my)) / sum((x-mx)^2)
Numerator = 211.00, Denominator = 28.00
Estimated slope = 7.5357 ~= 7.54
Interpretation: y increases by ~7.5 per unit x. | Slope ~= 7.54 |
A projectile launched at 17 m/s at 45 deg. Find range, max height, time of flight. (g=9.8) | v=17 m/s, theta=45 deg, g=9.8 m/s^2
v_x = v*cos(45) = 17*0.707 = 12.0 m/s
v_y = v*sin(45) = 17*0.707 = 12.0 m/s
Time of flight: T = 2*v_y/g = 2*12.0/9.8 = 2.5s
Range: R = v_x*T = 12.0*2.5 = 29.5m
Max height: H = v_y^2/(2g) = (12.0)^2/(2*9.8) = 7.4m | Range=29.5m, Max height=7.4m, Time=2.5s |
Write pseudocode for FizzBuzz up to 89. | Loop from 1 to 89:
- If i % 15 == 0: print 'FizzBuzz'
- Else if i % 3 == 0: print 'Fizz'
- Else if i % 5 == 0: print 'Buzz'
- Else: print i
Order matters: check 15 first (which is 3*5) to avoid double-matching. | Print 1..89, replacing multiples of 3 with 'Fizz', 5 with 'Buzz', both with 'FizzBuzz' |
Compute 5! recursively. | Recursive definition: n! = n * (n-1)!, 0! = 1
5! = 5 * 24
4! = 4 * 6
3! = 3 * 2
2! = 2 * 1
1! = 1
Therefore 5! = 120 | 120 |
Solve the quadratic equation: 2x^2 + -28x + 90 = 0 | Solve 2x^2 + -28x + 90 = 0.
Using the quadratic formula: x = [-b +/- sqrt(b^2 - 4ac)] / (2a)
b^2 - 4ac = (-28)^2 - 4(2)(90) = 784 - 720 = 64
sqrt(64) = 8
x = [28 +/- 8] / 4
x1 = [28 + 8] / 4 = 9 = 9
x2 = [28 - 8] / 4 = 5 = 5
Verification: plugging 9: 2*9^2 + -28*9 + 90 = 0 [OK]
Plugging 5: 2*5^2 + -28*5 + 90 = 0 [OK] | x = 9, x = 5 |
Data points (x,y): [(1, 56), (2, 60), (3, 66), (4, 77), (5, 83), (6, 88), (7, 98)]. Estimate the linear trend (slope). | n = 7 points
mean_x = 4.00, mean_y = 75.43
Using least squares: slope = sum((x-mx)(y-my)) / sum((x-mx)^2)
Numerator = 199.00, Denominator = 28.00
Estimated slope = 7.1071 ~= 7.11
Interpretation: y increases by ~7.1 per unit x. | Slope ~= 7.11 |
Solve the quadratic equation: 2x^2 + -38x + 180 = 0 | Solve 2x^2 + -38x + 180 = 0.
Using the quadratic formula: x = [-b +/- sqrt(b^2 - 4ac)] / (2a)
b^2 - 4ac = (-38)^2 - 4(2)(180) = 1444 - 1440 = 4
sqrt(4) = 2
x = [38 +/- 2] / 4
x1 = [38 + 2] / 4 = 10 = 10
x2 = [38 - 2] / 4 = 9 = 9
Verification: plugging 10: 2*10^2 + -38*10 + 180 = 0 [OK]
Plugging 9: 2*9^2 + -38*9 + 180 =... | x = 10, x = 9 |
Solve for x: 3x + 8 = 11 | I need to solve 3x + 8 = 11 for x.
First, subtract 8 from both sides: 3x = 11 - 8 = 3
Now divide both sides by 3: x = 3 / 3 = 1
Check: 3*1 + 8 = 3 + 8 = 11. Correct. | 1 |
Compute 3^8 mod 7 using modular exponentiation. | Using binary exponentiation.
