Found problems: 85335
1975 Poland - Second Round, 6
Let $ f(x) $ and $ g(x) $ be polynomials with integer coefficients. Prove that if for every integer value $ n $ the number $ g(n) $ is divisible by the number $ f(n) $, then $ g(x) = f(x)\cdot h(x) $, where $ h(x) $ is a polynomial,. Show with an example that the coefficients of the polynomial $ h(x) $ do not have to be integer.
1952 AMC 12/AHSME, 18
$ \log p \plus{} \log q \equal{} \log (p \plus{} q)$ only if:
$ \textbf{(A)}\ p \equal{} q \equal{} 0 \qquad\textbf{(B)}\ p \equal{} \frac {q^2}{1 \minus{} q} \qquad\textbf{(C)}\ p \equal{} q \equal{} 1$
$ \textbf{(D)}\ p \equal{} \frac {q}{q \minus{} 1} \qquad\textbf{(E)}\ p \equal{} \frac {q}{q \plus{} 1}$
2024 MMATHS, 10
In acute $\triangle{ABC},$ $AB=11$ and $CB=10.$ Points $E$ and $D$ are constructed such that $\angle{CBE}$ and $\angle{ABD}$ are right, and $ACEBD$ is a non-degenerate pentagon. Additionally, $\angle{AEB} \cong \angle{DCB}, AE=CD,$ and $ED=20.$ Given that $EA$ and $CD$ intersect at $P$ and $AP=4,$ find $CP^2.$
2023 BMT, 4
Let f$(x)$ be a continuous function over the real numbers such that for every integer $n$, $f(n) = n^2$ and $f(x) $ is linear over the interval $[n, n + 1]$. There exists a unique two-variable polynomial $g$ such that $g(x, \lfloor x \rfloor) = f(x)$ for all $x$. Compute $g(20, 23)$. (Here, $\lfloor x \rfloor$ is defined as the greatest integer less than or equal to $x$. For example, $\lfloor 2\rfloor = 2$ and $\lfloor -3.5 \rfloor = -4$.)
2013 Czech-Polish-Slovak Junior Match, 5
Let $a, b, c$ be positive real numbers for which $ab + ac + bc \ge a + b + c$. Prove that $a + b + c \ge 3$.
IV Soros Olympiad 1997 - 98 (Russia), 10.4
Solve the equation $$ \sqrt{\sqrt{2x^2+x-3}+2x^2-3}=x.$$
2010 APMO, 5
Find all functions $f$ from the set $\mathbb{R}$ of real numbers into $\mathbb{R}$ which satisfy for all $x, y, z \in \mathbb{R}$ the identity \[f(f(x)+f(y)+f(z))=f(f(x)-f(y))+f(2xy+f(z))+2f(xz-yz).\]
2017 Online Math Open Problems, 18
Let $p$ be an odd prime number less than $10^5$. Granite and Pomegranate play a game. First, Granite picks a integer $c \in \{2,3,\dots,p-1\}$.
Pomegranate then picks two integers $d$ and $x$, defines $f(t) = ct + d$, and writes $x$ on a sheet of paper.
Next, Granite writes $f(x)$ on the paper, Pomegranate writes $f(f(x))$, Granite writes $f(f(f(x)))$, and so on, with the players taking turns writing.
The game ends when two numbers appear on the paper whose difference is a multiple of $p$, and the player who wrote the most recent number wins. Find the sum of all $p$ for which Pomegranate has a winning strategy.
[i]Proposed by Yang Liu[/i]
2001 Mediterranean Mathematics Olympiad, 2
Find all integers $n$ for which the polynomial $p(x) = x^5 -nx -n -2$ can be represented as a product of two non-constant polynomials with integer coefficients.
2020 MBMT, 39
Let $f(x) = \sqrt{4x^2 - 4x^4}$. Let $A$ be the number of real numbers $x$ that satisfy
$$f(f(f(\dots f(x)\dots ))) = x,$$ where the function $f$ is applied to $x$ 2020 times. Compute $A \pmod {1000}$.
[i]Proposed by Timothy Qian[/i]
1956 AMC 12/AHSME, 45
A wheel with a rubber tire has an outside diameter of $ 25$ in. When the radius has been decreased a quarter of an inch, the number of revolutions in one mile will:
$ \textbf{(A)}\ \text{be increased about }2\% \qquad\textbf{(B)}\ \text{be increased about }1\%$
$ \textbf{(C)}\ \text{be increased about }20\% \qquad\textbf{(D)}\ \text{be increased about }\frac {1}{2}\% \qquad\textbf{(E)}\ \text{remain the same}$
2019 Junior Balkan Team Selection Tests - Romania, 1
If $a, b, c$ are real numbers such that a$b + bc + ca = 0$, prove the inequality $$2(a^2 + b^2 + c^2)(a^2b^2 + b^2c^2 + c^2a^2) \ge 27a^2b^2c^2$$
When does the equality hold ?
