Found problems: 1269
2011 Finnish National High School Mathematics Competition, 2
Find all integers $x$ and $y$ satisfying the inequality \[x^4-12x^2+x^2y^2+30\leq 0.\]
2009 ISI B.Math Entrance Exam, 3
Let $1,2,3,4,5,6,7,8,9,11,12,\cdots$ be the sequence of all positive integers which do not contain the digit zero. Write $\{a_n\}$ for this sequence. By comparing with a geometric series, show that $\sum_{k=1}^n \frac{1}{a_k} < 90$.
2009 Poland - Second Round, 3
For every integer $n\ge 3$ find all sequences of real numbers $(x_1,x_2,\ldots ,x_n)$ such that $\sum_{i=1}^{n}x_i=n$ and $\sum_{i=1}^{n} (x_{i-1}-x_i+x_{i+1})^2=n$, where $x_0=x_n$ and $x_{n+1}=x_1$.
2007 Tournament Of Towns, 4
Three nonzero real numbers are given. If they are written in any order as coefficients of a quadratic trinomial, then each of these trinomials has a real root. Does it follow that each of these trinomials has a positive root?
2001 Brazil Team Selection Test, Problem 1
given that p,q are two polynomials such that each one has at least one root and \[p(1+x+q(x)^2)=q(1+x+p(x)^2)\] then prove that p=q
1970 Regional Competition For Advanced Students, 4
Find all real solutions of the following set of equations:
\[72x^3+4xy^2=11y^3\]
\[27x^5-45x^4y-10x^2y^3=\frac{-143}{32}y^5\]
2006 Macedonia National Olympiad, 2
Determine all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that for all $x, y, z,$
\[f(x+y^2+z)=f(f(x))+yf(y)+f(z). \]
2006 QEDMO 3rd, 3
Find all functions $ f: \mathbb{R} \rightarrow \mathbb{R}$ such that for all real numbers $ x,y$:
$ x f(x)\minus{}yf(y)\equal{}(x\minus{}y)f(x\plus{}y)$.
2002 Germany Team Selection Test, 1
Let $P$ denote the set of all ordered pairs $ \left(p,q\right)$ of nonnegative integers. Find all functions $f: P \rightarrow \mathbb{R}$ satisfying
\[ f(p,q) \equal{} \begin{cases} 0 & \text{if} \; pq \equal{} 0, \\
1 \plus{} \frac{1}{2} f(p+1,q-1) \plus{} \frac{1}{2} f(p-1,q+1) & \text{otherwise} \end{cases}
\]
Compare IMO shortlist problem 2001, algebra A1 for the three-variable case.
1990 IMO Longlists, 80
Function $f(x, y): \mathbb N \times \mathbb N \to \mathbb Q$ satisfies the conditions:
(i) $f(1, 1) =1$,
(ii) $f(p + 1, q) + f(p, q + 1) = f(p, q)$ for all $p, q \in \mathbb N$, and
(iii) $qf(p + 1, q) = pf(p, q + 1)$ for all $p, q \in \mathbb N$.
Find $f(1990, 31).$
2008 South East Mathematical Olympiad, 2
Let $\{a_n\}$ be a sequence satisfying: $a_1=1$ and $a_{n+1}=2a_n+n\cdot (1+2^n),(n=1,2,3,\cdots)$. Determine the general term formula of $\{a_n\}$.
2014 Vietnam Team Selection Test, 1
Find all $ f:\mathbb{Z}\rightarrow\mathbb{Z} $ such that
\[ f(2m+f(m)+f(m)f(n))=nf(m)+m \] $ \forall m,n\in\mathbb{Z} $
1994 Korea National Olympiad, Problem 1
Let $ S$ be the set of nonnegative integers. Find all functions $ f,g,h: S\rightarrow S$ such that
$ f(m\plus{}n)\equal{}g(m)\plus{}h(n),$ for all $ m,n\in S$, and
$ g(1)\equal{}h(1)\equal{}1$.
2003 Bulgaria National Olympiad, 3
Given the sequence $\{y_n\}_{n=1}^{\infty}$ defined by $y_1=y_2=1$ and
\[y_{n+2} = (4k-5)y_{n+1}-y_n+4-2k, \qquad n\ge1\]
find all integers $k$ such that every term of the sequence is a perfect square.
1990 China Team Selection Test, 3
In set $S$, there is an operation $'' \circ ''$ such that $\forall a,b \in S$, a unique $a \circ b \in S$ exists. And
(i) $\forall a,b,c \in S$, $(a \circ b) \circ c = a \circ (b \circ c)$.
(ii) $a \circ b \neq b \circ a$ when $a \neq b$.
