Found problems: 15925
2020 Azerbaijan IMO TST, 3
Let $x_1, x_2, \dots, x_n$ be different real numbers. Prove that
\[\sum_{1 \leqslant i \leqslant n} \prod_{j \neq i} \frac{1-x_{i} x_{j}}{x_{i}-x_{j}}=\left\{\begin{array}{ll}
0, & \text { if } n \text { is even; } \\
1, & \text { if } n \text { is odd. }
\end{array}\right.\]
1998 IMO, 6
Determine the least possible value of $f(1998),$ where $f:\Bbb{N}\to \Bbb{N}$ is a function such that for all $m,n\in {\Bbb N}$,
\[f\left( n^{2}f(m)\right) =m\left( f(n)\right) ^{2}. \]
2014 Ukraine Team Selection Test, 11
Find all functions $f: R \to R$ that satisfy the condition $(f (x) - f (y)) (u - v) = (f (u) - f (v)) (x -y)$ for arbitrary real $x, y, u, v$ such that $x + y = u + v$.
2017 CMIMC Algebra, 2
For nonzero real numbers $x$ and $y$, define $x\circ y = \tfrac{xy}{x+y}$. Compute \[2^1\circ \left(2^2\circ \left(2^3\circ\cdots\circ\left(2^{2016}\circ 2^{2017}\right)\right)\right).\]
2018 Brazil Team Selection Test, 4
Given a set $S$ of positive real numbers, let $$\Sigma (S) = \Bigg\{ \sum_{x \in A} x : \emptyset \neq A \subset S \Bigg\}.$$
be the set of all the sums of elements of non-empty subsets of $S$. Find the least constant $L> 0$ with the following property: for every integer greater than $1$ and every set $S$ of $n$ positive real numbers, it is possible partition $\Sigma(S)$ into $n$ subsets $\Sigma_1,\ldots, \Sigma_n$ so that the ratio between the largest and smallest element of each $\Sigma_i$ is at most $L$.
2018 IMO Shortlist, A3
Given any set $S$ of positive integers, show that at least one of the following two assertions holds:
(1) There exist distinct finite subsets $F$ and $G$ of $S$ such that $\sum_{x\in F}1/x=\sum_{x\in G}1/x$;
(2) There exists a positive rational number $r<1$ such that $\sum_{x\in F}1/x\neq r$ for all finite subsets $F$ of $S$.
2017 Germany, Landesrunde - Grade 11/12, 1
Solve the equation \[ x^5+x^4+x^3+x^2=x+1 \] in $\mathbb{R}$.
Revenge ELMO 2023, 5
Complex numbers $a,b,w,x,y,z,p$ satisfy
\begin{align*}
\frac{(x-w)\lvert a-w \rvert}{(a-w)\lvert x-w \rvert}&=\text{(cyclic variants)};\\
\frac{(z-w)\lvert b-w \rvert}{(b-w)\lvert z-w \rvert}&=\text{(cyclic variants)};\\
p &= \frac{\sum_{\text{cyc}} \frac w{\lvert p-w \rvert}}{\sum_{\text{cyc}}\frac1{\lvert p-w \rvert}};
\end{align*}
where cyclic sums, equations, etc. are wrt $w,x,y,z$.
Prove that there exists a real $k$ such that
\[\sum_{\text{cyc}} \frac{(x-w)(a-w)}{\lvert x-w\rvert (p-w)}
=k\sum_{\text{cyc}} \frac{(z-w)(b-w)}{\lvert z-w\rvert(p-w)}.\]
[i]Neal Yan[/i]
1978 Czech and Slovak Olympiad III A, 1
Let $a_1,\ldots,a_n,b_1,\ldots,b_n$ be positive numbers. Show that
\[\sqrt{\left(a_1+\cdots+a_n\right)\left(b_1+\cdots+b_n\right)}\ge\sqrt{a_1b_1}+\cdots+\sqrt{a_nb_n}\]
and prove that equality holds if and only if
\[\frac{a_1}{b_1}=\cdots=\frac{a_n}{b_n}.\]
1986 IMO Longlists, 68
Consider the equation $x^4 + ax^3 + bx^2 + ax + 1 = 0$ with real coefficients $a, b$. Determine the number of distinct real roots and their multiplicities for various values of $a$ and $b$. Display your result graphically in the $(a, b)$ plane.
1976 All Soviet Union Mathematical Olympiad, 225
Given $4$ vectors $a,b,c,d$ in the plane, such that $a+b+c+d=0$. Prove the following inequality: $$|a|+|b|+|c|+|d| \ge |a+d|+|b+d|+|c+d|$$
2014 South africa National Olympiad, 2
Given that
\[\frac{a-b}{c-d}=2\quad\text{and}\quad\frac{a-c}{b-d}=3\]
for certain real numbers $a,b,c,d$, determine the value of
\[\frac{a-d}{b-c}.\]
Kvant 2021, M2641
Let $n>1$ be a given integer. The Mint issues coins of $n$ different values $a_1, a_2, ..., a_n$, where each $a_i$ is a positive integer (the number of coins of each value is unlimited). A set of values $\{a_1, a_2,..., a_n\}$ is called [i]lucky[/i], if the sum $a_1+ a_2+...+ a_n$ can be collected in a unique way (namely, by taking one coin of each value).
