Found problems: 1311
2010 District Olympiad, 2
Consider the sequence $ (x_n)_{n\ge 0}$ where $ x_n\equal{}2^{n}\minus{}1\ ,\ n\in \mathbb{N}$. Determine all the natural numbers $ p$ for which:
\[ s_p\equal{}x_0\plus{}x_1\plus{}x_2\plus{}...\plus{}x_p\]
is a power with natural exponent of $ 2$.
1979 IMO Longlists, 65
Given a function $f$ such that $f(x)\le x\forall x\in\mathbb{R}$ and $f(x+y)\le f(x)+f(y)\forall \{x,y\}\in\mathbb{R}$, prove that $f(x)=x\forall x\in\mathbb{R}$.
2012 Turkey Team Selection Test, 1
Let $S_r(n)=1^r+2^r+\cdots+n^r$ where $n$ is a positive integer and $r$ is a rational number. If $S_a(n)=(S_b(n))^c$ for all positive integers $n$ where $a, b$ are positive rationals and $c$ is positive integer then we call $(a,b,c)$ as [i]nice triple.[/i] Find all nice triples.
2014 ELMO Shortlist, 5
Let $\mathbb R^\ast$ denote the set of nonzero reals. Find all functions $f: \mathbb R^\ast \to \mathbb R^\ast$ satisfying \[ f(x^2+y)+1=f(x^2+1)+\frac{f(xy)}{f(x)} \] for all $x,y \in \mathbb R^\ast$ with $x^2+y\neq 0$.
[i]Proposed by Ryan Alweiss[/i]
2013 Czech-Polish-Slovak Match, 3
For each rational number $r$ consider the statement: If $x$ is a real number such that $x^2-rx$ and $x^3-rx$ are both rational, then $x$ is also rational.
[list](a) Prove the claim for $r \ge \frac43$ and $r \le 0$.
(b) Let $p,q$ be different odd primes such that $3p <4q$. Prove that the claim for $r=\frac{p}q$ does not hold.
[/list]
1997 Romania Team Selection Test, 1
Let $P(X),Q(X)$ be monic irreducible polynomials with rational coefficients. suppose that $P(X)$ and $Q(X)$ have roots $\alpha$ and $\beta$ respectively, such that $\alpha + \beta $ is rational. Prove that $P(X)^2-Q(X)^2$ has a rational root.
[i]Bogdan Enescu[/i]
2004 Germany Team Selection Test, 2
Find all functions $f: \Bbb{R}_{0}^{+}\rightarrow \Bbb{R}_{0}^{+}$ with the following properties:
(a) We have $f\left( xf\left( y\right) \right) \cdot f\left( y\right) =f\left( x+y\right)$ for all $x$ and $y$.
(b) We have $f\left(2\right) = 0$.
(c) For every $x$ with $0 < x < 2$, the value $f\left(x\right)$ doesn't equal $0$.
[b]NOTE.[/b] We denote by $\Bbb{R}_{0}^{+}$ the set of all non-negative real numbers.
2010 Indonesia TST, 3
Determine all real numbers $ a$ such that there is a function $ f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying \[ x\plus{}f(y)\equal{}af(y\plus{}f(x))\] for all real numbers $ x$ and $ y$.
[i]Hery Susanto, Malang[/i]
2008 ISI B.Math Entrance Exam, 3
Let $z$ be a complex number such that $z,z^2,z^3$ are all collinear in the complex plane . Show that $z$ is a real number .
2013 District Olympiad, 4
At the top of a piece of paper is written a list of distinctive natural numbers. To continue the list you must choose 2 numbers from the existent ones and write in the list the least common multiple of them, on the condition that it isn’t written yet. We can say that the list is closed if there are no other solutions left (for example, the list 2, 3, 4, 6 closes right after we add 12). Which is the maximum numbers which can be written on a list that had closed, if the list had at the beginning 10 numbers?
2003 Vietnam National Olympiad, 3
Let $\mathcal{F}$ be the set of all functions $f : (0,\infty)\to (0,\infty)$ such that $f(3x) \geq f( f(2x) )+x$ for all $x$. Find the largest $A$ such that $f(x) \geq A x$ for all $f\in\mathcal{F}$ and all $x$.
