benchmarks.wiki / Public workspace

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Omni-MATH / Tasks

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1–50 / 4406 tasks

Default config / test
Five identical empty buckets of 2-liter capacity stand at the vertices of a regular pentagon. Cinderella and her wicked Stepmother go through a sequence of rounds: At the beginning…
Answer published by the source
Identifiers
task_id
0000c3a0-8cb6-5a1d-82ca-a92804a8ed84
task_key
test--0000c3a0-8cb6-5a1d-82ca-a92804a8ed84
Default config / test
Lucas writes two distinct positive integers on a whiteboard. He decreases the…
Answer published by the source
Identifiers
task_id
000415ac-f144-5bd1-89e3-77cdb121266b
task_key
test--000415ac-f144-5bd1-89e3-77cdb121266b
Default config / test
A small village has n people. During their yearly elections, groups of three people come up to a stage and vote for someone in the village to be the new leader. After every possibl…
Answer published by the source
Identifiers
task_id
000bfd57-4d8d-5317-897c-5a586b3d08c0
task_key
test--000bfd57-4d8d-5317-897c-5a586b3d08c0
Default config / test
How many different graphs with 9 vertices exist where each vertex is connected…
Answer published by the source
Identifiers
task_id
00130d0e-0eee-58b0-ac46-075e87d25afa
task_key
test--00130d0e-0eee-58b0-ac46-075e87d25afa
Default config / test
Find the largest possible integer k, such that the following statement is true: Let 2009 arbitrary non-degenerated triangles be given. In every triangle the three sides are coloure…
Answer published by the source
Identifiers
task_id
00214950-956a-5d6d-ac65-e34f72a9572b
task_key
test--00214950-956a-5d6d-ac65-e34f72a9572b
Default config / test
Let ABCD be a tetrahedron such that its circumscribed sphere of radius R and its inscribed sphere of radius r are concentric. Given that AB=AC=1≤BC and R=4r, find BC².
Answer published by the source
Identifiers
task_id
00348d2c-0c54-5b19-9da9-25753f2ebaba
task_key
test--00348d2c-0c54-5b19-9da9-25753f2ebaba
Default config / test
How many six-digit multiples of 27 have only 3, 6, or 9 as their digits?
Answer published by the source
Identifiers
task_id
00384fdf-d354-5f45-af7a-8c1cefcc7425
task_key
test--00384fdf-d354-5f45-af7a-8c1cefcc7425
Default config / test
Find the maximum number of points X_(i) such that for each i, triangle A B X_{i} cong triangle C D X_{i}.
Answer published by the source
Identifiers
task_id
003a9615-038e-54d8-9847-20b75bc2d58a
task_key
test--003a9615-038e-54d8-9847-20b75bc2d58a
Default config / test
Compute the number of integers n∈{1,2,…,300} such that n is the product of two distinct primes, and is also the length of the longest leg of some nondegenerate right triangle with …
Answer published by the source
Identifiers
task_id
003e8a93-3901-551a-ba47-415a4fadb150
task_key
test--003e8a93-3901-551a-ba47-415a4fadb150
Default config / test
This question is unrelated to the graph shown in part a; instead, we consider a…
Answer published by the source
Identifiers
task_id
00402537-3170-5d15-b937-9fbbc1cc50b5
task_key
test--00402537-3170-5d15-b937-9fbbc1cc50b5
Default config / test
Fix a triangle ABC. We say that triangle XYZ is elegant if X lies on segment BC, Y lies on segment CA, Z lies on segment AB, and XYZ is similar to ABC (i.e., ∠A=∠X,∠B=∠Y,∠C=∠Z). Of…
Answer published by the source
Identifiers
task_id
005a532e-a316-557e-b440-c3d48aa7047e
task_key
test--005a532e-a316-557e-b440-c3d48aa7047e
Default config / test
What is 25% of 60?
