Found problems: 15925
1997 Bosnia and Herzegovina Team Selection Test, 1
Solve system of equation $$8(x^3+y^3+z^3)=73$$ $$2(x^2+y^2+z^2)=3(xy+yz+zx)$$ $$xyz=1$$ in set $\mathbb{R}^3$
2022 International Zhautykov Olympiad, 5
A polynomial $f(x)$ with real coefficients of degree greater than $1$ is given. Prove that there are infinitely many positive integers which cannot be represented in the form \[f(n+1)+f(n+2)+\cdots+f(n+k)\]
where $n$ and $k$ are positive integers.
OMMC POTM, 2023 11
Consider an infinite strictly increasing sequence of positive integers $a_1$, $a_2$,$...$ where for any real number $C$, there exists an integer $N$ where $a_k >Ck$ for any $k >N$. Do there necessarily exist inifinite many indices $k$ where $2a_k <a_{k-1}+a_{k+1}$ for any $0<i<k$?
2012 Bogdan Stan, 3
Consider $ 2011 $ positive real numbers $ a_1,a_2,\ldots ,a_{2011} . $ If they are in geometric progression, show that there exists a real number $ \lambda $ such that any $ i\in\{ 1,2,\ldots , 1005 \} $ implies $ \lambda =a_i\cdot a_{2012-i} . $ Disprove the converse.
[i]Teodor Radu[/i]
2024 Kazakhstan National Olympiad, 1
Positive integers $a,b,c$ satisfy the equations $a^2=b^3+ab$ and $c^3=a+b+c$. Prove that $a=bc$.
1991 Baltic Way, 8
Let $a, b, c, d, e$ be distinct real numbers. Prove that the equation
\[(x - a)(x - b)(x - c)(x - d) + (x - a)(x - b)(x - c)(x - e)\]
\[+(x - a)(x - b)(x - d)(x - e) + (x - a)(x - c)(x - d)(x - e)\]
\[+(x - b)(x - c)(x - d)(x - e) = 0\]
has four distinct real solutions.
2023 LMT Fall, 9
Find the least positive integer $k$ such that when $\frac{k}{2023}$ is written in simplest form, the sum of the numerator and denominator is divisible by $7$.
[i]Proposed byMuztaba Syed[/i]
2007 USA Team Selection Test, 6
For a polynomial $ P(x)$ with integer coefficients, $ r(2i \minus{} 1)$ (for $ i \equal{} 1,2,3,\ldots,512$) is the remainder obtained when $ P(2i \minus{} 1)$ is divided by $ 1024$. The sequence
\[ (r(1),r(3),\ldots,r(1023))
\]
is called the [i]remainder sequence[/i] of $ P(x)$. A remainder sequence is called [i]complete[/i] if it is a permutation of $ (1,3,5,\ldots,1023)$. Prove that there are no more than $ 2^{35}$ different complete remainder sequences.
2022 Brazil EGMO TST, 5
For a given value $t$, we consider number sequences $a_1, a_2, a_3,...$ such that $a_{n+1} =\frac{a_n + t}{a_n + 1}$ for all $n \ge 1$.
(a) Suppose that $t = 2$. Determine all starting values $a_1 > 0$ such that $\frac43 \le a_n \le \frac32$ holds for all $n \ge 2$.
(b) Suppose that $t = -3$. Investigate whether $a_{2020} = a_1$ for all starting values $a_1$ different from $-1$ and $1$.
2003 AIME Problems, 15
Let
\[P(x)=24x^{24}+\sum_{j=1}^{23}(24-j)(x^{24-j}+x^{24+j}). \]
Let $z_{1},z_{2},\ldots,z_{r}$ be the distinct zeros of $P(x),$ and let $z_{k}^{2}=a_{k}+b_{k}i$ for $k=1,2,\ldots,r,$ where $i=\sqrt{-1},$ and $a_{k}$ and $b_{k}$ are real numbers. Let
\[\sum_{k=1}^{r}|b_{k}|=m+n\sqrt{p}, \]
where $m,$ $n,$ and $p$ are integers and $p$ is not divisible by the square of any prime. Find $m+n+p.$
2023 Grosman Mathematical Olympiad, 3
Find all pairs of polynomials $p$, $q$ with complex coefficients so that
\[p(x)\cdot q(x)=p(q(x)).\]
2006 Thailand Mathematical Olympiad, 3
Let $P(x), Q(x)$ and $R(x)$ be polynomials satisfying the equation $2xP(x^3) + Q(-x -x^3) = (1 + x + x^2)R(x)$.
