Found problems: 15925
2023 Moldova Team Selection Test, 6
Show that if $2023$ real numbers $x_1,x_2,\dots,x_{2023}$ satisfy $x_1\geq x_2\geq\dots\geq x_{2023}\geq0,$ then $$x_1^2+3x_2^2+5x_3^2+\cdots+(2\cdot2023-1)\cdot x^2_{2023}\leq(x_1+x_2+\cdots+x_{2023})^2.$$ When does the equality take place?
MathLinks Contest 1st, 2
Let $f$ be a polynomial with real coefficients such that for each positive integer n the equation $f(x) = n$ has at least one rational solution. Find $f$.
2006 Turkey MO (2nd round), 3
Find all positive integers $n$ for which all coefficients of polynomial $P(x)$ are divisible by $7,$ where
\[P(x) = (x^2 + x + 1)^n - (x^2 + 1)^n - (x + 1)^n - (x^2 + x)^n + x^{2n} + x^n + 1.\]
2013 Peru MO (ONEM), 1
We define the polynomial $$P (x) = 2014x^{2013} + 2013x^{2012} +... + 4x^3 + 3x^2 + 2x.$$ Find the largest prime divisor of $P (2)$.
2004 Estonia National Olympiad, 5
Real numbers $a, b$ and $c$ satisfy $$\begin{cases} a^2 + b^2 + c^2 = 1 \\ a^3 + b^3 + c^3 = 1. \end{cases}$$ Find $a + b + c$.
2009 Estonia Team Selection Test, 1
For arbitrary pairwise distinct positive real numbers $a, b, c$, prove the inequality
$$\frac{(a^2- b^2)^3 + (b^2-c^2)^3+(c^2-a^2)^3}{(a- b)^3 + (b-c)^3+(c-a)^3}> 8abc$$
2003 AMC 12-AHSME, 25
Let $ f(x)\equal{}\sqrt{ax^2\plus{}bx}$. For how many real values of $ a$ is there at least one positive value of $ b$ for which the domain of $ f$ and the range of $ f$ are the same set?
$ \textbf{(A)}\ 0 \qquad
\textbf{(B)}\ 1 \qquad
\textbf{(C)}\ 2 \qquad
\textbf{(D)}\ 3 \qquad
\textbf{(E)}\ \text{infinitely many}$
2011 Iran MO (3rd Round), 5
Suppose that $n$ is a natural number. we call the sequence $(x_1,y_1,z_1,t_1),(x_2,y_2,z_2,t_2),.....,(x_s,y_s,z_s,t_s)$ of $\mathbb Z^4$ [b]good[/b] if it satisfies these three conditions:
[b]i)[/b] $x_1=y_1=z_1=t_1=0$.
[b]ii)[/b] the sequences $x_i,y_i,z_i,t_i$ be strictly increasing.
[b]iii)[/b] $x_s+y_s+z_s+t_s=n$. (note that $s$ may vary).
Find the number of good sequences.
[i]proposed by Mohammad Ghiasi[/i]
2019 ITAMO, 4
Let $\lfloor x \rfloor$ denote the greatest integer less than or equal to $x.$
Let $\lambda \geq 1$ be a real number and $n$ be a positive integer with the property that $\lfloor \lambda^{n+1}\rfloor, \lfloor \lambda^{n+2}\rfloor ,\cdots, \lfloor \lambda^{4n}\rfloor$ are all perfect squares$.$ Prove that $\lfloor \lambda \rfloor$ is a perfect square$.$
1980 IMO Shortlist, 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.
2021 CHMMC Winter (2021-22), 6
There is a unique degree-$10$ monic polynomial with integer coefficients $f(x)$ such that
$$f \left( \sum^9_{j=0}\sqrt[10]{2021^j}\right)= 0.$$
Find the remainder when $f(1)$ is divided by $1000$.
2021 Austrian MO Beginners' Competition, 1
The pages of a notebook are numbered consecutively so that the numbers $1$ and $2$ are on the second sheet, numbers $3$ and $4$, and so on. A sheet is torn out of this notebook. All of the remaining page numbers are addedand have sum $2021$.