Answer published by the source
Identifiers
task_id
006bd7b0-2905-53c3-9b3c-d3f20ee6e669
task_key
test--006bd7b0-2905-53c3-9b3c-d3f20ee6e669
Default config / test
Two unit squares S₁ and S₂ have horizontal and vertical sides. Let x be the minimum distance between a point in S₁ and a point in S₂, and let y be the maximum distance between a po…
Answer published by the source
Identifiers
task_id
00a63d8a-0368-5836-9647-12615900cadf
task_key
test--00a63d8a-0368-5836-9647-12615900cadf
Default config / test
Compute the number of ordered pairs of integers (x,y) such that x²+y²<2019 and x²+min(x,y)=y²+max(x,y)
Answer published by the source
Identifiers
task_id
00bc590f-0ba1-57c1-bb95-0232215955dc
task_key
test--00bc590f-0ba1-57c1-bb95-0232215955dc
Default config / test
Determine whether √(1001²+1)+√(1002²+1)+⋯+√(2000²+1) be a rational number or not?
Answer published by the source
Identifiers
task_id
00c7dc53-c6cb-5788-b79a-b8595fc26f68
task_key
test--00c7dc53-c6cb-5788-b79a-b8595fc26f68
Default config / test
What is the value of the expression 4+(3)/(10)+(9)/(1000)?
Answer published by the source
Identifiers
task_id
00c85df1-849c-59cb-a55c-b82b7950d93f
task_key
test--00c85df1-849c-59cb-a55c-b82b7950d93f
Default config / test
How many points does a sports team earn for 9 wins, 3 losses, and 4 ties, if…
Answer published by the source
Identifiers
task_id
00caefd9-ecf6-539f-8fe4-dc492181afaf
task_key
test--00caefd9-ecf6-539f-8fe4-dc492181afaf
Default config / test
If wxyz is a four-digit positive integer with w≠0, the layer sum of this integer equals wxyz+xyz+yz+z. If the layer sum of wxyz equals 2014, what is the value of w+x+y+z?
Answer published by the source
Identifiers
task_id
00d7d22e-238b-56c1-b493-6f52968874d2
task_key
test--00d7d22e-238b-56c1-b493-6f52968874d2
Default config / test
Find all integers n satisfying n≥2 and (σ(n))/(p(n)−1)=n, in which σ(n) denotes the sum of all positive divisors of n, and p(n) denotes the largest prime divisor of n.
Answer published by the source
Identifiers
task_id
00ed18ea-1f21-53cc-a643-225ff77a3451
task_key
test--00ed18ea-1f21-53cc-a643-225ff77a3451
Default config / test
If x=2, what is the value of 4x²−3x²?
Answer published by the source
Identifiers
task_id
00f6a2d4-113e-5632-882d-06f605ff20b5
task_key
test--00f6a2d4-113e-5632-882d-06f605ff20b5
Default config / test
If 2×2×3×3×5×6=5×6×n×n, what is a possible value of n?
Answer published by the source
Identifiers
task_id
00f71f2b-d785-52d1-a185-b198d0f70320
task_key
test--00f71f2b-d785-52d1-a185-b198d0f70320
Default config / test
Determine all pairs (x,y) of positive integers such that x²y+x+y is divisible by xy²+y+7.
Answer published by the source
Identifiers
task_id
010ad197-cc5a-5089-b411-4a2f05693c77
task_key
test--010ad197-cc5a-5089-b411-4a2f05693c77
Default config / test
While Travis is having fun on cubes, Sherry is hopping in the same manner on an…
Answer published by the source
Identifiers
task_id
010b940f-b02e-559b-ab39-23cd410d07fb
task_key
test--010b940f-b02e-559b-ab39-23cd410d07fb
Default config / test
We colour all the sides and diagonals of a regular polygon P with 43 vertices either red or blue in such a way that every vertex is an endpoint of 20 red segments and 22 blue segme…
Answer published by the source
Identifiers
task_id
011a9010-710d-527e-baab-25e9c13b7799
task_key
test--011a9010-710d-527e-baab-25e9c13b7799
Default config / test
Let ABC be a fixed acute triangle inscribed in a circle ω with center O . A variable point X is chosen on minor arc AB of ω , and segments CX and AB meet at D . Denote by O₁ and O₂…
Answer published by the source
Identifiers
task_id
0134743c-9694-50ae-a9e3-5bc775910948
task_key
test--0134743c-9694-50ae-a9e3-5bc775910948
Default config / test
Storage space on a computer is measured in gigabytes (GB) and megabytes (MB),…
Answer published by the source
Identifiers
task_id
013ac524-5bcf-57da-9df3-0b1a5bbb6617
task_key
test--013ac524-5bcf-57da-9df3-0b1a5bbb6617
Default config / test
Evaluate the sum …lnx denotes the natural logarithm of x.)
