Found problems: 15925
2010 AMC 10, 21
The polynomial $ x^3\minus{}ax^2\plus{}bx\minus{}2010$ has three positive integer zeros. What is the smallest possible value of $ a$?
$ \textbf{(A)}\ 78 \qquad
\textbf{(B)}\ 88 \qquad
\textbf{(C)}\ 98 \qquad
\textbf{(D)}\ 108 \qquad
\textbf{(E)}\ 118$
2020 Polish Junior MO First Round, 3.
Determine all triples $(x, y, z)$ of non-zero numbers such that
\[
xy(x + y) = yz(y + z) = zx(z + x).
\]
2018 Polish Junior MO First Round, 6
Positive integers $k, m, n$ satisfy the equation $m^2 + n = k^2 + k$. Show that $m \le n$.
1978 All Soviet Union Mathematical Olympiad, 257
Prove that there exists such an infinite sequence $\{x_i\}$, that for all $m$ and all $k$ ($m\ne k$) holds the inequality $$|x_m-x_k|>1/|m-k|$$
2015 Mathematical Talent Reward Programme, MCQ: P 4
Let $n$ be an odd integer. Placing no more than one $X$ in each cell of a $n \times n$ grid, what is the greatest number of $X$ 's that can be put on the grid without getting $n$ $X$'s together vertically, horizontally or diagonally?
[list=1]
[*] $2{{n}\choose {2}}$
[*] ${{n}\choose {2}}$
[*] $2n $
[*] $2{{n}\choose {2}}-1$
[/list]
2001 Moldova National Olympiad, Problem 8
If $a_1,a_2,\ldots,a_n$ are positive real numbers, prove the inequality
$$\dfrac1{\dfrac1{1+a_1}+\dfrac1{1+a_2}+\ldots+\dfrac1{1+a_n}}-\dfrac1{\dfrac1{a_1}+\dfrac1{a_2}+\ldots+\dfrac1{a_n}}\ge\frac1n.$$
STEMS 2023 Math Cat A, 8
For how many pairs of primes $(p, q)$, is $p^2 + 2pq^2 + 1$ also a prime?
1990 India National Olympiad, 1
Given the equation
\[ x^4 \plus{} px^3 \plus{} qx^2 \plus{} rx \plus{} s \equal{} 0\]
has four real, positive roots, prove that
(a) $ pr \minus{} 16s \geq 0$
(b) $ q^2 \minus{} 36s \geq 0$
with equality in each case holding if and only if the four roots are equal.
2010 Laurențiu Panaitopol, Tulcea, 1
Solve in the real numbers the equation $ \arcsin x=\lfloor 2x \rfloor . $
[i]Petre Guțescu[/i]
2016 China Girls Math Olympiad, 4
Let $n$ is a positive integers ,$a_1,a_2,\cdots,a_n\in\{0,1,\cdots,n\}$ . For the integer $j$ $(1\le j\le n)$ ,define $b_j$ is the number of elements in the set $\{i|i\in\{1,\cdots,n\},a_i\ge j\}$ .For example :When $n=3$ ,if $a_1=1,a_2=2,a_3=1$ ,then $b_1=3,b_2=1,b_3=0$ .
$(1)$ Prove that $$\sum_{i=1}^{n}(i+a_i)^2\ge \sum_{i=1}^{n}(i+b_i)^2.$$
$(2)$ Prove that $$\sum_{i=1}^{n}(i+a_i)^k\ge \sum_{i=1}^{n}(i+b_i)^k,$$
for the integer $k\ge 3.$
1962 Polish MO Finals, 1
Prove that if the numbers $ a_1, a_2,\ldots, a_n $ ($ n $ - natural number $ \geq 2 $) form an arithmetic progression, and none of them is zero, then
$$\frac{1}{a_1a_2} + \frac{1}{a_2a_3} + \ldots + \frac{1}{a_{n-1}a_n} = \frac{n-1}{a_1a_n}.$$
2018 India National Olympiad, 4
Find all polynomials with real coefficients $P(x)$ such that $P(x^2+x+1)$ divides $P(x^3-1)$.