(a) How many pages could the notebook originally have been?
(b) What page numbers can be on the torn sheet?
(Walther Janous)
1939 Moscow Mathematical Olympiad, 049
Let the product of two polynomials of a variable $x$ with integer coefficients be a polynomial with even coefficients not all of which are divisible by $4$. Prove that all the coefficients of one of the polynomials are even and that at least one of the coefficients of the other polynomial is odd.
2002 Vietnam Team Selection Test, 2
Find all polynomials $P(x)$ with integer coefficients such that the polynomial \[ Q(x)=(x^2+6x+10) \cdot P^2(x)-1 \] is the square of a polynomial with integer coefficients.
2019 New Zealand MO, 2
Find all real solutions to the equation $(x^2 + 3x + 1)^{x^2-x-6} = 1$.
2013 HMNT, 1
Two cars are driving directly towards each other such that one is twice as fast as the other. The distance between their starting points is $4$ miles. When the two cars meet, how many miles is the faster car from its starting point?
2023 Kyiv City MO Round 1, Problem 2
Positive integers $k$ and $n$ are given such that $3 \le k \le n$.Prove that among any $n$ pairwise distinct real numbers one can choose either $k$ numbers with positive sum, or $k-1$ numbers with negative sum.
[i]Proposed by Mykhailo Shtandenko[/i]
2005 Romania National Olympiad, 2
Let $f:[0,1)\to (0,1)$ a continous onto (surjective) function.
a) Prove that, for all $a\in(0,1)$, the function $f_a:(a,1)\to (0,1)$, given by $f_a(x) = f(x)$, for all $x\in(a,1)$ is onto;
b) Give an example of such a function.
2018 Purple Comet Problems, 19
Suppose that $a$ and $b$ are positive real numbers such that $3\log_{101}\left(\frac{1,030,301-a-b}{3ab}\right) = 3 - 2 \log_{101}(ab)$. Find $101 - \sqrt[3]{a}- \sqrt[3]{b}$.
1989 IMO Shortlist, 16
The set $ \{a_0, a_1, \ldots, a_n\}$ of real numbers satisfies the following conditions:
[b](i)[/b] $ a_0 \equal{} a_n \equal{} 0,$
[b](ii)[/b] for $ 1 \leq k \leq n \minus{} 1,$ \[ a_k \equal{} c \plus{} \sum^{n\minus{}1}_{i\equal{}k} a_{i\minus{}k} \cdot \left(a_i \plus{} a_{i\plus{}1} \right)\]
Prove that $ c \leq \frac{1}{4n}.$
2009 USA Team Selection Test, 9
Prove that for positive real numbers $x$, $y$, $z$, \[ x^3(y^2+z^2)^2 + y^3(z^2+x^2)^2+z^3(x^2+y^2)^2 \geq xyz\left[xy(x+y)^2 + yz(y+z)^2 + zx(z+x)^2\right].\] [i]Zarathustra (Zeb) Brady.[/i]
2010 Contests, 2
For each positive integer $n$, find the largest real number $C_n$ with the following property. Given any $n$ real-valued functions $f_1(x), f_2(x), \cdots, f_n(x)$ defined on the closed interval $0 \le x \le 1$, one can find numbers $x_1, x_2, \cdots x_n$, such that $0 \le x_i \le 1$ satisfying
\[|f_1(x_1)+f_2(x_2)+\cdots f_n(x_n)-x_1x_2\cdots x_n| \ge C_n\]
[i]Marko Radovanović, Serbia[/i]
1940 Moscow Mathematical Olympiad, 067
Which is greater: $300!$ or $100^{300}$?
2013 Bosnia And Herzegovina - Regional Olympiad, 1
If $x$ and $y$ are real numbers such that $x^{2013}+y^{2013}>x^{2012}+y^{2012}$, prove that $x^{2014}+y^{2014}>x^{2013}+y^{2013}$
2000 All-Russian Olympiad Regional Round, 10.5
Is there a function $f(x)$ defined for all $x \in R$ and for all $x, y \in R $ satisfying the inequality
$$|f(x + y) + \sin x + \sin y| < 2?$$