Found problems: 15925
2005 Postal Coaching, 25
Find all pairs of cubic equations $x^3 +ax^2 +bx +c =0$ and $x^3 +bx^2 + ax +c = 0$ where $a,b,c$ are integers, such that each equation has three integer roots and both the equations have exactly one common root.
2015 Saint Petersburg Mathematical Olympiad, 1
$x,y$ are real numbers such that $$x^2+y^2=1 , 20x^3-15x=3$$Find the value of $|20y^3-15y|$.(K. Tyshchuk)
2025 Romania Team Selection Tests, P3
Determine all polynomials $P{}$ with integer coefficients, satisfying $0 \leqslant P (n) \leqslant n!$ for all non-negative integers $n$.
[i]Andrei Chirita[/i]
1979 IMO Longlists, 60
Given the integer $n > 1$ and the real number $a > 0$ determine the maximum of $\sum_{i=1}^{n-1} x_i x_{i+1}$ taken over all nonnegative numbers $x_i$ with sum $a.$
2013 National Olympiad First Round, 35
What is the least positive integer $n$ such that $\overbrace{f(f(\dots f}^{21 \text{ times}}(n)))=2013$ where $f(x)=x+1+\lfloor \sqrt x \rfloor$? ($\lfloor a \rfloor$ denotes the greatest integer not exceeding the real number $a$.)
$
\textbf{(A)}\ 1214
\qquad\textbf{(B)}\ 1202
\qquad\textbf{(C)}\ 1186
\qquad\textbf{(D)}\ 1178
\qquad\textbf{(E)}\ \text{None of above}
$
IV Soros Olympiad 1997 - 98 (Russia), 9.8
The equation $P(x) = 0$, where $P(x) = x^2+bx+c$, has a single root, and the equation $P(P(P(x))) = 0$ has exactly three different roots. Solve the equation $P(P(P(x))) = 0.$
2018 Brazil Team Selection Test, 5
Prove: there are polynomials $S_1, S_2, \ldots$ in the variables $x_1, x_2, \ldots,y_1, y_2,\ldots$ with integer coefficients satisfying, for every integer $n \ge 1$, $$\sum_{d \mid n} d \cdot S_d ^{n/d}=\sum_{d \mid n} d \cdot (x_d ^{n/d}+y_d ^{n/d}) \quad (*)$$
Here, the sums run through the positive divisors $d$ of $n$.
For example, the first two polynomials are $S_1 = x_1 + y_1$ and $S_2 = x_2 + y_2 - x_1y_1$, which verify identity
$(*)$ for $n = 2$: $S_1^2 + 2S_2 = (x_1^2 + y_1^2) + 2 \cdot(x_2 + y_2)$.
2008 IMO Shortlist, 6
Let $ f: \mathbb{R}\to\mathbb{N}$ be a function which satisfies $ f\left(x \plus{} \dfrac{1}{f(y)}\right) \equal{} f\left(y \plus{} \dfrac{1}{f(x)}\right)$ for all $ x$, $ y\in\mathbb{R}$. Prove that there is a positive integer which is not a value of $ f$.
[i]Proposed by Žymantas Darbėnas (Zymantas Darbenas), Lithuania[/i]
2004 Nicolae Coculescu, 2
Solve in the real numbers the equation:
$$ \cos^2 \frac{(x-2)\pi }{4} +\cos\frac{(x-2)\pi }{3} =\log_3 (x^2-4x+6) $$
[i]Gheorghe Mihai[/i]
2022 Princeton University Math Competition, 9
In the complex plane, let $z_1, z_2, z_3$ be the roots of the polynomial $p(x) = x^3- ax^2 + bx - ab$. Find the number of integers $n$ between $1$ and $500$ inclusive that are expressible as $z^4_1 +z^4_2 +z^4_3$ for some choice of positive integers $a, b$.
PEN G Problems, 29
Let $p(x)=x^{3}+a_{1}x^{2}+a_{2}x+a_{3}$ have rational coefficients and have roots $r_{1}$, $r_{2}$, and $r_{3}$. If $r_{1}-r_{2}$ is rational, must $r_{1}$, $r_{2}$, and $r_{3}$ be rational?
2015 Federal Competition For Advanced Students, P2, 4
Let $x,y,z$ be positive real numbers with $x+y+z \ge 3$. Prove that
$\frac{1}{x+y+z^2} + \frac{1}{y+z+x^2} + \frac{1}{z+x+y^2} \le 1$
When does equality hold?
