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
Benchmark AI / Public workspace
Omni-MATH / Tasks
Choose a problem to read its original content, examples and discussion.
Option counts cover the whole public benchmark.
1–50 / 200 tasks
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
Let ∠XOY=(π)/(2); P is a point inside ∠XOY and we have OP=1;∠XOP=(π)/(6). A line passes P intersects the Rays OX and OY at M and N. Find the maximum value of OM+ON−MN.
Answer published by the source
Identifiers
- task_id
057c537c-be0e-5414-a65d-d7191e9e7e49- task_key
test--057c537c-be0e-5414-a65d-d7191e9e7e49
Default config / test
Given positive integers k,m,n such that 1≤k≤m≤n. Evaluate
Answer published by the source
Identifiers
- task_id
05d97329-05a5-5f80-aa88-4d3535e1ffac- task_key
test--05d97329-05a5-5f80-aa88-4d3535e1ffac
Default config / test
Find all positive integers a such that there exists a set X of 6 integers satisfying the following conditions: for every k=1,2,…,36 there exist x,y∈X such that ax+y−k is divisible …
Answer published by the source
Identifiers
- task_id
062dbf81-70a5-5fa8-b8b7-555fbdef3d41- task_key
test--062dbf81-70a5-5fa8-b8b7-555fbdef3d41
Default config / test
Find the smallest positive real constant a, such that for any three points A,B,C on the unit circle, there exists an equilateral triangle PQR with side length a such that all of …
Answer published by the source
Identifiers
- task_id
08a7a350-fb9e-5931-9719-b8ea47675c34- task_key
test--08a7a350-fb9e-5931-9719-b8ea47675c34
Default config / test
Let G be a simple graph with 100 vertices such that for each vertice u, there exists a vertice v∈N(u) and N left ( u right ) cap N left ( v right ) = o . Try to find the maximal …
Answer published by the source
Identifiers
- task_id
09f6759e-1e5e-5ae1-ab65-6014b7ab7f4c- task_key
test--09f6759e-1e5e-5ae1-ab65-6014b7ab7f4c
Default config / test
Let a,b,c,d,e≥−1 and a+b+c+d+e=5. Find the maximum and minimum value of S=(a+b)(b+c)(c+d)(d+e)(e+a).
Answer published by the source
Identifiers
- task_id
0a2f6002-3178-51e8-bc7d-93939135feae- task_key
test--0a2f6002-3178-51e8-bc7d-93939135feae
Default config / test
Find all primes p such that there exist positive integers x,y that satisfy x(y²−p)+y(x²−p)=5p .
Answer published by the source
Identifiers
- task_id
0c0e622a-38fa-507b-9c89-0c19e8cf5fe2- task_key
test--0c0e622a-38fa-507b-9c89-0c19e8cf5fe2
Default config / test
Find all positive integers x,y satisfying the equation 9(x²+y²+1)+2(3xy+2)=2005.
Answer published by the source
Identifiers
- task_id
0d9d1a0a-9d99-5fa2-b6b4-c38134700681- task_key
test--0d9d1a0a-9d99-5fa2-b6b4-c38134700681
Default config / test
On a given circle, six points A , B , C , D , E , and F are chosen at random, independently and uniformly with respect to arc length. Determine the probability that the two triangl…
Answer published by the source
Identifiers
- task_id
0f6ca6c4-9f0e-5798-8e25-25883c570134- task_key
test--0f6ca6c4-9f0e-5798-8e25-25883c570134
Default config / test
