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.
At the Oscar Awards 2025, 5 films were nominated for Best Production Design and 5 films were nominated for Best Cinematography. In fact, 3 films were nominated for both categories. What is the total number of films nominated for these two categories?
150 young adults were asked how they commuted to work. 125 said they took the underground and 93 said they cycled. Of all the people interviewed, 72 said they both cycled and took the underground. How many people do not cycle and do not take the underground?
Show that there are no more than 269 prime numbers less than or equal to 1000.
200 people were asked if they drank one of the following beverages regularly: tea, coffee and beer. 165 people said they drank at least one of these beverages. Funnily, for every choice of a pair of beverages, exactly 122 people said they drank at least one beverage out of the pair. The even stranger fact was that for each choice of a beverage, exactly 73 people admitted to drinking it.
How many people drink all three beverages?
Each student chooses at least one from the \(n\) different modules offered at a university. Let us number these modules as \(1,2,3,\dots,n\). For each natural number \(1\leq k\leq n\), we denote the number of students choosing the modules \(i_1,\dots,i_k\) by \(S(i_1,\dots,i_k)\). Give a formula for the number of students in terms of the numbers \(S(i_1,\dots,i_k)\).
As an example, if \(n = 5\), \(k=3\) and we look at \(i_1 = 4,i_2 = 2, i_3 =1\), then \(S(i_1,i_2,i_3) = S(4,2,1)\) is the number of students picking the modules \(1,2,4\).
There are \(n\) seats on a plane and each of the \(n\) passengers sat in the wrong seat. What is the total number of ways this could happen?