Benchmark AI / Public workspace
Omni-MATH
Mathematics
Omni-MATH evaluates mathematical reasoning on Olympiad-level problems. Its official dataset contains 4,428 problems accompanied by domain and difficulty information.
Japanese introduction
数学オリンピック水準の問題を通じて、数学的推論能力を評価するベンチマークです。公式データは4,428問を収録し、分野や難易度の情報を伴います。
omni-math
Full text: eligible public problems include text, images and discussion.
Tasks
Whole public benchmark.
- Public problems
- 200
- Formats
- text (200)
- Problems with images
- 0
- Answer availability
- Answer published by the source (200)
- Configs
- Default (200)
- Splits
- test (200)
- Awaiting a first discussion
- 200
- Determine whether √(1001²+1)+√(1002²+1)+⋯+√(2000²+1) be a rational number or not?
Answer published by the source
- Let n≥2 be a natural. Define
Answer published by the source
- 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
- Given positive integers k,m,n such that 1≤k≤m≤n. Evaluate
Answer published by the source
- 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
- 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
- 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
- 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
- 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
- Find all positive integers x,y satisfying the equation 9(x²+y²+1)+2(3xy+2)=2005.
Answer published by the source
- 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
- 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
- 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
- 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
- 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
- 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]
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- Determine all the roots , real or complex , of the system of simultaneous…
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- 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
- Attempt of a halfways nice solution.
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- 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
- 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
- Determine the triangle with sides a,b,c and circumradius R for which R(b+c)=a√(bc) .
Answer published by the source
- 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
- There are arbitrary 7 points in the plane. Circles are drawn through every 4…
Answer published by the source
- Find all triples (x,y,z) of real numbers that satisfy the system of equations
Answer published by the source
- 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
- 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
- For each positive integer n, let c(n) be the largest real number such that
Answer published by the source
- Find all functions f:(0,∞)→(0,∞) such that
Answer published by the source
- Let S_(r)=x^(r)+y^(r)+z^(r) with x,y,z real. It is known that if S₁=0 ,
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- Find all positive integers a,b,c and prime p satisfying that
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- Find all triplets of integers (a,b,c) such that the number
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- 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
- Determine whether or not there are any positive integral solutions of the…
Answer published by the source
- 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
- 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
- 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
- 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
- Find the maximum possible number of three term arithmetic progressions in a…
Answer published by the source
- 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
- 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
- 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
- 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
- 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
- Determine all f:R→R such that
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- 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
- 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…
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- 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…
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- Let P be a polynomial with integer coefficients such that P(0)=0 and
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- 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