Free Practice Questions for Concorso Normale Pisa

Exam-style questions and explanations by OpenExamPrep.

✓ No registration✓ No credit card

100+ Questions

100% Free

Loading practice questions...

Same family resources

Explore More Scuole Universitarie Superiori Admission Exams (Italy)

Continue into nearby exams from the same family. Each card keeps practice questions, study guides, flashcards, videos, and articles in one place.

Concorso di Ammissione Scuola Superiore Sant'Anna (Italy)

Practice Questions100 questions

Exam Review

Key Facts: Concorso Normale Pisa Exam

~60 places

Annual first-year collegiate places (Sciences & Humanities)

Scuola Normale Superiore Bando 2026

€0 fee

Free competitive admission entry

SNS Admissions Office

7 / 10

Minimum qualifying mark per written and oral test

Regolamento Concorso Ordinario

2 Written + 2 Oral

Official test format in candidate's disciplinary class

Bando Concorso Ordinario

~5% rate

Estimated admission acceptance rate

SNS Academic Records

100% Funded

Full tuition waiver, residential room, board, and research grant

SNS Collegio & Student Services

The Scuola Normale Superiore ordinary admission competition uses class- and subject-specific programs, two written tests, and two oral tests; science applicants also complete the specified CISIA TOLC prerequisite. Selection follows the annual call and merit ranking, with admitted students receiving the published collegiate benefits. This is an independent English-language multi-track MCQ adaptation.

Sample Concorso Normale Pisa Practice Questions

Try these sample questions to review concepts for the Concorso Normale Pisa exam. Each question includes a detailed explanation. Start the interactive quiz above for the full 100+ question experience with AI tutoring.

1What is the limit of the sequence a_n = (1 + 1/n^2)^n as n approaches infinity?

A.0

B.1

C.e

D.e^2

Explanation: Rewriting the sequence using logarithms yields ln(a_n) = n * ln(1 + 1/n^2). Using the standard Taylor expansion ln(1 + x) = x + O(x^2) as x approaches 0, we have ln(a_n) = n * (1/n^2 + O(1/n^4)) = 1/n + O(1/n^3), which tends to 0 as n approaches infinity. Therefore, lim a_n = e^0 = 1.

2If omega is a primitive complex cube root of unity (omega = e^(2*pi*i/3)), what is the value of the algebraic expression (1 - omega + omega^2) * (1 + omega - omega^2)?

A.1

B.3

C.4

D.omega

Explanation: Since omega is a cube root of unity distinct from 1, it satisfies 1 + omega + omega^2 = 0. From this identity, 1 + omega^2 = -omega and 1 + omega = -omega^2. Substituting these into the factors yields (-omega - omega) * (-omega^2 - omega^2) = (-2*omega) * (-2*omega^2) = 4 * omega^3 = 4 * 1 = 4.

3A set consists of 4 distinct letters addressed to 4 different recipients. If the letters are placed randomly into 4 pre-addressed envelopes, what is the exact number of derangements (configurations where no letter matches its intended envelope)?

A.6

B.14

C.12

D.9

Explanation: The number of derangements of n elements is given by D_n = n! * sum_{k=0}^n (-1)^k / k!. For n = 4, D_4 = 24 * (1 - 1 + 1/2 - 1/6 + 1/24) = 24 * (12/24 - 4/24 + 1/24) = 24 * (9/24) = 9.

4From an external point P, a secant line intersects a circle at points A and B such that PA = 4 and PB = 9. A tangent segment is drawn from P touching the circle at point T. What is the length of segment PT?

A.5

B.6

C.6.5

D.sqrt(65)

Explanation: By the Tangent-Secant Theorem (Power of a Point), the square of the length of the tangent segment from an external point equals the product of the lengths of the external secant segment and the entire secant segment: PT^2 = PA * PB. Here, PT^2 = 4 * 9 = 36, which gives PT = 6.

5Which of the following real functions is uniformly continuous on the unbounded interval [0, +infinity)?

A.f(x) = x^2

B.f(x) = sin(x^2)

C.f(x) = sqrt(x)

D.f(x) = x * sin(x)

Explanation: A continuous function on [0, +infinity) is uniformly continuous if its derivative is bounded or if it satisfies a Holder condition. For f(x) = sqrt(x), it is continuous on the compact set [0, 1] (hence uniformly continuous there by the Heine-Cantor theorem), and on [1, +infinity) its derivative is f'(x) = 1/(2*sqrt(x)) <= 1/2 (bounded derivative implies Lipschitz continuity, which implies uniform continuity). Alternatively, |sqrt(x) - sqrt(y)| <= sqrt(|x - y|) holds for all x, y >= 0.

6What is the maximum possible order of an element in the symmetric group S_5 on 5 symbols?

A.5

B.6

C.10

D.12

Explanation: The order of a permutation written as a product of disjoint cycles is the least common multiple (lcm) of the cycle lengths, where the sum of cycle lengths is at most 5. The partitions of 5 are: 5 (order 5), 4+1 (order 4), 3+2 (order lcm(3,2) = 6), 3+1+1 (order 3), 2+2+1 (order lcm(2,2) = 2), 2+1+1+1 (order 2), and 1+1+1+1+1 (order 1). The maximum order is therefore lcm(3, 2) = 6.

