Found problems: 15925
2005 India IMO Training Camp, 2
Find all functions $ f: \mathbb{N^{*}}\to \mathbb{N^{*}}$ satisfying
\[ \left(f^{2}\left(m\right)+f\left(n\right)\right) \mid \left(m^{2}+n\right)^{2}\]
for any two positive integers $ m$ and $ n$.
[i]Remark.[/i] The abbreviation $ \mathbb{N^{*}}$ stands for the set of all positive integers:
$ \mathbb{N^{*}}=\left\{1,2,3,...\right\}$.
By $ f^{2}\left(m\right)$, we mean $ \left(f\left(m\right)\right)^{2}$ (and not $ f\left(f\left(m\right)\right)$).
[i]Proposed by Mohsen Jamali, Iran[/i]
1984 Czech And Slovak Olympiad IIIA, 6
Let f be a function from the set Z of all integers into itself, that satisfies the condition for all $m \in Z$,
$$f(f(m)) =-m. \ \ (1)$$
Then:
(a) $f$ is a mutually unique mapping, i.e. a simple mapping of the set $Z$ onto the set $Z$ ,
(b) for all $m \in Z$ holds that $f(-m) = -f(m)$ ,
(c) $f(m) = 0$ if and only if $m = 0$ .
Prove these statements and construct an example of a mapping f that satisfies condition (1).
2007 All-Russian Olympiad, 5
Two numbers are written on each vertex of a convex $100$-gon. Prove that it is possible to remove a number from each vertex so that the remaining numbers on any two adjacent vertices are different.
[i]F. Petrov [/i]
2009 All-Russian Olympiad, 1
Find all value of $ n$ for which there are nonzero real numbers $ a, b, c, d$ such that after expanding and collecting similar terms, the polynomial $ (ax \plus{} b)^{100} \minus{} (cx \plus{} d)^{100}$ has exactly $ n$ nonzero coefficients.
2010 Albania National Olympiad, 4
The sequence of Fibonnaci's numbers if defined from the two first digits $f_1=f_2=1$ and the formula $f_{n+2}=f_{n+1}+f_n$, $\forall n \in N$.
[b](a)[/b] Prove that $f_{2010} $ is divisible by $10$.
[b](b)[/b] Is $f_{1005}$ divisible by $4$?
Albanian National Mathematical Olympiad 2010---12 GRADE Question 4.
1989 IMO Longlists, 7
Prove that $ \forall n > 1, n \in \mathbb{N}$ the equation \[ \sum^n_{k\equal{}1} \frac{x^k}{k!} \plus{} 1 \equal{} 0\] has no rational roots.
2019 LIMIT Category A, Problem 1
A can contains a mixture of two liquids A and B in the ratio $7:5$. When $9$ litres of the mixture are drawn and replaced by the same amount of liquid $B$, the ratio of $A$ and $B$ becomes $7:9$. How many litres of liquid A was contained in the can initially?
$\textbf{(A)}~18$
$\textbf{(B)}~19$
$\textbf{(C)}~20$
$\textbf{(D)}~\text{None of the above}$
2010 Today's Calculation Of Integral, 578
Find the range of $ k$ for which the following inequality holds for $ 0\leq x\leq 1$.
\[ \int_0^x \frac {dt}{\sqrt {(3 \plus{} t^2)^3}}\geq k\int _0^x \frac {dt}{\sqrt {3 \plus{} t^2}}\]
If necessary, you may use $ \ln 3 \equal{} 1.10$.
1982 Bundeswettbewerb Mathematik, 3
Given that $a_1, a_2, . . . , a_n$ are nonnegative real numbers with $a_1 + \cdots + a_n = 1$, prove that the expression
$$ \frac{a_1}{1+a_2 +a_3 +\cdots +a_n }\; +\; \frac{a_2}{1+a_1 +a_3 +\cdots +a_n }\; +\; \cdots \; +\, \frac{a_n }{1+a_1 +a_2+\cdots +a_{n-1} }$$
attains its minimum, and determine this minimum.
1996 Spain Mathematical Olympiad, 4
For each real value of $p$, find all real solutions of the equation $\sqrt{x^2 - p}+2\sqrt{x^2-1} = x$.
1967 IMO Longlists, 21
Without using tables, find the exact value of the product:
\[P = \prod^7_{k=1} \cos \left(\frac{k \pi}{15} \right).\]
2018 Hanoi Open Mathematics Competitions, 7
Let $\{u_n\}_ {n\ge 1}$ be given sequence satisfying the conditions: $u_1 = 0$, $u_2 = 1$, $u_{n+1} = u_{n-1} + 2n - 1$ for $n \ge 2$.
