Found problems: 15925
1991 IberoAmerican, 5
Let $P(x,\, y)=2x^{2}-6xy+5y^{2}$. Let us say an integer number $a$ is a value of $P$ if there exist integer numbers $b$, $c$ such that $P(b,\, c)=a$.
a) Find all values of $P$ lying between 1 and 100.
b) Show that if $r$ and $s$ are values of $P$, then so is $rs$.
2011 Princeton University Math Competition, B2
Prove for irrational number $\alpha$ and positive integer $n$ that \[ \left( \alpha + \sqrt{\alpha^2 - 1} \right)^{1/n} + \left(\alpha - \sqrt{\alpha^2 - 1} \right)^{1/n} \] is irrational.
2014 AMC 12/AHSME, 19
There are exactly $N$ distinct rational numbers $k$ such that $|k|<200$ and \[5x^2+kx+12=0\] has at least one integer solution for $x$. What is $N$?
$\textbf{(A) }6\qquad
\textbf{(B) }12\qquad
\textbf{(C) }24\qquad
\textbf{(D) }48\qquad
\textbf{(E) }78\qquad$
2003 Singapore Team Selection Test, 3
Determine all functions $f : Z\to Z$, where $Z$ is the set of integers, such that $$f(m + f(f(n))) = -f(f(m + 1)) - n$$ for all integers $m$ and $n$.
1989 IMO Longlists, 78
Let $ P(x)$ be a polynomial with integer coefficients such that \[ P(m_1) \equal{} P(m_2) \equal{} P(m_3) \equal{} P(m_4) \equal{} 7\] for given distinct integers $ m_1,m_2,m_3,$ and $ m_4.$ Show that there is no integer m such that $ P(m) \equal{} 14.$
2021 Harvard-MIT Mathematics Tournament., 4
Suppose that $P(x, y, z)$ is a homogeneous degree 4 polynomial in three variables such that $P(a, b, c) = P(b, c, a)$ and $P(a, a, b) = 0$ for all real $a$, $b$, and $c$. If $P(1, 2, 3) = 1$, compute $P(2, 4, 8)$.
Note: $P(x, y, z)$ is a homogeneous degree $4$ polynomial if it satisfies $P(ka, kb, kc) = k^4P(a, b, c)$ for all real $k, a, b, c$.
2023 Iran MO (3rd Round), 3
For numbers $a,b \in \mathbb{R}$ we consider the sets:
$$A=\{a^n | n \in \mathbb{N}\} , B=\{b^n | n \in \mathbb{N}\}$$
Find all $a,b > 1$ for which there exists two real , non-constant polynomials $P,Q$ with positive leading coefficients st for each $r \in \mathbb{R}$:
$$ P(r) \in A \iff Q(r) \in B$$
2000 Iran MO (3rd Round), 3
Prove that for every natural number $ n$ there exists a polynomial $ p(x)$ with
integer coefficients such that$ p(1),p(2),...,p(n)$ are distinct powers of $ 2$ .
2000 239 Open Mathematical Olympiad, 3
For all positive real numbers $a_1, a_2, \dots, a_n$, prove that
$$
\frac{a_1\! +\! a_2}{2} \cdot \frac{a_2\! +\! a_3}{2} \cdot \dots \cdot
\frac{a_n\! +\! a_1}{2} \leq \frac{a_1\!+\!a_2\!+\!a_3}{2 \sqrt{2}} \cdot
\frac{a_2\!+\!a_3\!+\!a_4}{2 \sqrt{2}} \cdot \dots \cdot
\frac{a_n\!+\!a_1\!+\!a_2}{2 \sqrt{2}}.$$
2006 China Team Selection Test, 2
Given positive integer $n$, find the biggest real number $C$ which satisfy the condition that if the sum of the reciprocals of a set of integers (They can be the same.) that are greater than $1$ is less than $C$, then we can divide the set of numbers into no more than $n$ groups so that the sum of reciprocals of every group is less than $1$.
2012 HMNT, 10
Let $\alpha$ and $\beta$ be reals. Find the least possible value of $$(2 \cos \alpha + 5 \sin \beta - 8)^2 + (2 \sin \alpha + 5 \cos \beta - 15)^2.$$
1983 Putnam, A4
Let $k$ be a positive integer and let $m=6k-1$. Let
$$S(m)=\sum_{j=1}^{2k-1}(-1)^{j+1}\binom m{3j-1}.$$Prove that $S(m)$ is never zero.