2015 Abels Math Contest (Norwegian MO) Final, 1a
Find all triples $(x, y, z) \in R^3$ satisfying the equations $\begin{cases} x^2 + 4y^2 = 4zx \\
y^2 + 4z^2 = 4xy \\
z^2 + 4x^2 = 4yz \end{cases}$
2020 Jozsef Wildt International Math Competition, W15
Show that the number$$4\sin\frac{\pi}{34}\left(\sin\frac{3\pi}{34}+\sin\frac{7\pi}{34}+\sin\frac{11\pi}{34}+\sin\frac{15\pi}{34}\right)$$
is an integer and determine it.
III Soros Olympiad 1996 - 97 (Russia), 9.7
Solve the system of equations:
$$\begin{cases} xy+zu=14
\\ xz+yu=11
\\ xu+yz=10
\\ x+y+z+u=10
\end{cases}$$
1972 Polish MO Finals, 3
Prove that there is a polynomial $P(x)$ with integer coefficients such that for all $x$ in the interval $\left[ \frac{1}{10}
, \frac{9}{10}\right]$ we have $$\left|P(x) -\frac12 \right| < \frac{ 1}{1000 }.$$
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]
2020 Latvia TST, 1.2
Sequences $(a_n), (b_n)$ are defined by $ a_1 = 1, b_1 = 2$, $a_{n+1} = \frac{ 1 + a_n + a_nb_n}{b_n}$, $ b_{n+1} = \frac{ 1 +b_n+ a_nb_n}{a_n}$ for all positive integers $n$. Prove that $a_{2020} < 5$.
1972 IMO Longlists, 2
Find all real values of the parameter $a$ for which the system of equations
\[x^4 = yz - x^2 + a,\]
\[y^4 = zx - y^2 + a,\]
\[z^4 = xy - z^2 + a,\]
has at most one real solution.
1953 Moscow Mathematical Olympiad, 256
Find roots of the equation
$$1 -\frac{x}{1}+ \frac{x(x - 1)}{2!} -... +\frac{ (-1)^nx(x-1)...(x - n + 1)}{n!}= 0$$
2011 Regional Competition For Advanced Students, 2
Determine all triples $(x,y,z)$ of real numbers such that the following system of equations holds true:
\begin{align*}2^{\sqrt[3]{x^2}}\cdot 4^{\sqrt[3]{y^2}}\cdot 16^{\sqrt[3]{z^2}}&=128\\
\left(xy^2+z^4\right)^2&=4+\left(xy^2-z^4\right)^2\mbox{.}\end{align*}
2012 Harvard-MIT Mathematics Tournament, 7
Let $\otimes$ be a binary operation that takes two positive real numbers and returns a positive real number. Suppose further that $\otimes$ is continuous, commutative $(a\otimes b=b\otimes a)$, distributive across multiplication $(a\otimes(bc)=(a\otimes b)(a\otimes c))$, and that $2\otimes 2=4$. Solve the equation $x\otimes y=x$ for $y$ in terms of $x$ for $x>1$.
Today's calculation of integrals, 769
In $xyz$ space, find the volume of the solid expressed by $x^2+y^2\leq z\le \sqrt{3}y+1.$
2016 ITAMO, 5
Let $x_0,x_1,x_2,\ldots$ be a sequence of rational numbers defined recursively as follows: $x_0$ can be any rational number and, for $n\ge 0$,
\[
x_{n+1}=\begin{cases} \left|\frac{x_n}2-1\right| & \text{if the numerator of }x_n\text{ is even}, \\
\left|\frac1{x_n}-1\right| & \text{if the numerator of }x_n\text{ is odd},\end{cases}
\]
where by numerator of a rational number we mean the numerator of the fraction in its lowest terms. Prove that for any value of $x_0$:
(a) the sequence contains only finitely many distinct terms;
(b) the sequence contains exactly one of the numbers $0$ and $2/3$ (namely, either there exists an index $k$ such that $x_k=0$, or there exists an index $m$ such that $x_m=2/3$, but not both).
2013 Iran MO (3rd Round), 4
Prime $p=n^2 +1$ is given. Find the sets of solutions to the below equation:
\[x^2 - (n^2 +1)y^2 = n^2.\]
(25 points)