Found problems: 6
2024-IMOC, A7
Given positive integers $n$, $P_1$, $P_2$, …$P_n$ and two sets
\[B=\{ (a_1,a_2,…,a_n)|a_i=0 \vee 1,\ \forall i \in \mathbb{N} \}, S=\{ (x_1,x_2,…,x_n)|1 \leq x_i \leq P_i \wedge x_i \in \mathbb{N} ,\ \forall i \in \mathbb{N} \}\]
A function $f:S \to \mathbb{Z}$ is called [b]Real[/b], if and only if for any positive integers $(y_1,y_2,…,y_n)$ and positive integer $a$ which satisfied $ 1 \leq y_i \leq P_i-a$ $\forall i \in \mathbb{N}$, we always have:
\begin{align*}
\sum_{(a_1,a_2,…,a_n) \in B \wedge 2| \sum_{i=1}^na_i}f(y+a \times a_1,y+a \times a_2,……,y+a \times a_n)&>\\
\sum_{(a_1,a_2,…,a_n) \in B \wedge 2 \nmid \sum_{i=1}^na_i}f(y+a \times a_1,y+a \times a_2,……,y+a \times a_n)&.
\end{align*}
Find the minimum of $\sum_{i_1=1}^{P_1}\sum_{i_2=1}^{P_2}....\sum_{i_n=1}^{P_n}|f(i_1,i_2,...,i_n)|$, where $f$ is a [b]Real[/b] function.
[i]Proposed by tob8y[/i]
1996 Tuymaada Olympiad, 2
Given a finite set of real numbers $A$, not containing $0$ and $1$ and possessing the property: if the number a belongs to $A$, then numbers $\frac{1}{a}$ and $1-a$ also belong to $A$. How many numbers are in the set $A$?
2004 German National Olympiad, 1
Find all real numbers $x,y$ satisfying the following system of equations
\begin{align*}
x^4 +y^4 & =17(x+y)^2 \\
xy & =2(x+y).
\end{align*}
2024 India Regional Mathematical Olympiad, 4
Let $a_1,a_2,a_3,a_4$ be real numbers such that $a_1^2 + a_2^2 + a_3^2 + a_4^2 = 1$. Show that there exist $i,j$ with $ 1 \leq i < j \leq 4$, such that $(a_i - a_j)^2 \leq \frac{1}{5}$.
1975 Bundeswettbewerb Mathematik, 1
Let $a, b, c, d$ be distinct positive real numbers. Prove that if one of the numbers $c, d$ lies between $a$ and $b$, or one of $a, b$ lies between $c$ and $d$, then
$$\sqrt{(a+b)(c+d)} >\sqrt{ab} +\sqrt{cd}$$
and that otherwise, one can choose $a, b, c, d$ so that this inequality is false.
1998 German National Olympiad, 6a
Find all real pairs $(x,y)$ that solve the system of equations \begin{align} x^5 &= 21x^3+y^3
\\ y^5 &= x^3+21y^3. \end{align}