Found problems: 15925
2014 ELMO Shortlist, 4
Find all triples $(f,g,h)$ of injective functions from the set of real numbers to itself satisfying
\begin{align*}
f(x+f(y)) &= g(x) + h(y) \\
g(x+g(y)) &= h(x) + f(y) \\
h(x+h(y)) &= f(x) + g(y)
\end{align*}
for all real numbers $x$ and $y$. (We say a function $F$ is [i]injective[/i] if $F(a)\neq F(b)$ for any distinct real numbers $a$ and $b$.)
[i]Proposed by Evan Chen[/i]
2008 Swedish Mathematical Competition, 3
The function $f(x)$ has the property that $\frac{f(x)}{x}$ is increasing for $x>0$. Show that
\[
f(x)+f(y) \leq f(x+y) \qquad , \qquad \text{for all } x,y>0
\]
2018 India IMO Training Camp, 3
Let $a_n, b_n$ be sequences of positive reals such that,$$a_{n+1}= a_n + \frac{1}{2b_n}$$ $$b_{n+1}= b_n + \frac{1}{2a_n}$$ for all $n\in\mathbb N$.
Prove that, $\text{max}\left(a_{2018}, b_{2018}\right) >44$.
2019 Caucasus Mathematical Olympiad, 3
Find all positive integers $n\geqslant 2$ such that there exists a permutation $a_1$, $a_2$, $a_3$, \ldots, $a_{2n}$ of the numbers $1, 2, 3, \ldots, 2n$ satisfying $$a_1\cdot a_2 + a_3\cdot a_4 + \ldots + a_{2n-3} \cdot a_{2n-2} = a_{2n-1} \cdot a_{2n}.$$
1967 IMO Longlists, 43
The equation
\[x^5 + 5 \lambda x^4 - x^3 + (\lambda \alpha - 4)x^2 - (8 \lambda + 3)x + \lambda \alpha - 2 = 0\]
is given. Determine $\alpha$ so that the given equation has exactly (i) one root or (ii) two roots, respectively, independent from $\lambda.$
I Soros Olympiad 1994-95 (Rus + Ukr), 11.8
A polynomial with rational coefficients is called [i]integer[/i], if it takes integer values for all integer values of the variable. For an integer polynomial $P$, consider the sequence $(-1)^{P(1)},(-1)^{P(2)},(-1)^{P(3)},...$
a) Prove that this sequence is periodic, the period of which is some power of two (i.e. for some integer $k$ and for all natural $i$, the $i$-th and ($i+2^k$)th members of the sequence are equal).
b) Prove that for any periodic sequence consisting of $(- 1)$ and $ 1$ and with a period of some power of two, there exists a integer, polynomial P for which this sequence is $(-1)^{P(1)},(-1)^{P(2)},(-1)^{P(3)},...$
2018 IFYM, Sozopol, 1
$A = \{a_1, a_2, . . . , a_k\}$ is a set of positive integers for which the sum of some (we can have only one number too) different numbers from the set is equal to a different number i.e. there $2^k - 1$ different sums of different numbers from $A$. Prove that the following inequality holds:
$\frac{1}{a_1}+\frac{1}{a_2}+...+\frac{1}{a_k}<2$
Oliforum Contest IV 2013, 4
Let $p,q$ be integers such that the polynomial $x^2+px+q+1$ has two positive integer roots. Show that $p^2+q^2$ is composite.
2023 Bangladesh Mathematical Olympiad, P10
Let all possible $2023$-degree real polynomials: $P(x)=x^{2023}+a_1x^{2022}+a_2x^{2021}+\cdots+a_{2022}x+a_{2023}$,
where $P(0)+P(1)=0$, and the polynomial has 2023 real roots $r_1, r_2,\cdots r_{2023}$ [not necessarily distinct] so that $0\leq r_1,r_2,\cdots r_{2023}\leq1$. What is the maximum value of $r_1 \cdot r_2 \cdots r_{2023}?$
The Golden Digits 2024, P2
Find all the functions $\varphi:\mathbb{Z}[x]\to\mathbb{Z}[x]$ such that $\varphi(x)=x,$ any integer polynomials $f, g$ satisfy $\varphi(f+g)=\varphi(f)+\varphi(g)$ and $\varphi(f)$ is a perfect power if and only if $f{}$ is a perfect power.
[i]Note:[/i] A polynomial $f\in \mathbb{Z}[x]$ is a perfect power if $f = g^n$ for some $g\in \mathbb{Z}[x]$ and $n\geqslant 2.$
[i]Proposed by Pavel Ciurea[/i]
2022 LMT Spring, 4
Kevin runs uphill at a speed that is $4$ meters per second slower than his speed when he runs downhill. Kevin takes a total of $80$ seconds to run up and down a hill on one path. Given that the path is $300$ meters long (he travels $600$ meters total), find how long Kevin takes to run up the hill in seconds.
2020 CHMMC Winter (2020-21), 7
Consider the polynomial $x^3-3x^2+10$. Let $a, b, c$ be its roots. Compute $a^2b^2c^2 + a^2b^2 + b^2c^2 + c^2a^2 + a^2 + b^2 + c^2$.
