Found problems: 1311
2010 ISI B.Stat Entrance Exam, 5
Let $A$ be the set of all functions $f:\mathbb{R} \to \mathbb{R}$ such that $f(xy)=xf(y)$ for all $x,y \in \mathbb{R}$.
(a) If $f \in A$ then show that $f(x+y)=f(x)+f(y)$ for all $x,y \in \mathbb{R}$
(b) For $g,h \in A$, define a function $g\circ h$ by $(g \circ h)(x)=g(h(x))$ for $x \in \mathbb{R}$. Prove that $g \circ h$ is in $A$ and is equal to $h \circ g$.
2008 Indonesia MO, 4
Find all function $ f: \mathbb{N}\rightarrow\mathbb{N}$ satisfy $ f(mn)\plus{}f(m\plus{}n)\equal{}f(m)f(n)\plus{}1$ for all natural number $ n$
2000 Brazil National Olympiad, 3
Define $f$ on the positive integers by $f(n) = k^2 + k + 1$, where $n=2^k(2l+1)$ for some $k,l$ nonnegative integers.
Find the smallest $n$ such that $f(1) + f(2) + ... + f(n) \geq 123456$.
2009 Kazakhstan National Olympiad, 1
Let $S_n$ be number of ordered sets of natural numbers $(a_1;a_2;....;a_n)$ for which $\frac{1}{a_1}+\frac{1}{a_2}+....+\frac{1}{a_n}=1$. Determine
1)$S_{10} mod(2)$.
2)$S_7 mod(2)$.
(1) is first problem in 10 grade, (2)- third in 9 grade.
1990 India National Olympiad, 1
Given the equation
\[ x^4 \plus{} px^3 \plus{} qx^2 \plus{} rx \plus{} s \equal{} 0\]
has four real, positive roots, prove that
(a) $ pr \minus{} 16s \geq 0$
(b) $ q^2 \minus{} 36s \geq 0$
with equality in each case holding if and only if the four roots are equal.
2006 ISI B.Stat Entrance Exam, 10
Consider a function $f$ on nonnegative integers such that $f(0)=1, f(1)=0$ and $f(n)+f(n-1)=nf(n-1)+(n-1)f(n-2)$ for $n \ge 2$. Show that
\[\frac{f(n)}{n!}=\sum_{k=0}^n \frac{(-1)^k}{k!}\]
1989 Iran MO (2nd round), 3
Let $\{a_n\}_{n \geq 1}$ be a sequence in which $a_1=1$ and $a_2=2$ and
\[a_{n+1}=1+a_1a_2a_3 \cdots a_{n-1}+(a_1a_2a_3 \cdots a_{n-1} )^2 \qquad \forall n \geq 2.\]
Prove that
\[\lim_{n \to \infty} \biggl( \frac{1}{a_1}+\frac{1}{a_2}+\frac{1}{a_3}+\cdots + \frac{1}{a_n} \biggr) =2\]
2023 Dutch IMO TST, 4
Find all functions $f: \mathbb{Q^+} \rightarrow \mathbb{Q}$ satisfying $f(x)+f(y)= \left(f(x+y)+\frac{1}{x+y} \right) (1-xy+f(xy))$ for all $x, y \in \mathbb{Q^+}$.
2009 India IMO Training Camp, 5
Let $ f(x)$and $ g(y)$ be two monic polynomials of degree=$ n$ having complex coefficients.
We know that there exist complex numbers $ a_i,b_i,c_i \forall 1\le i \le n$, such that
$ f(x)\minus{}g(y)\equal{}\prod_{i\equal{}1}^n{(a_ix\plus{}b_iy\plus{}c_i)}$.
Prove that there exists $ a,b,c\in\mathbb{C}$ such that
$ f(x)\equal{}(x\plus{}a)^n\plus{}c\text{ and }g(y)\equal{}(y\plus{}b)^n\plus{}c$.
2000 Baltic Way, 20
For every positive integer $n$, let
\[x_n=\frac{(2n+1)(2n+3)\cdots (4n-1)(4n+1)}{(2n)(2n+2)\cdots (4n-2)(4n)}\]
Prove that $\frac{1}{4n}<x_n-\sqrt{2}<\frac{2}{n}$.
2009 Romania Team Selection Test, 2
Let $n$ and $k$ be positive integers. Find all monic polynomials $f\in \mathbb{Z}[X]$, of degree $n$, such that $f(a)$ divides $f(2a^k)$ for $a\in \mathbb{Z}$ with $f(a)\neq 0$.
2010 Indonesia TST, 3
Let $ a_1,a_2,\dots$ be sequence of real numbers such that $ a_1\equal{}1$, $ a_2\equal{}\dfrac{4}{3}$, and \[ a_{n\plus{}1}\equal{}\sqrt{1\plus{}a_na_{n\minus{}1}}, \quad \forall n \ge 2.\] Prove that for all $ n \ge 2$, \[ a_n^2>a_{n\minus{}1}^2\plus{}\dfrac{1}{2}\] and \[ 1\plus{}\dfrac{1}{a_1}\plus{}\dfrac{1}{a_2}\plus{}\dots\plus{}\dfrac{1}{a_n}>2a_n.\]
[i]Fajar Yuliawan, Bandung[/i]
2010 Iran MO (3rd Round), 7
[b]interesting function[/b]
$S$ is a set with $n$ elements and $P(S)$ is the set of all subsets of $S$ and
$f : P(S) \rightarrow \mathbb N$
is a function with these properties:
for every subset $A$ of $S$ we have $f(A)=f(S-A)$.
for every two subsets of $S$ like $A$ and $B$ we have
$max(f(A),f(B))\ge f(A\cup B)$
prove that number of natural numbers like $x$ such that there exists $A\subseteq S$ and $f(A)=x$ is less than $n$.
time allowed for this question was 1 hours and 30 minutes.
