Found problems: 15925
2016 Thailand Mathematical Olympiad, 9
A real number $a \ne 0$ is given. Determine all functions $f : R \to R$ satisfying $f(x)f(y) + f(x + y) = axy$ for all real numbers $x, y$.
I Soros Olympiad 1994-95 (Rus + Ukr), 11.1
Let the function $f:R \to R$ satisfies the following conditions:
1) for all $x, y\in R$, $ f(x +y) = f(x) +f(y)$
2)$ f(1)=1$
3) for all $x \ne 0$ , $ f(1/x) =\frac{f(x)}{x^2}$
Prove that for all $x \in R$, $f(x) = x$.
2005 Greece Team Selection Test, 3
Let the polynomial $P(x)=x^3+19x^2+94x+a$ where $a\in\mathbb{N}$. If $p$ a prime number, prove that no more than three numbers of the numbers $P(0), P(1),\ldots, P(p-1)$ are divisible by $p$.
2016 USAMO, 4
Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that for all real numbers $x$ and $y$,
$$(f(x)+xy)\cdot f(x-3y)+(f(y)+xy)\cdot f(3x-y)=(f(x+y))^2.$$
2005 Estonia Team Selection Test, 4
Find all pairs $(a, b)$ of real numbers such that the roots of polynomials $6x^2 -24x -4a$ and $x^3 + ax^2 + bx - 8$ are all non-negative real numbers.
2011 ELMO Shortlist, 2
Find all functions $f:\mathbb{R}^+\to\mathbb{R}^+$ such that whenever $a>b>c>d>0$ and $ad=bc$,
\[f(a+d)+f(b-c)=f(a-d)+f(b+c).\]
[i]Calvin Deng.[/i]
2015 Mathematical Talent Reward Programme, MCQ: P 14
$z=x+i y$ where $x$ and $y$ are two real numbers. Find the locus of the point $(x, y)$ in the plane, for which $\frac{z+i}{z-i}$ is purely imaginary (that is, it is of the form $i b$ where $b$ is a real number). [Here, $i=\sqrt{-1}$
[list=1]
[*] A straight line
[*] A circle
[*] A parabole
[*] None of these
[/list]
2001 AIME Problems, 11
In a rectangular array of points, with 5 rows and $N$ columns, the points are numbered consecutively from left to right beginning with the top row. Thus the top row is numbered 1 through $N,$ the second row is numbered $N+1$ through $2N,$ and so forth. Five points, $P_1, P_2, P_3, P_4,$ and $P_5,$ are selected so that each $P_i$ is in row $i.$ Let $x_i$ be the number associated with $P_i.$ Now renumber the array consecutively from top to bottom, beginning with the first column. Let $y_i$ be the number associated with $P_i$ after the renumbering. It is found that $x_1=y_2,$ $x_2=y_1,$ $x_3=y_4,$ $x_4=y_5,$ and $x_5=y_3.$ Find the smallest possible value of $N.$
2025 Austrian MO Regional Competition, 1
Let $n \geqslant 3$ be a positive integer. Furthermore, let $x_1, x_2,\ldots, x_n \in [0, 2]$ be real numbers subject to $x_1 + x_2 +\cdots + x_n = 5$. Prove the inequality$$x_1^2 + x_2^2 + \cdots + x_n^2 \leqslant 9.$$When does equality hold?
[i](Walther Janous)[/i]
LMT Accuracy Rounds, 2022 S4
Kevin runs uphill at a speed that is $4$ meters per second slower than his speed when he runs downhill. Kevin takes a total of $80$ seconds to run up and down a hill on one path. Given that the path is $300$ meters long (he travels $600$ meters total), find how long Kevin takes to run up the hill in seconds.
2017 ISI Entrance Examination, 3
Suppose $f:\mathbb{R} \to \mathbb{R}$ is a function given by
$$f(x) =\begin{cases} 1 & \mbox{if} \ x=1 \\ e^{(x^{10}-1)}+(x-1)^2\sin\frac1{x-1} & \mbox{if} \ x\neq 1\end{cases}$$
(a) Find $f'(1)$
(b) Evaluate $\displaystyle \lim_{u\to\infty} \left[100u-u\sum_{k=1}^{100} f\left(1+\frac{k}{u}\right)\right]$.
1939 Moscow Mathematical Olympiad, 043
Solve the system $\begin{cases} 3xyz -x^3 - y^3-z^3 = b^3 \\
x + y+ z = 2b \\
x^2 + y^2-z^2 = b^2
\end{cases}$ in $C$
1986 Vietnam National Olympiad, 3
Suppose $ M(y)$ is a polynomial of degree $ n$ such that $ M(y) \equal{} 2^y$ for $ y \equal{} 1, 2, \ldots, n \plus{} 1$. Compute $ M(n \plus{} 2)$.
