Alex writes natural numbers in a row: \(123456789101112...\) Counting from the beginning, in what places do the digits \(555\) first appear? For example, \(101\) first appears in the 10th, 11th and 12th places.
Frodo ventures into a thick fog where he is to meet his three companions: Sam, Merry, and Pippin. He can tell they are standing in a row before him – one on the left, one in the middle, and one on the right – but he cannot see who is who. To help, he asks each of them to speak.
Remember: Sam always lies, Merry always tells the truth, and Pippin sometimes lies and sometimes tells the truth.
Here is what Frodo hears:
The one on the left says: “Merry is next to me.”
The one on the right says: “The person who just spoke is Pippin.”
The one in the middle says: “On my left is Sam.”
Can you work out who is standing on Frodo’s left, in the middle, and on the right?
Using areas of squares and rectangles, show that for any positive
real numbers \(a\) and \(b\), \((a+b)^2 =
a^2+2ab+b^2\).
The identity above is true for any real numbers, not necessarily
positive, in fact in order to prove it the usual way one only needs to
remember that multiplication is commutative and the distributive
property of addition and multiplication:
\(a\times b = b\times a\);
\((a+b)\times c = a\times c + b\times c\).
Annie found a prime number \(p\) to which you can add \(4\) to make it a perfect square. What is the value of \(p\)?
Let \(a\) and \(b\) be positive real numbers. Using areas
of rectangles and squares, show that \(a^2 -
b^2 = (a-b) \times (a+b)\).
Try to prove it in two ways, one geometric and one algebraic.
Let \(a\) and \(b\) be positive real numbers. Using volumes
of cubes and parallelepipeds, show that \((a+b)^3 = a^3 +3a^2b+3ab^2 +b^3\).
Hint: Place the cubes with sides \(a\)
and \(b\) along the same diagonal.
The real numbers \(a,b,c\) are non-zero and satisfy the following equations: \[\left\{ \begin{array}{l} a^2 +a = b^2 \\ b^2 +b = c^2 \\ c^2 +c = a^2. \end{array} \right.\] Show that \((a-b)(b-c)(c-a)=1\).
A five-digit number is called indecomposable if it is not decomposed into the product of two three-digit numbers. What is the largest number of indecomposable five-digit numbers that can come in a row?
Find the representation of \((a+b)^n\) as the sum of \(X_{n,k}a^kb^{n-k}\) for general \(n\). Here by \(X_{n,k}\) we denote coefficients that depend only on \(k\) and \(n\).
The positive real numbers \(a, b, c, x, y\) satisfy the following system of equations: \[\left\{ \begin{aligned} x^2 + xy + y^2 = a^2\\ y^2 + yz + z^2 = b^2\\ x^2 + xz + z^2 = c^2 \end{aligned} \right.\]
Find the value of \(xy + yz + xz\) in terms of \(a, b,\) and \(c.\)