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Sample OBM Practice Questions

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1What is the remainder when 7^2026 is divided by 100?

A.49

B.01

C.07

D.43

Explanation: Since gcd(7, 100) = 1, we can compute the multiplicative order or use Euler's Totient Theorem with φ(100) = 40. Notice that 7^4 = 2401 ≡ 1 (mod 100), meaning the period modulo 100 is 4. Reducing the exponent modulo 4 gives 2026 = 4 × 506 + 2 ≡ 2 (mod 4). Therefore, 7^2026 ≡ 7^2 = 49 (mod 100).

2Find the smallest positive integer x satisfying the system of congruences: x ≡ 2 (mod 3), x ≡ 3 (mod 5), and x ≡ 2 (mod 7).

A.8

B.17

C.23

D.38

Explanation: From x ≡ 2 (mod 3) and x ≡ 2 (mod 7), since gcd(3, 7) = 1, we immediately have x ≡ 2 (mod 21). Thus x can be written as x = 21k + 2 for some integer k. Substituting into the remaining congruence gives 21k + 2 ≡ 3 (mod 5), which simplifies to k + 2 ≡ 3 (mod 5), so k ≡ 1 (mod 5). The smallest non-negative integer k is 1, yielding x = 21(1) + 2 = 23.

3How many ordered pairs of integers (x, y) with x ≤ y satisfy the equation 1/x + 1/y = 1/2026?

A.5

B.9

C.18

D.10

Explanation: Multiplying by 2026xy and rearranging yields (x - 2026)(y - 2026) = 2026^2. The prime factorization is 2026 = 2 × 1013 (with 1013 prime), so 2026^2 = 2^2 × 1013^2, which has d(2026^2) = (2+1)(2+1) = 9 positive divisors. For positive pairs x, y > 0, each pair of complementary positive factors (d, 2026^2/d) gives a solution, yielding (9 + 1)/2 = 5 solutions with x ≤ y. For integer solutions with x < 0 < y, setting u = 2026 - x > 0 and v = y - 2026 > 0 gives -uv = -2026^2, requiring u to be a divisor of 2026^2 strictly greater than 2026, which yields 4 negative solutions. Total pairs with x ≤ y = 5 + 4 = 9.

4What is the highest power of 7 that divides 1000! (that is, the 7-adic valuation v_7(1000!))?

A.142

B.162

C.163

D.164

Explanation: By Legendre's Formula, the p-adic valuation of n! is given by v_p(n!) = ∑_{k=1}^∞ ⌊n / p^k⌋. Computing each successive term for p = 7 and n = 1000: ⌊1000/7⌋ = 142, ⌊1000/49⌋ = 20, ⌊1000/343⌋ = 2, and ⌊1000/2401⌋ = 0. Summing these values gives v_7(1000!) = 142 + 20 + 2 = 164.

5Determine the highest power of 2 that divides 3^1024 - 1 (that is, v_2(3^1024 - 1)).

A.10

B.11

C.13

D.12

Explanation: By the Lifting the Exponent Lemma (LTE) for p = 2 with odd integers x, y and even n, we have v_2(x^n - y^n) = v_2(x - 1) + v_2(x + 1) + v_2(n) - 1. Here x = 3, y = 1, and n = 1024 = 2^10. Computing the valuations: v_2(3 - 1) = v_2(2) = 1, v_2(3 + 1) = v_2(4) = 2, and v_2(1024) = 10. Substituting these into LTE gives v_2(3^1024 - 1) = 1 + 2 + 10 - 1 = 12.

6What is the remainder when 99! is divided by the prime 101?

A.1

B.100

C.0

D.50

Explanation: By Wilson's Theorem, for any prime p, (p - 1)! ≡ -1 ≡ p - 1 (mod p). For p = 101, this gives 100! ≡ -1 (mod 101). Expanding the factorial: 100! = 100 × 99! ≡ (-1) × 99! (mod 101). Equating the two expressions gives (-1) × 99! ≡ -1 (mod 101), which immediately yields 99! ≡ 1 (mod 101).

7How many quadratic residues (mod 71) are in the set of non-zero residue classes {1, 2, 3, ..., 70}?

A.70

B.36

C.35

D.34

Explanation: For any odd prime p, the multiplicative group (Z/pZ)* is cyclic of order p - 1. Under the squaring map x ↦ x^2 (mod p), the kernel consists of {1, -1}, which has order 2. Therefore, by the First Isomorphism Theorem, the image (the subgroup of non-zero quadratic residues) has size exactly (p - 1)/2. For p = 71, (71 - 1)/2 = 35.

