benchmarks.wiki / Public workspace

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MATH / Tasks

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1–50 / 5000 tasks

algebra / test
Melanie ran at 5 m/s for 400 meters, then ran at 4 m/s for the next 800 meters, and finally ran at 8 m/s for the final 400 meters. What was her average speed, in meters per second,…
Answer published by the source
Identifiers
task_id
0061482d-e313-5559-9311-7ba4c9d08f65
task_key
algebra--test--0061482d-e313-5559-9311-7ba4c9d08f65
algebra / test
How much greater than 27² is 43² ?
Answer published by the source
Identifiers
task_id
006f327a-4390-5e1c-9e95-e81db9bf02fb
task_key
algebra--test--006f327a-4390-5e1c-9e95-e81db9bf02fb
algebra / test
Given the functions f(x)=(x+5)/(3) and g(x)=(1)/(f^(−1)(x)+1), find the value of g(3).
Answer published by the source
Identifiers
task_id
00cc7bda-6478-5134-880f-ee85c86b454d
task_key
algebra--test--00cc7bda-6478-5134-880f-ee85c86b454d
algebra / test
Point A lies somewhere within or on the square which has opposite corners at (0,0) and (2,2). Point B lies somewhere within or on the square which has opposite corners at points …
Answer published by the source
Identifiers
task_id
012d9160-f2ea-53fc-b6a4-e061af915f90
task_key
algebra--test--012d9160-f2ea-53fc-b6a4-e061af915f90
algebra / test
The difference of the roots of the quadratic equation x²+bx+c=0 is |b−2c|. If c≠0, then find c in terms of b.
Answer published by the source
Identifiers
task_id
014cec44-3541-523d-85d5-bfbb7cf7c09a
task_key
algebra--test--014cec44-3541-523d-85d5-bfbb7cf7c09a
algebra / test
Rationalize the denominator of (2)/(root[3](4)+root[3](32)). The answer can be written in the form of (root[3](A))/(B), where A and B are positive integers. Find the minimum possib…
Answer published by the source
Identifiers
task_id
015d5541-a4f1-5fdf-80ad-4660359811b9
task_key
algebra--test--015d5541-a4f1-5fdf-80ad-4660359811b9
algebra / test
How many different values are represented by the following quantities? 3^(−2),9^(−2),27^(−(2)/(3)),9⋅81^(−1),243^(−(4)/(5))
Answer published by the source
Identifiers
task_id
019f16ea-b86f-5c02-afe5-fc8f8c545656
task_key
algebra--test--019f16ea-b86f-5c02-afe5-fc8f8c545656
algebra / test
If displaystyle{f(x)=x^{(x+1)}(x+2)^{(x+3)}}, then find the value of f(0)+f(−1)+f(−2)+f(−3).
Answer published by the source
Identifiers
task_id
01e5ab1a-8e09-51a3-81b9-25268f32ef58
task_key
algebra--test--01e5ab1a-8e09-51a3-81b9-25268f32ef58
algebra / test
What is the 2003rd term of the sequence of odd numbers 1, 3, 5, 7, …?
Answer published by the source
Identifiers
task_id
02087be0-b1d5-5456-93a5-7934f31bca49
task_key
algebra--test--02087be0-b1d5-5456-93a5-7934f31bca49
algebra / test
If x, y, and z are real numbers for which …xyz?
Answer published by the source
Identifiers
task_id
0282fa0e-3598-5eb2-888a-d3d188862f02
task_key
algebra--test--0282fa0e-3598-5eb2-888a-d3d188862f02
algebra / test
Let f(x) = left{ begin{array}{cl} 2x + 1 & text{if } x le 3, 8 - 4x & text{if } x > 3. end{array} right.Find the sum of all values of x such that f(x)=0.
Answer published by the source
Identifiers
task_id
0293d861-f612-570c-b729-202aa5c2aabe
task_key
algebra--test--0293d861-f612-570c-b729-202aa5c2aabe
counting_and_probability / test
There are 360 people in my school. 15 take calculus, physics, and chemistry, and…
Answer published by the source
Identifiers
task_id
003cdc23-49c3-5563-a635-7b21b1b11c3d
task_key
counting~5fand~5fprobability--test--003cdc23-49c3-5563-a635-7b21b1b11c3d
counting_and_probability / test
If we flip 6 coins, what is the probability of getting at least 2 heads?
