Found problems: 15925
2016 CMIMC, 5
The parabolas $y=x^2+15x+32$ and $x = y^2+49y+593$ meet at one point $(x_0,y_0)$. Find $x_0+y_0$.
1986 Traian Lălescu, 2.1
Find the real values $ m\in\mathbb{R} $ such that all solutions of the equation
$$ 1=2mx(2x-1)(2x-2)(2x-3) $$
are real.
2023 Israel TST, P1
Find all functions $f:\mathbb{R}\to \mathbb{R}$ such that for all $x, y\in \mathbb{R}$ the following holds:
\[f(x)+f(y)=f(xy)+f(f(x)+f(y))\]
2005 Regional Competition For Advanced Students, 4
Prove: if an infinte arithmetic sequence ($ a_n\equal{}a_0\plus{}nd$) of positive real numbers contains two different powers of an integer $ a>1$, then the sequence contains an infinite geometric sequence ($ b_n\equal{}b_0q^n$) of real numbers.
2008 Thailand Mathematical Olympiad, 5
Let $P(x)$ be a polynomial of degree $2008$ with the following property: all roots of $P$ are real, and for all real $a$, if $P(a) = 0$ then $P(a+ 1) = 1$. Prove that P must have a repeated root.
2012 China Northern MO, 2
Positive integers $x_1,x_2,...,x_n$ ($n \in N_+$) satisfy $x_1^2 +x_2^2+...+x_n^2=111$, find the maximum possible value of $S =\frac{x_1 +x_2+...+x_n}{n}$.
2021 BMT, 9
Compute the sum of the positive integers $n \le 100$ for which the polynomial $x^n + x + 1$ can be written as the product of at least $2$ polynomials of positive degree with integer coefficients.
2003 Alexandru Myller, 1
Let be a natural number $ n, $ a positive real number $ \lambda , $ and a complex number $ z. $ Prove the following inequalities.
$$ 0\le -\lambda +\frac{1}{n}\sum_{\stackrel{w\in\mathbb{C}}{w^n=1 }} \left| z-\lambda w \right|\le |z| $$
[i]Gheorghe Iurea[/i]
2000 Manhattan Mathematical Olympiad, 1
Prove there exists no polynomial $f(x)$, with integer coefficients, such that $f(7) = 11$ and $f(11) = 13$.
2019 India IMO Training Camp, P3
Let $n\ge 2$ be an integer. Solve in reals:
\[|a_1-a_2|=2|a_2-a_3|=3|a_3-a_4|=\cdots=n|a_n-a_1|.\]
2013 AMC 10, 8
What is the value of \[\frac{2^{2014}+2^{2012}}{2^{2014}-2^{2012}}?\]
$ \textbf{(A)}\ -1\qquad\textbf{(B)}\ 1\qquad\textbf{(C)}\ \frac{5}{3}\qquad\textbf{(D)}\ 2013\qquad\textbf{(E)}\ 2^{4024} $
2019 Serbia JBMO TST, 2
If a b c positive reals smaller than 1, prove:
a+b+c+2abc>ab+bc+ca+2(abc)^(1/2)
2019 Romania Team Selection Test, 3
Let be three positive integers $ a,b,c $ and a function $ f:\mathbb{N}\longrightarrow\mathbb{N} $ defined as
$$ f(n)=\left\{ \begin{matrix} n-a, & n>c\\ f\left( f(n+b) \right) ,& n\le c \end{matrix} \right. . $$
Determine the number of fixed points this function has.
2022 Dutch IMO TST, 3
For real numbers $x$ and $y$ we define $M(x, y)$ to be the maximum of the three numbers $xy$, $(x- 1)(y - 1)$, and $x + y - 2xy$. Determine the smallest possible value of $M(x, y)$ where $x$ and $y$ range over all real numbers satisfying $0 \le x, y \le 1$.