8How many primitive roots modulo 29 exist in the residue system {1, 2, ..., 28}?

A.6

B.12

C.14

D.28

Explanation: The multiplicative group (Z/pZ)* is cyclic of order p - 1 for any prime p. The generators of a cyclic group of order n are elements whose power is coprime to n, so there are exactly φ(n) generators. For p = 29, the number of primitive roots is φ(p - 1) = φ(28). Factoring 28 = 2^2 × 7, we compute φ(28) = 28 × (1 - 1/2) × (1 - 1/7) = 28 × (1/2) × (6/7) = 12.

9What is the largest integer postage amount that CANNOT be formed using only 5-cent and 7-cent stamps (the Frobenius coin problem for {5, 7})?

A.19

B.23

C.24

D.35

Explanation: For any two coprime positive integers a and b, Sylvester's theorem on the Frobenius coin problem states that the largest integer that cannot be expressed as ax + by with non-negative integers x, y ≥ 0 is given by g(a, b) = ab - a - b. For a = 5 and b = 7, gcd(5, 7) = 1, so g(5, 7) = 5 × 7 - 5 - 7 = 35 - 12 = 23.

10Compute gcd(2^2024 - 1, 2^2032 - 1).

A.1

B.15

C.127

D.255

Explanation: A fundamental number theoretic identity establishes that for any integer a ≥ 2 and positive integers m, n, gcd(a^m - 1, a^n - 1) = a^(gcd(m, n)) - 1. Here a = 2, m = 2024, and n = 2032. Using the Euclidean algorithm: gcd(2024, 2032) = gcd(2024, 2032 - 2024) = gcd(2024, 8). Since 2024 = 8 × 253, gcd(2024, 8) = 8. Thus, gcd(2^2024 - 1, 2^2032 - 1) = 2^8 - 1 = 256 - 1 = 255.

About the OBM Exam

The Olimpíada Brasileira de Matemática (OBM) is Brazil's premier national mathematics competition, established in 1979 by the Sociedade Brasileira de Matemática (SBM) in collaboration with the Instituto de Matemática Pura e Aplicada (IMPA). Structured across four competitive tiers—Nível 1 (6th and 7th grades of Ensino Fundamental), Nível 2 (8th and 9th grades of Ensino Fundamental), Nível 3 (Ensino Médio / High School), and Nível Universitário (Undergraduate students)—the OBM serves as the official national selection and training pathway for the Brazilian delegations representing the country at elite international competitions, including the International Mathematical Olympiad (IMO), European Girls' Mathematical Olympiad (EGMO), Iberoamerican Mathematical Olympiad (OIM), Romanian Master of Mathematics (RMM), and International Mathematics Competition for University Students (IMC). Under official SBM regulations, Levels 1–3 use purely discursive proof-based examinations, while Nível Universitário includes a preliminary objective paper. This bank provides an accessible multiple-choice study adaptation of classic OBM lemmas, calculations, and problem-solving strategies.

Assessment

Under the 48th OBM regulation (2026), school tiers (Níveis 1–3) compete in a single-phase national written proof exam (Fase Única): Nível 1 sits one session with 5 discursive problems (4h30, 250 points), while Níveis 2 and 3 sit two consecutive days with 3 discursive problems per 4h30 day (300 points). Nível Universitário runs in two phases: the 7ª Competição Elon Lages Lima de Matemática serves as Phase 1 (objective, 15 MCQs, 1h30, in person), followed by a Phase 2 discursive proof exam over two consecutive days (3 questions per 4h30 day).

Time Limit

Nível 1: one 4h30 session; Níveis 2–3 and University Phase 2: two days of 4h30 each; University Phase 1 (Elon Lages): 1h30

Passing Score

Jury-awarded medal score thresholds (Gold, Silver, Bronze, Honorable Mention) and Olympic team selection criteria

Exam Fee

Free (Gratuito) (Sociedade Brasileira de Matemática (SBM) & IMPA)

OBM Exam Content Outline

25%

Teoria dos Números Avançada

Diophantine equations, modular arithmetic, Fermat's Little Theorem, Euler's totient, primitive roots, quadratic residues, integer polynomials, and p-adic valuations (LTE)

25%

Geometria Olímpica Sintética

Cyclic quadrilaterals, power of a point, radical axis, Menelaus and Ceva theorems, homothety, inversion, projective configurations, and trigonometry in geometry

20%

Álgebra Olímpica e Polinômios

Classical inequalities (AM-GM, Cauchy-Schwarz, Jensen, Muirhead, Schur), functional equations, polynomial roots, Vieta's formulas, sequences, and recurrences