Leonard Giugiuc
2022 MMATHS, 10
Suppose that $A_1A_2A_3$ is a triangle with $A_1A_2 = 16$ and $A_1A_3 = A_2A_3 = 10$. For each integer $n \ge 4$, set An to be the circumcenter of triangle $A_{n-1}A_{n-2}A_{n-3}$. There exists a unique point $Z$ lying in the interiors of the circumcircles of triangles $A_kA_{k+1}A_{k+2}$ for all integers $k \ge 1$. If $ZA^2_1+ ZA^2_2+ ZA^2_3+ ZA^2_4$ can be expressed as $\frac{a}{b}$ for positive integers $a, b$ with $gcd(a, b) = 1$, find $a + b$.
2012 ELMO Problems, 4
Let $a_0,b_0$ be positive integers, and define $a_{i+1}=a_i+\lfloor\sqrt{b_i}\rfloor$ and $b_{i+1}=b_i+\lfloor\sqrt{a_i}\rfloor$ for all $i\ge0$. Show that there exists a positive integer $n$ such that $a_n=b_n$.
[i]David Yang.[/i]
2025 Vietnam National Olympiad, 6
Let $a,b,c$ be non-negative numbers such that $a+b+c=3.$ Prove that
\[\sqrt{3a^3+4bc+b+c}+\sqrt{3b^3+4ca+c+a}+\sqrt{3c^3+4ab+a+b} \geqslant 9.\]
2012 South East Mathematical Olympiad, 3
For composite number $n$, let $f(n)$ denote the sum of the least three divisors of $n$, and $g(n)$ the sum of the greatest two divisors of $n$. Find all composite numbers $n$, such that $g(n)=(f(n))^m$ ($m\in N^*$).
2019 ASDAN Math Tournament, 8
Let triangle $\vartriangle AEF$ be inscribed in a square $ABCD$ such that $E$ lies on $BC$ and $F$ lies on $CD$. If $\angle EAF = 45^o$ and $\angle BEA = 70^o$, compute $\angle CF E$.
1988 Iran MO (2nd round), 2
In a cyclic quadrilateral $ABCD$, let $I,J$ be the midpoints of diagonals $AC, BD$ respectively and let $O$ be the center of the circle inscribed in $ABCD.$ Prove that $I, J$ and $O$ are collinear.
2020 Yasinsky Geometry Olympiad, 2
An equilateral triangle $BDE$ is constructed on the diagonal $BD$ of the square $ABCD$, and the point $C$ is located inside the triangle $BDE$. Let $M$ be the midpoint of $BE$. Find the angle between the lines $MC$ and $DE$.
(Dmitry Shvetsov)
1997 AIME Problems, 2
The nine horizontal and nine vertical lines on an $8\times8$ checkerboard form $r$ rectangles, of which $s$ are squares. The number $s/r$ can be written in the form $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$
2014 Contests, 3
Let $n$ be a positive integer. Show that there are positive real numbers $a_0, a_1, \dots, a_n$ such that for each choice of signs the polynomial
$$\pm a_nx^n\pm a_{n-1}x^{n-1} \pm \dots \pm a_1x \pm a_0$$
has $n$ distinct real roots.
(Proposed by Stephan Neupert, TUM, München)
2020 OMpD, 2
A pile of $2020$ stones is given. Arnaldo and Bernaldo play the following game: In each move, it is allowed to remove $1, 4, 16, 64, ...$ (any power of $4$) stones from the pile. They make their moves alternately, and the player who can no longer play loses. If Arnaldo is the first to play, who has the winning strategy?
2014 AMC 12/AHSME, 6
The difference between a two-digit number and the number obtained by reversing its digits is $5$ times the sum of the digits of either number. What is the sum of the two digit number and its reverse?
$\textbf{(A) }44\qquad
\textbf{(B) }55\qquad
\textbf{(C) }77\qquad
\textbf{(D) }99\qquad
\textbf{(E) }110$
2015 Middle European Mathematical Olympiad, 1
Find all surjective functions $f:\mathbb{N}\to\mathbb{N}$ such that for all positive integers $a$ and $b$, exactly one of the following equations is true:
\begin{align*}
f(a)&=f(b), <br /> \\
f(a+b)&=\min\{f(a),f(b)\}.
\end{align*}
[i]Remarks:[/i] $\mathbb{N}$ denotes the set of all positive integers. A function $f:X\to Y$ is said to be surjective if for every $y\in Y$ there exists $x\in X$ such that $f(x)=y$.
2021 All-Russian Olympiad, 6
In tetrahedron $ABCS$ no two edges have equal length. Point $A'$ in plane $BCS$ is symmetric to $S$ with respect to the perpendicular bisector of $BC$. Points $B'$ and $C'$ are defined analagously. Prove that planes $ABC, AB'C', A'BC'$ abd $A'B'C$ share a common point.