1) Calculate $u_5$.
2) Calculate $u_{100} + u_{101}$.
2016 Mathematical Talent Reward Programme, SAQ: P 1
Show that there exist a polynomial $P(x)$ whose one cofficient is $\frac{1}{2016}$ and remaining cofficients are rational numbers, such that $P(x)$ is an integer for any integer $x$ .
1903 Eotvos Mathematical Competition, 2
For a given pair of values $x$ and $y$ satisfying $x = \sin \alpha , y = \sin \beta$ , there can be four different values of $z = \sin( \alpha +\beta )$.
(a) Set up a relation between $x, y$ and $z$ not involving trigonometric functions or radicals.
(b) Find those pairs of values $(x, y)$ for which $z = \sin (\alpha +\beta )$ takes on fewer than four distinct values.
2018 Romania National Olympiad, 2
Let $a, b, c, d$ be natural numbers such that $a + b + c + d = 2018$. Find the minimum value of the expression:
$$E = (a-b)^2 + 2(a-c)^2 + 3(a-d)^2+4(b-c)^2 + 5(b-d)^2 + 6(c-d)^2.$$
1988 Czech And Slovak Olympiad IIIA, 2
If for the coefficients of equation $x^3+ax^2+bx+c=0$ whose roots are all real, holds, $a^2= 2(b+1)$ then $|a-c|\le 2$. Prove it.
KoMaL A Problems 2017/2018, A. 706
Find all positive integer $k$s for which such $f$ exists and unique:
$f(mn)=f(n)f(m)$ for $n, m \in \mathbb{Z^+}$
$f^{n^k}(n)=n$ for all $n \in \mathbb{Z^+}$ for which $f^x (n)$ means the n times operation of function $f$(i.e. $f(f(...f(n))...)$)
2018 Thailand TSTST, 5
Find all triples of real numbers $(a, b, c)$ satisfying $$a+b+c=14, \quad a^2+b^2+c^2=84,\quad a^3+b^3+c^3=584.$$
1994 Baltic Way, 4
Is there an integer $n$ such that $\sqrt{n-1}+\sqrt{n+1}$ is a rational number?
2014 NIMO Problems, 8
Define the function $\xi : \mathbb Z^2 \to \mathbb Z$ by $\xi(n,k) = 1$ when $n \le k$ and $\xi(n,k) = -1$ when $n > k$, and construct the polynomial \[ P(x_1, \dots, x_{1000}) = \prod_{n=1}^{1000} \left( \sum_{k=1}^{1000} \xi(n,k)x_k \right). \]
(a) Determine the coefficient of $x_1x_2 \dots x_{1000}$ in $P$.
(b) Show that if $x_1, x_2, \dots, x_{1000} \in \left\{ -1,1 \right\}$ then $P(x_1,x_2,\dots,x_{1000}) = 0$.
[i]Proposed by Evan Chen[/i]
2016 Brazil Team Selection Test, 4
Suppose that a sequence $a_1,a_2,\ldots$ of positive real numbers satisfies \[a_{k+1}\geq\frac{ka_k}{a_k^2+(k-1)}\] for every positive integer $k$. Prove that $a_1+a_2+\ldots+a_n\geq n$ for every $n\geq2$.
1982 IMO Longlists, 16
Let $p(x)$ be a cubic polynomial with integer coefficients with leading coefficient $1$ and with one of its roots equal to the product of the other two. Show that $2p(-1)$ is a multiple of $p(1)+p(-1)-2(1+p(0)).$
2011 Princeton University Math Competition, A6 / B7
A sequence of real numbers $\{a_n\}_{n = 1}^\infty (n=1,2,...)$ has the following property:
\begin{align*}
6a_n+5a_{n-2}=20+11a_{n-1}\ (\text{for }n\geq3).
\end{align*}
The first two elements are $a_1=0, a_2=1$. Find the integer closest to $a_{2011}$.
1949-56 Chisinau City MO, 53
Solve the equation: $\sqrt[3]{a+\sqrt{x}}+\sqrt[3]{a-\sqrt{x}}=\sqrt[3]{b}$
2010 Polish MO Finals, 2
Positive rational number $a$ and $b$ satisfy the equality
\[a^3 + 4a^2b = 4a^2 + b^4.\]
Prove that the number $\sqrt{a}-1$ is a square of a rational number.