7What is the exact value of the definite integral I = int_0^(pi/2) ln(sin x) dx?

A.-pi * ln(2)

B.-ln(2)

C.-(pi / 4) * ln(2)

D.-(pi / 2) * ln(2)

Explanation: By symmetry, I = int_0^(pi/2) ln(cos x) dx. Adding the two gives 2I = int_0^(pi/2) ln(sin x * cos x) dx = int_0^(pi/2) ln((sin 2x)/2) dx = int_0^(pi/2) ln(sin 2x) dx - int_0^(pi/2) ln(2) dx. Substituting u = 2x turns int_0^(pi/2) ln(sin 2x) dx into (1/2) int_0^pi ln(sin u) du = int_0^(pi/2) ln(sin u) du = I. Thus 2I = I - (pi/2)*ln(2), which simplifies to I = -(pi/2)*ln(2).

8In triangle ABC, points D, E, and F lie on sides BC, CA, and AB respectively. According to Ceva's Theorem, what is the necessary and sufficient condition for the cevians AD, BE, and CF to be concurrent?

A.(BD / DC) + (CE / EA) + (AF / FB) = 1

B.(BD / DC) * (CE / EA) * (AF / FB) = 1

C.(BD / BC) * (CE / CA) * (AF / AB) = 1

D.(BD * DC) + (CE * EA) + (AF * FB) = 0

Explanation: Ceva's Theorem states that three lines connecting the vertices of a triangle to points on the opposite sides are concurrent if and only if the product of the ratios of the directed segments along the perimeter equals 1: (BD / DC) * (CE / EA) * (AF / FB) = 1 (or +1 using standard non-directed side ratios for interior points).

9What is the volume of a regular tetrahedron whose edges all have length a?

A.(a^3 * sqrt(2)) / 12

B.(a^3 * sqrt(3)) / 6

C.(a^3 * sqrt(2)) / 6

D.(a^3 * sqrt(3)) / 12

Explanation: The base is an equilateral triangle with area A = (sqrt(3)/4) * a^2. The height h is found by considering the right triangle formed by an edge of length a and the circumradius of the base R = a / sqrt(3): h = sqrt(a^2 - a^2/3) = a * sqrt(2/3). The volume is V = (1/3) * A * h = (1/3) * (sqrt(3)/4 * a^2) * (sqrt(2)/sqrt(3) * a) = (a^3 * sqrt(2)) / 12.

10What are the last two decimal digits of the integer 7^2024?

A.01

B.07

C.43

D.49

Explanation: Finding the last two digits is equivalent to evaluating 7^2024 modulo 100. Since gcd(7, 100) = 1, we can apply Euler's totient theorem. The totient of 100 is phi(100) = 100 * (1 - 1/2) * (1 - 1/5) = 40. Therefore, 7^40 is congruent to 1 (mod 100). Dividing the exponent 2024 by 40 gives 2024 = 40 * 50 + 24. We compute 7^2 = 49, 7^4 = 2401 = 1 (mod 100). Since 4 divides 2024, 7^2024 = (7^4)^506 = 1^506 = 1 (mod 100), giving '01'.

About the Concorso Normale Pisa Exam

The Concorso Ordinario di Ammissione at the Scuola Normale Superiore di Pisa is Italy's most prestigious and selective public collegiate entrance competition. Founded by Napoleon in 1810 as a branch of the Ecole Normale Superieure in Paris, the Normale admits approximately 60 first-year undergraduate students annually across its two historic academic classes: the Classe di Scienze (Mathematics, Physics, Chemistry, Biology) and the Classe di Lettere e Filosofia (History, Philosophy, Classical Philology/Literature). Successful candidates receive full collegiate benefits, including free tuition, room, board at the collegiate residences in Pisa, and dedicated research stipends while concurrently pursuing university degree programs at the Universita di Pisa. The official competition consists of two rigorous written examinations followed by two comprehensive oral colloquia before the academic commissions, with science candidates completing a CISIA TOLC preselection requirement. This practice question bank is an independent educational prep resource created by OpenExamPrep. It adapts core syllabus concepts across advanced mathematics, physics, chemistry, biology, history, philosophy, and classical philology into an English-language multiple-choice practice format. It is not an official examination paper, past competition simulation, or authorized translation, and OpenExamPrep is not affiliated with, endorsed by, or sponsored by the Scuola Normale Superiore di Pisa or CISIA.

Exam sponsor: Scuola Normale Superiore di Pisa. The requirements and fees below concern the certification or admission exam, separate from our free practice resources.

Assessment

The ordinary admission competition uses class- and subject-specific published programs, with two written and two oral tests; science applicants also satisfy the stated CISIA TOLC prerequisite. Eligibility, scoring, dates, and ranking are governed by the annual call rather than one fictional shared paper. This is an independent English-language multi-track MCQ adaptation, not a simulation of proofs, essays, or oral performance.