Prove that:
a.) $\forall a,b,c \in S$, $(a \circ b) \circ c = a \circ c$.
b.) If $S = \{1,2, \ldots, 1990\}$, try to define an operation $'' \circ ''$ in $S$ with the above properties.
1988 IMO Longlists, 38
[b]i.)[/b] The polynomial $x^{2 \cdot k} + 1 + (x+1)^{2 \cdot k}$ is not divisible by $x^2 + x + 1.$ Find the value of $k.$
[b]ii.)[/b] If $p,q$ and $r$ are distinct roots of $x^3 - x^2 + x - 2 = 0$ the find the value of $p^3 + q^3 + r^3.$
[b]iii.)[/b] If $r$ is the remainder when each of the numbers 1059, 1417 and 2312 is divided by $d,$ where $d$ is an integer greater than one, then find the value of $d-r.$
[b]iv.)[/b] What is the smallest positive odd integer $n$ such that the product of
\[ 2^{\frac{1}{7}}, 2^{\frac{3}{7}}, \ldots, 2^{\frac{2 \cdot n + 1}{7}} \]
is greater than 1000?
2010 India IMO Training Camp, 9
Let $A=(a_{jk})$ be a $10\times 10$ array of positive real numbers such that the sum of numbers in row as well as in each column is $1$.
Show that there exists $j<k$ and $l<m$ such that
\[a_{jl}a_{km}+a_{jm}a_{kl}\ge \frac{1}{50}\]
2006 USAMO, 4
Find all positive integers $n$ such that there are $k \geq 2$ positive rational numbers $a_1, a_2, \ldots, a_k$ satisfying $a_1 + a_2 + \ldots + a_k = a_1 \cdot a_2 \cdots a_k = n.$
2002 Turkey MO (2nd round), 3
Let $n$ be a positive integer and let $T$ denote the collection of points $(x_1, x_2, \ldots, x_n) \in \mathbb R^n$ for which there exists a permutation $\sigma$ of $1, 2, \ldots , n$ such that $x_{\sigma(i)} - x_{\sigma(i+1) } \geq 1$ for each $i=1, 2, \ldots , n.$ Prove that there is a real number $d$ satisfying the following condition:
For every $(a_1, a_2, \ldots, a_n) \in \mathbb R^n$ there exist points $(b_1, \ldots, b_n)$ and $(c_1,\ldots, c_n)$ in $T$ such that, for each $i = 1, . . . , n,$
\[a_i=\frac 12 (b_i+c_i) , \quad |a_i - b_i| \leq d, \quad \text{and} \quad |a_i - c_i| \leq d.\]
2012 China Second Round Olympiad, 7
Find the sum of all integers $n$ satisfying the following inequality:
\[\frac{1}{4}<\sin\frac{\pi}{n}<\frac{1}{3}.\]
2005 China Team Selection Test, 3
Let $\alpha$ be given positive real number, find all the functions $f: N^{+} \rightarrow R$ such that $f(k + m) = f(k) + f(m)$ holds for any positive integers $k$, $m$ satisfying $\alpha m \leq k \leq (\alpha + 1)m$.
2004 South East Mathematical Olympiad, 8
Determine the number of ordered quadruples $(x, y, z, u)$ of integers, such that
\[\dfrac{x-y}{x+y}+\dfrac{y-z}{y+z}+\dfrac{z-u}{z+u}>0 \textrm{ and } 1\le x,y,z,u\le 10.\]
2011 Brazil National Olympiad, 1
We call a number [i]pal[/i] if it doesn't have a zero digit and the sum of the squares of the digits is a perfect square. For example, $122$ and $34$ are pal but $304$ and $12$ are not pal. Prove that there exists a pal number with $n$ digits, $n > 1$.
2002 China Team Selection Test, 1
Let $P_n(x)=a_0 + a_1x + \cdots + a_nx^n$, with $n \geq 2$, be a real-coefficient polynomial. Prove that if there exists $a > 0$ such that
\begin{align*}
P_n(x) = (x + a)^2 \left( \sum_{i=0}^{n-2} b_i x^i \right),
\end{align*}
where $b_i$ are positive real numbers, then there exists some $i$, with $1 \leq i \leq n-1$, such that \[a_i^2 - 4a_{i-1}a_{i+1} \leq 0.\]
2012 Federal Competition For Advanced Students, Part 2, 1
Given a sequence $<a_1,a_2,a_3,\cdots >$ of real numbers, we define $m_n$ as the arithmetic mean of the numbers $a_1$ to $a_n$ for $n\in\mathbb{Z}^+$.
If there is a real number $C$, such that
\[ (i-j)m_k+(j-k)m_i+(k-i)m_j=C\]
for every triple $(i,j,k)$ of distinct positive integers, prove that the sequence $<a_1,a_2,a_3,\cdots >$ is an arithmetic progression.