Found problems: 15925
1965 AMC 12/AHSME, 13
Let $ n$ be the number of number-pairs $ (x,y)$ which satisfy $ 5y \minus{} 3x \equal{} 15$ and $ x^2 \plus{} y^2 \le 16$. Then $ n$ is:
$ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ \text{more than two, but finite} \qquad \textbf{(E)}\ \text{greater than any finite number}$
2005 Serbia Team Selection Test, 3
Find all polynomial with real coefficients such that:
P(x^2+1)=P(x)^2+1
1979 Poland - Second Round, 1
Given are the points $A$ and $B$ on the edge of a circular pool. The athlete has to get from point $A$ to point $B$ by walking along the edge of the pool or swimming in the pool; he can change the way he moves many times. How should an athlete move to get from point A to B in the shortest time, given that he moves twice as slowly in water as on land?
2015 VJIMC, 3
[b]Problem 3[/b]
Let $ P(x) = x^{2015} -2x^{2014}+1$ and $ Q(x) = x^{2015} -2x^{2014}-1$. Determine for each of the polynomials
$P$ and $Q$ whether it is a divisor of some nonzero polynomial $c_0 + c_{1}x +\ldots + c_{n}x^n$
n whose coefficients $c_i$ are all in the set $ \{ -1, 1\}$.
2002 Romania Team Selection Test, 2
Let $n\geq 4$ be an integer, and let $a_1,a_2,\ldots,a_n$ be positive real numbers such that \[ a_1^2+a_2^2+\cdots +a_n^2=1 . \] Prove that the following inequality takes place \[ \frac{a_1}{a_2^2+1}+\cdots +\frac{a_n}{a_1^2+1} \geq \frac{4}{5}\left( a_1 \sqrt{a_1}+\cdots +a_n \sqrt{a_n} \right)^2 . \] [i]Bogdan Enescu, Mircea Becheanu[/i]
1969 IMO Shortlist, 24
$(GBR 1)$ The polynomial $P(x) = a_0x^k + a_1x^{k-1} + \cdots + a_k$, where $a_0,\cdots, a_k$ are integers, is said to be divisible by an integer $m$ if $P(x)$ is a multiple of $m$ for every integral value of $x$. Show that if $P(x)$ is divisible by $m$, then $a_0 \cdot k!$ is a multiple of $m$. Also prove that if $a, k,m$ are positive integers such that $ak!$ is a multiple of $m$, then a polynomial $P(x)$ with leading term $ax^k$can be found that is divisible by $m.$
2019 Dutch IMO TST, 2
Determine all $4$-tuples $(a,b, c, d)$ of positive real numbers satisfying $a + b +c + d = 1$ and
$\max (\frac{a^2}{b},\frac{b^2}{a}) \cdot \max (\frac{c^2}{d},\frac{d^2}{c}) = (\min (a + b, c + d))^4$
2013 JBMO Shortlist, 2
$\boxed{\text{A2}}$ Find the maximum value of $|\sqrt{x^2+4x+8}-\sqrt{x^2+8x+17}|$ where $x$ is a real number.
2005 China Northern MO, 3
Let positive numbers $a_1, a_2, ..., a_{3n}$ $(n \geq 2)$ constitute an arithmetic progression with common difference $d > 0$. Prove that among any $n + 2$ terms in this progression, there exist two terms $a_i, a_j$ $(i \neq j)$ satisfying $1 < \frac{|a_i - a_j|}{nd} < 2$.
2025 CMIMC Algebra/NT, 2
I plotted the graphs $y=(x-0)^2, y=(x-5)^2, \ldots, y=(x-45)^2.$ I also draw a line $y=k,$ and notice that it intersects the parabolas at exactly $19$ distinct points. What is $k$?
2012 IFYM, Sozopol, 6
Find all triples $(x,y,z)$ of real numbers satisfying the system of equations
$\left\{\begin{matrix} 3(x+\frac{1}{x})=4(y+\frac{1}{y})=5(z+\frac{1}{z}),\\ xy+yz+zx=1.\end{matrix}\right.$
2005 AIME Problems, 7
In quadrilateral $ABCD$, $BC=8$, $CD=12$, $AD=10$, and $m\angle A= m\angle B = 60^\circ$. Given that $AB=p + \sqrt{q}$, where $p$ and $q$ are positive integers, find $p+q$.
