Found problems: 1269
1984 Vietnam National Olympiad, 1
$(a)$ Find a polynomial with integer coefficients of the smallest degree having $\sqrt{2} + \sqrt[3]{3}$ as a root.
$(b)$ Solve $1 +\sqrt{1 + x^2}(\sqrt{(1 + x)^3}-\sqrt{(1- x)^3}) = 2\sqrt{1 - x^2}$.
2014 India IMO Training Camp, 2
For $j=1,2,3$ let $x_{j},y_{j}$ be non-zero real numbers, and let $v_{j}=x_{j}+y_{j}$.Suppose that the following statements hold:
$x_{1}x_{2}x_{3}=-y_{1}y_{2}y_{3}$
$x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=y_{1}^{2}+y_{2}^{2}+y_{3}^2$
$v_{1},v_{2},v_{3}$ satisfy triangle inequality
$v_{1}^{2},v_{2}^{2},v_{3}^{2}$ also satisfy triangle inequality.
Prove that exactly one of $x_{1},x_{2},x_{3},y_{1},y_{2},y_{3}$ is negative.
2010 IFYM, Sozopol, 8
Solve this equation with $x \in R$:
$x^3-3x=\sqrt{x+2}$
1985 USAMO, 5
Let $a_1,a_2,a_3,\cdots$ be a non-decreasing sequence of positive integers. For $m\ge1$, define $b_m=\min\{n: a_n \ge m\}$, that is, $b_m$ is the minimum value of $n$ such that $a_n\ge m$. If $a_{19}=85$, determine the maximum value of \[a_1+a_2+\cdots+a_{19}+b_1+b_2+\cdots+b_{85}.\]
2007 Irish Math Olympiad, 1
Let $ r,s,$ and $ t$ be the roots of the cubic polynomial: $ p(x)\equal{}x^3\minus{}2007x\plus{}2002.$
Determine the value of: $ \frac{r\minus{}1}{r\plus{}1}\plus{}\frac{s\minus{}1}{s\plus{}1}\plus{}\frac{t\minus{}1}{t\plus{}1}$.
2010 Canada National Olympiad, 3
Three speed skaters have a friendly "race" on a skating oval. They all start from the same point and skate in the same direction, but with different speeds that they maintain throughout the race. The slowest skater does $1$ lap per minute, the fastest one does $3.14$ laps per minute, and the middle one does $L$ laps a minute for some $1 < L < 3.14$. The race ends at the moment when all three skaters again come together to the same point on the oval (which may differ from the starting point.) Determine the number of different choices for $L$ such that exactly $117$ passings occur before the end of the race.
Note: A passing is defined as when one skater passes another one. The beginning and the end of the race when all three skaters are together are not counted as passings.
2002 Iran MO (2nd round), 5
Let $\delta$ be a symbol such that $\delta \neq 0$ and $\delta^2 = 0$. Define $\mathbb R[\delta] = \{a + b \delta | a, b \in \mathbb R\}$, where $a+ b \delta = c+ d \delta$ if and only if $a = c$ and $b = d$, and define
\[(a + b \delta) + (c + d \delta) = (a + c) + (b + d) \delta,\]\[(a + b \delta) \cdot (c + d \delta) = ac + (ad + bc) \delta.\]
Let $P(x)$ be a polynomial with real coefficients. Show that $P(x)$ has a multiple real root if and only if $P(x)$ has a non-real root in $\mathbb R[\delta].$
1989 IMO Longlists, 79
Given two natural numbers $ w$ and $ n,$ the tower of $ n$ $ w's$ is the natural number $ T_n(w)$ defined by
\[ T_n(w) = w^{w^{\cdots^{w}}},\]
with $ n$ $ w's$ on the right side. More precisely, $ T_1(w) = w$ and $ T_{n+1}(w) = w^{T_n(w)}.$ For example, $ T_3(2) = 2^{2^2} = 16,$ $ T_4(2) = 2^{16} = 65536,$ and $ T_2(3) = 3^3 = 27.$ Find the smallest tower of $ 3's$ that exceeds the tower of $ 1989$ $ 2's.$ In other words, find the smallest value of $ n$ such that $ T_n(3) > T_{1989}(2).$ Justify your answer.
1993 Vietnam Team Selection Test, 2
Let an integer $k > 1$ be given. For each integer $n > 1$, we put
\[f(n) = k \cdot n \cdot \left(1-\frac{1}{p_1}\right) \cdot \left(1-\frac{1}{p_2}\right) \cdots \left(1-\frac{1}{p_r}\right)\]
where $p_1, p_2, \ldots, p_r$ are all distinct prime divisors of $n$. Find all values $k$ for which the sequence $\{x_m\}$ defined by $x_0 = a$ and $x_{m+1} = f(x_m), m = 0, 1, 2, 3, \ldots$ is bounded for all integers $a > 1$.
2012 Indonesia TST, 1
Suppose a function $f : \mathbb{Z}^+ \rightarrow \mathbb{Z}^+$ satisfies $f(f(n)) + f(n+1) = n+2$ for all positive integer $n$. Prove that $f(f(n)+n) = n+1$ for all positive integer $n$.
BIMO 2022, Open
Given $k\ge 2$, for which polynomials $P\in \mathbb{Z}[X]$ does there exist a function $h:\mathbb{N}\rightarrow\mathbb{N}$ with $h^{(k)}(n)=P(n)$?