Answer published by the source
Identifiers
task_id
00b4b747-f110-5e44-a3ed-7983d04849d7
task_key
counting~5fand~5fprobability--test--00b4b747-f110-5e44-a3ed-7983d04849d7
counting_and_probability / test
In the row of Pascal's triangle that starts with 1 and then 10, what is the next…
Answer published by the source
Identifiers
task_id
00b56435-4bcb-5720-bff6-9f435f0c77a3
task_key
counting~5fand~5fprobability--test--00b56435-4bcb-5720-bff6-9f435f0c77a3
counting_and_probability / test
Mikka wants to order a pizza with two different toppings. He has 8 different…
Answer published by the source
Identifiers
task_id
017bfa0a-0fdc-536f-b4af-ce77c7bf4489
task_key
counting~5fand~5fprobability--test--017bfa0a-0fdc-536f-b4af-ce77c7bf4489
counting_and_probability / test
Two jokers are added to a 52 card deck and the entire stack of 54 cards is shuffled randomly. What is the expected number of cards that will be strictly between the two jokers?
Answer published by the source
Identifiers
task_id
019e1bea-7bbc-51e4-abfb-a31aeee5587b
task_key
counting~5fand~5fprobability--test--019e1bea-7bbc-51e4-abfb-a31aeee5587b
counting_and_probability / test
How many different triangles can be formed using three vertices of a hexagon as…
Answer published by the source
Identifiers
task_id
01b80e6d-e352-5f75-bd1d-127f944cb02f
task_key
counting~5fand~5fprobability--test--01b80e6d-e352-5f75-bd1d-127f944cb02f
counting_and_probability / test
Mr. and Mrs. Lopez have three children. When they get into their family car, two…
Answer published by the source
Identifiers
task_id
0211a9a5-e8fd-54e1-9927-393717c0b769
task_key
counting~5fand~5fprobability--test--0211a9a5-e8fd-54e1-9927-393717c0b769
geometry / test
In the diagram below, we have sin∠RPQ=(7)/(25). What is cos∠RPS? [asy] pair R,P,Q,SS; SS = (-2,0); P = (0,0); Q = (2,0); R = rotate(aSin(7/25))*(1.5,0); dot("S",SS,S); dot("Q",Q,S)…
Answer published by the source
Identifiers
task_id
005eb71b-9faa-53f4-ac9a-97ecc6d24abf
task_key
geometry--test--005eb71b-9faa-53f4-ac9a-97ecc6d24abf
geometry / test
In △XYZ, ∠Z=90^(∘), cosY=(7)/(25), and XY=25. What is tanX?
Answer published by the source
Identifiers
task_id
00626890-fd2d-5e73-90dc-75ed3408d093
task_key
geometry--test--00626890-fd2d-5e73-90dc-75ed3408d093
geometry / test
In convex hexagon ABCDEF, all six sides are congruent, ∠A and ∠D are right angles, and ∠B, ∠C, ∠E, and ∠F are congruent. The area of the hexagonal region is 2116(√(2)+1). Find AB.
Answer published by the source
Identifiers
task_id
006a8925-2dc1-5795-82bb-a6bb2a03b77e
task_key
geometry--test--006a8925-2dc1-5795-82bb-a6bb2a03b77e
geometry / test
An equiangular octagon has four sides of length 1 and four sides of length (√(2))/(2), arranged so that no two consecutive sides have the same length. What is the area of the octag…
Answer published by the source
Identifiers
task_id
00d66142-a921-517f-bd02-07f2ce4f441b
task_key
geometry--test--00d66142-a921-517f-bd02-07f2ce4f441b
geometry / test
A water-filled spherical balloon lands on a sidewalk, momentarily flattening to…
Answer published by the source
Identifiers
task_id
016dccae-7668-510d-abe2-0b99d8fc495f
task_key
geometry--test--016dccae-7668-510d-abe2-0b99d8fc495f
geometry / test
A company makes a six-sided hollow aluminum container in the shape of a…
Answer published by the source
Identifiers
task_id
028207f6-21ba-570a-92c0-a754b3d5e193
task_key
geometry--test--028207f6-21ba-570a-92c0-a754b3d5e193
intermediate_algebra / test
Let P be a point on the hyperbola x²+8xy+7y²=225. Find the shortest possible distance from the origin to P.