2012 Romania Team Selection Test, 1
Let $m$ and $n$ be two positive integers greater than $1$. Prove that there are $m$ positive integers $N_1$ , $\ldots$ , $N_m$ (some of them may be equal) such that \[\sqrt{m}=\sum_{i=1}^m{(\sqrt{N_i}-\sqrt{N_i-1})^{\frac{1}{n}}.}\]
2015 Romanian Master of Mathematics, 3
A finite list of rational numbers is written on a blackboard. In an [i]operation[/i], we choose any two numbers $a$, $b$, erase them, and write down one of the numbers \[
a + b, \; a - b, \; b - a, \; a \times b, \; a/b \text{ (if $b \neq 0$)}, \; b/a \text{ (if $a \neq 0$)}.
\] Prove that, for every integer $n > 100$, there are only finitely many integers $k \ge 0$, such that, starting from the list \[ k + 1, \; k + 2, \; \dots, \; k + n, \] it is possible to obtain, after $n - 1$ operations, the value $n!$.
2013 District Olympiad, 4
Let $n\in {{\mathbb{N}}^{*}}$. Prove that $2\sqrt{{{2}^{n}}}\cos \left( n\arccos \frac{\sqrt{2}}{4} \right)$ is an odd integer.
2008 International Zhautykov Olympiad, 2
A polynomial $ P(x)$ with integer coefficients is called good,if it can be represented as a sum of cubes of several polynomials (in variable $ x$) with integer coefficients.For example,the polynomials $ x^3 \minus{} 1$ and $ 9x^3 \minus{} 3x^2 \plus{} 3x \plus{} 7 \equal{} (x \minus{} 1)^3 \plus{} (2x)^3 \plus{} 2^3$ are good.
a)Is the polynomial $ P(x) \equal{} 3x \plus{} 3x^7$ good?
b)Is the polynomial $ P(x) \equal{} 3x \plus{} 3x^7 \plus{} 3x^{2008}$ good?
Justify your answers.
1992 Baltic Way, 10
Find all fourth degree polynomial $ p(x)$ such that the following four conditions are satisfied:
(i) $ p(x)\equal{}p(\minus{}x)$ for all $ x$,
(ii) $ p(x)\ge0$ for all $ x$,
(iii) $ p(0)\equal{}1$
(iv) $ p(x)$ has exactly two local minimum points $ x_1$ and $ x_2$ such that $ |x_1\minus{}x_2|\equal{}2$.
2010 ISI B.Math Entrance Exam, 7
We are given $a,b,c \in \mathbb{R}$ and a polynomial $f(x)=x^3+ax^2+bx+c$ such that all roots (real or complex) of $f(x)$ have same absolute value. Show that $a=0$ iff $b=0$.
1990 IMO Longlists, 53
Find the real solution(s) for the system of equations
\[\begin{cases}x^3+y^3 &=1\\x^5+y^5 &=1\end{cases}\]
2003 District Olympiad, 2
Find all functions $\displaystyle f : \mathbb N^\ast \to M$ such that
\[ \displaystyle 1 + f(n) f(n+1) = 2 n^2 \left( f(n+1) - f(n) \right), \, \forall n \in \mathbb N^\ast , \]
in each of the following situations:
(a) $\displaystyle M = \mathbb N$;
(b) $\displaystyle M = \mathbb Q$.
[i]Dinu Şerbănescu[/i]
1981 Canada National Olympiad, 4
$P(x),Q(x)$ are two polynomials such that $P(x)=Q(x)$ has no real solution, and $P(Q(x))\equiv Q(P(x))\forall x\in\mathbb{R}$. Prove that $P(P(x))=Q(Q(x))$ has no real solution.
2012 Kyoto University Entry Examination, 3
When real numbers $x,\ y$ moves in the constraint with $x^2+xy+y^2=6.$
Find the range of $x^2y+xy^2-x^2-2xy-y^2+x+y.$
30 points
1996 All-Russian Olympiad, 4
Show that if the integers $a_1$; $\dots$ $a_m$ are nonzero and for each $k =0; 1; \dots ;n$ ($n < m - 1$),
$a_1 + a_22^k + a_33^k + \dots + a_mm^k = 0$; then the sequence $a_1, \dots, a_m$ contains at least $n+1$ pairs of consecutive terms having opposite signs.
[i]O. Musin[/i]
2010 Slovenia National Olympiad, 1
For a real number $t$ and positive real numbers $a,b$ we have
\[2a^2-3abt+b^2=2a^2+abt-b^2=0\]
Find $t.$