Pi Day Maths Quiz for Olympiad Aspirants: Practice Questions & Solutions

The Pi Day Maths Quiz for Olympiad Aspirants is designed to help students practice challenging mathematical problems and strengthen their problem-solving skills. This quiz includes questions from important Olympiad topics such as Geometry, Algebra, Number Theory, and Combinatorics, which are commonly tested in exams like the Indian Olympiad Qualifier in Mathematics and Joint Entrance Examination Advanced. 

By solving these problems and reviewing their solutions, students can improve logical reasoning, develop critical thinking abilities, and gain confidence for high-level mathematics competitions.

Maths Olympiad Challenges

This session provides a "real Olympiad feel" through problem-solving practice, crucial for developing mathematical understanding and critical thinking skills. Physics Wallah faculty continuously develop resources, including questions from various foreign Olympiads (HMMT, ARML, CMIMC), to enhance learning. Upcoming programs like IQQM 2.0 further support student preparation.

Problem 1: Geometry - Rectangle and Intersecting Circle

Problem Statement:

A rectangle ABCD intersects a circle at four points E, F, G, and H. Given AE = 3, DH = 4, and GH = 5, find the value of EF.

Pedagogical Insight:

Participation in Olympiads offers numerous opportunities for self-improvement and future academic paths, including exams like IOQM, RMO, INMO, and JEE Advanced. These platforms allow students to prove their capabilities in various mathematical disciplines.

Answer: 7 (Option D)

Problem 2: Algebra – Sum of Digits

Problem Statement:

Find the sum of the digits of a given number (implies a specific number was provided in the visual problem, requiring a logical rather than brute-force approach).

Pedagogical Insight:

The European Girls' Mathematical Olympiad (EGMO) is a significant international opportunity for female students, highlighting diverse competitive avenues.

Answer: 58 (Option D)

Problem 3: Number Theory/Arithmetic – Addition Puzzle

Problem Statement:

Given an addition puzzle where three distinct numbers A, B, and C are added, resulting in a specific sum (implied by the puzzle format), find the sum A + B + C. The values A, B, and C are pair-wise distinct.

Pedagogical Insight:

The curriculum is enriched by aggregating questions from international Olympiads. The teaching methodology emphasizes problem practice over extensive theoretical lectures to foster critical thinking and boost mathematical understanding.

Answer: 17 (Option A)

Problem 4: Number Theory – Smallest Positive Integer

Problem Statement:

Find the smallest positive integer that ends with the digits 17, is divisible by 17, and whose sum of digits is 17.

Solution Approach (Hint):

The constraint that the sum of digits is 17 is key for derivation and verification.

Pedagogical Insight:

Students who excel in Olympiads demonstrate strong critical thinking skills. Maintaining continuous effort is crucial, as improvement manifests over time through perseverance.

Answer: 15317 (Option B)

Problem 5: Number Theory/Combinatorics – 2011-Digit Number

Problem Statement:

In a 2011-digit number, each pair of consecutive digits forms a multiple of either 17 or 23. The last digit is 1. Find the first digit of this number.

Solution Approach:

1. Identify multiples of 17 or 23 that end in 1.

2. Work backward from the last digit (1).

3. Construct a chain of digits where each consecutive pair satisfies the condition.

4. Determine the first digit of this 2011-digit chain.

Pedagogical Insight:

Olympiad participation builds problem-solving strategies, time management, and the ability to overcome challenges, which are highly beneficial for competitive exams like JEE Advanced. The core skill developed is critical thinking.

Answer: D (Specific digit not mentioned, only the option)

Problem 6: Algebra – Exponential Equation

Problem Statement:

Given an equation involving nested exponents (e.g., $4^{2^{3^x}} = 2^{2^{81}}$), find the value of x. The order of operations for such exponential towers is from top to bottom.

Solution Approach:

The strategy involves standardizing the base and equating exponents:

1. Standardize the base: Convert 4 to $2^2$.

• $4^{2^{3^x}} = (2^2)^{2^{3^x}} = 2^{2 \cdot 2^{3^x}} = 2^{2^{3^x+1}}$.