Answer published by the source
Identifiers
task_id
001f87ab-8774-5a62-a181-15032ed4d64a
task_key
intermediate~5falgebra--test--001f87ab-8774-5a62-a181-15032ed4d64a
intermediate_algebra / test
Let z and w be complex numbers such that |z|=2 and |w|=5. Find the largest possible value of |z+w|.
Answer published by the source
Identifiers
task_id
00735d98-0350-58a0-9af0-6c0f84a787e7
task_key
intermediate~5falgebra--test--00735d98-0350-58a0-9af0-6c0f84a787e7
intermediate_algebra / test
Find the infinite sum of
Answer published by the source
Identifiers
task_id
008c82f6-f4a7-5117-8fe8-78a0848b0970
task_key
intermediate~5falgebra--test--008c82f6-f4a7-5117-8fe8-78a0848b0970
intermediate_algebra / test
Let p(x) be a polynomial with positive leading coefficient such that
Answer published by the source
Identifiers
task_id
009e0c3f-2eed-55a0-8c04-c97c015c1c94
task_key
intermediate~5falgebra--test--009e0c3f-2eed-55a0-8c04-c97c015c1c94
intermediate_algebra / test
Let a and b be real numbers such that a>2b>0. Find the minimum value of 3a³√(3)+(2)/(ab−2b²).
Answer published by the source
Identifiers
task_id
00e134c2-0298-5ba4-93f4-8f6a27edb684
task_key
intermediate~5falgebra--test--00e134c2-0298-5ba4-93f4-8f6a27edb684
intermediate_algebra / test
Find the sum of the real roots of x⁴−80x−36=0.
Answer published by the source
Identifiers
task_id
00fa9b9b-8ced-5f38-b4f5-acc07179992a
task_key
intermediate~5falgebra--test--00fa9b9b-8ced-5f38-b4f5-acc07179992a
intermediate_algebra / test
Find the minimum value of the function
Answer published by the source
Identifiers
task_id
015b67f0-d503-538b-aa13-478e5b78fe69
task_key
intermediate~5falgebra--test--015b67f0-d503-538b-aa13-478e5b78fe69
intermediate_algebra / test
The range of the function f(x) is [−3,5]. Let g(x)=[f(x)]². Find the range of the function g(x).
Answer published by the source
Identifiers
task_id
01869d83-7d73-5c83-8779-a78ce376903e
task_key
intermediate~5falgebra--test--01869d83-7d73-5c83-8779-a78ce376903e
intermediate_algebra / test
Compute
Answer published by the source
Identifiers
task_id
0199f2d8-c181-5c09-9534-9861c04060f3
task_key
intermediate~5falgebra--test--0199f2d8-c181-5c09-9534-9861c04060f3
intermediate_algebra / test
At a competition with N players, the number of players given elite status is equal to 2^(1+⌊log₂(N−1)⌋)−N. Suppose that 19 players are given elite status. What is the sum of the tw…
Answer published by the source
Identifiers
task_id
01d1ae60-864d-552f-a68c-509c8abed372
task_key
intermediate~5falgebra--test--01d1ae60-864d-552f-a68c-509c8abed372
intermediate_algebra / test
If log₃6≈1.63, what is the value of log₉4, accurate to two decimal places?
Answer published by the source
Identifiers
task_id
0228b086-c0a6-5531-830d-15ea81032f96
task_key
intermediate~5falgebra--test--0228b086-c0a6-5531-830d-15ea81032f96
intermediate_algebra / test
Determine the complex number z satisfying the equation 3z+4i(z)¯=1−8i. Note that (z)¯ denotes the conjugate of z.
Answer published by the source
Identifiers
task_id
0228d2e3-d5a6-5ee2-a5df-d602aacf2af7
task_key
intermediate~5falgebra--test--0228d2e3-d5a6-5ee2-a5df-d602aacf2af7
intermediate_algebra / test
For how many positive integers m, 1≤m≤1000, does there exist at least one positive integer n such that m⋅n≤m+n?
