Found problems: 15925
2000 Czech And Slovak Olympiad IIIA, 4
For which quadratic polynomials $f(x)$ does there exist a quadratic polynomial $g(x)$ such that the equations $g(f(x)) = 0$ and $f(x)g(x) = 0$ have the same roots, which are mutually distinct and form an arithmetic progression?
2019 Dutch IMO TST, 3
Find all functions $f : Z \to Z$ satisfying the following two conditions:
(i) for all integers $x$ we have $f(f(x)) = x$,
(ii) for all integers $x$ and y such that $x + y$ is odd, we have $f(x) + f(y) \ge x + y$.
2006 South africa National Olympiad, 6
Consider the function $f$ defined by
\[f(n)=\frac{1}{n}\left (\left \lfloor\frac{n}{1}\right \rfloor+\left \lfloor\frac{n}{2}\right \rfloor+\cdots+\left \lfloor\frac{n}{n}\right \rfloor \right )\]
for all positive integers $n$. (Here $\lfloor x\rfloor$ denotes the greatest integer less than or equal to $x$.) Prove that
(a) $f(n+1)>f(n)$ for infinitely many $n$.
(b) $f(n+1)<f(n)$ for infinitely many $n$.
2014 BMT Spring, 3
Emma is seated on a train traveling at a speed of $120$ miles per hour. She notices distance markers are placed evenly alongside the track, with a constant distance $x$ between any two consecutive ones, and during a span of 6 minutes, she sees precisely $11$ markers pass by her. Determine the difference (in miles) between the largest and smallest possible values of $x$.
2010 Contests, 1
a) Factorize $xy - x - y + 1$.
b) Prove that if integers $a$ and $b$ satisfy $ |a + b| > |1 + ab|$, then $ab = 0$.
2022 Junior Balkan Team Selection Tests - Moldova, 10
Solve in the set $R$ the equation
$$2 \cdot [x] \cdot \{x\} = x^2 - \frac32 \cdot x - \frac{11}{16}$$
where $[x]$ and $\{x\}$ represent the integer part and the fractional part of the real number $x$, respectively.
2020 Moldova Team Selection Test, 5
Let $n$ be a natural number. Find all solutions $x$ of the system of equations $$\left\{\begin{matrix} sinx+cosx=\frac{\sqrt{n}}{2}\\tg\frac{x}{2}=\frac{\sqrt{n}-2}{3}\end{matrix}\right.$$ On interval $\left[0,\frac{\pi}{4}\right).$
2017 Thailand TSTST, 6
Find all polynomials $f$ with real coefficients such that for all reals $x, y, z$ such that $x+y+z =0$, the following relation holds: $$f(xy) + f(yz) + f(zx) = f(xy + yz + zx).$$
2016 Canadian Mathematical Olympiad Qualification, 2
Let $P = (7, 1)$ and let $O = (0, 0)$.
(a) If $S$ is a point on the line $y = x$ and $T$ is a point on the horizontal $x$-axis so that $P$ is on the line segment $ST$, determine the minimum possible area of triangle $OST$.
(b) If $U$ is a point on the line $y = x$ and $V$ is a point on the horizontal $x$-axis so that $P$ is on the line segment $UV$, determine the minimum possible perimeter of triangle $OUV$.
2013 AMC 12/AHSME, 6
Real numbers $x$ and $y$ satisfy the equation $x^2+y^2=10x-6y-34$. What is $x+y$?
$ \textbf{(A) }1\qquad\textbf{(B) }2\qquad\textbf{(C) }3\qquad\textbf{(D) }6\qquad\textbf{(E) }8 $
1992 IMO Longlists, 31
Let $ f(x) \equal{} x^8 \plus{} 4x^6 \plus{} 2x^4 \plus{} 28x^2 \plus{} 1.$ Let $ p > 3$ be a prime and suppose there exists an integer $ z$ such that $ p$ divides $ f(z).$ Prove that there exist integers $ z_1, z_2, \ldots, z_8$ such that if \[ g(x) \equal{} (x \minus{} z_1)(x \minus{} z_2) \cdot \ldots \cdot (x \minus{} z_8),\] then all coefficients of $ f(x) \minus{} g(x)$ are divisible by $ p.$
2004 Harvard-MIT Mathematics Tournament, 10
There exists a polynomial $P$ of degree $5$ with the following property: if $z$ is a complex number such that $z^5+2004z=1$, then $P(z^2)=0$. Calculate the quotient $\tfrac{P(1)}{P(-1)}$.
