Problems

Darya has made eleven pancakes, each has an area of \(1\), but each pancake can have a different shape. She places them carefully on a plate of area \(6\). Show that if the pancakes fit entirely inside the plate, then there must be two pancakes that overlap by at least an area of \(1/11\).

Four points \(A,B,C,D\) are chosen on the sides of a square of side length \(1\). The quadrilateral with vertices \(A,B,C,D\) has side lengths \(a,b,c,d\) as in the picture below. Show that \(2\leq a^2+b^2+c^2+d^2\leq 4\).

Let \(a, b, c\) be numbers such that \(a^2 + b^2 + c^2 = 1\). Show that \[-\frac12 \leq ab + bc + ac \leq 1.\]

An ordered triple of numbers is given. It is permitted to perform the following operation on the triple: to change two of them, say \(a\) and \(b\), to \(\frac{a+b}{\sqrt{2}}\) and \(\frac{a-b}{\sqrt{2}}\). Is it possible to obtain the triple \((1,\sqrt{2},1+\sqrt{2})\) from the triple \((2,\sqrt{2},\frac{1}{\sqrt{2}})\) using this operation?

(USAMO 1997) Let \(p_1, p_2, p_3,\dots\) be the prime numbers listed in increasing order, and let \(0 < x_0 < 1\) be a real number between 0 and 1. For each positive integer \(k\), define \[x_k = \begin{cases} 0 & \text{ if } x_{k-1} = 0 \\ \left\{\frac{p_k}{x_{k-1}} \right\} & \text{ if } x_{k-1} \neq 0 \end{cases}\] where \(\{x\}\) denotes the fractional part of \(x\). For example, \(\{2.53\} = 0.53\) and \(\{3.1415926...\} = 0.1415926...\). Find, with proof, all \(x_0\) satisfying \(0 <x_0 <1\) for which the sequence \(x_0, x_1, x_2,\dots\) eventually becomes 0.

Take the number \(2026^{2026}\). We remove the leading digit and add it to the remaining number. This action is repeated until there are exactly \(10\) digits left. Show that there must be two digits that are the same in the end.

Every even number is not prime. The number \(9\) is not an even number, therefore it is not not prime, i.e: the number \(9\) is prime.

In a school there are three sports clubs, which we call \(A\), \(B\), and \(C\).

A student argues as follows:

“To find how many people attend at least one club, we can add the number of people in each club. However, students who attend all three clubs get counted three times. To fix this, we should subtract them twice. Therefore, the number of people who attend at least one club is \[\text{people in }A+\text{people in }B+\text{people in }C -2\times(\text{people in all three clubs}).\]”

Is this reasoning correct?

The diagram shows a triangle drawn on a square grid. The area of the shaded triangle is \(9~\text{cm}^2\). What is the area of one of the little squares of the grid?

The first \(2026\) prime numbers are multiplied. How many zeroes are at the end of this resulting number?