Found problems: 15925
2017 AMC 12/AHSME, 3
Suppose that $x$ and $y$ are nonzero real numbers such that \[\frac{3x+y}{x-3y}= -2.\] What is the value of \[\frac{x+3y}{3x-y}?\]
$\textbf{(A) } {-3} \qquad \textbf{(B) } {-1} \qquad \textbf{(C) } 1 \qquad \textbf{(D) }2 \qquad \textbf{(E) } 3$
2010 Morocco TST, 3
Let $G$ be a non-empty set of non-constant functions $f$ such that $f(x)=ax + b$ (where $a$ and $b$ are two reals) and satisfying the following conditions:
1) if $f \in G$ and $g \in G$ then $gof \in G$,
2) if $f \in G$ then $f^ {-1} \in G$,
3) for all $f \in G$ there exists $x_f \in \mathbb{R}$ such that $f(x_f)=x_f$.
Prove that there is a real $k$ such that for all $f \in G$ we have $f(k)=k$
1988 Swedish Mathematical Competition, 3
Show that if $x_1+x_2+x_3 = 0$ for real numbers $x_1,x_2,x_3$, then $x_1x_2+x_2x_3+x_3x_1\le 0$.
Find all $n \ge 4$ for which $x_1+x_2+...+x_n = 0$ implies $x_1x_2+x_2x_3+...+x_{n-1}x_n+x_nx_1 \le 0$.
1995 Poland - First Round, 9
A polynomial with integer coefficients when divided by $x^2-12x+11$ gives the remainder $990x-889$. Prove that the polynomial has no integer roots.
2010 Iran MO (3rd Round), 3
suppose that $G<S_n$ is a subgroup of permutations of $\{1,...,n\}$ with this property that for every $e\neq g\in G$ there exist exactly one $k\in \{1,...,n\}$ such that $g.k=k$. prove that there exist one $k\in \{1,...,n\}$ such that for every $g\in G$ we have $g.k=k$.(20 points)
2023 German National Olympiad, 1
Determine all pairs $(m,n)$ of integers with $n \ge m$ satisfying the equation
\[n^3+m^3-nm(n+m)=2023.\]
2008 Hanoi Open Mathematics Competitions, 3
Find the coefficient of $x$ in the expansion of $(1 + x)(1 - 2x)(1 + 3x)(1 - 4x) ...(1 - 2008x)$.
2016 German National Olympiad, 1
Find all real pairs $(a,b)$ that solve the system of equation \begin{align*} a^2+b^2 &= 25, \\ 3(a+b)-ab &= 15. \end{align*} [i](German MO 2016 - Problem 1)[/i]
2011 Mathcenter Contest + Longlist, 9 sl13
Let $a,b,c\in\mathbb{R^+}$ If $3=a+b+c\le 3abc$ , prove that $$\frac{1}{\sqrt{2a+1}}+ \frac{1}{\sqrt{2b+1}}+\frac{1}{\sqrt{2c+1}}\le \left( \frac32\right)^{3/2}$$
[i](Real Matrik)[/i]
1970 IMO Longlists, 37
Solve the set of simultaneous equations
\begin{align*}
v^2+ w^2+ x^2+ y^2 &= 6 - 2u, \\
u^2+ w^2+ x^2+ y^2 &= 6 - 2v, \\
u^2+ v^2+ x^2+ y^2 &= 6- 2w, \\
u^2+ v^2+ w^2+ y^2 &= 6 - 2x, \\
u^2+ v^2+ w^2+ x^2 &= 6- 2y.
\end{align*}
1988 IMO Longlists, 49
Let $ f(n)$ be a function defined on the set of all positive integers and having its values in the same set. Suppose that $ f(f(n) \plus{} f(m)) \equal{} m \plus{} n$ for all positive integers $ n,m.$ Find the possible value for $ f(1988).$
2020 Canada National Olympiad, 1
There are $n \ge 3$ distinct positive real numbers. Show that there are at most $n-2$ different integer power of three that can be written as the sum of three distinct elements from these $n$ numbers.
1971 Swedish Mathematical Competition, 1
Show that
\[
\left(1 + a + a^2\right)^2 < 3\left(1 + a^2 + a^4\right)
\]
for real $a \neq 1$.
