Found problems: 3597
1969 Yugoslav Team Selection Test, Problem 2
Let $f(x)$ and $g(x)$ be degree $n$ polynomials, and $x_0,x_1,\ldots,x_n$ be real numbers such that
$$f(x_0)=g(x_0),f'(x_1)=g'(x_1),f''(x_2)=g''(x_2),\ldots,f^{(n)}(x_n)=g^{(n)}(x_n).$$Prove that $f(x)=g(x)$ for all $x$.
2005 Taiwan TST Round 1, 1
Let $f(x)=Ax^2+Bx+C$, $g(x)=ax^2+bx+c$ be two quadratic polynomial functions with real coefficients that satisfy the relation \[|f(x)| \ge |g(x)|\] for all real $x$. Prove that $|b^2-4ac| \le |B^2-4AC|.$
My solution was nearly complete...
2008 Vietnam Team Selection Test, 2
Find all values of the positive integer $ m$ such that there exists polynomials $ P(x),Q(x),R(x,y)$ with real coefficient satisfying the condition: For every real numbers $ a,b$ which satisfying $ a^m-b^2=0$, we always have that $ P(R(a,b))=a$ and $ Q(R(a,b))=b$.
1986 AIME Problems, 2
Evaluate the product \[(\sqrt 5+\sqrt6+\sqrt7)(-\sqrt 5+\sqrt6+\sqrt7)(\sqrt 5-\sqrt6+\sqrt7)(\sqrt 5+\sqrt6-\sqrt7).\]
1998 Bulgaria National Olympiad, 2
The polynomials $P_n(x,y), n=1,2,... $ are defined by \[P_1(x,y)=1, P_{n+1}(x,y)=(x+y-1)(y+1)P_n(x,y+2)+(y-y^2)P_n(x,y)\] Prove that $P_{n}(x,y)=P_{n}(y,x)$ for all $x,y \in \mathbb{R}$ and $n $.
2011 Romanian Master of Mathematics, 2
Determine all positive integers $n$ for which there exists a polynomial $f(x)$ with real coefficients, with the following properties:
(1) for each integer $k$, the number $f(k)$ is an integer if and only if $k$ is not divisible by $n$;
(2) the degree of $f$ is less than $n$.
[i](Hungary) Géza Kós[/i]
2011 Bulgaria National Olympiad, 2
Let $f_1(x)$ be a polynomial of degree $2$ with the leading coefficient positive and $f_{n+1}(x) =f_1(f_n(x))$ for $n\ge 1.$ Prove that if the equation $f_2(x)=0$ has four different non-positive real roots, then for arbitrary $n$ then $f_n(x)$ has $2^n$ different real roots.
2024 IFYM, Sozopol, 2
Let \(m,n\) and \(a\) be positive integers. Lumis has \(m\) cards, each with the number \(n\) written on it, and an infinite number of cards with each of the symbols addition, subtraction, multiplication, division, opening, and closing brackets. Umbra has composed an arithmetic expression with them, whose value is a positive integer less than \(\displaystyle\frac{n}{2^m}\). Prove that if \(n\) is replaced everywhere by \(a\), then the resulting expression will have the same value as before or will be undefined due to division by zero.
1989 AIME Problems, 8
Assume that $x_1,x_2,\ldots,x_7$ are real numbers such that
\[ \begin{array}{r} x_1+4x_2+9x_3+16x_4+25x_5+36x_6+49x_7=1\,\,\,\,\,\,\,\, \\ 4x_1+9x_2+16x_3+25x_4+36x_5+49x_6+64x_7=12\,\,\,\,\, \\ 9x_1+16x_2+25x_3+36x_4+49x_5+64x_6+81x_7=123. \\ \end{array} \] Find the value of \[16x_1+25x_2+36x_3+49x_4+64x_5+81x_6+100x_7.\]
2005 SNSB Admission, 3
Let $ f:\mathbb{C}\longrightarrow\mathbb{C} $ be an holomorphic function which has the property that there exist three positive real numbers $ a,b,c $ such that $ |f(z)|\geqslant a|z|^b , $ for any complex numbers $ z $ with $ |z|\geqslant c. $
Prove that $ f $ is polynomial with degree at least $ \lceil b\rceil . $
2001 District Olympiad, 3
Consider a continuous function $f:[0,1]\rightarrow \mathbb{R}$ such that for any third degree polynomial function $P:[0,1]\to [0,1]$, we have
\[\int_0^1f(P(x))dx=0\]
Prove that $f(x)=0,\ (\forall)x\in [0,1]$.
[i]Mihai Piticari[/i]
1983 IMO Longlists, 66
Let $ a$, $ b$ and $ c$ be the lengths of the sides of a triangle. Prove that
\[ a^{2}b(a \minus{} b) \plus{} b^{2}c(b \minus{} c) \plus{} c^{2}a(c \minus{} a)\ge 0.
\]
Determine when equality occurs.