Time Limit

Multi-day written and oral examination sessions in Pisa

Passing Score

Minimum 7/10 in each written test to qualify for oral exams, minimum 7/10 in oral exams; final admission determined by merit ranking within available quota (~5% acceptance rate)

Exam / Certification Fees

€0 (free competitive entry; requires TOLC prerequisite for sciences)

Exam sponsor website

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

Official sources

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.

25%

Classe di Scienze: Advanced Mathematics

Real analysis, limits of sequences and functions, single- and multivariable calculus, abstract algebra, group theory, combinatorics, number theory, and Euclidean geometry

25%

Classe di Scienze: Advanced Physics

Classical mechanics, central forces, Lagrangian formulations, thermodynamics, electrostatics, magnetostatics, Maxwell's equations, wave optics, and special relativity

20%

Classe di Scienze: Chemistry & Biology

Chemical thermodynamics, reaction kinetics, electrochemistry, organic reaction mechanisms, stereochemistry, molecular biology, DNA replication, gene regulation, and cellular bioenergetics

30%

Classe di Lettere e Filosofia: Humanities, History & Philosophy

Ancient Greek and Roman history, modern and contemporary European history, ancient and modern philosophical systems, epistemology, ethics, classical philology, and Italian literature

Preparing for the Concorso Normale Pisa Exam

What You Need to Know

Passing score: Minimum 7/10 in each written test to qualify for oral exams, minimum 7/10 in oral exams; final admission determined by merit ranking within available quota (~5% acceptance rate)
Assessment: The ordinary admission competition uses class- and subject-specific published programs, with two written and two oral tests; science applicants also satisfy the stated CISIA TOLC prerequisite. Eligibility, scoring, dates, and ranking are governed by the annual call rather than one fictional shared paper. This is an independent English-language multi-track MCQ adaptation, not a simulation of proofs, essays, or oral performance.
Time limit: Multi-day written and oral examination sessions in Pisa
Exam / certification fees: €0 (free competitive entry; requires TOLC prerequisite for sciences) 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

Concorso Normale Pisa: Suggested Study Strategy

1Focus on fundamental principles and problem-solving derivations rather than rote formula memorization: Normale written problems require creative insight and rigorous multi-step proofs.

2For Classe di Scienze applicants: practice challenging mathematical and physical problems from past olympiads and SNS competition archives, developing fluency in real analysis, Newtonian and Lagrangian mechanics, and electromagnetism.

3For Classe di Lettere applicants: cultivate rigorous critical argumentation, combining precise historical contextualization with philosophical rigor and philological attention to primary texts.

4Prepare for oral colloquia: practice explaining your problem-solving steps aloud, demonstrating intellectual flexibility, methodological clarity, and receptiveness to faculty hints.

5Verify deadlines, eligible TOLC test dates, and documentation requirements in the official Bando di Concorso published annually on the official SNS portal (sns.it).

Frequently Asked Questions

What is the Scuola Normale Superiore and what is the Concorso Ordinario?

The Scuola Normale Superiore di Pisa is a prestigious Italian public university institution of special status (Scuola Universitaria Superiore ad ordinamento speciale). The Concorso Ordinario is its annual nationwide competitive entrance examination that selects high-achieving students for collegiate admission. Admitted normalisti study concurrently at the University of Pisa and the Normale, receiving free room, board, tuition waivers, and advanced internal seminars.

What is the structure of the official competitive examination?

The official competition consists of two written tests and two oral colloquia conducted in Pisa before faculty evaluation commissions in the candidate's chosen academic class. For science candidates, passing an external CISIA TOLC prerequisite is required to validate application eligibility. Each written test is graded on a scale out of 10; candidates must achieve at least 7/10 on each written paper to be admitted to the oral phase, where a minimum 7/10 is also required. Final admission is granted strictly in order of merit within the statutory class quota.

What are the costs and financial benefits of attending the Normale?

Participation in the Concorso Ordinario is completely free of charge (€0 exam/application fee). All students admitted to the Scuola Normale Superiore receive a full scholarship covering complete university tuition fees, free residential lodging in collegiate halls of residence in Pisa, free daily meals at the student dining halls, and an annual study allowance.

What languages are used in the official competition and how does this practice bank relate to it?

The official written papers and oral colloquia of the Scuola Normale Superiore are administered primarily in Italian (officialLanguages: ['it']), in accordance with the annual call (Bando di Concorso). This practice question bank is an independent English-language multiple-choice adaptation designed to assist students in reviewing and mastering fundamental disciplinary concepts across the syllabus. It is not an official examination, past paper replication, or affiliated test simulation.

How competitive is admission to the Scuola Normale Superiore?

The Normale admission competition is among the most selective in Europe, with an acceptance rate typically around 5%. Approximately 60 total first-year places are awarded annually (distributed roughly equally between the Classe di Scienze and Classe di Lettere e Filosofia) from hundreds of top applicants across Italy and abroad.