Answer published by the source
Identifiers
task_id
0287e16a-fef3-5826-831f-c95209b868b5
task_key
intermediate~5falgebra--test--0287e16a-fef3-5826-831f-c95209b868b5
intermediate_algebra / test
Find the real roots of
Answer published by the source
Identifiers
task_id
028d43d0-0d56-542f-a6c2-70d07c696c4e
task_key
intermediate~5falgebra--test--028d43d0-0d56-542f-a6c2-70d07c696c4e
number_theory / test
What is the residue modulo 13 of the sum of the modulo 13 inverses of the first 12 positive integers? Express your answer as an integer from 0 to 12, inclusive.
Answer published by the source
Identifiers
task_id
017cceec-ebfd-5171-a9d2-b7bcc9128491
task_key
number~5ftheory--test--017cceec-ebfd-5171-a9d2-b7bcc9128491
number_theory / test
What integer n satisfies 0≤n<101 and 123456equiv npmod {101}~?
Answer published by the source
Identifiers
task_id
019ea047-2923-5a97-8121-8f5a6ad2a008
task_key
number~5ftheory--test--019ea047-2923-5a97-8121-8f5a6ad2a008
number_theory / test
What is the value of a+b+c+d+e+f for the decimal representation of (4)/(37)+(3)/(11)+(23)/(9)=2.abcdef…?
Answer published by the source
Identifiers
task_id
024d5d34-149e-51d2-9d02-a959b1fa3709
task_key
number~5ftheory--test--024d5d34-149e-51d2-9d02-a959b1fa3709
prealgebra / test
The five-digit number N=14,9AB is divisible by 12. Given that A and B are nonzero digits, what is the smallest possible value of N?
Answer published by the source
Identifiers
task_id
005b719c-965c-50a1-bc9a-fe769806ecec
task_key
prealgebra--test--005b719c-965c-50a1-bc9a-fe769806ecec
prealgebra / test
The table shows some of the results of a survey by radio station KAMC. What…
Answer published by the source
Identifiers
task_id
00c4edbb-645b-5395-aafa-209280f3b539
task_key
prealgebra--test--00c4edbb-645b-5395-aafa-209280f3b539
prealgebra / test
Simplify √(32670).
Answer published by the source
Identifiers
task_id
00de2422-5b05-5291-a75b-597169712bc2
task_key
prealgebra--test--00de2422-5b05-5291-a75b-597169712bc2
prealgebra / test
What is the greatest common divisor of 1313 and 1001?
Answer published by the source
Identifiers
task_id
01203d37-c306-5560-a11c-99fc3d9ac0fd
task_key
prealgebra--test--01203d37-c306-5560-a11c-99fc3d9ac0fd
prealgebra / test
How many positive integers are less than 200π?
Answer published by the source
Identifiers
task_id
02a6db23-c6bf-58d6-a628-ed95f9be6498
task_key
prealgebra--test--02a6db23-c6bf-58d6-a628-ed95f9be6498
precalculus / test
If |z|=5, and |w|=2, what is the minimum value of |z−w|?
Answer published by the source
Identifiers
task_id
0037973a-be03-5ed9-8490-df18d9dacab7
task_key
precalculus--test--0037973a-be03-5ed9-8490-df18d9dacab7
precalculus / test
Find all values of k, for which there exists a nonzero vector v such that begin{pmatrix} 2 & -2 & 1 2 & -3 & 2 -1 & 2 & 0 end{pmatrix} mathbf{v} = k mathbf{v}.
Answer published by the source
Identifiers
task_id
02065204-8ee9-52f1-baa5-6918230451d2
task_key
precalculus--test--02065204-8ee9-52f1-baa5-6918230451d2
precalculus / test
Find the matrix that corresponds to rotating about the origin by an angle of…
Answer published by the source
Identifiers
task_id
02247ee2-1a67-5eb6-a573-00778d9d9338
task_key
precalculus--test--02247ee2-1a67-5eb6-a573-00778d9d9338
precalculus / test
A line is parameterized by
Answer published by the source
Identifiers
task_id
02a10f9b-31fa-5b0a-851f-ffaab22b9bbb
task_key
precalculus--test--02a10f9b-31fa-5b0a-851f-ffaab22b9bbb