9 responses

1. To find the minimum of the given expression, we can use the Cauchy-Schwarz inequality and the fact that the sum of squares of positive real numbers is minimized when the numbers are proportional.

   Let’s consider the sum: S = √(1^2 + a₁^2) + √(2^2 + a₂^2) + … + √(8^2 + a₈^2)

   Using the Cauchy-Schwarz inequality, we have: S² ≤ (1 + a₁² + 2 + a₂² + … + 8 + a₈²) × (1 + 1 + … + 1 + 1) = (1 + 2 + … + 8 + a₁² + a₂² + … + a₈²) × 8

   Since a₁, a₂, …, a₈ are positive real numbers that add up to 27, we can rewrite the above inequality as: S² ≤ (1 + 2 + … + 8 + 27) × 8 = (36) × 8 = 288

   Now, let’s consider the case of equality in the Cauchy-Schwarz inequality. This occurs when the numbers 1/a₁, 1/a₂, …, 1/a₈ are proportional to a₁, a₂, …, a₈, respectively. In other words, we want to find positive constants k₁, k₂, …, k₈ such that:

   1/a₁ = k₁ × a₁ 1/a₂ = k₂ × a₂ … 1/a₈ = k₈ × a₈

   Rearranging the above equations, we get:

   a₁² = 1/k₁ a₂² = 1/k₂ … a₈² = 1/k₈

   Since we want to find the minimum of the expression, we can assume that the values of a₁, a₂, …, a₈ that minimize the expression satisfy the equality condition above.

   Now, since the sum of a₁, a₂, …, a₈ is 27, we have:

   a₁ + a₂ + … + a₈ = 27

   Substituting the above equations, we get:

   √(1^2 + a₁^2) + √(2^2 + a₂^2) + … + √(8^2 + a₈^2) = √(1^2 + 1/k₁) + √(2^2 + 1/k₂) + … + √(8^2 + 1/k₈)

   To minimize this expression, we need to find the values of k₁, k₂, …, k₈ such that the sum is minimized. Since k₁, k₂, …, k₈ are positive constants, their reciprocals 1/k₁, 1/k₂, …, 1/k₈ should be proportional to the squares of the positive integers 1, 2, …, 8.

   Therefore, the minimum of the given expression can be achieved when a₁, a₂, …, a₈ are rational numbers, but not necessarily rational themselves. The values of a₁, a₂, …, a₈ that minimize the expression should satisfy the equality condition derived above.

   Vivek Kumar – June 28th, 2023

2. 45？

   benben – August 11th, 2023

3. A beautiful problem which demonstrates the importance of shifting perspective in mathematics.

   Algebraic methods are unnecessary here; this is a geometry problem about minimizing the total length of eight lines joined end-to-end, so that the overall zigzag path connects opposite corners of a 36 by 27 rectangle. To find this minimum we simply take the diagonal line. Considering similarity to a 3-4-5 right triangle, we see that this diagonal has length 45. By similarity again we can easily calculate the values of the a_i’s and observe that they are all rational.

   The positivity condition can be dropped, because allowing the zigzag path to go up and down does not change the fact that the diagonal is the shortest path between opposite corners.

   Slight variation problem for readers:

   Suppose \(a_1, \ldots, a_8 \) are real numbers which add to \(27\). And suppose that \(b_1, \ldots,b_8\) are real numbers which add to \(60\). Find the minimum of

   \( \sqrt{1^2 + a_1^2 + b_1^2} + \sqrt{2^2 + a_2^2 + b_2^2} + \cdots + \sqrt{8^2 + a_8^2 + b_8^2}. \)

   Richard E – August 14th, 2023

4. To find the minimum of the given expression:

   √(1^2 + a1^2) + √(2^2 + a2^2) + ⋯ + √(8^2 + a8^2)

   we can use the Cauchy-Schwarz inequality, which states that for any two vectors u and v in an inner product space:

   (u · v)^2 ≤ (u · u) * (v · v)

   In this case, we can consider two vectors u and v, where:

   u = (1, 2, 3, 4, 5, 6, 7, 8) v = (a1, a2, a3, a4, a5, a6, a7, a8)

   Now, let’s compute u · u and v · v:

   u · u = 1^2 + 2^2 + 3^2 + 4^2 + 5^2 + 6^2 + 7^2 + 8^2 = 204

   v · v = a1^2 + a2^2+ a3^2 + a4^2 + a5^2 + a6^2 + a7^2 + a8^2

   According to the Cauchy-Schwarz inequality, we have:

   (u · v)^2 ≤ (u · u) * (v · v)

   [(1 * a1 + 2 * a2 + 3 * a3 + 4 * a4 + 5 * a5 + 6 * a6 + 7 * a7 + 8 * a8)^2] ≤ 204 * (a1^2 + a2^2 + a3^2 + a4^2 + a5^2 + a6^2 + a7^2 + a8^2)

   Now, let’s substitute the sum of ai’s, which is 27:

   (1 * a1 + 2 * a2 + 3 * a3 + 4 * a4 + 5 * a5 + 6 * a6 + 7 * a7 + 8 * a8)^2 ≤ 204 * 27

   (1 * a1 + 2 * a2 + 3 * a3 + 4 * a4 + 5 * a5 + 6 * a6 + 7 * a7 + 8 * a8)^2 ≤ 5508

   Now, let’s minimize the left-hand side of the inequality to find the minimum value of the given expression. To minimize it, we should choose the values of ai that maximize the left-hand side of the inequality.

   The values that maximize the left-hand side of the inequality are the values of ai such that they are proportional to the corresponding values in vector u. In other words:

   a1 = 1 * k a2 = 2 * k a3 = 3 * k a4 = 4 * k a5 = 5 * k a6 = 6 * k a7 = 7 * k a8 = 8 * k

   where k is a positive constant.

   Now, let’s plug these values into the inequality:

   [(1 * (1k) + 2 * (2k) + 3 * (3k) + 4 * (4k) + 5 * (5k) + 6 * (6k) + 7 * (7k) + 8 * (8k))^2] ≤ 5508

   [(1k + 4k + 9k + 16k + 25k + 36k + 49k + 64k)^2] ≤ 5508

   [(204k)^2] ≤ 5508

   (204^2)k^2 ≤ 5508

   k^2 ≤ 5508 / (204^2)

   k^2 ≤ 0.1324

   k ≤ √0.1324

   k ≤ 0.364

   Now, let’s find the minimum value of the given expression by plugging in these values of ai:

   √(1^2 + (1 * 0.364)^2) + √(2^2 + (2 * 0.364)^2) + ⋯ + √(8^2 + (8 * 0.364)^2)

   Calculating this sum gives us the minimum value of the expression. This minimum value cannot be rational since it involves square roots of irrational numbers. However, we have found the values of ai that minimize the expression, and they are not rational.

   xyz – August 17th, 2023

5. I would really like to understand how maths works here.

   Suppose we have some \sqrt{a}, the how does that appear?

   Varun – November 13th, 2023

6. This is the way I did it. I don’t know the Cauchy Schwarz inequality but I did it by thinking of each one as a hypotenuse of a triangle. And drew 8 triangles with one corner on the origin and they all lied on the x-axis. Then reasoned that to minimise the sum of all the hypotenuses I would need them all to lie on the same line. Then I yield the answer of 45, and all of them can be rational.

   Daniel Fostiak – November 13th, 2023

7. Would you be able to post solutions to q4-6 before interviews this year? Thank you

   HD – November 13th, 2023

8. Could we not consider a more trigonometric approach? ai=(i)tan(θ)i. We then simplify to get secθ1 2sec θ2 3sec θ3………8sec θ 8 and the use the fact that summation from 0 to 8 of itan(θi)=27 always?

   B) – November 13th, 2023

9. The Lagrangian of the constrained optimization problem is given by \[ L(a_1, …, a_8, \lambda, s_1, …, s_8, \lambda_1, …, \lambda_8) = \sum_{i=1}^8 \sqrt{i^2 a_i^2} \lambda \left(\sum_{i=1}^8 a_i - 27 \right) \sum_{i=1}^8 \lambda_i \left(a_i - s_i^2\right) \] Note that \( \lambda \neq 0 \) since the minimum over positives \( a_i \) is approached as \( a_i \to 0 \), which does not satisfy the constraint on the sum. Now, setting the partial derivatives wrt \( a_i \) to \( 0 \), we get \[ \frac{a_i}{\sqrt{i^2 a_i^2}} \lambda = 0 \] which solves to \[ a_i^2 = \frac{\lambda^2 i^2}{1 - \lambda^2} \] Hence, we have \( a_i \propto i \), yielding \( a_i = 3i/4 \). The sum can be calculated to be \(45 \).

   JS – November 13th, 2023