Found problems: 1311
2010 Iran MO (3rd Round), 2
prove the third sylow theorem: suppose that $G$ is a group and $|G|=p^em$ which $p$ is a prime number and $(p,m)=1$. suppose that $a$ is the number of $p$-sylow subgroups of $G$ ($H<G$ that $|H|=p^e$). prove that $a|m$ and $p|a-1$.(Hint: you can use this: every two $p$-sylow subgroups are conjugate.)(20 points)
2000 Iran MO (3rd Round), 3
Suppose $f : \mathbb{N} \longrightarrow \mathbb{N}$ is a function that satisfies $f(1) = 1$ and
$f(n + 1) =\{\begin{array}{cc} f(n)+2&\mbox{if}\ n=f(f(n)-n+1),\\f(n)+1& \mbox{Otherwise}\end {array}$
$(a)$ Prove that $f(f(n)-n+1)$ is either $n$ or $n+1$.
$(b)$ Determine$f$.
1984 Balkan MO, 4
Let $a,b,c$ be positive real numbers. Find all real solutions $(x,y,z)$ of the system:
\[ ax+by=(x-y)^{2}
\\ by+cz=(y-z)^{2}
\\ cz+ax=(z-x)^{2}\]
2010 IberoAmerican Olympiad For University Students, 7
(a) Prove that, for any positive integers $m\le \ell$ given, there is a positive integer $n$ and positive integers $x_1,\cdots,x_n,y_1,\cdots,y_n$ such that the equality \[ \sum_{i=1}^nx_i^k=\sum_{i=1}^ny_i^k\] holds for every $k=1,2,\cdots,m-1,m+1,\cdots,\ell$, but does not hold for $k=m$.
(b) Prove that there is a solution of the problem, where all numbers $x_1,\cdots,x_n,y_1,\cdots,y_n$ are distinct.
[i]Proposed by Ilya Bogdanov and Géza Kós.[/i]
2014 Contests, 2
Find all real non-zero polynomials satisfying $P(x)^3+3P(x)^2=P(x^{3})-3P(-x)$ for all $x\in\mathbb{R}$.
2002 JBMO ShortLists, 7
Consider integers $ a_i,i\equal{}\overline{1,2002}$ such that
$ a_1^{ \minus{} 3} \plus{} a_2^{ \minus{} 3} \plus{} \ldots \plus{} a_{2002}^{ \minus{} 3} \equal{} \frac {1}{2}$
Prove that at least 3 of the numbers are equal.
1977 Canada National Olympiad, 6
Let $0 < u < 1$ and define
\[u_1 = 1 + u, \quad u_2 = \frac{1}{u_1} + u, \quad \dots, \quad u_{n + 1} = \frac{1}{u_n} + u, \quad n \ge 1.\]
Show that $u_n > 1$ for all values of $n = 1$, 2, 3, $\dots$.
2012 Bulgaria National Olympiad, 3
We are given a real number $a$, not equal to $0$ or $1$. Sacho and Deni play the following game. First is Sasho and then Deni and so on (they take turns). On each turn, a player changes one of the “*” symbols in the equation:
\[*x^4+*x^3+*x^2+*x^1+*=0\]
with a number of the type $a^n$, where $n$ is a whole number. Sasho wins if at the end the equation has no real roots, Deni wins otherwise. Determine (in term of $a$) who has a winning strategy
2014 Contests, 2
Determine all the functions $f : \mathbb{R}\rightarrow\mathbb{R}$ that satisfies the following.
$f(xf(x)+f(x)f(y)+y-1)=f(xf(x)+xy)+y-1$
2010 IberoAmerican Olympiad For University Students, 4
Let $p(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$ be a monic polynomial of degree $n>2$, with real coefficients and all its roots real and different from zero. Prove that for all $k=0,1,2,\cdots,n-2$, at least one of the coefficients $a_k,a_{k+1}$ is different from zero.
2000 Iran MO (3rd Round), 1
Let $n$ be a positive integer. Suppose $S$ is a set of ordered $n-\mbox{tuples}$ of
nonnegative integers such that, whenever $(a_1,\dots,an)\in S$ and $b_i$ are nonnegative integers with$b_i\le a_i$, the $n-\text{tuple}$ $(b_1,\dots,b_n)$ is also in $S$. If $h_m$
is the number of elements of $S$ with the sum of components equal to$m$,
prove that $h_m$ is a polynomial in $m$ for all sufficiently large$m$.
2007 Korea - Final Round, 6
Let f:N→N be a function satisfying $ kf(n)\le f(kn)\le kf(n) \plus{} k \minus{} 1$ for all $ k, n\in N$.
(a)Prove that $ f(a) \plus{} f(b)\le f(a \plus{} b)\le f(a) \plus{} f(b) \plus{} 1$ for all $ a, b\in N$.
(b)If $ f$ satisfies $ f(2007n)\le 2007f(n) \plus{} 200$ for every $ n\in N$, show that there exists $ c\in N$ such that $ f(2007c) \equal{} 2007f(c)$.
