Found problems: 15925
2011 ISI B.Math Entrance Exam, 1
Given $a,x\in\mathbb{R}$ and $x\geq 0$,$a\geq 0$ . Also $\sin(\sqrt{x+a})=\sin(\sqrt{x})$ . What can you say about $a$??? Justify your answer.
2010 Germany Team Selection Test, 3
Let $f$ be any function that maps the set of real numbers into the set of real numbers. Prove that there exist real numbers $x$ and $y$ such that \[f\left(x-f(y)\right)>yf(x)+x\]
[i]Proposed by Igor Voronovich, Belarus[/i]
2015 IFYM, Sozopol, 7
Determine all polynomials $P(x)$ with real coefficients such that $(x+1)P(x-1)-(x-1)P(x)$ is a constant polynomial.
2004 District Olympiad, 2
The real numbers $a, b, c, d$ satisfy $a > b > c > d$ and
$$a + b + c + d = 2004 \,\,\, and \,\,\, a^2 - b^2 + c^2 - d^2 = 2004.$$
Answer, with proof, to the following questions:
a) What is the smallest possible value of $a$?
b) What is the number of possible values of $a$?
2008 Harvard-MIT Mathematics Tournament, 10
Determine the number of $ 8$-tuples of nonnegative integers $ (a_1,a_2,a_3,a_4,b_1,b_2,b_3,b_4)$ satisfying $ 0\le a_k\le k$, for each $ k \equal{} 1,2,3,4$, and $ a_1 \plus{} a_2 \plus{} a_3 \plus{} a_4 \plus{} 2b_1 \plus{} 3b_2 \plus{} 4b_3 \plus{} 5b_4 \equal{} 19$.
2008 JBMO Shortlist, 1
If for the real numbers $x, y,z, k$ the following conditions are valid, $x \ne y \ne z \ne x$ and
$x^3 +y^3 +k(x^2 +y^2) = y^3 +z^3 +k(y^2 +z^2) = z^3 +x^3 +k(z^2 +x^2) = 2008$, fi nd the product $xyz$.
2015 CHMMC (Fall), 3
A trio of lousy salespeople charge increasing prices on tomatoes as you buy more. The first charges you $x^1_1$ dollars for the $x_1$[i]th [/i]tomato you buy from him, the second charges $x^2_2$ dollars for the $x_2$[i]th[/i] tomato, and the third charges $x^3_3$ dollars for the $x_3$[i]th [/i]tomato. If you want to buy $100$ tomatoes for as cheap as possible, how many should you buy from the first salesperson?
2003 China Team Selection Test, 1
$m$ and $n$ are positive integers. Set $A=\{ 1, 2, \cdots, n \}$. Let set $B_{n}^{m}=\{ (a_1, a_2 \cdots, a_m) \mid a_i \in A, i= 1, 2, \cdots, m \}$ satisfying:
(1) $|a_i - a_{i+1}| \neq n-1$, $i=1,2, \cdots, m-1$; and
(2) at least three of $a_1, a_2, \cdots, a_m$ ($m \geq 3$) are pairwise distince.
Find $|B_n^m|$ and $|B_6^3|$.
2009 India IMO Training Camp, 5
Let $ f(x)$and $ g(y)$ be two monic polynomials of degree=$ n$ having complex coefficients.
We know that there exist complex numbers $ a_i,b_i,c_i \forall 1\le i \le n$, such that
$ f(x)\minus{}g(y)\equal{}\prod_{i\equal{}1}^n{(a_ix\plus{}b_iy\plus{}c_i)}$.
Prove that there exists $ a,b,c\in\mathbb{C}$ such that
$ f(x)\equal{}(x\plus{}a)^n\plus{}c\text{ and }g(y)\equal{}(y\plus{}b)^n\plus{}c$.
2009 Princeton University Math Competition, 2
Given that $P(x)$ is the least degree polynomial with rational coefficients such that
\[P(\sqrt{2} + \sqrt{3}) = \sqrt{2},\] find $P(10)$.
2017 China Girls Math Olympiad, 4
Partition $\frac1{2002},\frac1{2003},\frac1{2004},\ldots,\frac{1}{2017}$ into two groups. Define $A$ the sum of the numbers in the first group, and $B$ the sum of the numbers in the second group. Find the partition such that $|A-B|$ attains it minimum and explains the reason.
1971 Swedish Mathematical Competition, 5
Show that \[
\max\limits_{|x|\leq t} |1 - a \cos x| \geq \tan^2 \frac{t}{2}
\]
for $a$ positive and $t \in (0, \frac{\pi}{2})$.
