Found problems: 15925
VI Soros Olympiad 1999 - 2000 (Russia), 11.2
A bus and a cyclist left town $A$ at $10$ o'clock in the same direction, and a motorcyclist left town $B$ to meet them $15$ minutes later. The bus drove past the pedestrian at $10$ o'clock $30$ minutes, met the motorcyclist at $11$ o'clock and arrived in the city of $B$ at $12$ o'clock. The motorcyclist met the cyclist $15$ minutes after meeting the bus and another $15$ minutes later caught up with the pedestrian. At what time did the cyclist and the pedestrian meet? (The speeds and directions of movement of all participants were equal, the pedestrian and the motorcyclist were moving in the direction of city $A$.)
2010 Romania Team Selection Test, 2
Let $n$ be a positive integer number and let $a_1, a_2, \ldots, a_n$ be $n$ positive real numbers. Prove that $f : [0, \infty) \rightarrow \mathbb{R}$, defined by
\[f(x) = \dfrac{a_1 + x}{a_2 + x} + \dfrac{a_2 + x}{a_3 + x} + \cdots + \dfrac{a_{n-1} + x}{a_n + x} + \dfrac{a_n + x}{a_1 + x}, \]
is a decreasing function.
[i]Dan Marinescu et al.[/i]
2003 China Team Selection Test, 3
Let $ \left(x_{n}\right)$ be a real sequence satisfying $ x_{0}=0$, $ x_{2}=\sqrt[3]{2}x_{1}$, and $ x_{n+1}=\frac{1}{\sqrt[3]{4}}x_{n}+\sqrt[3]{4}x_{n-1}+\frac{1}{2}x_{n-2}$ for every integer $ n\geq 2$, and such that $ x_{3}$ is a positive integer. Find the minimal number of integers belonging to this sequence.
1962 AMC 12/AHSME, 12
When $ \left ( 1 \minus{} \frac{1}{a} \right ) ^6$ is expanded the sum of the last three coefficients is:
$ \textbf{(A)}\ 22 \qquad
\textbf{(B)}\ 11 \qquad
\textbf{(C)}\ 10 \qquad
\textbf{(D)}\ \minus{}10 \qquad
\textbf{(E)}\ \minus{}11$
2020 Dutch BxMO TST, 3
Find all functions $f: R \to R$ that satisfy
$$f (x^2y) + 2f (y^2) =(x^2 + f (y)) \cdot f (y)$$ for all $x, y \in R$
2004 Iran MO (3rd Round), 20
$ p(x)$ is a polynomial in $ \mathbb{Z}[x]$ such that for each $ m,n\in \mathbb{N}$ there is an integer $ a$ such that $ n\mid p(a^m)$. Prove that $0$ or $1$ is a root of $ p(x)$.
2024 Poland - Second Round, 1
Does there exist a rational $x_1$, such that all members of the sequence $x_1, x_2, \ldots, x_{2024}$ defined by $x_{n+1}=x_n+\sqrt{x_n^2-1}$ for $n=1, 2, \ldots, 2023$ are greater than $1$ and rational?
2006 Austrian-Polish Competition, 2
Find all polynomials $P(x)$ with real coefficients satisfying the equation \[(x+1)^{3}P(x-1)-(x-1)^{3}P(x+1)=4(x^{2}-1) P(x)\] for all real numbers $x$.
1985 Traian Lălescu, 2.2
Let $ a,b,c\in\mathbb{R} , E=(a-b)^2(b-c)^2(c-a)^2, $ and $ S_k=a^k+b^k+c^k,\forall k\in\{ 1,2,3,4\} . $
Write $ E $ in terms of $ S_k. $
1997 All-Russian Olympiad, 1
Do there exist two quadratic trinomials $ax^2 +bx+c$ and $(a+1)x^2 +(b + 1)x + (c + 1)$ with integer coeficients, both of which have two integer roots?
[i]N. Agakhanov[/i]
2019 PUMaC Algebra A, 8
For real numbers $a$ and $b$, define the sequence $\{x_{a,b}(n)\}$ as follows: $x_{a,b}(1)=a$, $x_{a,b}(2)=b$, and for $n>1$, $x_{a,b}(n+1)=(x_{a+b}(n-1))^2+(x_{a,b}(n))^2$. For real numbers $c$ and $d$, define the sequence $\{y_{c,d}(n)\}$ as follows: $y_{c,d}(1)=c$, $y_{c,d}(2)=d$, and for $n>1$, $y_{c,d}(n+1)=(y_{c,d}(n-1)+y_{c,d}(n))^2$. Call $(a,b,c)$ a good triple if there exists $d$ such that for all $n$ sufficiently large, $y_{c,d}(n)=(x_{a,b}(n))^2$. For some $(a,b)$ there are exactly three values of $c$ that make $(a,b,c)$ a good triple. Among these pairs $(a,b)$, compute the maximum value of $\lfloor 100(a+b)\rfloor$.