1994 Baltic Way, 1
Let $a\circ b=a+b-ab$. Find all triples $(x,y,z)$ of integers such that
\[(x\circ y)\circ z +(y\circ z)\circ x +(z\circ x)\circ y=0\]
2013 China Team Selection Test, 3
Find all positive real numbers $r<1$ such that there exists a set $\mathcal{S}$ with the given properties:
i) For any real number $t$, exactly one of $t, t+r$ and $t+1$ belongs to $\mathcal{S}$;
ii) For any real number $t$, exactly one of $t, t-r$ and $t-1$ belongs to $\mathcal{S}$.
2004 Poland - Second Round, 1
Positive real numbers $a,b,c,d$ satisfy the equalities
\[a^3+b^3+c^3=3d^3\\ b^4+c^4+d^4=3a^4\\ c^5+d^5+a^5=3b^5. \]
Prove that $a=b=c=d$.
1995 Vietnam National Olympiad, 2
Find all poltnomials $ P(x)$ with real coefficients satisfying: For all $ a>1995$, the number of real roots of $ P(x)\equal{}a$ (incuding multiplicity of each root) is greater than 1995, and every roots are greater than 1995.
2000 JBMO ShortLists, 11
Prove that for any integer $n$ one can find integers $a$ and $b$ such that
\[n=\left[ a\sqrt{2}\right]+\left[ b\sqrt{3}\right] \]
2001 Federal Competition For Advanced Students, Part 2, 2
Determine all triples of positive real numbers $(x, y, z)$ such that
\[x+y+z=6,\]\[\frac 1x + \frac 1y + \frac 1z = 2 - \frac{4}{xyz}.\]
2006 Taiwan National Olympiad, 3
$f(x)=x^3-6x^2+17x$. If $f(a)=16, f(b)=20$, find $a+b$.
2014 ELMO Shortlist, 7
Find all positive integers $n$ with $n \ge 2$ such that the polynomial \[ P(a_1, a_2, ..., a_n) = a_1^n+a_2^n + ... + a_n^n - n a_1 a_2 ... a_n \] in the $n$ variables $a_1$, $a_2$, $\dots$, $a_n$ is irreducible over the real numbers, i.e. it cannot be factored as the product of two nonconstant polynomials with real coefficients.
[i]Proposed by Yang Liu[/i]
2010 Mathcenter Contest, 1
A function $ f: R^3\rightarrow R$ for all reals $ a,b,c,d,e$ satisfies a condition:
\[ f(a,b,c)\plus{}f(b,c,d)\plus{}f(c,d,e)\plus{}f(d,e,a)\plus{}f(e,a,b)\equal{}a\plus{}b\plus{}c\plus{}d\plus{}e\]
Show that for all reals $ x_1,x_2,\ldots,x_n$ ($ n\geq 5$) equality holds:
\[ f(x_1,x_2,x_3)\plus{}f(x_2,x_3,x_4)\plus{}\ldots \plus{}f(x_{n\minus{}1},x_n,x_1)\plus{}f(x_n,x_1,x_2)\equal{}x_1\plus{}x_2\plus{}\ldots\plus{}x_n\]
1989 Romania Team Selection Test, 1
Prove that $\sqrt {1+\sqrt {2+\ldots +\sqrt {n}}}<2$, $\forall n\ge 1$.
1998 Brazil Team Selection Test, Problem 3
Find all functions $f: \mathbb N \to \mathbb N$ for which
\[ f(n) + f(n+1) = f(n+2)f(n+3)-1996\]
holds for all positive integers $n$.
2009 Bosnia Herzegovina Team Selection Test, 3
Let $n$ be a positive integer and $x$ positive real number such that none of numbers $x,2x,\dots,nx$ and none of $\frac{1}{x},\frac{2}{x},\dots,\frac{\left\lfloor nx\right\rfloor }{x}$ is an integer. Prove that \[
\left\lfloor x\right\rfloor +\left\lfloor 2x\right\rfloor +\dots+\left\lfloor nx\right\rfloor +\left\lfloor \frac{1}{x}\right\rfloor +\left\lfloor \frac{2}{x}\right\rfloor +\dots+\left\lfloor \frac{\left\lfloor nx\right\rfloor }{x}\right\rfloor =n\left\lfloor nx\right\rfloor \]
2010 Contests, 2
Let $\mathbb{N}_0$ and $\mathbb{Z}$ be the set of all non-negative integers and the set of all integers, respectively. Let $f:\mathbb{N}_0\rightarrow\mathbb{Z}$ be a function defined as
\[f(n)=-f\left(\left\lfloor\frac{n}{3}\right\rfloor \right)-3\left\{\frac{n}{3}\right\} \]
where $\lfloor x \rfloor$ is the greatest integer smaller than or equal to $x$ and $\{ x\}=x-\lfloor x \rfloor$. Find the smallest integer $n$ such that $f(n)=2010$.