2012 IMO Shortlist, N5
For a nonnegative integer $n$ define $\operatorname{rad}(n)=1$ if $n=0$ or $n=1$, and $\operatorname{rad}(n)=p_1p_2\cdots p_k$ where $p_1<p_2<\cdots <p_k$ are all prime factors of $n$. Find all polynomials $f(x)$ with nonnegative integer coefficients such that $\operatorname{rad}(f(n))$ divides $\operatorname{rad}(f(n^{\operatorname{rad}(n)}))$ for every nonnegative integer $n$.
2011 Bogdan Stan, 1
Find the natural numbers $ n $ which have the property that
$$ 2011=\left| \mathbb{Q}\cap\bigcup_{k=1}^n\left\{ x\in\mathbb{R} | 1+k^2x^2=2k\left( x-\lfloor x\rfloor \right) \right\} \right| . $$
[i]Marian Teler[/i]
2021 BMT, 7
Let $z_1, z_2, ..., z_{2020}$ be the roots of the polynomial $z^{2020} + z^{2019} +...+ z + 1$. Compute
$$\sum^{2020}_{i=1} \frac{1}{1 -z^{2020}_i}.$$
2018 Brazil Undergrad MO, 5
Consider the set $A = \left\{\frac{j}{4}+\frac{100}{j}|j=1,2,3,..\right\} $ What is the smallest number that belongs to the $ A $ set?
2003 All-Russian Olympiad Regional Round, 10.5
Find all $x$ for which the equation $ x^2 + y^2 + z^2 + 2xyz = 1$ (relative to $z$) has a valid solution for any $y$.
2023 BMT, 24
Define the sequence $s_0$, $s_1$, $s_2$,$ . . .$ by $s_0 = 0$ and $s_n = 3s_{n-1}+2$ for $n \ge 1$. The monic polynomial $f(x)$ defined as $$f(x) =\frac{1}{s_{2023}} \sum^{32}_{k=0} s_{2023+k}x^{32-k}$$ can be factored uniquely (up to permutation) as the product of $16$ monic quadratic polynomials $p_1$, $p_2$, $....$, $p_{16}$ with real coefficients, where $p_i(x) = x^2 + a_ix + b_i$ for $1\le i \le 16$. Compute the integer $N$ that minimizes
$$\left|N - \sum^{16}_{k=1} (a_k + b_k)\right|.$$
2005 Tournament of Towns, 4
For any function $f(x)$, define $f^1(x) = f(x)$ and $f^n (x) = f(f^{n-1}(x))$ for any integer $n \ge 2$. Does there exist a quadratic polynomial $f(x)$ such that the equation $f^n(x) = 0$ has exactly $2^n$ distinct real roots for every positive integer $n$?
[i](6 points)[/i]
2014 Germany Team Selection Test, 2
Let $\mathbb{Z} _{>0}$ be the set of positive integers. Find all functions $f: \mathbb{Z} _{>0}\rightarrow \mathbb{Z} _{>0}$ such that
\[ m^2 + f(n) \mid mf(m) +n \]
for all positive integers $m$ and $n$.
1951 Polish MO Finals, 3
Prove that if $ a > 0 $, $ b > 0 $, $ c > 0 $, then the inequality holds
$$ ab (a + b) + bc (b + c) + ca (c + a) \geq 6abc.$$
1952 Moscow Mathematical Olympiad, 221
Prove that if for any positive $p$ all roots of the equation $ax^2 + bx + c + p = 0$ are real and positive then $a = 0$.
2019 Hanoi Open Mathematics Competitions, 9
Let $a$ and $b$ be positive real numbers with $a > b$. Find the smallest possible values of $$S = 2a +3 +\frac{32}{(a - b)(2b +3)^2}$$
1995 Turkey Team Selection Test, 1
Given real numbers $b \geq a>0$, find all solutions of the system
\begin{align*}
&x_1^2+2ax_1+b^2=x_2,\\
&x_2^2+2ax_2+b^2=x_3,\\
&\qquad\cdots\cdots\cdots\\
&x_n^2+2ax_n+b^2=x_1.
\end{align*}
1999 Chile National Olympiad, 7
Let $f$ be a function defined on the set of positive integers , and with values in the same set, which satisfies:
$\bullet$ $f (n + f (n)) = 1$ for all $n\ge 1$.
$\bullet$ $f (1998) = 2$
Find the lowest possible value of the sum $f (1) + f (2) +... + f (1999)$, and find the formula of $f$ for which this minimum is satisfied,