Found problems: 15925
1991 Arnold's Trivium, 65
Find the mean value of the function $\ln r$ on the circle $(x - a)^2 + (y-b)^2 = R^2$ (of the function $1/r$ on the sphere).
2006 MOP Homework, 4
Determine if there exists a strictly increasing sequence of positive integers $a_1$, $a_2$, ... such that $a_n \le n^3$ for every positive integer $n$ and that every positive integer can be written uniquely as the difference of two terms in the sequence.
2020 Centroamerican and Caribbean Math Olympiad, 5
Let $P(x)$ be a polynomial with real non-negative coefficients. Let $k$ be a positive integer and $x_1, x_2, \dots, x_k$ positive real numbers such that $x_1x_2\cdots x_k=1$. Prove that
$$P(x_1)+P(x_2)+\cdots+P(x_k)\geq kP(1).$$
2001 India IMO Training Camp, 1
Complex numbers $\alpha$ , $\beta$ , $\gamma$ have the property that $\alpha^k +\beta^k +\gamma^k$ is an integer for every natural number $k$. Prove that the polynomial \[(x-\alpha)(x-\beta )(x-\gamma )\] has integer coefficients.
2015 BMT Spring, 10
Quadratics $g(x) = ax^2 + bx + c$ and $h(x) = dx^2 + ex + f$ are such that the six roots of $g,h$, and $g - h$ are distinct real numbers (in particular, they are not double roots) forming an arithmetic progression in some order. Determine all possible values of $a/d$.
2004 Croatia National Olympiad, Problem 4
The sequence $1,2,3,4,0,9,6,9,4,8,7,\ldots$ is formed so that each term, starting from the fifth, is the units digit of the sum of the previous four.
(a) Do the digits $2,0,0,4$ occur in the sequence in this order?
(b) Will the initial digits $1,2,3,4$ ever occur again in this order?
1999 Federal Competition For Advanced Students, Part 2, 2
Given a real number $A$ and an integer $n$ with $2 \leq n \leq 19$, find all polynomials $P(x)$ with real coefficients such that $P(P(P(x))) = Ax^n +19x+99$.
1991 Spain Mathematical Olympiad, 3
What condition must be satisfied by the coefficients $u,v,w$ if the roots of the polynomial $x^3 -ux^2+vx-w$ are the sides of a triangle
2021 Malaysia IMONST 1, 13
Jasmin has a mobile phone that runs on a battery. When the battery is dead, it takes $2$ hours to recharge it fully, if she is not using the phone. If she uses the phone while recharging, $75\%$ of the charge obtained is immediately consumed and the remaining is stored in the battery. One day, her battery died. Jasmin took $2$ hours $30$ minutes to recharge the battery fully. For how many minutes did she use the phone while recharging?
2021 IMO, 6
Let $m\ge 2$ be an integer, $A$ a finite set of integers (not necessarily positive) and $B_1,B_2,...,B_m$ subsets of $A$. Suppose that, for every $k=1,2,...,m$, the sum of the elements of $B_k$ is $m^k$. Prove that $A$ contains at least $\dfrac{m}{2}$ elements.
Russian TST 2021, P1
Suppose that $a,b,c,d$ are positive real numbers satisfying $(a+c)(b+d)=ac+bd$. Find the smallest possible value of
$$\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}.$$
[i]Israel[/i]
2012 Poland - Second Round, 1
$f,g:\mathbb{R}\rightarrow\mathbb{R}$ find all $f,g$ satisfying $\forall x,y\in \mathbb{R}$:
\[g(f(x)-y)=f(g(y))+x.\]
2005 China Team Selection Test, 3
Let $n$ be a positive integer, and $a_j$, for $j=1,2,\ldots,n$ are complex numbers. Suppose $I$ is an arbitrary nonempty subset of $\{1,2,\ldots,n\}$, the inequality $\left|-1+ \prod_{j\in I} (1+a_j) \right| \leq \frac 12$ always holds.
Prove that $\sum_{j=1}^n |a_j| \leq 3$.
1990 Tournament Of Towns, (257) 1
Prove that for all natural $n$ there exists a polynomial $P(x)$ divisible by $(x-1)^n$ such that its degree is not greater than $2^n$ and each of its coefficients is equal to $1$, $0$ or $-1$.