8 in binary = 1000
bit 3 (2^3): multiply by 2 -> 2
Result: 3^8 mod 7 = 2 | 2 |
DNA strand: TACTGGGCAAATC. Give the complementary strand. | DNA base-pairing rules: A-T, C-G
Original: TACTGGGCAAATC
T -> A
A -> T
C -> G
T -> A
G -> C
G -> C
G -> C
C -> G
A -> T
A -> T
A -> T
T -> A
C -> G
Complement: ATGACCCGTTTAG | ATGACCCGTTTAG |
Data points (x,y): [(1, 40), (2, 45), (3, 53), (4, 60), (5, 64), (6, 72), (7, 76)]. Estimate the linear trend (slope). | n = 7 points
mean_x = 4.00, mean_y = 58.57
Using least squares: slope = sum((x-mx)(y-my)) / sum((x-mx)^2)
Numerator = 173.00, Denominator = 28.00
Estimated slope = 6.1786 ~= 6.18
Interpretation: y increases by ~6.2 per unit x. | Slope ~= 6.18 |
How many moles are in 97g of Cl? (Atomic mass: 35.45 g/mol) | Moles = mass / atomic mass
Moles = 97g / 35.45 g/mol = 2.736248 mol
Rounded: 2.7362 mol | 2.7362 mol |
Solve for x: 5x + 20 = 60 | I need to solve 5x + 20 = 60 for x.
First, subtract 20 from both sides: 5x = 60 - 20 = 40
Now divide both sides by 5: x = 40 / 5 = 8
Check: 5*8 + 20 = 40 + 20 = 60. Correct. | 8 |
Solve for x: 4x + 8 = 28 | I need to solve 4x + 8 = 28 for x.
First, subtract 8 from both sides: 4x = 28 - 8 = 20
Now divide both sides by 4: x = 20 / 4 = 5
Check: 4*5 + 8 = 20 + 8 = 28. Correct. | 5 |
DNA strand: GCCTTCGCTC. Give the complementary strand. | DNA base-pairing rules: A-T, C-G
Original: GCCTTCGCTC
G -> C
C -> G
C -> G
T -> A
T -> A
C -> G
G -> C
C -> G
T -> A
C -> G
Complement: CGGAAGCGAG | CGGAAGCGAG |
54-year-old parent has a 11-year-old child. In how many years will the parent be exactly twice as old as the child? | Let x = number of years. Then parent age = 54+x, child age = 11+x.
Equation: 54+x = 2(11+x)
54+x = 22+2x
54-22 = x
32 = x
Check: in 32 years, parent=86, child=43. Ratio=2.0 ~= 2 [OK] | 32 |
A circle has radius 9. What is its area? (Use pi = 3.14159) | Area of a circle = pi * r^2
r = 9, r^2 = 81
Area = 3.14159 * 81 = 254.46879
Rounded to 2 decimal places: 254.47 | 254.47 |
42-year-old parent has a 17-year-old child. In how many years will the parent be exactly twice as old as the child? | Let x = number of years. Then parent age = 42+x, child age = 17+x.
Equation: 42+x = 2(17+x)
42+x = 34+2x
42-34 = x
8 = x
Check: in 8 years, parent=50, child=25. Ratio=2.0 ~= 2 [OK] | 8 |
Given nums = [10, 2, 10, 3, 12], target = 13. Find two indices that sum to target. | Use hash map for O(n) solution.
Target = 13
i=0, v=10, need=3 -> add to map
i=1, v=2, need=11 -> add to map
i=2, v=10, need=3 -> add to map
i=3, v=3, need=10 -> found at index 2
i=4, v=12, need=1 -> add to map
Result: [2, 3] | [2, 3] |
Reverse the string 'ecfjgbgfc' manually. | String: 'ecfjgbgfc'
Iterate from end to start:
index 8: 'c'
index 7: 'f'
index 6: 'g'
index 5: 'b'
index 4: 'g'
index 3: 'j'
index 2: 'f'
index 1: 'c'
index 0: 'e'
Result: 'cfgbgjfce' | cfgbgjfce |
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