1992 AIME Problems, 8
For any sequence of real numbers $A=(a_1,a_2,a_3,\ldots)$, define $\Delta A$ to be the sequence $(a_2-a_1,a_3-a_2,a_4-a_3,\ldots)$, whose $n^\text{th}$ term is $a_{n+1}-a_n$. Suppose that all of the terms of the sequence $\Delta(\Delta A)$ are $1$, and that $a_{19}=a_{92}=0$. Find $a_1$.
2005 China Team Selection Test, 3
Let $n$ be a positive integer, and $a_j$, for $j=1,2,\ldots,n$ are complex numbers. Suppose $I$ is an arbitrary nonempty subset of $\{1,2,\ldots,n\}$, the inequality $\left|-1+ \prod_{j\in I} (1+a_j) \right| \leq \frac 12$ always holds.
Prove that $\sum_{j=1}^n |a_j| \leq 3$.
2005 Georgia Team Selection Test, 4
Find all polynomials with real coefficients, for which the equality
\[ P(2P(x)) \equal{} 2P(P(x)) \plus{} 2(P(x))^{2}\]
holds for any real number $ x$.
2018 Abels Math Contest (Norwegian MO) Final, 3a
Find all polynomials $P$ such that $P(x)+3P(x+2)=3P(x+1)+P(x+3)$ for all real numbers $x$.
1996 Bulgaria National Olympiad, 3
The quadratic polynomials $f$ and $g$ with real coefficients are such that if $g(x)$ is an integer for some $x>0$, then so is $f(x)$. Prove that there exist integers $m,n$ such that $f(x)=mg(x)+n$ for all $x$.
1998 Federal Competition For Advanced Students, Part 2, 2
Let $P(x) = x^3 - px^2 + qx - r$ be a cubic polynomial with integer roots $a, b, c$.
[b](a)[/b] Show that the greatest common divisor of $p, q, r$ is equal to $1$ if the greatest common divisor of $a, b, c$ is equal to $1$.
[b](b)[/b] What are the roots of polynomial $Q(x) = x^3-98x^2+98sx-98t$ with $s, t$ positive integers.
1958 February Putnam, A1
If $a_0 , a_1 ,\ldots, a_n$ are real number satisfying
$$ \frac{a_0 }{1} + \frac{a_1 }{2} + \ldots + \frac{a_n }{n+1}=0,$$
show that the equation $a_n x^n + \ldots +a_1 x+a_0 =0$ has at least one real root.
2004 Austrian-Polish Competition, 10
For each polynomial $Q(x)$ let $M(Q)$ be the set of non-negative integers $x$ with $0 < Q(x) < 2004.$ We consider polynomials $P_n(x)$ of the form
\[P_n(x) = x^n + a_1 \cdot x^{n-1} + \ldots + a_{n-1} \cdot x + 1\]
with coefficients $a_i \in \{ \pm1\}$ for $i = 1, 2, \ldots, n-1.$
For each $n = 3^k, k > 0$ determine:
a.) $m_n$ which represents the maximum of elements in $M(P_n)$ for all such polynomials $P_n(x)$
b.) all polynomials $P_n(x)$ for which $|M(P_n)| = m_n.$
2016 Balkan MO Shortlist, N5
A positive integer is called [i]downhill[/i] if the digits in its decimal representation form a nonstrictly decreasing sequence from left to right. Suppose that a polynomial $P(x)$ with rational coefficients takes on an integer value for each downhill positive integer $x$. Is it necessarily true that $P(x)$ takes on an integer value for each integer $x$?
2018-2019 Winter SDPC, 1
Let $r_1$, $r_2$, $r_3$ be the distinct real roots of $x^3-2019x^2-2020x+2021=0$. Prove that $r_1^3+r_2^3+r_3^3$ is an integer multiple of $3$.
Russian TST 2017, P1
For any positive integer $k$, denote the sum of digits of $k$ in its decimal representation by $S(k)$. Find all polynomials $P(x)$ with integer coefficients such that for any positive integer $n \geq 2016$, the integer $P(n)$ is positive and $$S(P(n)) = P(S(n)).$$
[i]Proposed by Warut Suksompong, Thailand[/i]
2010 Contests, 2
For any set $A=\{a_1,a_2,\cdots,a_m\}$, let $P(A)=a_1a_2\cdots a_m$. Let $n={2010\choose99}$, and let $A_1, A_2,\cdots,A_n$ be all $99$-element subsets of $\{1,2,\cdots,2010\}$. Prove that $2010|\sum^{n}_{i=1}P(A_i)$.
1980 IMO, 2
In a rectangular coordinate system we call a horizontal line parallel to the $x$ -axis triangular if it intersects the curve with equation \[y = x^4 + px^3 + qx^2 + rx + s\] in the points $A,B,C$ and $D$ (from left to right) such that the segments $AB, AC$ and $AD$ are the sides of a triangle. Prove that the lines parallel to the $x$ - axis intersecting the curve in four distinct points are all triangular or none of them is triangular.