(D. Fomin, Leningrad)
2009 Hungary-Israel Binational, 3
Does there exist a pair $ (f; g)$ of strictly monotonic functions, both from $ \mathbb{N}$ to $ \mathbb{N}$, such that \[ f(g(g(n))) < g(f(n))\] for every $ n \in\mathbb{N}$?
2018 MOAA, 5
Mr. DoBa likes to listen to music occasionally while he does his math homework. When he listens to classical music, he solves one problem every $3$ minutes. When he listens to rap music, however, he only solves one problem every $5$ minutes. Mr. DoBa listens to a playlist comprised of $60\%$ classical music and $40\%$ rap music. Each song is exactly $4$ minutes long. Suppose that the expected number of problems he solves in an hour does not depend on whether or not Mr. DoBa is listening to music at any given moment, and let $m$ the average number of problems Mr. DoBa solves per minute when he is not listening to music. Determine the value of $1000m$.
2003 Alexandru Myller, 1
[b]1)[/b] Show that there exist quadratic polynoms $ P\in\mathbb{R}[X] $ whose composition with themselves have $
1,2 $ and $ 3 $ as their fixed points.
[b]2)[/b] Prove that the polynoms referred to at [b]1)[/b] are not integer.
[i]Gheorghe Iurea[/i]
2010 Paenza, 2
A polynomial $f$ with integer coefficients is written on the blackboard. The teacher is a mathematician who has $3$ kids: Andrew, Beth and Charles. Andrew, who is $7$, is the youngest, and Charles is the oldest. When evaluating the polynomial on his kids' ages he obtains:
[list]$f(7) = 77$
$f(b) = 85$, where $b$ is Beth's age,
$f(c) = 0$, where $c$ is Charles' age.[/list]
How old is each child?
2012 Romania Team Selection Test, 3
Let $a_1$ , $\ldots$ , $a_n$ be positive integers and $a$ a positive integer that is greater than $1$ and is divisible by the product $a_1a_2\ldots a_n$. Prove that $a^{n+1}+a-1$ is not divisible by the product $(a+a_1-1)(a+a_2-1)\ldots(a+a_n-1)$.
1999 Italy TST, 3
(a) Find all strictly monotone functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that
\[f(x+f(y))=f(x)+y\quad\text{for all real}\ x,y. \]
(b) If $n>1$ is an integer, prove that there is no strictly monotone function $f:\mathbb{R}\rightarrow\mathbb{R}$ such that
\[ f(x+f(y))=f(x)+y^n\quad \text{for all real}\ x, y.\]
1987 IMO Shortlist, 9
Does there exist a set $M$ in usual Euclidean space such that for every plane $\lambda$ the intersection $M \cap \lambda$ is finite and nonempty ?
[i]Proposed by Hungary.[/i]
[hide="Remark"]I'm not sure I'm posting this in a right Forum.[/hide]
2019 Philippine MO, 1
Find all functions $f : R \to R$ such that $f(2xy) + f(f(x + y)) = xf(y) + yf(x) + f(x + y)$ for all real numbers $x$ and $y$.
2001 South africa National Olympiad, 2
Find all triples $(x,y,z)$ of real numbers that satisfy \[ \begin{aligned} & x\left(1 - y^2\right)\left(1 - z^2\right) + y\left(1 - z^2\right)\left(1 - x^2\right) + z\left(1 - x^2\right)\left(1 - y^2\right) \\ & = 4xyz \\ & = 4(x + y + z). \end{aligned} \]
1994 French Mathematical Olympiad, Problem 3
Let us define a function $f:\mathbb N\to\mathbb N_0$ by $f(1)=0$ and, for all $n\in\mathbb N$,
$$f(2n)=2f(n)+1,\qquad f(2n+1)=2f(n).$$Given a positive integer $p$, define a sequence $(u_n)$ by $u_0=p$ and $u_{k+1}=f(u_k)$ whenever $u_k\ne0$.
(a) Prove that, for each $p\in\mathbb N$, there is a unique integer $v(p)$ such that $u_{v(p)}=0$.
(b) Compute $v(1994)$. What is the smallest integer $p>0$ for which $v(p)=v(1994)$.
(c) Given an integer $N$, determine the smallest integer $p$ such that $v(p)=N$.
1972 Yugoslav Team Selection Test, Problem 1
Given non-zero real numbers $u,v,w,x,y,z$, how many different possibilities are there for the signs of these numbers if
$$(u+ix)(v+iy)(w+iz)=i?$$