20%

Combinatória Olímpica e Grafos

Pigeonhole principle, double counting, graph theory (Eulerian/Hamiltonian, bipartite, coloring), invariants and monovariants, and the extremal principle

10%

Raciocínio Lógico e Estratégias

Combinatorial games, winning strategies, parity arguments, grid tiling invariants, and probabilistic method basics

How to Pass the OBM Exam

What You Need to Know

Passing score: Jury-awarded medal score thresholds (Gold, Silver, Bronze, Honorable Mention) and Olympic team selection criteria
Assessment: Under the 48th OBM regulation (2026), school tiers (Níveis 1–3) compete in a single-phase national written proof exam (Fase Única): Nível 1 sits one session with 5 discursive problems (4h30, 250 points), while Níveis 2 and 3 sit two consecutive days with 3 discursive problems per 4h30 day (300 points). Nível Universitário runs in two phases: the 7ª Competição Elon Lages Lima de Matemática serves as Phase 1 (objective, 15 MCQs, 1h30, in person), followed by a Phase 2 discursive proof exam over two consecutive days (3 questions per 4h30 day).
Time limit: Nível 1: one 4h30 session; Níveis 2–3 and University Phase 2: two days of 4h30 each; University Phase 1 (Elon Lages): 1h30
Exam fee: Free (Gratuito)

Keys to Passing

Work through all 100 available questions
Review every answer and explanation
Track weak areas and revisit them
Use our AI tutor for tough concepts

OBM Study Tips from Top Performers

1Master rigorous proof-writing: partial credit in OBM is heavily dependent on complete, deductive arguments and lemma justifications rather than raw answers.

2Work systematically through past Phase 3 problems and articles in SBM's official journal, Revista Eureka!.

3Build mastery across the core four Olympiad domains: Number Theory (modular arithmetic, LTE, Diophantine equations), Synthetic Geometry (cyclic quads, radical axis, power of a point), Algebra (inequalities, functional equations, Vieta's formulas), and Combinatorics (invariants, graphs, extremal principle).

4Participate in specialized Olympic training programs such as POTI (Polos Olímpicos de Treinamento Intensivo) and IMPA's PIC.

5Simulate timed 4.5-hour problem sets with 3 challenging proof problems to build concentration, speed, and analytical stamina.

Frequently Asked Questions

What is the OBM and who organizes it?

The Olimpíada Brasileira de Matemática (OBM) is Brazil's highest-tier national mathematics competition, organized by the Sociedade Brasileira de Matemática (SBM) in partnership with the Instituto de Matemática Pura e Aplicada (IMPA).

What are the competition levels of the OBM?

OBM is organized into four levels: Nível 1 (6th and 7th grades of Ensino Fundamental), Nível 2 (8th and 9th grades of Ensino Fundamental), Nível 3 (Ensino Médio / High School), and Nível Universitário (undergraduate university students).

Are OBM problems multiple-choice or proof-based?

According to the official SBM competition regulations, Levels 1, 2, and 3 are entirely written, proof-based discursive competitions where full mathematical justifications are required. The Nível Universitário route includes a preliminary objective (multiple-choice) first phase followed by a discursive second phase. This online preparation bank adapts key competition lemmas, computations, and theorems into high-yield multiple-choice questions for structured study.

How do school students qualify for the OBM?

Under the 48th OBM (2026) regulation, school students (Níveis 1–3) qualify for the single-phase national exam (Fase Única) primarily through exceptional performance in the national OBMEP or through other qualification routes defined in the annual SBM regulation, such as affiliated regional Olympiads.

What international competitions does the OBM select teams for?

Top medalists in OBM are invited to the Olympic Training Week (Semana Olímpica) to undergo selection tests for Brazil's official national teams at the International Mathematical Olympiad (IMO), European Girls' Mathematical Olympiad (EGMO), Iberoamerican Mathematical Olympiad (OIM), and International Mathematics Competition for University Students (IMC).

How is the OBM school-level exam scored in 2026?

The Fase Única is graded by specialized juries appointed by the SBM National Olympiad Commission. Under the 2026 regulation, each Nível 1 problem is worth 50 points (250 total across 5 problems), and each Níveis 2–3 problem is worth 50 points (300 total across 6 problems over two days), with scores reflecting mathematical correctness, rigor, and completeness.

Is there any cost to participate in the OBM?

No. Participation in the OBM is 100% free of charge (gratuito) for all eligible students and participating schools throughout Brazil.

100+ Free OBM Practice Questions

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