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Free Practice Questions for Slovak Math Olympiad Cat A

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Slovak Physics Olympiad — Category A (Fyzikálna olympiáda, kategória A)

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Key Facts: Slovak Math Olympiad Cat A Exam

The 76th year of the Slovak Mathematical Olympiad takes place in the 2026/2027 school year.

NIVaM organizes the competition and the Slovak Mathematical Olympiad Committee is its expert guarantor.

Category A has a home round, a school round, a regional round and a national round.

The home round has 6 tasks graded 2, 1 or 0 points.

The school round has 3 tasks and the regional round 4 tasks, each worth 6 points and solved in 4 hours.

The national round has 6 tasks worth 7 points each, solved in two 4.5-hour sessions.

The 2026/2027 national round of category A is scheduled for 14–17 March 2027.

About 20 students are invited to a selection camp; the top 6 go to the IMO and the next 6 not in their final year to MEMO.

Tasks are set in Slovak or in the minority language of instruction of the student's school.

Mathematical Olympiad category A (2026/2027) runs from a 6-task home round (due 1 December 2026) through a school round on 8 December 2026 and a regional round on 12 January 2027 (4 tasks, 4 hours, success from 12 of 24 points) to a two-day national round on 14–17 March 2027.

Sample Slovak Math Olympiad Cat A Practice Questions

Try these sample questions to review concepts for the Slovak Math Olympiad Cat A exam. Each question includes a detailed explanation. Start the interactive quiz above for the full 100+ question experience with AI tutoring.