1996 Vietnam Team Selection Test, 3
Find all reals $a$ such that the sequence $\{x(n)\}$, $n=0,1,2, \ldots$ that satisfy: $x(0)=1996$ and $x_{n+1} = \frac{a}{1+x(n)^2}$ for any natural number $n$ has a limit as n goes to infinity.
2006 India Regional Mathematical Olympiad, 7
Let $ X$ be the set of all positive integers greater than or equal to $ 8$ and let $ f: X\rightarrow X$ be a function such that $ f(x\plus{}y)\equal{}f(xy)$ for all $ x\ge 4, y\ge 4 .$ if $ f(8)\equal{}9$, determine $ f(9) .$
2013 Albania Team Selection Test, 3
Solve the function $f: \Re \to \Re$:
\[ f( x^{3} )+ f(y^{3}) = (x+y)(f(x^{2} )+f(y^{2} )-f(xy))\]
1986 China Team Selection Test, 2
Let $ a_1$, $ a_2$, ..., $ a_n$ and $ b_1$, $ b_2$, ..., $ b_n$ be $ 2 \cdot n$ real numbers. Prove that the following two statements are equivalent:
[b]i)[/b] For any $ n$ real numbers $ x_1$, $ x_2$, ..., $ x_n$ satisfying $ x_1 \leq x_2 \leq \ldots \leq x_ n$, we have $ \sum^{n}_{k \equal{} 1} a_k \cdot x_k \leq \sum^{n}_{k \equal{} 1} b_k \cdot x_k,$
[b]ii)[/b] We have $ \sum^{s}_{k \equal{} 1} a_k \leq \sum^{s}_{k \equal{} 1} b_k$ for every $ s\in\left\{1,2,...,n\minus{}1\right\}$ and $ \sum^{n}_{k \equal{} 1} a_k \equal{} \sum^{n}_{k \equal{} 1} b_k$.
2005 Federal Competition For Advanced Students, Part 2, 2
Find all real $a,b,c,d,e,f$ that satisfy the system
$4a = (b + c + d + e)^4$
$4b = (c + d + e + f)^4$
$4c = (d + e + f + a)^4$
$4d = (e + f + a + b)^4$
$4e = (f + a + b + c)^4$
$4f = (a + b + c + d)^4$
2001 China Team Selection Test, 3
For a given natural number $k > 1$, find all functions $f:\mathbb{R} \to \mathbb{R}$ such that for all $x, y \in \mathbb{R}$, $f[x^k + f(y)] = y +[f(x)]^k$.
1983 Canada National Olympiad, 2
For each $r\in\mathbb{R}$ let $T_r$ be the transformation of the plane that takes the point $(x, y)$ into the point $(2^r x; r2^r x+2^r y)$. Let $F$ be the family of all such transformations (i.e. $F = \{T_r : r\in\mathbb{R}\}$). Find all curves $y = f(x)$ whose graphs remain unchanged by every transformation in $F$.
2008 Moldova National Olympiad, 9.6
find x and y in R
$\begin{array}{l} (\frac{1}{{\sqrt[3]{x}}} + \frac{1}{{\sqrt[3]{y}}})(\frac{1}{{\sqrt[3]{x}}} + 1)(\frac{1}{{\sqrt[3]{y}}} + 1) = 18 \\ \frac{1}{x} + \frac{1}{y} = 9 \\ \end{array}$
1976 IMO Longlists, 43
Prove that if for a polynomial $P(x, y)$, we have
\[P(x - 1, y - 2x + 1) = P(x, y),\]
then there exists a polynomial $\Phi(x)$ with $P(x, y) = \Phi(y - x^2).$
1976 IMO Longlists, 40
Let $g(x)$ be a fixed polynomial with real coefficients and define $f(x)$ by $f(x) =x^2 + xg(x^3)$. Show that $f(x)$ is not divisible by $x^2 - x + 1$.
2010 Tuymaada Olympiad, 1
Baron Münchausen boasts that he knows a remarkable quadratic triniomial with positive coefficients. The trinomial has an integral root; if all of its coefficients are increased by $1$, the resulting trinomial also has an integral root; and if all of its coefficients are also increased by $1$, the new trinomial, too, has an integral root. Can this be true?
1993 China Team Selection Test, 2
Let $n \geq 2, n \in \mathbb{N}$, $a,b,c,d \in \mathbb{N}$, $\frac{a}{b} + \frac{c}{d} < 1$ and $a + c \leq n,$ find the maximum value of $\frac{a}{b} + \frac{c}{d}$ for fixed $n.$
1989 IMO Longlists, 13
Let $ n \leq 44, n \in \mathbb{N}.$ Prove that for any function $ f$ defined over $ \mathbb{N}^2$ whose images are in the set $ \{1, 2, \ldots , n\},$ there are four ordered pairs $ (i, j), (i, k), (l, j),$ and $ (l, k)$ such that \[ f(i, j) \equal{} f(i, k) \equal{} f(l, j) \equal{} f(l, k),\] in which $ i, j, k, l$ are chosen in such a way that there are natural numbers $ m, p$ that satisfy \[ 1989m \leq i < l < 1989 \plus{} 1989m\] and \[ 1989p \leq j < k < 1989 \plus{} 1989p.\]
2003 China Team Selection Test, 1
Find all functions $f: \mathbb{Z}^+\to \mathbb{R}$, which satisfies $f(n+1)\geq f(n)$ for all $n\geq 1$ and $f(mn)=f(m)f(n)$ for all $(m,n)=1$.