2019 India PRMO, 16
A pen costs $\mathrm{Rs.}\, 13$ and a note book costs $\mathrm{Rs.}\, 17$. A school spends exactly $\mathrm{Rs.}\, 10000$ in the year $2017-18$ to buy $x$ pens and $y$ note books such that $x$ and $y$ are as close as possible (i.e., $|x-y|$ is minimum). Next year, in $2018-19$, the school spends a little more than $\mathrm{Rs.}\, 10000$ and buys $y$ pens and $x$ note books. How much [b]more[/b] did the school pay?
2016 Baltic Way, 10
Let $a_{0,1}, a_{0,2}, . . . , a_{0, 2016}$ be positive real numbers. For $n\geq 0$ and $1 \leq k < 2016$ set $$a_{n+1,k} = a_{n,k} +\frac{1}{2a_{n,k+1}} \ \ \text{and} \ \ a_{n+1,2016} = a_{n,2016} +\frac{1}{2a_{n,1}}.$$
Show that $\max_{1\leq k \leq 2016} a_{2016,k} > 44.$
2024 Bangladesh Mathematical Olympiad, P7
Find all functions $f:\mathbb{N} \to \mathbb{N}$ such that\[f\left(\Big \lceil \frac{f(m)}{n} \Big \rceil\right)=\Big \lceil \frac{m}{f(n)} \Big \rceil\]for all $m,n \in \mathbb{N}$.
[i]Proposed by Md. Ashraful Islam Fahim[/i]
2014 Contests, 1a
Assume that $x, y \ge 0$. Show that $x^2 + y^2 + 1 \le \sqrt{(x^3 + y + 1)(y^3 + x + 1)}$.
2016 BMT Spring, 10
Define $T_n =\sum^{n}){i=1} i(n + 1 - i)$. Find $\lim_{n\to \infty} \frac{T_n}{n^3}$.
2013 Estonia Team Selection Test, 3
Let $x_1,..., x_n$ be non-negative real numbers, not all of which are zeros.
(i) Prove that
$$1 \le \frac{\left(x_1+\frac{x_2}{2}+\frac{x_3}{3}+...+\frac{x_n}{n}\right)(x_1+2x_2+3x_3+...+nx_n)}{(x_1+x_2+x_3+...+x_n)^2} \le \frac{(n+1)^2}{4n}$$
(ii) Show that, for each $n > 1$, both inequalities can hold as equalities.
2022 Iran Team Selection Test, 5
Find all $C\in \mathbb{R}$ such that every sequence of integers $\{a_n\}_{n=1}^{\infty}$ which is bounded from below and for all $n\geq 2$ satisfy $$0\leq a_{n-1}+Ca_n+a_{n+1}<1$$ is periodic.
Proposed by Navid Safaei
1963 All Russian Mathematical Olympiad, 038
Find such real $p, q, a, b$, that for all $x$ an equality is held: $$(2x-1)^{20} - (ax+b)^{20} = (x^2+px+q)^{10}$$
1994 IMO Shortlist, 3
Let $ S$ be the set of all real numbers strictly greater than −1. Find all functions $ f: S \to S$ satisfying the two conditions:
(a) $ f(x \plus{} f(y) \plus{} xf(y)) \equal{} y \plus{} f(x) \plus{} yf(x)$ for all $ x, y$ in $ S$;
(b) $ \frac {f(x)}{x}$ is strictly increasing on each of the two intervals $ \minus{} 1 < x < 0$ and $ 0 < x$.
2013 Abels Math Contest (Norwegian MO) Final, 1b
The sequence $a_1, a_2, a_3,...$ is defined so that $a_1 = 1$ and $a_{n+1} =\frac{a_1 + a_2 + ...+ a_n}{n}+1$ for $n \ge 1$. Show that for every positive real number $b$ we can find $a_k$ so that $a_k < bk$.
2012 Abels Math Contest (Norwegian MO) Final, 4b
Positive numbers $b_1, b_2,..., b_n$ are given so that $b_1 + b_2 + ...+ b_n \le 10$.
Further, $a_1 = b_1$ and $a_m = sa_{m-1} + b_m$ for $m > 1$, where $0 \le s < 1$.
Show that $a^2_1 + a^2_2 + ... + a^2_n \le \frac{100}{1 - s^2} $