1Let r_1, r_2, and r_3 be the roots of the cubic polynomial P(x) = x^3 - 4x^2 + 6x - 2. What is the exact value of r_1^2 + r_2^2 + r_3^2?
A.4
B.6
C.8
D.12
Explanation: By Vieta's formulas for P(x) = x^3 - 4x^2 + 6x - 2, the elementary symmetric polynomials are e_1 = r_1 + r_2 + r_3 = 4 and e_2 = r_1 r_2 + r_2 r_3 + r_3 r_1 = 6. Using the identity r_1^2 + r_2^2 + r_3^2 = e_1^2 - 2e_2, we evaluate 4^2 - 2(6) = 16 - 12 = 4.
2What is the remainder when the polynomial P(x) = x^2026 - x^2025 + 3 is divided by x^2 - 1?
A.x + 2
B.-2x + 5
C.3x + 1
D.-x + 4
Explanation: Since the divisor x^2 - 1 is quadratic, the remainder has the form R(x) = ax + b. By the polynomial division theorem, P(1) = R(1) and P(-1) = R(-1). We compute P(1) = 1^2026 - 1^2025 + 3 = 3, so a + b = 3. Next, P(-1) = (-1)^2026 - (-1)^2025 + 3 = 1 - (-1) + 3 = 5, so -a + b = 5. Adding the two linear equations yields 2b = 8, so b = 4, and consequently a = -1. Therefore, the remainder is -x + 4.
3Let x and y be positive real numbers such that x + 2y = 8. What is the maximum possible value of the product xy?
A.8
B.6
C.12
D.16
Explanation: Applying the AM-GM inequality to the positive terms x and 2y gives sqrt(x * 2y) <= (x + 2y) / 2 = 8 / 2 = 4. Squaring both sides yields 2xy <= 16, which implies xy <= 8. Equality holds when x = 2y = 4, yielding the positive real pair (x, y) = (4, 2), for which xy = 8.
4Real numbers x and y satisfy x + y = 5 and x^3 + y^3 = 35. What is the value of x^2 + y^2?
A.11
B.15
C.17
D.13
Explanation: Using the algebraic factorization x^3 + y^3 = (x + y)(x^2 - xy + y^2) = (x + y)((x + y)^2 - 3xy), we substitute the given values: 35 = 5 * (25 - 3xy). Dividing by 5 gives 7 = 25 - 3xy, so 3xy = 18 and xy = 6. Then x^2 + y^2 = (x + y)^2 - 2xy = 5^2 - 2(6) = 25 - 12 = 13.
5For real numbers a and b satisfying a^2 + b^2 = 5, what is the maximum possible value of the linear expression 2a + 3b?
A.sqrt(55)
B.sqrt(65)
C.9
D.5*sqrt(3)
Explanation: By the Cauchy-Schwarz inequality in R^2, (2a + 3b)^2 <= (2^2 + 3^2)(a^2 + b^2) = (4 + 9) * 5 = 13 * 5 = 65. Equality holds when the vectors (a, b) and (2, 3) are positively proportional, namely a / 2 = b / 3 = sqrt(5 / 13). Taking the positive square root gives the maximum value sqrt(65).
6A function f: R \ {0} -> R satisfies f(x) + 2f(1/x) = 3x for all non-zero real numbers x. What is the value of f(2)?
A.-1
B.1
C.3/2
D.-2
Explanation: Substitute x = 2 into the given equation: f(2) + 2f(1/2) = 6. Now substitute x = 1/2 into the equation: f(1/2) + 2f(2) = 3(1/2) = 3/2. Multiplying this second equation by 2 yields 2f(1/2) + 4f(2) = 3. Subtracting the first equation from this doubled equation eliminates f(1/2): (4f(2) - f(2)) = 3 - 6, which gives 3f(2) = -3, hence f(2) = -1.
7How many distinct real roots does the polynomial equation x^4 - 5x^2 + 4 = 0 possess?
A.1
B.2
C.3
D.4
Explanation: Treating the equation as quadratic in u = x^2, we have u^2 - 5u + 4 = 0, which factors as (u - 1)(u - 4) = 0. This gives two positive solutions for u: u = 1 and u = 4. Substituting back u = x^2 yields x^2 = 1 (giving x = 1 and x = -1) and x^2 = 4 (giving x = 2 and x = -2). All four roots are real and distinct.
8For all strictly positive real numbers a and b, what is the minimum value of the expression (a + b)(1/a + 1/b)?
A.4
B.2
C.8
D.16
Explanation: Expanding the product yields (a + b)(1/a + 1/b) = 1 + a/b + b/a + 1 = 2 + (a/b + b/a). By the AM-GM inequality, for any positive real ratio t = a/b > 0, t + 1/t >= 2, with equality if and only if t = 1 (meaning a = b). Therefore, the expression is always at least 2 + 2 = 4.
9Let r_1, r_2, r_3, r_4 be the roots of the quartic polynomial P(x) = x^4 - 6x^3 + 13x^2 - 12x + 4. What is the value of 1/r_1 + 1/r_2 + 1/r_3 + 1/r_4?
A.2
B.4
C.6
D.3
Explanation: By Vieta's formulas, the sum of reciprocals of the roots of a quartic polynomial P(x) = a_4 x^4 + a_3 x^3 + a_2 x^2 + a_1 x + a_0 is given by -a_1 / a_0. Here a_1 = -12 and a_0 = 4, so the sum equals -(-12) / 4 = 12 / 4 = 3.
10A continuous function f: R -> R satisfies f(x + y) = f(x) + f(y) + 2xy for all real x and y. If f(1) = 4, what is the value of f(5)?
A.40
B.35
C.45
D.50
Explanation: Define an auxiliary function g(x) = f(x) - x^2. Then g(x + y) = f(x + y) - (x + y)^2 = (f(x) + f(y) + 2xy) - (x^2 + 2xy + y^2) = (f(x) - x^2) + (f(y) - y^2) = g(x) + g(y). Thus g satisfies Cauchy's functional equation on R. Since f is continuous, g is continuous, so g(x) = cx for some constant c. We are given f(1) = 4, so g(1) = 4 - 1^2 = 3, which implies c = 3. Hence f(x) = x^2 + 3x. Evaluating at x = 5 gives f(5) = 5^2 + 3(5) = 25 + 15 = 40.

About the Slovak Math Olympiad Cat A Exam

Independent study practice for category A of the Slovak Mathematical Olympiad (Matematická olympiáda), the national mathematics competition for students in the last two years of secondary school. In its 76th year (2026/2027) the competition is organized by NIVaM, with the Slovak Mathematical Olympiad Committee as expert guarantor, under the Mathematical Olympiad organizational rules and the ministry's directive No. 19/2024 on competitions. Category A has a home round with 6 tasks, a 4-hour school round with 3 tasks, a 4-hour regional round with 4 tasks, and a two-day national round for about 40 contestants with 6 tasks worth 7 points each. Results of the national round and the year-round Kvalifikácia lead to a selection camp of about 20 students, from which the IMO and MEMO teams are chosen. The 100 practice questions cover algebra and inequalities, number theory, combinatorics and geometry. This bank is an English-language, four-option multiple-choice study adaptation for checking results and techniques; it is not an official problem set and does not replace writing complete proofs.

Exam sponsor: NIVaM (Národný inštitút vzdelávania a mládeže) as organizer, with the Slovak Mathematical Olympiad Committee (SK MO) as expert guarantor, under the Ministry of Education's directive No. 19/2024 on competitions. The requirements and fees below concern the certification or admission exam, separate from our free practice resources.