2013 BMT Spring, 9
Sequences $x_n$ and $y_n$ satisfy the simultaneous relationships $x_k = x_{k+1} + y_{k+1}$ and $x_k > y_k$ for all $k \ge 1$. Furthermore, either $y_k = y_{k+1}$ or $y_k = x_{k+1}$. If $x_1 = 3 + \sqrt2$, $x_3 = 5 -\sqrt2$, and $y_1 = y_5$, evaluate $$(y_1)^2 + (y_2)^2 + (y_3)^2 + . . .$$
2015 BMT Spring, P2
Let $f(x)$ be a nonconstant monic polynomial of degree $n$ with rational coefficents that is irreducible, meaning it cannot be factored into two nonconstant rational polynomials. Find and prove a formula for the number of monic complex polynomials that divide $f$.
2010 May Olympiad, 2
In stage $0$ the numbers are written: $1 , 1$.
In stage $1$ the sum of the numbers is inserted: $1, 2, 1$.
In stage $2$, between each pair of numbers from the previous stage, the sum of them is inserted: $1, 3, 2, 3, 1$.
One more stage: $1, 4, 3, 5, 2, 5, 3, 4, 1$.
How many numbers are there in stage $10$?
What is the sum of all the numbers in stage $10$?
1998 Austrian-Polish Competition, 5
Determine all pairs $(a, b)$ of positive integers for which the equation $x^3 - 17x^2 + ax - b^2 = 0$ has three integer roots (not necessarily different).
PEN A Problems, 3
Let $a$ and $b$ be positive integers such that $ab+1$ divides $a^{2}+b^{2}$. Show that \[\frac{a^{2}+b^{2}}{ab+1}\] is the square of an integer.
2020 DMO Stage 1, 3.
[b]Q.[/b] Determine all the functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that
$$f(x) \geqslant x+1, \forall\ x \in \mathbb{R}\quad \text{and}\quad f(x+y) \geqslant f(x) f(y), \forall\ x, y \in \mathbb{R}$$
[i]Proposed by TuZo[/i]
1998 All-Russian Olympiad, 6
A binary operation $*$ on real numbers has the property that $(a * b) * c = a+b+c$ for all $a$, $b$, $c$. Prove that $a * b = a+b$.
2009 Math Prize For Girls Problems, 7
Compute the value of the expression
\[ 2009^4 \minus{} 4 \times 2007^4 \plus{} 6 \times 2005^4 \minus{} 4 \times 2003^4 \plus{} 2001^4 \, .\]
2012 EGMO, 3
Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that \[f\left( {yf(x + y) + f(x)} \right) = 4x + 2yf(x + y)\] for all $x,y\in\mathbb{R}$.
[i]Netherlands (Birgit van Dalen)[/i]
2003 Switzerland Team Selection Test, 1
Real numbers $x,y,a$ satisfy the equations $$x+y = x^3 +y^3 = x^5 +y^5 = a$$
Find all possible values of $a$.
2012 Saint Petersburg Mathematical Olympiad, 4
$x_1,...,x_n$ are reals and $x_1^2+...+x_n^2=1$
Prove, that exists such $y_1,...,y_n$ and $z_1,...,z_n$ such that $|y_1|+...+|y_n| \leq 1$; $max(|z_1|,...,|z_n|) \leq 1$ and $2x_i=y_i+z_i$ for every $i$
2009 Romania National Olympiad, 1
[b]a)[/b] Show that two real numbers $ x,y>1 $ chosen so that $ x^y=y^x, $ are equal or there exists a positive real number $ m\neq 1 $ such that $ x=m^{\frac{1}{m-1}} $ and $ y=m^{\frac{m}{m-1}} . $
[b]b)[/b] Solve in $ \left( 1,\infty \right)^2 $ the equation: $ x^y+x^{x^{y-1}}=y^x+y^{y^{x-1}} . $
2023 Simon Marais Mathematical Competition, A4
Let $x_0, x_1, x_2 \dots$ be a sequence of positive real numbers such that for all $n \geq 0$, $$x_{n+1} = \dfrac{(n^2+1)x_n^2}{x_n^3+n^2}$$ For which values of $x_0$ is this sequence bounded?