2010 IberoAmerican, 1
The arithmetic, geometric and harmonic mean of two distinct positive integers are different numbers. Find the smallest possible value for the arithmetic mean.
2023 Saint Petersburg Mathematical Olympiad, 1
Let $f(x), g(x)$ be real polynomials of degrees $2$ and $3$, respectively. Could it happen that $f(g(x))$ has $6$ distinct roots, which are powers of $2$?
2002 Argentina National Olympiad, 6
Let $P_1,P_2,\ldots ,P_n$, be infinite arithmetic progressions of positive integers, of differences $d_1,d_2,\ldots ,d_n$, respectively. Prove that if every positive integer appears in at least one of the $n$ progressions then one of the differences $d_i$ divides the least common multiple of the remaining $n-1$ differences.
Note: $P_i=\left \{ a_i,a_i+d_i,a_i+2d_i,a_i+3d_i,a_i+4d_i,\cdots \right \}$ with $ a_i$ and $d_i$ positive integers.
2025 India STEMS Category B, 1
Let $\mathcal{P}$ be the set of all polynomials with coefficients in $\{0, 1\}$. Suppose $a, b$ are non-zero integers such that for every $f \in \mathcal{P}$ with $f(a)\neq 0$, we have $f(a) \mid f(b)$. Prove that $a=b$.
[i]Proposed by Shashank Ingalagavi and Krutarth Shah[/i]
1994 Argentina National Olympiad, 2
For what positive integer values of $x$ is $x^4 + 6x^3 + 11x^2 + 3x + 31$ a perfect square?
STEMS 2023 Math Cat A, 7
Suppose a biased coin gives head with probability $\dfrac{2}{3}$. The coin is tossed repeatedly, if
it shows heads then player $A$ rolls a fair die, otherwise player $B$ rolls the same die. The process
ends when one of the players get a $6$, and that player is declared the winner.
If the probability that $A$ will win is given by $\dfrac{m}{n}$ where $m,n$ are coprime, then what is the value of $m^2n$?
2019 PUMaC Algebra A, 2
Let $f(x)=x^2+4x+2$. Let $r$ be the difference between the largest and smallest real solutions of the equation $f(f(f(f(x))))=0$. Then $r=a^{\frac{p}{q}}$ for some positive integers $a$, $p$, $q$ so $a$ is square-free and $p,q$ are relatively prime positive integers. Compute $a+p+q$.
2002 All-Russian Olympiad, 3
Prove that for every integer $n > 10000$ there exists an integer $m$ such that it can be written as the sum of two squares, and $0<m-n<3\sqrt[4]n$.
2020 Balkan MO Shortlist, A1
Denote $\mathbb{Z}_{>0}=\{1,2,3,...\}$ the set of all positive integers. Determine all functions $f:\mathbb{Z}_{>0}\rightarrow \mathbb{Z}_{>0}$ such that, for each positive integer $n$,
$\hspace{1cm}i) \sum_{k=1}^{n}f(k)$ is a perfect square, and
$\vspace{0.1cm}$
$\hspace{1cm}ii) f(n)$ divides $n^3$.
[i]Proposed by Dorlir Ahmeti, Albania[/i]
2001 Iran MO (2nd round), 1
Find all polynomials $P$ with real coefficients such that $\forall{x\in\mathbb{R}}$ we have:
\[ P(2P(x))=2P(P(x))+2(P(x))^2. \]
2009 IMO Shortlist, 5
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]
2000 Swedish Mathematical Competition, 2
$p(x)$ is a polynomial such that $p(y^2+1) = 6y^4 - y^2 + 5$. Find $p(y^2-1)$.
2005 Bundeswettbewerb Mathematik, 2
Let be $x$ a rational number.
Prove: There are only finitely many triples $(a,b,c)$ of integers with $a<0$ and $b^2-4ac=5$ such that $ax^2+bx+c$ is positive.