1994 Czech And Slovak Olympiad IIIA, 4
Let $a_1,a_2,...$ be a sequence of natural numbers such that for each $n$, the product $(a_n - 1)(a_n- 2)...(a_n - n^2)$ is a positive integral multiple of $n^{n^2-1}$. Prove that for any finite set $P$ of prime numbers the following inequality holds:
$$\sum_{p\in P}\frac{1}{\log_p a_p}< 1$$
1989 IMO Longlists, 2
An accurate 12-hour analog clock has an hour hand, a minute hand, and a second hand that are aligned at 12:00 o’clock and make one revolution in 12 hours, 1 hour, and 1 minute, respectively. It is well known, and not difficult to prove, that there is no time when the three hands are equally spaced around the clock, with each separating angle $ \frac{2 \cdot \pi}{3}.$ Let $ f(t), g(t), h(t)$ be the respective absolute deviations of the separating angles from \frac{2 \cdot \pi}{3} at $ t$ hours after 12:00 o’clock. What is the minimum value of $ max\{f(t), g(t), h(t)\}?$
2021 Caucasus Mathematical Olympiad, 1
Let $a$, $b$, $c$ be real numbers such that $a^2+b=c^2$, $b^2+c=a^2$, $c^2+a=b^2$. Find all possible values of $abc$.
2018 JBMO Shortlist, A4
Let $k > 1, n > 2018$ be positive integers, and let $n$ be odd. The nonzero rational numbers $x_1,x_2,\ldots,x_n$ are not all equal and satisfy $$x_1+\frac{k}{x_2}=x_2+\frac{k}{x_3}=x_3+\frac{k}{x_4}=\ldots=x_{n-1}+\frac{k}{x_n}=x_n+\frac{k}{x_1}$$
Find:
a) the product $x_1 x_2 \ldots x_n$ as a function of $k$ and $n$
b) the least value of $k$, such that there exist $n,x_1,x_2,\ldots,x_n$ satisfying the given conditions.
2015 İberoAmerican, 3
Let $\alpha$ and $\beta$ be the roots of $x^{2} - qx + 1$, where $q$ is a rational number larger than $2$. Let $s_1 = \alpha + \beta$, $t_1 = 1$, and for all integers $n \geq 2$:
$s_n = \alpha^n + \beta^n$
$t_n = s_{n-1} + 2s_{n-2} + \cdot \cdot \cdot + (n - 1)s_{1} + n$
Prove that, for all odd integers $n$, $t_n$ is the square of a rational number.
2018 Junior Balkan Team Selection Tests - Romania, 1
Prove that the equation $x^2+y^2+z^2 = x+y+z+1$ has no rational solutions.
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 \[a_i^2 - 4a_{i-1}a_{i+1} \leq 0.\]
2005 Taiwan TST Round 2, 1
Let $a,b$ be two constants within the open interval $(0,\frac{1}{2})$. Find all continous functions $f(x)$ such that \[f(f(x))=af(x)+bx\] holds for all real $x$.
This is much harder than the problems we had in the 1st TST...
1957 Moscow Mathematical Olympiad, 355
a) A student takes a subway to an Olympiad, pays one ruble and gets his change. Prove that if he takes a tram (street car) on his way home, he will have enough coins to pay the fare without change.
b) A student is going to a club. (S)he takes a tram, pays one ruble and gets the change. Prove that on the way back by a tram (s)he will be able to pay the fare without any need to change.
Note: In $1957$, the price of a subway ticket was $50$ kopeks, that of a tram ticket $30$ kopeks, the denominations of the coins were $1, 2, 3, 5, 10, 15$, and $20$ kopeks. ($1$ rouble = $100$ kopeks.)
Gheorghe Țițeica 2024, P1
Let $E(x,y)=\frac{(1+x)(1+y)(1+xy)}{(1+x^2)(1+y^2)}$. Find the minimum and maximum value of $E$ on $\mathbb{R}^2$.
[i]Dorel Miheț[/i]
2009 Saint Petersburg Mathematical Olympiad, 7
$f(x)=x^2+x$
$b_1,...,b_{10000}>0$ and $|b_{n+1}-f(b_n)|\leq \frac{1}{1000}$ for $n=1,...,9999$
Prove, that there is such $a_1>0$ that $a_{n+1}=f(a_n);n=1,...,9999$ and $|a_n-b_n|<\frac{1}{100}$
1991 India National Olympiad, 6
(i) Determine the set of all positive integers $n$ for which $3^{n+1}$ divides $2^{3^n} + 1$;
(ii) Prove that $3^{n+2}$ does not divide $2^{3^n} + 1$ for any positive integer $n$.