(a) Prove that there exists a lucky set of values $\{a_1, a_2, ..., a_n\}$ with $$a_1+ a_2+...+ a_n < n \cdot 2^n.$$
(b) Prove that every lucky set of values $\{a_1, a_2,..., a_n\}$ satisfies $$a_1+ a_2+...+ a_n >n \cdot 2^{n-1}.$$
Proposed by Ilya Bogdanov
2018 USA Team Selection Test, 2
Find all functions $f\colon \mathbb{Z}^2 \to [0, 1]$ such that for any integers $x$ and $y$,
\[f(x, y) = \frac{f(x - 1, y) + f(x, y - 1)}{2}.\]
[i]Proposed by Yang Liu and Michael Kural[/i]
2024 Chile National Olympiad., 4
Find all pairs \((x, y)\) of real numbers that satisfy the system
\[
(x + 1)(x^2 + 1) = y^3 + 1
\]
\[
(y + 1)(y^2 + 1) = x^3 + 1
\]
2016 Costa Rica - Final Round, F1
Let $a, b$ and $c$ be real numbers, and let $f (x) = ax^2 + bx + c$ and $g (x) = cx^2 + bx + a$ functions such that $| f (-1) | \le 1$, $| f (0) | \le 1$ and $| f (1) | \le 1$. Show that if $-1 \le x \le 1$, then $| f (x) | \le \frac54$ and $| g (x) | \le 2$.
2015 Polish MO Finals, 2
Let $P$ be a polynomial with real coefficients. Prove that if for some integer $k$ $P(k)$ isn't integral, then there exist infinitely many integers $m$, for which $P(m)$ isn't integral.
1959 AMC 12/AHSME, 30
$A$ can run around a circular track in $40$ seconds. $B$, running in the opposite direction, meets $A$ every $15$ seconds. What is $B$'s time to run around the track, expressed in seconds?
$ \textbf{(A)}\ 12\frac12 \qquad\textbf{(B)}\ 24\qquad\textbf{(C)}\ 25\qquad\textbf{(D)}\ 27\frac12\qquad\textbf{(E)}\ 55 $
2012 Switzerland - Final Round, 2
Determine all functions $f : R \to R$ such that for all $x, y\in R$ holds $$f (f(x) + 2f(y)) = f(2x) + 8y + 6.$$
1968 German National Olympiad, 1
Determine all ordered quadruples of real numbers $(x_1, x_2, x_3, x_4)$ for which the following system of equations exists, is fulfilled:
$$x_1 + ax_2 + x_3 = b $$
$$x_2 + ax_3 + x_4 = b $$
$$x_3 + ax_4 + x_1 = b $$
$$x_4 + ax_1 + x_2 = b$$
Here $a$ and $b$ are real numbers (case distinction!).
1986 Tournament Of Towns, (114) 1
For which natural number $k$ does $\frac{k^2}{1.001^k}$ attain its maximum value?
IV Soros Olympiad 1997 - 98 (Russia), 11.7
Solve the inequality $$\log_{\frac12} x\ge 16^x$$
2006 Germany Team Selection Test, 2
Find all functions $ f: \mathbb{R}\to\mathbb{R}$ such that $ f(x+y)+f(x)f(y)=f(xy)+2xy+1$ for all real numbers $ x$ and $ y$.
[i]Proposed by B.J. Venkatachala, India[/i]
1999 Switzerland Team Selection Test, 8
Find all $n$ for which there are real numbers $0 < a_1 \le a_2 \le ... \le a_n$ with
$$\begin{cases} \sum_{k=1}^{n}a_k = 96 \\ \\ \sum_{k=1}^{n}a_k^2 = 144 \\ \\ \sum_{k=1}^{n}a_k^3 = 216 \end{cases}$$
1995 Bulgaria National Olympiad, 5
Let $A = \{1,2,...,m + n\}$, where $m,n$ are positive integers, and let the function f : $A \to A$ be defined by:
$f(m) = 1$, $f(m+n) = m+1$ and $f(i) = i+1$ for all the other $i$.
(a) Prove that if $m$ and $n$ are odd, then there exists a function $g : A \to A$ such that $g(g(a)) = f(a)$ for all $a \in A$.
(b) Prove that if $m$ is even, then there is a function $g : A\to A$ such that $g(g(a))=f(a)$ for all $a \in A$ is and only if $n = m$.