Given positive integers n,k such that n≥4k, find the minimal value λ=λ(n,k) such that for any positive reals a₁,a₂,…,a_(n), we have …
Answer published by the source
Identifiers
- task_id
10300b69-0435-59bf-ad83-4384eb3164c2- task_key
test--10300b69-0435-59bf-ad83-4384eb3164c2
Default config / test
Let R denote a non-negative rational number. Determine a fixed set of integers a,b,c,d,e,f , such that for every choice of R , |(aR²+bR+c)/(dR²+eR+f)−root[3](2)|<|R−root[3](2)|
Answer published by the source
Identifiers
- task_id
129db43e-aec1-54bc-ac15-1733f9461a63- task_key
test--129db43e-aec1-54bc-ac15-1733f9461a63
Default config / test
Consider the assertion that for each positive integer n≥2 , the remainder upon dividing 2^(2^(n)) by 2^(n)−1 is a power of 4. Either prove the assertion or find (with proof) a coun…
Answer published by the source
Identifiers
- task_id
12e4d184-cf57-53d2-bbce-5c4149c86352- task_key
test--12e4d184-cf57-53d2-bbce-5c4149c86352
Default config / test
Let f:N→N be a function satisfying the following conditions: (1) f(1)=1; (2) ∀n∈N, 3f(n)f(2n+1)=f(2n)(1+3f(n)); (3) ∀n∈N, f(2n)<6f(n). Find all solutions of equation f(k)+f(l)=293,…
Answer published by the source
Identifiers
- task_id
13d65ac7-0bf0-571a-b0a6-b73b2e6f1df0- task_key
test--13d65ac7-0bf0-571a-b0a6-b73b2e6f1df0
Default config / test
Find all ordered triples of primes (p,q,r) such that p∣q^(r)+1, q∣r^(p)+1, r∣p^(q)+1. [i]Reid Barton[/i]
Answer published by the source
Identifiers
- task_id
15e65c69-7ccb-57a2-9b9d-e8363d6d69c5- task_key
test--15e65c69-7ccb-57a2-9b9d-e8363d6d69c5
Default config / test
Determine all the roots , real or complex , of the system of simultaneous…
Answer published by the source
Identifiers
- task_id
16815bdc-8e98-5ee6-890d-07ea44b93aea- task_key
test--16815bdc-8e98-5ee6-890d-07ea44b93aea
Default config / test
Determine whether or not there exist positive integers a and b such that a does not divide b^n minus{} n for all positive integers n.
Answer published by the source
Identifiers
- task_id
19d1ef5b-081f-5651-b106-aabf05dac19b- task_key
test--19d1ef5b-081f-5651-b106-aabf05dac19b
Default config / test
Attempt of a halfways nice solution.
Answer published by the source
Identifiers
- task_id
1a9c7eab-4eff-5847-8421-40bd5ba0367e- task_key
test--1a9c7eab-4eff-5847-8421-40bd5ba0367e
Default config / test
101 people, sitting at a round table in any order, had 1,2,...,101 cards, respectively. A transfer is someone give one card to one of the two people adjacent to him. Find the small…
Answer published by the source
Identifiers
- task_id
1d3f97a3-fb71-58d0-a690-9d7b6a44bc90- task_key
test--1d3f97a3-fb71-58d0-a690-9d7b6a44bc90
Default config / test
Fix positive integers k,n. A candy vending machine has many different colours of candy, where there are 2n candies of each colour. A couple of kids each buys from the vending machi…
Answer published by the source
Identifiers
- task_id
1e1ea259-e2f3-5dee-a8a0-9747193bc2de- task_key
test--1e1ea259-e2f3-5dee-a8a0-9747193bc2de
Default config / test
Determine the triangle with sides a,b,c and circumradius R for which R(b+c)=a√(bc) .