Answer published by the source
Identifiers
task_id
01506210-337d-5ff1-a7b2-c4b0d03a3264
task_key
test--01506210-337d-5ff1-a7b2-c4b0d03a3264
Default config / test
There is a city with n citizens. The city wants to buy [i]sceptervirus[/i] tests with which it is possible to analyze the samples of several people at the same time. The result of …
Answer published by the source
Identifiers
task_id
0167ceb8-5cc3-5b56-b0d7-20865e262b99
task_key
test--0167ceb8-5cc3-5b56-b0d7-20865e262b99
Default config / test
The country Dreamland consists of 2016 cities. The airline Starways wants to…
Answer published by the source
Identifiers
task_id
016a2013-be1c-5fff-b6df-c06cb455d127
task_key
test--016a2013-be1c-5fff-b6df-c06cb455d127
Default config / test
Kelvin the Frog has a pair of standard fair 8-sided dice (each labelled from 1…
Answer published by the source
Identifiers
task_id
016ee311-57b0-5d18-ba8c-3e1df4e25e70
task_key
test--016ee311-57b0-5d18-ba8c-3e1df4e25e70
Default config / test
Evaluate ∑_(k=1)^(∞)((−1)^(k−1))/(k)∑_(n=0)^(∞)(1)/(k2^(n)+1).
Answer published by the source
Identifiers
task_id
019bcd8c-3a41-553e-b718-999f3696db73
task_key
test--019bcd8c-3a41-553e-b718-999f3696db73
Default config / test
The real numbers x,y and z satisfy the three equations x+y=7, xz=−180, and (x+y+z)²=4. If S is the sum of the two possible values of y, what is −S?
Answer published by the source
Identifiers
task_id
01aa389e-d6ae-5513-ad27-3ea46ec1ee89
task_key
test--01aa389e-d6ae-5513-ad27-3ea46ec1ee89
Default config / test
Mark writes the expression √(d) for each positive divisor d of 8 ! on the board. Seeing that these expressions might not be worth points on HMMT, Rishabh simplifies each expression…
Answer published by the source
Identifiers
task_id
01b0c3cb-1483-5297-a6ab-7ef00a1ccff4
task_key
test--01b0c3cb-1483-5297-a6ab-7ef00a1ccff4
Default config / test
Let ABCD be an isosceles trapezoid with AB=1,BC=DA=5,CD=7. Let P be the intersection of diagonals AC and BD, and let Q be the foot of the altitude from D to BC. Let PQ intersect AB…
Answer published by the source
Identifiers
task_id
01b3edf0-2876-5a06-aaac-a067fd6db28b
task_key
test--01b3edf0-2876-5a06-aaac-a067fd6db28b
Default config / test
Determine the value of ∑_(n=1)^(∞)ln(1+(1)/(n))⋅ln(1+(1)/(2n))⋅ln(1+(1)/(2n+1)).