Assessment

Performance-based assessment

Time Limit

School and regional rounds 4 hours; national round 4.5 hours on each of two days

Passing Score

Half of the available points in the school and regional rounds (12 of 24 in the regional round); in the national round about 20 contestants are declared successful solvers in gold, silver and bronze bands.

Exam / Certification Fees

Free; the competition is organized by NIVaM for students registered by their schools.

Exam sponsor website

Fees, eligibility, and exam policies can change. Confirm them with the exam sponsor before applying or paying.

Our practice resources: topics covered

We aim to reflect publicly available exam outlines and topic information in our study resources. Coverage, format, and difficulty may differ from the actual exam, and we cannot guarantee that every detail is accurate or current. Confirm exam requirements, fees, and policies with the official exam sponsor.

No official weighting; 28 of 100 practice questions

Algebra a nerovnosti (Algebra and Inequalities)

Polynomials and Vieta's formulas, functional equations, sequences and recurrences, systems of equations, and classical inequalities (AM-GM, Cauchy-Schwarz, Jensen, Schur).

No official weighting; 24 of 100 practice questions

Teória čísel (Number Theory)

Divisibility, gcd and lcm, congruences, Fermat's and Wilson's theorems, orders and primitive roots, Diophantine equations, and p-adic valuations.

No official weighting; 24 of 100 practice questions

Kombinatorika (Combinatorics)

Counting, inclusion-exclusion, the pigeonhole principle, invariants, graph theory, recurrences and classical counting results such as Catalan numbers and Burnside's lemma.

No official weighting; 24 of 100 practice questions

Geometria (Geometry)

Triangle centres and classical theorems (Ceva, Menelaus, Ptolemy), circles and power of a point, the nine-point circle, inversion, homothety and spiral similarity, and coordinate and complex-number methods.

Preparing for the Slovak Math Olympiad Cat A Exam

What You Need to Know

  • Passing score: Half of the available points in the school and regional rounds (12 of 24 in the regional round); in the national round about 20 contestants are declared successful solvers in gold, silver and bronze bands.
  • Assessment: Performance-based assessment
  • Time limit: School and regional rounds 4 hours; national round 4.5 hours on each of two days
  • Exam / certification fees: Free; the competition is organized by NIVaM for students registered by their schools. Official sources

Using Our Practice Resources

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

Slovak Math Olympiad Cat A: Suggested Study Strategy

1Solve past Slovak Mathematical Olympiad problems (archived on skmo.sk) and write full solutions, because points are awarded for complete, justified arguments rather than answers.
2Build a toolkit of standard techniques: substitutions in functional equations, AM-GM and Cauchy-Schwarz, modular arithmetic, invariants and the extremal principle.
3Practise geometry with careful diagrams and learn when to switch to coordinates, complex numbers or inversion.
4Train for the 4-hour regional round by attempting four unseen problems in one sitting and reviewing which ideas you missed.
5Join the year-round Kvalifikácia run by the Slovak Mathematical Olympiad Committee, which also counts towards invitations to the selection camp.

Frequently Asked Questions

Who can compete in category A of the Mathematical Olympiad?

Category A is for students who will finish secondary school in the current school year or one year later, typically the 3rd and 4th years of four-year schools, septima and oktáva of eight-year gymnázia, or the 4th and 5th years of five-year gymnázia. Students may also compete in a higher category than their own, and the school registers them in EduPage.

What are the rounds and dates of category A in 2026/2027?

Home-round solutions (6 tasks) are due by 1 December 2026, the school round (3 tasks, 4 hours) is on 8 December 2026, the regional round (4 tasks, 4 hours) is on 12 January 2027, and the national round for about 40 contestants takes place on 14–17 March 2027 with 6 tasks solved in two 4.5-hour sessions.

How are solutions scored and who counts as successful?

Home-round tasks are graded 2, 1 or 0 points, and successful solvers score at least 1 point on at least 4 tasks. School- and regional-round tasks are worth 6 points each, and the success threshold is half of the available points. National-round tasks are worth 7 points each; the committee declares about 20 successful solvers in gold, silver and bronze bands.

Is this practice bank an official Mathematical Olympiad problem set?

No. This is independent practice by OpenExamPrep: an English-language, four-option multiple-choice study adaptation of olympiad topics in algebra, number theory, combinatorics and geometry. It is not an official problem set or translation, and the real competition requires complete written proofs.