1974 IMO Longlists, 40
Three players $A,B$ and $C$ play a game with three cards and on each of these $3$ cards it is written a positive integer, all $3$ numbers are different. A game consists of shuffling the cards, giving each player a card and each player is attributed a number of points equal to the number written on the card and then they give the cards back. After a number $(\geq 2)$ of games we find out that A has $20$ points, $B$ has $10$ points and $C$ has $9$ points. We also know that in the last game B had the card with the biggest number. Who had in the first game the card with the second value (this means the middle card concerning its value).
1967 IMO Shortlist, 1
Let $a_1,\ldots,a_8$ be reals, not all equal to zero. Let
\[ c_n = \sum^8_{k=1} a^n_k\]
for $n=1,2,3,\ldots$. Given that among the numbers of the sequence $(c_n)$, there are infinitely many equal to zero, determine all the values of $n$ for which $c_n = 0.$
2008 Bundeswettbewerb Mathematik, 2
Represent the number $ 2008$ as a sum of natural number such that the addition of the reciprocals of the summands yield 1.
1980 IMO Longlists, 4
Determine all positive integers $n$ such that the following statement holds: If a convex polygon with with $2n$ sides $A_1 A_2 \ldots A_{2n}$ is inscribed in a circle and $n-1$ of its $n$ pairs of opposite sides are parallel, which means if the pairs of opposite sides
\[(A_1 A_2, A_{n+1} A_{n+2}), (A_2 A_3, A_{n+2} A_{n+3}), \ldots , (A_{n-1} A_n, A_{2n-1} A_{2n})\]
are parallel, then the sides \[ A_n A_{n+1}, A_{2n} A_1\] are parallel as well.
2017 Brazil Team Selection Test, 4
Find all functions $f:(0,\infty)\rightarrow (0,\infty)$ such that for any $x,y\in (0,\infty)$, $$xf(x^2)f(f(y)) + f(yf(x)) = f(xy) \left(f(f(x^2)) + f(f(y^2))\right).$$
1958 Polish MO Finals, 4
Prove that if $ k $ is a natural number, then
$$ (1 + x)(1 + x^2) (1 + x^4) \ldots (1 + x^{2^k}) =1 + x + x^2 + x^3+ \ldots + x^m$$
where $ m $ is a natural number dependent on $ k $; determine $ m $.
1988 Flanders Math Olympiad, 1
show that the polynomial $x^4+3x^3+6x^2+9x+12$ cannot be written as the product of 2 polynomials of degree 2 with integer coefficients.
2020 Bulgaria Team Selection Test, 5
Given is a function $f:\mathbb{R}\rightarrow \mathbb{R}$ such that $|f(x+y)-f(x)-f(y)|\leq 1$.
Prove the existence of an additive function $g:\mathbb{R}\rightarrow \mathbb{R}$ (that is $g(x+y)=g(x)+g(y)$) such that $|f(x)-g(x)|\leq 1$ for any $x \in \mathbb{R}$
2005 Austrian-Polish Competition, 2
Determine all polynomials $P$ with integer coefficients satisfying
\[P(P(P(P(P(x)))))=x^{28}\cdot P(P(x))\qquad \forall x\in\mathbb{R}\]
2022 Princeton University Math Competition, A3 / B5
Find the number of real solutions $(x,y)$ to the system of equations:
$$\begin{cases}
\sin(x^2-y) = 0 \\
|x|+|y|=2\pi
\end{cases}$$
2004 CentroAmerican, 2
Define the sequence $(a_n)$ as follows: $a_0=a_1=1$ and for $k\ge 2$, $a_k=a_{k-1}+a_{k-2}+1$.
Determine how many integers between $1$ and $2004$ inclusive can be expressed as $a_m+a_n$ with $m$ and $n$ positive integers and $m\not= n$.
2012 Iran MO (3rd Round), 5
Let $p$ be an odd prime number and let $a_1,a_2,...,a_n \in \mathbb Q^+$ be rational numbers. Prove that \[\mathbb Q(\sqrt[p]{a_1}+\sqrt[p]{a_2}+...+\sqrt[p]{a_n})=\mathbb Q(\sqrt[p]{a_1},\sqrt[p]{a_2},...,\sqrt[p]{a_n}).\]
2025 Euler Olympiad, Round 1, 2
Find all five-digit numbers that satisfy the following conditions:
1. The number is a palindrome.
2. The middle digit is twice the value of the first digit.
3. The number is a perfect square.
[i]Proposed by Tamar Turashvili, Georgia [/i]
1993 Cono Sur Olympiad, 1
On a table there is a pile with $ T$ tokens which incrementally shall be converted into piles with three tokens each. Each step is constituted of selecting one pile removing one of its tokens. And then the remaining pile is separated into two piles. Is there a sequence of steps that can accomplish this process?
a.) $ T \equal{} 1000$ (Cono Sur)
b.) $ T \equal{} 2001$ (BWM)