(Karl Czakler)
2023 239 Open Mathematical Olympiad, 6
An arrangement of 12 real numbers in a row is called [i]good[/i] if for any four consecutive numbers the arithmetic mean of the first and last numbers is equal to the product of the two middle numbers. How many good arrangements are there in which the first and last numbers are 1, and the second number is the same as the third?
1991 China National Olympiad, 2
Given $I=[0,1]$ and $G=\{(x,y)|x,y \in I\}$, find all functions $f:G\rightarrow I$, such that $\forall x,y,z \in I$ we have:
i. $f(f(x,y),z)=f(x,f(y,z))$;
ii. $f(x,1)=x, f(1,y)=y$;
iii. $f(zx,zy)=z^kf(x,y)$.
($k$ is a positive real number irrelevant to $x,y,z$.)
2023 Macedonian Balkan MO TST, Problem 4
Let $f$ be a non-zero function from the set of positive integers to the set of non-negative integers such that for all positive integers $a$ and $b$ we have
$$2f(ab)=(b+1)f(a)+(a+1)f(b).$$
Prove that for every prime number $p$ there exists a prime $q$ and positive integers $x_{1}$, ..., $x_{n}$ and $m \geq 0$ so that
$$\frac{f(q^{p})}{f(q)} = (px_{1}+1) \cdot ... \cdot (px_{n}+1) \cdot p^{m},$$
where the integers $px_{1}+1$,..., $px_{n}+1$ are all prime.
[i]Authored by Nikola Velov[/i]
1998 Romania Team Selection Test, 3
Let $n$ be a positive integer and $\mathcal{P}_n$ be the set of integer polynomials of the form $a_0+a_1x+\ldots +a_nx^n$ where $|a_i|\le 2$ for $i=0,1,\ldots ,n$. Find, for each positive integer $k$, the number of elements of the set $A_n(k)=\{f(k)|f\in \mathcal{P}_n \}$.
[i]Marian Andronache[/i]
1991 Brazil National Olympiad, 3
Given $k > 0$, the sequence $a_n$ is defined by its first two members and \[ a_{n+2} = a_{n+1} + \frac{k}{n}a_n \]
a)For which $k$ can we write $a_n$ as a polynomial in $n$?
b) For which $k$ can we write $\frac{a_{n+1}}{a_n} = \frac{p(n)}{q(n)}$? ($p,q$ are polynomials in $\mathbb R[X]$).
1996 Baltic Way, 12
Let $S$ be a set of integers containing the numbers $0$ and $1996$. Suppose further that any integer root of any non-zero polynomial with coefficients in $S$ also belongs to $S$. Prove that $-2$ belongs to $S$.
2000 District Olympiad (Hunedoara), 1
Solve in the set of $ 2\times 2 $ integer matrices the equation
$$ X^2-4\cdot X+4\cdot\left(\begin{matrix}1\quad 0\\0\quad 1\end{matrix}\right) =\left(\begin{matrix}7\quad 8\\12\quad 31\end{matrix}\right) . $$
2021 USEMO, 5
Given a polynomial $p(x)$ with real coefficients, we denote by $S(p)$ the sum of the squares of its coefficients. For example $S(20x+ 21)=20^2+21^2=841$.
Prove that if $f(x)$, $g(x)$, and $h(x)$ are polynomials with real coefficients satisfying the indentity $f(x) \cdot g(x)=h(x)^ 2$, then $$S(f) \cdot S(g) \ge S(h)^2$$
[i]Proposed by Bhavya Tiwari[/i]
2018 PUMaC Algebra B, 3
Let
$$a_k = 0.\overbrace{0 \ldots 0}^{k - 1 \: 0's} 1 \overbrace{0 \ldots 0}^{k - 1 \: 0's} 1$$
The value of $\sum_{k = 1}^\infty a_k$ can be expressed as a rational number $\frac{p}{q}$ in simplest form. Find $p + q$.
1966 IMO Shortlist, 40
For a positive real number $p$, find all real solutions to the equation
\[\sqrt{x^2 + 2px - p^2} -\sqrt{x^2 - 2px - p^2} =1.\]
2014 Israel National Olympiad, 7
Find one real value of $x$ satisfying $\frac{x^7}{7}=1+\sqrt[7]{10}x\left(x^2-\sqrt[7]{10}\right)^2$.
1987 India National Olympiad, 1
Given $ m$ and $ n$ as relatively prime positive integers greater than one, show that
\[ \frac{\log_{10} m}{\log_{10} n}\]
is not a rational number.
1998 India National Olympiad, 2
Let $a$ and $b$ be two positive rational numbers such that $\sqrt[3] {a} + \sqrt[3]{b}$ is also a rational number. Prove that $\sqrt[3]{a}$ and $\sqrt[3] {b}$ themselves are rational numbers.