2017 Purple Comet Problems, 17
The expression $\left(1 + \sqrt[6]{26 + 15\sqrt3} -\sqrt[6]{26 - 15\sqrt3}\right)^6= m + n\sqrt{p}$ , where $m, n$, and $p$ are positive integers, and $p$ is not divisible by the square of any prime. Find $m + n + p$.
2009 Putnam, B2
A game involves jumping to the right on the real number line. If $ a$ and $ b$ are real numbers and $ b>a,$ the cost of jumping from $ a$ to $ b$ is $ b^3\minus{}ab^2.$ For what real numbers $ c$ can one travel from $ 0$ to $ 1$ in a finite number of jumps with total cost exactly $ c?$
2019 Durer Math Competition Finals, 13
Let $k > 1$ be a positive integer and $n \ge 2019$ be an odd positive integer. The non-zero rational numbers $x_1, x_2,..., x_n$ are not all equal, and satisfy the following chain of equalities:
$$x_1 +\frac{k}{x_2}= x_2 +\frac{k}{x_3}= x_3 +\frac{k}{x_4}= ... = x_{n-1} +\frac{k}{x_n}= x_n +\frac{k}{x_1}.$$
What is the smallest possible value of $k$?
2024 Thailand Mathematical Olympiad, 7
Let $m$ and $n$ be positive integers for which $n\leq m\leq 2n$. Find the number of all complex solutions $(z_1,z_2,...,z_m)$ that satisfy
$$z_1^7+z_2^7+...+z_m^7=n$$
Such that $z_k^3-2z_k^2+2z_k-1=0$ for all $k=1,2,...,m$.
2017 IMEO, 4
Let $a,b,c$ be positive real numbers such that $abc=1$. Prove that
$$\sqrt{\frac{a^3}{1+bc}}+\sqrt{\frac{b^3}{1+ac}}+\sqrt{\frac{c^3}{1+ab}}\geq 2$$
Are there any triples $(a,b,c)$, for which the equality holds?
[i]Proposed by Konstantinos Metaxas.[/i]
2012 District Olympiad, 4
For all odd natural numbers $ n, $ prove that
$$ \left|\sum_{j=0}^{n-1} (a+ib)^j\right|\in\mathbb{Q} , $$
where $ a,b\in\mathbb{Q} $ are two numbers such that $ 1=a^2+b^2. $
2015 Spain Mathematical Olympiad, 1
On the graph of a polynomial with integer coefficients, two points are chosen with integer coordinates. Prove that if the distance between them is an integer, then the segment that connects them is parallel to the horizontal axis.
2017 Iran Team Selection Test, 3
Find all functions $f: \mathbb {R}^+ \times \mathbb {R}^+ \to \mathbb {R}^+$ that satisfy the following conditions for all positive real numbers $x,y,z:$
$$f\left ( f(x,y),z \right )=x^2y^2f(x,z)$$
$$f\left ( x,1+f(x,y) \right ) \ge x^2 + xyf(x,x)$$
[i]Proposed by Mojtaba Zare, Ali Daei Nabi[/i]
2010 Balkan MO Shortlist, A2
Let the sequence $(a_n)_{n \in \mathbb{N}}$, where $\mathbb{N}$ denote the set of natural numbers, is given with $a_1=2$ and $a_{n+1}$ $=$ $a_n^2$ $-$ $a_n+1$. Find the minimum real number $L$, such that for every $k$ $\in$ $\mathbb{N}$
\begin{align*} \sum_{i=1}^k \frac{1}{a_i} < L \end{align*}
STEMS 2023 Math Cat A, 5
Consider a polynomial $P(x) \in \mathbb{R}[x]$, with degree $2023$, such that $P(\sin^2(x))+P(\cos^2(x)) =1$ for all $x \in \mathbb{R}$. If the sum of all roots of $P$ is equal to $\dfrac{p}{q}$ with $p, q$ coprime, then what is the product $pq$?
2017 Thailand TSTST, 5
Prove that for all polynomials $P \in \mathbb{R}[x]$ and positive integers $n$, $P(x)-x$ divides $P^n(x)-x$ as polynomials.
1970 Czech and Slovak Olympiad III A, 6
Determine all real $x$ such that \[\sqrt{\tan(x)-1}\,\Bigl(\log_{\tan(x)}\bigl(2+4\cos^2(x)-2\bigr)\Bigr)\ge0.\]