2007 South africa National Olympiad, 2
Consider the equation $ x^4 \equal{} ax^3 \plus{} bx^2 \plus{} cx \plus{} 2007$, where $ a,b,c$ are real numbers. Determine the largest value of $ b$ for which this equation has exactly three distinct solutions, all of which are integers.
2024 Belarus Team Selection Test, 4.1
Six integers $a,b,c,d,e,f$ satisfy:
$\begin{cases}
ace+3ebd-3bcf+3adf=5 \\
bce+acf-ade+3bdf=2
\end{cases}$
Find all possible values of $abcde$
[i]D. Bazyleu[/i]
2016 Indonesia TST, 2
Determine all triples of real numbers $(x, y, z)$ which satisfy the following system of equations:
\[ \begin{cases} x+y+z=0 \\ x^3+y^3+z^3 = 90 \\ x^5+y^5+z^5=2850. \end{cases} \]
1988 IMO Longlists, 28
Find a necessary and sufficient condition on the natural number $ n$ for the equation
\[ x^n \plus{} (2 \plus{} x)^n \plus{} (2 \minus{} x)^n \equal{} 0
\]
to have a integral root.
2023 Indonesia TST, 1
Let $(a_n)_{n\geq 1}$ be a sequence of positive real numbers with the property that
$$(a_{n+1})^2 + a_na_{n+2} \leq a_n + a_{n+2}$$
for all positive integers $n$. Show that $a_{2022}\leq 1$.
2013 Canada National Olympiad, 1
Determine all polynomials $P(x)$ with real coefficients such that
\[(x+1)P(x-1)-(x-1)P(x)\]
is a constant polynomial.
2017 Pan-African Shortlist, A4
Find all functions $f : R\rightarrow R$ such that $f ( f (x)+y) = f (x^2 -y)+4 f (x)y$ for all $x,y \in R$
.
1987 AIME Problems, 12
Let $m$ be the smallest integer whose cube root is of the form $n+r$, where $n$ is a positive integer and $r$ is a positive real number less than $1/1000$. Find $n$.
2013 AMC 12/AHSME, 4
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} $
1955 AMC 12/AHSME, 19
Two numbers whose sum is $ 6$ and the absolute value of whose difference is $ 8$ are roots of the equation:
$ \textbf{(A)}\ x^2\minus{}6x\plus{}7\equal{}0 \qquad
\textbf{(B)}\ x^2\minus{}6x\minus{}7\equal{}0 \qquad
\textbf{(C)}\ x^2\plus{}6x\minus{}8\equal{}0 \\
\textbf{(D)}\ x^2\minus{}6x\plus{}8\equal{}0 \qquad
\textbf{(E)}\ x^2\plus{}6x\minus{}7\equal{}0$
1982 Putnam, A2
For positive real $x$, let
$$B_n(x)=1^x+2^x+\ldots+n^x.$$Prove or disprove the convergence of
$$\sum_{n=2}^\infty\frac{B_n(\log_n2)}{(n\log_2n)^2}.$$
2009 Brazil Team Selection Test, 2
Be $x_1, x_2, x_3, x_4, x_5$ be positive reais with $x_1x_2x_3x_4x_5=1$. Prove that
$$\frac{x_1+x_1x_2x_3}{1+x_1x_2+x_1x_2x_3x_4}+\frac{x_2+x_2x_3x_4}{1+x_2x_3+x_2x_3x_4x_5}+\frac{x_3+x_3x_4x_5}{1+x_3x_4+x_3x_4x_5x_1}+\frac{x_4+x_4x_5x_1}{1+x_4x_5+x_4x_5x_1x_2}+\frac{x_5+x_5x_1x_2}{1+x_5x_1+x_5x_1x_2x_3} \ge \frac{10}{3}$$
1994 USAMO, 5
Let $\, |U|, \, \sigma(U) \,$ and $\, \pi(U) \,$ denote the number of elements, the sum, and the product, respectively, of a finite set $\, U \,$ of positive integers. (If $\, U \,$ is the empty set, $\, |U| = 0, \, \sigma(U) = 0, \, \pi(U) = 1$.) Let $\, S \,$ be a finite set of positive integers. As usual, let $\, \binom{n}{k} \,$ denote $\, n! \over k! \, (n-k)!$. Prove that \[ \sum_{U \subseteq S} (-1)^{|U|} \binom{m - \sigma(U)}{|S|} = \pi(S) \] for all integers $\, m \geq \sigma(S)$.