Problems

Michael used different numbers \(\{0,1,2,3,4,5,6,7,8,9\}\) to put in the circles in the picture below, without using any one of them twice. Inside each triangle he wrote down either the sum or the product of the numbers at its vertices. Then he erased the numbers in the circles. Which numbers need to be written in circles so that the condition is satisfied?

Solve the puzzle: \[\textrm{AC}\times\textrm{CC}\times\textrm{K} = 2002.\] Different letters correspond to different digits, identical letters correspond to identical digits.

• We call two figures congruent if their corresponding sides and angles are equal. Let \(ABD\) an \(A'B'D'\) be two right-angled triangles with right angle \(D\). Then if \(AD=A'D'\) and \(AB=A'B'\) then the triangles \(ABD\) and \(A'B'D'\) are congruent.

• It follows from the previous statement that if two lines \(AB\) and \(CD\) are parallel than angles \(BCD\) and \(CBA\) are equal.

We prove the other two assertions from the description:

• The sum of all internal angles of a triangle is also \(180^{\circ}\).

• In an isosceles triangle (which has two sides of equal lengths), two angles touching the third side are equal.

In the triangle \(ABC\) the sides are compared as following: \(AC>BC>AB\). Prove that the angles are compared as follows: \(\angle B > \angle A > \angle C\).

Find all solutions of the puzzle \(HE \times HE = SHE\). Different letters denote different digits, while the same letters correspond to the same digits.

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 can meet either Sam, or Pippin, or Merry in the fog. One day everyone came out to meet Frodo, but the fog was thick, and Frodo could not see where everyone was, so he asked each of his friends to introduce themselves.
The one who from Frodo’s perspective was on the left, said: "Merry is next to me."
The one on Frodo’s right said: "The person who just spoke is Pippin."
Finally, the one in the center announced, "On my left is Sam."
Identify who stood where, knowing that Sam always lies, Pippin sometimes lies, and Merry never lies?

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 \(p\)?