Answer published by the source
Identifiers
- task_id
1e22b975-eb90-5d0a-8d9a-5b578f7b3536- task_key
test--1e22b975-eb90-5d0a-8d9a-5b578f7b3536
Default config / test
A number n is [i]interesting[/i] if 2018 divides d(n) (the number of positive divisors of n). Determine all positive integers k such that there exists an infinite arithmetic progre…
Answer published by the source
Identifiers
- task_id
1e794897-9b4c-55cf-ab7a-59cebcd47e83- task_key
test--1e794897-9b4c-55cf-ab7a-59cebcd47e83
Default config / test
There are arbitrary 7 points in the plane. Circles are drawn through every 4…
Answer published by the source
Identifiers
- task_id
1ea20bad-1e08-544d-aa1f-55d6a0754c1f- task_key
test--1ea20bad-1e08-544d-aa1f-55d6a0754c1f
Default config / test
Find all triples (x,y,z) of real numbers that satisfy the system of equations
Answer published by the source
Identifiers
- task_id
1f0c4419-a430-5937-9733-fb0d39d51475- task_key
test--1f0c4419-a430-5937-9733-fb0d39d51475
Default config / test
Given integer n≥2. Find the minimum value of λ, satisfy that for any real numbers a₁, a₂, ⋯, a_(n) and b, …
Answer published by the source
Identifiers
- task_id
20654564-5f56-5fe3-a6c3-e90e08235770- task_key
test--20654564-5f56-5fe3-a6c3-e90e08235770
Default config / test
Consider pairs (f,g) of functions from the set of nonnegative integers to itself such that [list] [*]f(0)≥f(1)≥f(2)≥…≥f(300)≥0 [*]f(0)+f(1)+f(2)+…+f(300)≤300 [*]for any 20 nonnegat…
Answer published by the source
Identifiers
- task_id
225b144d-1d53-54ef-98f0-550be7fc611f- task_key
test--225b144d-1d53-54ef-98f0-550be7fc611f
Default config / test
For each positive integer n, let c(n) be the largest real number such that
Answer published by the source
Identifiers
- task_id
24a735d9-5a21-56cf-a9f3-2f15a7fb03ae- task_key
test--24a735d9-5a21-56cf-a9f3-2f15a7fb03ae
Default config / test
Find all functions f:(0,∞)→(0,∞) such that
Answer published by the source
Identifiers
- task_id
2572f509-74bd-51fe-a1b1-ca79075592ef- task_key
test--2572f509-74bd-51fe-a1b1-ca79075592ef
Default config / test
Let S_(r)=x^(r)+y^(r)+z^(r) with x,y,z real. It is known that if S₁=0 ,
Answer published by the source
Identifiers
- task_id
25bf8255-7cd3-5acf-8611-cdc5a25a1bac- task_key
test--25bf8255-7cd3-5acf-8611-cdc5a25a1bac
Default config / test
Find all positive integers a,b,c and prime p satisfying that
Answer published by the source
Identifiers
- task_id
27b40f4a-6700-5985-9c32-21f2e7f364f7- task_key
test--27b40f4a-6700-5985-9c32-21f2e7f364f7
Default config / test
Find all triplets of integers (a,b,c) such that the number
Answer published by the source
Identifiers
- task_id
28895e10-f4e8-516c-8ce7-27a7ae3c698a- task_key
test--28895e10-f4e8-516c-8ce7-27a7ae3c698a
Default config / test
A social club has 2k+1 members, each of whom is fluent in the same k languages. Any pair of members always talk to each other in only one language. Suppose that there were no three…
Answer published by the source
Identifiers
- task_id
28ddc7bf-2103-5e64-9121-de2dfabf7e0d- task_key
test--28ddc7bf-2103-5e64-9121-de2dfabf7e0d
Default config / test
Determine whether or not there are any positive integral solutions of the…
Answer published by the source
Identifiers
- task_id
2aa982bd-557d-594d-9589-c18f8281592d- task_key
test--2aa982bd-557d-594d-9589-c18f8281592d
Default config / test
Find the largest positive integer m which makes it possible to color several cells of a 70×70 table red such that [list] [*] There are no two red cells satisfying: the two rows in …
Answer published by the source
Identifiers
- task_id
2c04cc8e-b892-5852-923e-077ecd1fee56- task_key
test--2c04cc8e-b892-5852-923e-077ecd1fee56
Default config / test
Find all nonnegative integer solutions (x,y,z,w) of the equation2^(x)⋅3^(y)−5^(z)⋅7^(w)=1.
Answer published by the source
Identifiers
- task_id
2c6998ba-1bd0-5778-bab7-feaa4bd3040c- task_key
test--2c6998ba-1bd0-5778-bab7-feaa4bd3040c
Default config / test
Find all functions f,g:R→R such that f(x+yg(x))=g(x)+xf(y) for x,y∈R.
