Found problems: 15925
2009 Kosovo National Mathematical Olympiad, 2
Solve the equation:
$x^2+2xcos(x-y)+1=0$
1988 All Soviet Union Mathematical Olympiad, 482
Let $m, n, k$ be positive integers with $m \ge n$ and $1 + 2 + ... + n = mk$. Prove that the numbers $1, 2, ... , n$ can be divided into $k$ groups in such a way that the sum of the numbers in each group equals $m$.
1992 Chile National Olympiad, 2
For a finite set of naturals $(C)$, the product of its elements is going to be noted $P(C)$. We are going to define $P (\phi) = 1$. Calculate the value of the expression $$\sum_{C \subseteq \{1,2,...,n\}} \frac{1}{P(C)}$$
1961 All Russian Mathematical Olympiad, 011
Prove that for three arbitrary infinite sequences, of natural numbers $a_1,a_2,...,a_n,... $ , $b_1,b_2,...,b_n,... $, $c_1,c_2,...,c_n,...$ there exist numbers $p$ and $q$ such, that $a_p \ge a_q$, $b_p \ge b_q$ and $c_p \ge c_q$.
VMEO III 2006 Shortlist, A3
For positive real numbers $x,y,z$ that satisfy $ xy + yz + zx + xyz=4$, prove that
$$\frac{x+y+z}{xy+yz+zx}\le 1+\frac{5}{247}\cdot \left( (x-y)^2+(y-z)^2+(z-x)^2\right)$$
2001 Abels Math Contest (Norwegian MO), 1a
Suppose that $a, b, c$ are real numbers such that $a + b + c> 0$, and so the equation $ax^2 + bx + c = 0$ has no real solutions. Show that $c> 0$.
2023 Romania National Olympiad, 2
Determine functions $f : \mathbb{R} \rightarrow \mathbb{R},$ with property that
\[
f(f(x)) + y \cdot f(x) \le x + x \cdot f(f(y)),
\]
for every $x$ and $y$ are real numbers.
1989 Dutch Mathematical Olympiad, 5
$\forall k\in N \,\,\, \exists n(k) \in N, a(k):0<a(k)<1 [(1+\sqrt2)^{2k+1}=n(k)+a(k)]$
Prove: $(n(k) + a(k))a(k) = 1$, for all $k \in N$, and calculate $\lim_{k \to \infty }a(k)$
1989 IMO Longlists, 91
For $ \phi: \mathbb{N} \mapsto \mathbb{Z}$ let us define \[ M_{\phi} \equal{} \{f: \mathbb{N} \mapsto \mathbb{Z}, f(x) > f(\phi(x)), \forall x \in \mathbb{N} \}.\] Prove that if $ M_{\phi_1} \equal{} M_{\phi_2} \neq \emptyset,$ then $ \phi_1 \equal{} \phi_2.$ Does this property remain true if \[ M_{\phi} \equal{} \{f: \mathbb{N} \mapsto \mathbb{N}, f(x) > f(\phi(x)), \forall x \in \mathbb{N} \}?\]
2010 Junior Balkan Team Selection Tests - Romania, 3
Let $a, b, c$ be real numbers with the property as $ab + bc + ca = 1$. Show that:
$$\frac {(a + b) ^ 2 + 1} {c ^ 2 + 2} + \frac {(b + c) ^ 2 + 1} {a ^ 2 + 2} + \frac {(c + a) ^ 2 + 1} {b ^ 2 + 2} \ge 3 $$.
1998 Romania National Olympiad, 1
Let $n$ be a positive integer and $x_1,x_2,...,x_n$ be integer numbers such that $$x_1^2+x_2^2+...+x_n^2+ n^3 \le (2n - 1)(x_1+x_2+...+x_n ) + n^2$$ .
Show that :
a) $x_1,x_2,...,x_n$ are non-negative integers
b) the number $x_1+x_2+...+x_n+n+1$ is not a perfect square.
2015 239 Open Mathematical Olympiad, 4
On a circle $4$ points are chosen and for each point we wrote the multiple of its distances to the rest. Could the written numbers be $1,2,3, 4$ in some order?
2016 Saudi Arabia BMO TST, 3
Does there exist a polynomial $P(x)$ with integral coefficients such that
a) $P(\sqrt[3]{25 }+ \sqrt[3]{5}) = 220\sqrt[3]{25} + 284\sqrt[3]{5}$ ?
b) $P(\sqrt[3]{25 }+ \sqrt[3]{5}) = 1184\sqrt[3]{25} + 1210\sqrt[3]{5}$ ?