2. Equate exponents: Given $2^{2^{3^x+1}} = 2^{2^{81}}$, we equate the top exponents: $3^x+1 = 81$.

3. Solve for x: This simplifies further, if the example was $4^{2^x} = 2^{2^{81}}$, then $(2^2)^{2^x} = 2^{2 \cdot 2^x} = 2^{2^{x+1}}$. Equating to $2^{2^{81}}$ yields $x+1=81 \implies x=80$. (Following the provided lecture example and derivation which led to $2x+1=81 \implies x=40$, assuming a different structure than the stated example implies). Based on the final provided solution derivation, for $2^{2^{2x+1}}$ equal to $2^{2^{81}}$, then $2x+1=81 \implies x=40$.

Pedagogical Insight:

To cover academic backlog, students should commit to at least two lectures per day with two Daily Practice Problems (DPPs). This intense effort can significantly reduce backlog within a short timeframe.

Answer: 40 (Option A)

Problem 7: Number Theory/Combinatorics – Triplets of Positive Integers

Problem Statement:

How many triplets of positive integers (a, b, c) exist such that an implicit equation simplifies to a ratio of A and B, and $a+b+c=12$?

Solution Approach:

The equation simplifies to a ratio of A and B being 11:1 (i.e., A/B = 11/1). The actual calculation by the lecturer proceeds with the condition that $A+B+C \le 30$.

1. Case 1: A = 11, B = 1.

• The sum of A and B is 12. For $A+B+C \le 30$, C can range from 1 to 18 (i.e., $12+C \le 30 \implies C \le 18$). This yields 18 possible triplets.

2. Case 2: A = 22, B = 2.

• The sum of A and B is 24. For $A+B+C \le 30$, C can range from 1 to 6 (i.e., $24+C \le 30 \implies C \le 6$). This yields 6 possible triplets.

The total number of triplets is $18 + 6 = 24$.

Pedagogical Insight:

Students must trust their own problem-solving abilities and focus on individual effort, fostering self-reliance and consistency for competition success.

Answer: 24 (Option A)

Problem 8: Combinatorics – Colouring Problem

Problem Statement:

All six sides of a convex hexagon $A_1A_2A_3A_4A_5A_6$ are colored Red. Each of its diagonals is either Blue or Red. Find the number of such colourings such that each triangle $A_iA_jA_k$ has at least one Red side.

Pedagogical Insight:

Colouring Problems and Tiling Problems are important components of Combinatorics. This field often requires a personalized strategy, manual counting, and self-confidence. Developing one's own strategy is key in Game Theory and Combinatorics.

Answer:

The problem was declared a bonus question due to an error, as the correct answer, 392, was not among the options.

Problem 9: Number Theory/Algebra - Positive Integer A with Appended Digits

Problem Statement:

Three digits were appended to the end of a positive integer 'A'. The resulting number was equal to the sum of the natural numbers from 1 to A (i.e., $A(A+1)/2$). Find all possible values of 'A'.

Pedagogical Insight:

Consistency in effort is a crucial factor for success in competitive exams.

Answer: 199 (Option D)

Problem 10: Combinatorics - Theater Seating Arrangement

Problem Statement:

Ten people are seated in a row in a theatre. After a break, they rearrange themselves such that only two people remain in their original positions. The remaining eight people move to positions adjacent to their former positions (either immediately to their left or right). In how many ways could they have done this?

Pedagogical Insight:

It is never too late to begin and excel in Olympiad preparation. Dedicated effort can lead to significant improvement and success, demonstrating that any level of participation fosters growth.

Answer: 62 (Option A)

Problem 11: Geometry – Rectangle with Inscribed Triangles and In-circles

Problem Statement:

Let ABCD be a rectangle. Let S be the midpoint of the side CD. The in-circle of triangle ASD has a radius of 3. The in-circle of triangle BSC also has a radius of 3. A third triangle (e.g., ABS or ACS) has an in-radius of 4. Find the side lengths of the rectangle.

Pedagogical Insight:

Students often prefer challenging problems, highlighting a strong desire for intellectual stimulation. The faculty also maintains a commitment to continuous learning, actively seeking and solving new problems to provide high-quality instructional content.

Resource Recommendation: "The Maths Olympiad Launchpad" Book

Our book, "The Maths Olympiad Launchpad", contains many exam-oriented problems. While designed for initial Olympiad aspirants, it is particularly beneficial for students who identify specific weak topics like Number Theory, Geometry, Algebra, or Combinatorics and struggle directly with the IOQM level. We recommend starting with this book for foundational understanding to achieve significant improvement.

Book Link