2018 Brazil Undergrad MO, 21
Consider $ p (x) = x ^ n + a_ {n-1} x ^ {n-1} + ... + a_ {1} x + 1 $ a polynomial of positive real coefficients, degree $ n \geq 2 $ e with $ n $ real roots. Which of the following statements is always true?
a) $ p (2) <2 (2 ^ {n-1} +1) $ (b) $ p (1) <3 $ c) $ p (1)> 2 ^ n $ d) $ p (3 ) <3 (2 ^ {n-1} -2) $
2006 IMO Shortlist, 5
If $a,b,c$ are the sides of a triangle, prove that
\[\frac{\sqrt{b+c-a}}{\sqrt{b}+\sqrt{c}-\sqrt{a}}+\frac{\sqrt{c+a-b}}{\sqrt{c}+\sqrt{a}-\sqrt{b}}+\frac{\sqrt{a+b-c}}{\sqrt{a}+\sqrt{b}-\sqrt{c}}\leq 3 \]
[i]Proposed by Hojoo Lee, Korea[/i]
1939 Moscow Mathematical Olympiad, 046
Solve the equation $\sqrt{a-\sqrt{a+ x}} = x$ for $x$.
2001 Tuymaada Olympiad, 4
Natural numbers $1, 2, 3,.., 100$ are contained in the union of $N$ geometric progressions (not necessarily with integer denominations). Prove that $N \ge 31$
1988 Poland - Second Round, 2
Given real numbers $ x_i $, $ y_i $ ($ i = 1, 2, \ldots, n $) such that $$ \qquad x_1 \geq x_2 \geq \ldots \geq x_n \geq 0, \ \ y_1 > y_2 > \ldots > y_n \geq 0,$$
and $$ \prod_{i=1}^k x_i \geq \prod_{i=1}^k y_i, \ \ \text{ for } \ \ k=1,2,\ldots, n.$$
Prove that
$$
\sum_{i=1}^n x_i > \sum_{i=1}^n y_i.$$
2016 Saudi Arabia IMO TST, 3
Find all functions $f : R \to R$ such that $x[f(x + y) - f (x - y)] = 4y f (x)$ for any real numbers $x, y$.
2017 NZMOC Camp Selection Problems, 8
Find all possible real values for $a, b$ and $c$ such that
(a) $a + b + c = 51$,
(b) $abc = 4000$,
(c) $0 < a \le 10$ and $c \ge 25$.
2007 Estonia Math Open Senior Contests, 4
The Fibonacci sequence is determined by conditions $ F_0 \equal{} 0, F1 \equal{} 1$, and $ F_k\equal{}F_{k\minus{}1}\plus{}F_{k\minus{}2}$ for all $ k \ge 2$. Let $ n$ be a positive integer and let $ P(x) \equal{} a_mx^m \plus{}. . .\plus{} a_1x\plus{} a_0$ be a polynomial that satisfies the following two conditions:
(1) $ P(F_n) \equal{} F_{n}^{2}$ ;
(2) $ P(F_k) \equal{} P(F_{k\minus{}1}) \plus{} P(F_{k\minus{}2}$ for all $ k \ge 2$.
Find the sum of the coefficients of P.
2014 Contests, 1
Find the smallest possible value of the expression \[\left\lfloor\frac{a+b+c}{d}\right\rfloor+\left\lfloor\frac{b+c+d}{a}\right\rfloor+\left\lfloor\frac{c+d+a}{b}\right\rfloor+\left\lfloor\frac{d+a+b}{c}\right\rfloor\]
in which $a,~ b,~ c$, and $d$ vary over the set of positive integers.
(Here $\lfloor x\rfloor$ denotes the biggest integer which is smaller than or equal to $x$.)
1949-56 Chisinau City MO, 50
Prove the inequality: $ctg \frac{a}{2}> 1 + ctg a$ for $0 <a <\frac{\pi}{2}$
2009 Canadian Mathematical Olympiad Qualification Repechage, 9
Suppose that $m$ and $k$ are positive integers. Determine the number of sequences $x_1, x_2, x_3, \dots , x_{m-1}, x_m$ with
[list]
[*]$x_i$ an integer for $i = 1, 2, 3, \dots , m$,
[*]$1\le x_i \le k$ for $i = 1, 2, 3, \dots , m$,
[*]$x_1\neq x_m$, and
[*]no two consecutive terms equal.[/list]
1956 Putnam, B7
The polynomials $P(z)$ and $Q(z)$ with complex coefficients have the same set of numbers for their zeros but possibly different multiplicities. The same is true for the polynomials
$$P(z)+1 \;\; \text{and} \;\; Q(z)+1.$$
Prove that $P(z)=Q(z).$
2010 Laurențiu Panaitopol, Tulcea, 3
Let be two polynoms $ P,Q\in\mathbb{R} [X] $ having the property that
$$ \left| \{ n\in\{ 0\}\cup\mathbb{N} | P(n)\le Q(n) \} \right| =\left| \{ n\in\{ 0\}\cup\mathbb{N} | P(n)\ge Q(n) \} \right| =\infty .$$
Show that $ P=Q. $
[i]Laurențiu Panaitopol[/i]