2020 Kosovo National Mathematical Olympiad, 4
Let $a_0$ be a fixed positive integer. We define an infinite sequence of positive integers $\{a_n\}_{n\ge 1}$ in an inductive way as follows: if we are given the terms $a_0,a_1,...a_{n-1}$ , then $a_n$ is the smallest positive integer such that $\sqrt[n]{a_0\cdot a_1\cdot ...\cdot a_n}$ is a positive integer. Show that the sequence $\{a_n\}_{n\ge 1}$ is eventually constant.
[b]Note:[/b] The sequence $\{a_n\}_{n\ge 1}$ is eventually constant if there exists a positive integer $k$ such that $a_n=c$, for every $n\ge k$.
1997 Romania National Olympiad, 1
Let $k$ be an integer number and $P(X)$ be the polynomial $$P(X) = X^{1997}-X^{1995} +X^2-3kX+3k+1$$
Prove that:
a) the polynomial has no integer root;
β) the numbers $P(n)$ and $P(n) + 3$ are relatively prime, for every integer $n$.
2019 Saudi Arabia JBMO TST, 5
Let non-integer real numbers $a, b,c,d$ are given, such that the sum of each $3$ of them is integer. May it happen that $ab + cd$ is an integer.
2020 Azerbaijan IMO TST, 1
Let $u_1, u_2, \dots, u_{2019}$ be real numbers satisfying \[u_{1}+u_{2}+\cdots+u_{2019}=0 \quad \text { and } \quad u_{1}^{2}+u_{2}^{2}+\cdots+u_{2019}^{2}=1.\] Let $a=\min \left(u_{1}, u_{2}, \ldots, u_{2019}\right)$ and $b=\max \left(u_{1}, u_{2}, \ldots, u_{2019}\right)$. Prove that
\[
a b \leqslant-\frac{1}{2019}.
\]
2010 JBMO Shortlist, 3
Find all pairs $(x,y)$ of real numbers such that $ |x|+ |y|=1340$ and $x^{3}+y^{3}+2010xy= 670^{3}$ .
2022 Greece Junior Math Olympiad, 1
(a) Find the value of the real number $k$, for which the polynomial $P(x)=x^3-kx+2$ has the number $2$ as a root. In addition, for the value of $k$ you will find, write this polynomial as the product of two polynomials with integer coefficients.
(b) If the positive real numbers $a,b$ satisfy the equation $$2a+b+\frac{4}{ab}=10,$$ find the maximum possible value of $a$.
MOAA Team Rounds, TO2
The Den has two deals on chicken wings. The first deal is $4$ chicken wings for $3$ dollars, and the second deal is $11$ chicken wings for $ 8$ dollars. If Jeremy has $18$ dollars, what is the largest number of chicken wings he can buy?
1940 Putnam, A6
Let $f(x)$ be a polynomial of degree $n$ such that $f(x)^{p}$ is divisible by $f'(x)^{q}$ for some positive integers $p,q$.
Prove that $f(x)$ is divisible by $f'(x)$ and that $f(x)$ has a single root of multiplicity $n$.
2024 Junior Balkan Team Selection Tests - Romania, P3
Determine all positive integers $a,b,c,d,e,f$ satisfying the following condition: for any two of them, $x{}$ and $y{},$ two of the remaining numbers, $z{}$ and $t{},$ satisfy $x/y=z/t.$
[i]Cristi Săvescu[/i]
1994 Romania TST for IMO, 3:
Let $a_1, a_2, . . ., a_n$ be a finite sequence of $0$ and $1$. Under any two consecutive terms of this sequence $0$ is written if the digits are equal and $1$ is written otherwise. This way a new sequence of length $n -1$ is obtained.
By repeating this procedure $n - 1$ times one obtains a triangular table of $0$ and $1$. Find the maximum possible number of ones that can appear on this table
2015 India Regional MathematicaI Olympiad, 4
Find all three digit natural numbers of the form $(abc)_{10}$ such that $(abc)_{10}$, $(bca)_{10}$ and $(cab)_{10}$ are in geometric progression. (Here $(abc)_{10}$ is representation in base $10$.)
1988 IMO Longlists, 38
[b]i.)[/b] The polynomial $x^{2 \cdot k} + 1 + (x+1)^{2 \cdot k}$ is not divisible by $x^2 + x + 1.$ Find the value of $k.$
[b]ii.)[/b] If $p,q$ and $r$ are distinct roots of $x^3 - x^2 + x - 2 = 0$ the find the value of $p^3 + q^3 + r^3.$
[b]iii.)[/b] If $r$ is the remainder when each of the numbers 1059, 1417 and 2312 is divided by $d,$ where $d$ is an integer greater than one, then find the value of $d-r.$
[b]iv.)[/b] What is the smallest positive odd integer $n$ such that the product of
\[ 2^{\frac{1}{7}}, 2^{\frac{3}{7}}, \ldots, 2^{\frac{2 \cdot n + 1}{7}} \]
is greater than 1000?