Show that $x - 1$ divides $P(x) - Q(x)$.
2008 Austria Beginners' Competition, 2
Determine all real numbers $x$ satisfying $$x \lfloor x \lfloor x \rfloor \rfloor =\sqrt2.$$
1996 Tournament Of Towns, (488) 1
Prove that if $a, b$ and $c$ are positive numbers such that
$$a^2 + b^2 - ab = c^2,$$
then $(a - c)(b - c) < 0.$
(A Egorov)
I Soros Olympiad 1994-95 (Rus + Ukr), 11.8
Let's write down a segment of a series of integers from $0$ to $1995$. Among the numbers written out, two have been crossed out. Let's consider the longest arithmetic progression contained among the remaining $1994$ numbers. Let $K$ be the length of the progression. Which two numbers must be crossed out so that the value of $K$ is the smallest?
1984 All Soviet Union Mathematical Olympiad, 389
Given a sequence $\{x_n\}$, $$x_1 = x_2 = 1, x_{n+2} = x^2_{n+1} - \frac{x_n}{2}$$ Prove that the sequence has limit and find it.
2019 China Team Selection Test, 3
Find all positive integer $n$, such that there exists $n$ points $P_1,\ldots,P_n$ on the unit circle , satisfying the condition that for any point $M$ on the unit circle, $\sum_{i=1}^n MP_i^k$ is a fixed value for
\\a) $k=2018$
\\b) $k=2019$.
2006 Regional Competition For Advanced Students, 4
Let $ <h_n>$ $ n\in\mathbb N$ a harmonic sequence of positive real numbers (that means that every $ h_n$ is the harmonic mean of its two neighbours $ h_{n\minus{}1}$ and $ h_{n\plus{}1}$ : $ h_n\equal{}\frac{2h_{n\minus{}1}h_{n\plus{}1}}{h_{n\minus{}1}\plus{}h_{n\plus{}1}}$)
Show that: if the sequence includes a member $ h_j$, which is the square of a rational number, it includes infinitely many members $ h_k$, which are squares of rational numbers.
2004 India IMO Training Camp, 2
Let $P(x) = x^4 + ax^3 + bx^2 + cx + d$ and $Q(x) = x^2 + px + q$be two real polynomials. Suppose that there exista an interval $(r,s)$ of length greater than $2$ SUCH THAT BOTH $P(x)$ AND $Q(x)$ ARE nEGATIVE FOR $X \in (r,s)$ and both are positive for $x > s$ and $x<r$. Show that there is a real $x_0$ such that $P(x_0) < Q(x_0)$
1951 AMC 12/AHSME, 22
The values of $ a$ in the equation: $ \log_{10}(a^2 \minus{} 15a) \equal{} 2$ are:
$ \textbf{(A)}\ \frac {15\pm\sqrt {233}}{2} \qquad\textbf{(B)}\ 20, \minus{} 5 \qquad\textbf{(C)}\ \frac {15 \pm \sqrt {305}}{2}$
$ \textbf{(D)}\ \pm20 \qquad\textbf{(E)}\ \text{none of these}$
2007 Spain Mathematical Olympiad, Problem 1
Let $a_0, a_1, a_2, a_3, a_4$ be five positive numbers in the arithmetic progression with a difference $d$. Prove that $a^3_2 \leq \frac{1}{10}(a^3_0 + 4a^3_1 + 4a^3_3 + a^3_4).$
1963 Dutch Mathematical Olympiad, 3
Twenty numbers $a_1,a_2,..,a_{20}$ satisfy:
$$a_k \ge 7k \,\,\,\,\, for \,\,\,\,\, k = 1,2,..., 20$$
$$a_1+a_2+...+a_{20}=1518$$
Prove that among the numbers $k = 1,2,... ,20$ there are no more than seventeen, for which $a_k \ge 20k -2k^2$.
Kvant 2021, M2681
Find all pairs of distinct rational numbers $(a,b)$ such that $a^a=b^b$.
[i]Proposed by I. Dorofeev[/i]
1988 Czech And Slovak Olympiad IIIA, 5
Find all numbers $a \in (-2, 2)$ for which the polynomial $x^{154}-ax^{77}+1$ is a multiple of the polynomial $x^{14}-ax^{7}+1$.
2008 Bulgarian Autumn Math Competition, Problem 8.1
Solve the equation $|x-m|+|x+m|=x$ depending on the value of the parameter $m$.