Answer published by the source
Identifiers
- task_id
2cea3741-58a0-537e-bfce-e93aeed163fa- task_key
test--2cea3741-58a0-537e-bfce-e93aeed163fa
Default config / test
Given positive integers n and k, n>k²>4. In a n×n grid, a k[i]-group[/i] is a set of k unit squares lying in different rows and different columns. Determine the maximal possible N,…
Answer published by the source
Identifiers
- task_id
2e9656c9-20cb-5391-9c26-c26f8bef5d79- task_key
test--2e9656c9-20cb-5391-9c26-c26f8bef5d79
Default config / test
Find the maximum possible number of three term arithmetic progressions in a…
Answer published by the source
Identifiers
- task_id
3029afd0-de3a-5160-b40d-abc31fafede7- task_key
test--3029afd0-de3a-5160-b40d-abc31fafede7
Default config / test
Given positive integer n≥5 and a convex polygon P, namely A₁A₂...A_(n). No diagonals of P are concurrent. Proof that it is possible to choose a point inside every quadrilateral …
Answer published by the source
Identifiers
- task_id
303330a5-8331-5ec7-bc2b-2701f9c8dd74- task_key
test--303330a5-8331-5ec7-bc2b-2701f9c8dd74
Default config / test
Determine all integers k such that there exists infinitely many positive integers n [b]not[/b] satisfying n+k |binom{2n}{n}
Answer published by the source
Identifiers
- task_id
30e6067a-25bc-50f1-8d67-9f74c414b3b9- task_key
test--30e6067a-25bc-50f1-8d67-9f74c414b3b9
Default config / test
For a given integer n≥2, let a₀,a₁,…,a_(n) be integers satisfying 0=a₀<a₁<…<a_(n)=2n−1. Find the smallest possible number of elements in the set {a_(i)+a_(j)∣0≤i≤j≤n}.
Answer published by the source
Identifiers
- task_id
3191ff2a-47ac-5a01-a7b5-c6be16813981- task_key
test--3191ff2a-47ac-5a01-a7b5-c6be16813981
Default config / test
Find a real number t such that for any set of 120 points P₁,…P₁₂₀ on the boundary of a unit square, there exists a point Q on this boundary with |P₁Q|+|P₂Q|+⋯+|P₁₂₀Q|=t.
Answer published by the source
Identifiers
- task_id
36d5e112-ee69-5dd2-bdde-c1b2ed89146a- task_key
test--36d5e112-ee69-5dd2-bdde-c1b2ed89146a
Default config / test
Let the circumcenter of triangle ABC be O. H_(A) is the projection of A onto BC. The extension of AO intersects the circumcircle of BOC at A'. The projections of A' onto AB,AC are …
Answer published by the source
Identifiers
- task_id
372dfa89-570a-5d00-9ac1-4516b2e4d443- task_key
test--372dfa89-570a-5d00-9ac1-4516b2e4d443
Default config / test
Determine all f:R→R such that
Answer published by the source
Identifiers
- task_id
388acca2-8bfe-5df5-9a66-fee9a914dbb4- task_key
test--388acca2-8bfe-5df5-9a66-fee9a914dbb4
Default config / test
Given a square ABCD whose side length is 1, P and Q are points on the sides AB and AD. If the perimeter of APQ is 2 find the angle PCQ.
Answer published by the source
Identifiers
- task_id
3a44eece-3976-5726-ae3a-ffcee9f96176- task_key
test--3a44eece-3976-5726-ae3a-ffcee9f96176
Default config / test
Two incongruent triangles ABC and XYZ are called a pair of [i]pals[/i] if they satisfy the following conditions: (a) the two triangles have the same area; (b) let M and W be the re…
Answer published by the source
Identifiers
- task_id
3c49a3a1-ed51-5fe5-808d-eb6109819e04- task_key
test--3c49a3a1-ed51-5fe5-808d-eb6109819e04
Default config / test
For which positive integers m does there exist an infinite arithmetic sequence of integers a₁,a₂,⋯ and an infinite geometric sequence of integers g₁,g₂,⋯ satisfying the following p…
Answer published by the source
Identifiers
- task_id
3dc3c87e-01f9-57d6-8a85-112cc4339b09- task_key
test--3dc3c87e-01f9-57d6-8a85-112cc4339b09
Default config / test
Let P be a polynomial with integer coefficients such that P(0)=0 and
Answer published by the source
Identifiers
- task_id
3e2c8498-8e66-5ea3-ac80-150b0ac26849- task_key
test--3e2c8498-8e66-5ea3-ac80-150b0ac26849
Default config / test
Two rational numbers (m)/(n) and (n)/(m) are written on a blackboard, where m and n are relatively prime positive integers. At any point, Evan may pick two of the numbers x and y w…
Answer published by the source
Identifiers
- task_id
3f5d9e33-0756-5fa0-a621-5dc5629414be- task_key
test--3f5d9e33-0756-5fa0-a621-5dc5629414be