Answer published by the source
Identifiers
task_id
01d637ce-4125-5f97-92d2-bac25a85bd8e
task_key
test--01d637ce-4125-5f97-92d2-bac25a85bd8e
Default config / test
Alice starts with the number 0. She can apply 100 operations on her number. In…
Answer published by the source
Identifiers
task_id
01ddc601-eb6b-5cd9-b22b-449c6ae18c16
task_key
test--01ddc601-eb6b-5cd9-b22b-449c6ae18c16
Default config / test
Find the number of integers n such that 1+leftlfloorfrac{100 n}{101}rightrfloor=leftlceilfrac{99 n}{100}rightrceil
Answer published by the source
Identifiers
task_id
01dee348-c6e5-594e-93f7-a4f7fea28771
task_key
test--01dee348-c6e5-594e-93f7-a4f7fea28771
Default config / test
Let n≥2 be a natural. Define
Answer published by the source
Identifiers
task_id
020494e4-81cf-5d35-b218-c120f87007b2
task_key
test--020494e4-81cf-5d35-b218-c120f87007b2
Default config / test
What is the maximum number of colours that can be used to paint an 8×8 chessboard so that every square is painted in a single colour, and is adjacent , horizontally, vertically but…
Answer published by the source
Identifiers
task_id
022e10ee-94ab-5b2a-b49b-4ef5025283c3
task_key
test--022e10ee-94ab-5b2a-b49b-4ef5025283c3
Default config / test
Two different integers u and v are written on a board. We perform a sequence of steps. At each step we do one of the following two operations: (i) If a and b are different integers…
Answer published by the source
Identifiers
task_id
02331d68-582e-5c47-947e-89ec72cc2c9b
task_key
test--02331d68-582e-5c47-947e-89ec72cc2c9b
Default config / test
Minimize the function F(x₁,x₂,⋯,x_(n))=∑_(i=1)^(n)|x_(i)| subject to the condition that after redistribution we should have at each A_(i),a_(i)−x₁+x_(i−1)=N for i∈{1,2,…,n} where …
Answer published by the source
Identifiers
task_id
0233537f-ac0d-571b-ac52-d3523adf8c2b
task_key
test--0233537f-ac0d-571b-ac52-d3523adf8c2b
Default config / test
Consider an isosceles triangle T with base 10 and height 12. Define a sequence ω₁,ω₂,… of circles such that ω₁ is the incircle of T and ω_(i+1) is tangent to ω_(i) and both legs of…
Answer published by the source
Identifiers
task_id
02353cd4-8059-525e-abd6-877bf1ecad61
task_key
test--02353cd4-8059-525e-abd6-877bf1ecad61
Default config / test
Let P and Q be points on line l with PQ=12. Two circles, ω and Ω, are both tangent to l at P and are externally tangent to each other. A line through Q intersects ω at A and B, wit…
Answer published by the source
Identifiers
task_id
023661c3-c71e-5e38-a72e-4609f5b8a375
task_key
test--023661c3-c71e-5e38-a72e-4609f5b8a375
Default config / test
Let n>1 be a positive integer. Each unit square in an n×n grid of squares is colored either black or white, such that the following conditions hold: - Any two black squares can be …
Answer published by the source
Identifiers
task_id
023666b5-2b19-54df-bda8-8f880b114cfc
task_key
test--023666b5-2b19-54df-bda8-8f880b114cfc
Default config / test
How many positive integers at most 420 leave different remainders when divided…
Answer published by the source
Identifiers
task_id
02406c8f-a9e3-561f-b83d-162703497245
task_key
test--02406c8f-a9e3-561f-b83d-162703497245
Default config / test
Find all the triples of positive integers (a,b,c) for which the number
Answer published by the source
Identifiers
task_id
0242cc66-7e11-565e-a1f0-0da2e4fe00c3
task_key
test--0242cc66-7e11-565e-a1f0-0da2e4fe00c3
Default config / test
Fifty numbers have an average of 76. Forty of these numbers have an average of…
Answer published by the source
Identifiers
task_id
0247b344-2817-5eb4-87d9-49801a278baf
task_key
test--0247b344-2817-5eb4-87d9-49801a278baf
Default config / test
Abbot writes the letter A on the board. Every minute, he replaces every occurrence of A with AB and every occurrence of B with BA, hence creating a string that is twice as long. Af…
Answer published by the source
Identifiers
task_id
0252ed2c-7539-5c66-b4ee-ff43ee1f27a8
task_key
test--0252ed2c-7539-5c66-b4ee-ff43ee1f27a8
Default config / test
For each positive integer n , find the number of n -digit positive integers that satisfy both of the following conditions: • no two consecutive digits are equal, and • the last dig…
Answer published by the source
Identifiers
task_id
02604070-c7ed-5efd-8443-84accec17451
task_key
test--02604070-c7ed-5efd-8443-84accec17451
Default config / test
In 12 years, Janice will be 8 times as old as she was 2 years ago. How old is…
Answer published by the source
Identifiers
task_id
02691c3a-6dad-5973-803b-9a1f0266dc33
task_key
test--02691c3a-6dad-5973-803b-9a1f0266dc33