1988 Balkan MO, 2
Find all polynomials of two variables $P(x,y)$ which satisfy
\[P(a,b) P(c,d) = P (ac+bd, ad+bc), \forall a,b,c,d \in \mathbb{R}.\]
2007 IberoAmerican Olympiad For University Students, 5
Determine all pairs of polynomials $f,g\in\mathbb{C}[x]$ with complex coefficients such that the following equalities hold for all $x\in\mathbb{C}$:
$f(f(x))-g(g(x))=1+i$
$f(g(x))-g(f(x))=1-i$
2014 China National Olympiad, 3
Prove that: there exists only one function $f:\mathbb{N^*}\to\mathbb{N^*}$ satisfying:
i) $f(1)=f(2)=1$;
ii)$f(n)=f(f(n-1))+f(n-f(n-1))$ for $n\ge 3$.
For each integer $m\ge 2$, find the value of $f(2^m)$.
2012 ELMO Shortlist, 4
Let $a_0,b_0$ be positive integers, and define $a_{i+1}=a_i+\lfloor\sqrt{b_i}\rfloor$ and $b_{i+1}=b_i+\lfloor\sqrt{a_i}\rfloor$ for all $i\ge0$. Show that there exists a positive integer $n$ such that $a_n=b_n$.
[i]David Yang.[/i]
2009 Baltic Way, 1
A polynomial $p(x)$ of degree $n\ge 2$ has exactly $n$ real roots, counted with multiplicity. We know that the coefficient of $x^n$ is $1$, all the roots are less than or equal to $1$, and $p(2)=3^n$. What values can $p(1)$ take?
2012 China Team Selection Test, 3
$n$ being a given integer, find all functions $f\colon \mathbb{Z} \to \mathbb{Z}$, such that for all integers $x,y$ we have $f\left( {x + y + f(y)} \right) = f(x) + ny$.
2002 Iran MO (3rd Round), 18
Find all continious $f: \mathbb R\longrightarrow\mathbb R$ that for any $x,y$ \[f(x)+f(y)+f(xy)=f(x+y+xy)\]
2009 Indonesia TST, 3
Find all function $ f: \mathbb{R} \rightarrow \mathbb{R}$ such that
\[ f(x \plus{} y)(f(x) \minus{} y) \equal{} xf(x) \minus{} yf(y)
\]
for all $ x,y \in \mathbb{R}$.
2005 Turkey Team Selection Test, 1
Find all functions $ f :\mathbb{R}_{0}^{+}\mapsto\mathbb{R}_{0}^{+} $ satisfying the conditions $4f(x)\geq 3x$ and $f(4f(x)-3x)=x$ for all $x\geq 0$ .
2008 Korea - Final Round, 3
Determine all functions $f : \mathbb{R}^+\rightarrow\mathbb{R}$ that satisfy the following
$f(1)=2008$, $|{f(x)}| \le x^2+1004^2$, $f\left (x+y+\frac{1}{x}+\frac{1}{y}\right )=f\left (x+\frac{1}{y}\right )+f\left (y+\frac{1}{x}\right ).$
1987 IMO Longlists, 15
Let $a_1, a_2, a_3, b_1, b_2, b_3, c_1, c_2, c_3$ be nine strictly positive real numbers. We set
\[S_1 = a_1b_2c_3, \quad S_2 = a_2b_3c_1, \quad S_3 = a_3b_1c_2;\]\[T_1 = a_1b_3c_2, \quad T_2 = a_2b_1c_3, \quad T_3 = a_3b_2c_1.\]
Suppose that the set $\{S1, S2, S3, T1, T2, T3\}$ has at most two elements.
Prove that
\[S_1 + S_2 + S_3 = T_1 + T_2 + T_3.\]
2013 India IMO Training Camp, 3
Let $h \ge 3$ be an integer and $X$ the set of all positive integers that are greater than or equal to $2h$. Let $S$ be a nonempty subset of $X$ such that the following two conditions hold:
[list]
[*]if $a + b \in S$ with $a \ge h, b \ge h$, then $ab \in S$;
[*]if $ab \in S$ with $a \ge h, b \ge h$, then $a + b \in S$.[/list]
Prove that $S = X$.
2010 JBMO Shortlist, 1
The real numbers $a$, $b$, $c$, $d$ satisfy simultaneously the equations
\[abc -d = 1, \ \ \ bcd - a = 2, \ \ \ cda- b = 3, \ \ \ dab - c = -6.\] Prove that $a + b + c + d \not = 0$.
2012 Puerto Rico Team Selection Test, 1
Let $x, y$ and $z$ be consecutive integers such that
\[\frac 1x+\frac 1y+\frac 1z >\frac{1}{45}.\]
Find the maximum value of $x + y + z$.