Found problems: 15925
2014 Cuba MO, 2
Let $a$ and $b$ be real numbers with $0 \le a, b \le 1$.
(a) Prove that $ \frac{a}{b+1} +\frac{b}{a+1} \le 1.$
(b) Find the case of equality.
2003 All-Russian Olympiad, 4
A sequence $(a_n)$ is defined as follows: $a_1 = p$ is a prime number with exactly $300$ nonzero digits, and for each $n \geq 1, a_{n+1}$ is the decimal period of $1/a_n$ multiplies by $2$. Determine $a_{2003}.$
1995 IMO Shortlist, 4
Suppose that $ x_1, x_2, x_3, \ldots$ are positive real numbers for which \[ x^n_n \equal{} \sum^{n\minus{}1}_{j\equal{}0} x^j_n\] for $ n \equal{} 1, 2, 3, \ldots$ Prove that $ \forall n,$ \[ 2 \minus{} \frac{1}{2^{n\minus{}1}} \leq x_n < 2 \minus{} \frac{1}{2^n}.\]
2010 India IMO Training Camp, 9
Let $A=(a_{jk})$ be a $10\times 10$ array of positive real numbers such that the sum of numbers in row as well as in each column is $1$.
Show that there exists $j<k$ and $l<m$ such that
\[a_{jl}a_{km}+a_{jm}a_{kl}\ge \frac{1}{50}\]
2022 Balkan MO Shortlist, A6
Determine all functions $f : \mathbb{R}^2 \to\mathbb {R}$ for which \[f(A)+f(B)+f(C)+f(D)=0,\]whenever $A,B,C,D$ are the vertices of a square with side-length one.
[i]Ilir Snopce[/i]
2008 China Team Selection Test, 3
Let $ n>m>1$ be odd integers, let $ f(x)\equal{}x^n\plus{}x^m\plus{}x\plus{}1$. Prove that $ f(x)$ can't be expressed as the product of two polynomials having integer coefficients and positive degrees.
2016 BMT Spring, 3
Α half-mile long train is traveling at a speed of $90$ miles per hour. As it enters a $1$ mile long tunnel, Steve starts running from the back of the train to the front of the train at a speed of $10$ miles per hour. When Steve is out of the tunnel, he stops running. How far along the train has Steve run in miles?
2018 Korea Winter Program Practice Test, 1
Find all functions $f : \mathbb{R} \rightarrow \mathbb{R}$ satisfying the following conditions :
1) $f(x+y)-f(x)-f(y) \in \{0,1\} $ for all $x,y \in \mathbb{R}$
2) $\lfloor f(x) \rfloor = \lfloor x \rfloor $ for all real $x$.
1985 IberoAmerican, 3
Find all the roots $ r_{1}$, $ r_{2}$, $ r_{3}$ y $ r_{4}$ of the equation $ 4x^{4}\minus{}ax^{3}\plus{}bx^{2}\minus{}cx\plus{}5 \equal{} 0$, knowing that they are real, positive and that:
\[ \frac{r_{1}}{2}\plus{}\frac{r_{2}}{4}\plus{}\frac{r_{3}}{5}\plus{}\frac{r_{4}}{8}\equal{} 1.\]
1969 Swedish Mathematical Competition, 4
Define $g(x)$ as the largest value of$ |y^2 - xy|$ for $y$ in $[0, 1]$. Find the minimum value of $g$ (for real $x$).
2022 AMC 10, 16
The roots of the polynomial $10x^3 - 39x^2 + 29x - 6$ are the height, length, and width of a rectangular box (right rectangular prism. A new rectangular box is formed by lengthening each edge of the original box by 2 units. What is the volume of the new box?
$\textbf{(A) }\frac{24}{5}\qquad\textbf{(B) }\frac{42}{5}\qquad\textbf{(C) }\frac{81}{5}\qquad\textbf{(D) }30\qquad\textbf{(E) }48$
III Soros Olympiad 1996 - 97 (Russia), 9.1
Is rational or irrational,the number
$$\left(\dfrac{2}{\sqrt[3]{25}+\sqrt[3]{15}+\sqrt[3]{9}}+\dfrac{1}{\sqrt[3]{9}+\sqrt[3]{6}+\sqrt[3]{4}}\right) \times \left(\sqrt[3]{25}+\sqrt[3]{10}+\sqrt[3]{4}\right)?$$
2011 Abels Math Contest (Norwegian MO), 3b
Find all functions $f$ from the real numbers to the real numbers such that $f(xy) \le \frac12 \left(f(x) + f(y) \right)$ for all real numbers $x$ and $y$.
2003 All-Russian Olympiad Regional Round, 8.2
A beetle crawls along each of two intersecting straight lines at constant speeds, without changing direction. It is known that projections of the beetles on the $OX$ axis never coincide (neither in the past nor in the future). Prove that the projections of the beetles on the $OY$ axis will necessarily coincide or have coincided before.
[hide=oroginal wording] По каждой из двух пересекающихся прямых с постоянными скоростями, не меняя направления, ползет по жуку. Известно, что проекции жуков на ось OX никогда не совпадают (ни в прошлом, ни в будущем). Докажите, что проекции жуков на ось OY обязательно совпадут или совпадали раньше.[/hide]
2014 China Team Selection Test, 3
Let the function $f:N^*\to N^*$ such that
[b](1)[/b] $(f(m),f(n))\le (m,n)^{2014} , \forall m,n\in N^*$;
[b](2)[/b] $n\le f(n)\le n+2014 , \forall n\in N^*$
Show that: there exists the positive integers $N$ such that $ f(n)=n $, for each integer $n \ge N$.
(High School Affiliated to Nanjing Normal University )
2024 Simon Marais Mathematical Competition, A3
Let $W$ be a fixed positive integer. Let $S$ be the set of all pairs $(a, b)$ of positive integers such that $a \neq b$. For each $(a, b) \in S$, let $m(a,b)$ be the largest integer satisfying
\[
m(a, b) \leq \frac{1 + na}{1 + nb}
\]
for all integers $n \geq 1$.
(a) For each $(a, b) \in S$, prove that there exists a positive integer $k$ such that
\[
m(a,b) \leq \frac{1 + na}{W + nb}
\]
for all $n \geq k$.
(b) For each $(a, b) \in S$, let $k(a,b)$ be the smallest value of $k$ that satisfies the condition of part (a). Determine $\max \{k(a,b) \mid (a,b) \in S \}$ or prove that it does not exist.
2000 China Team Selection Test, 1
Let $F$ be the set of all polynomials $\Gamma$ such that all the coefficients of $\Gamma (x)$ are integers and $\Gamma (x) = 1$ has integer roots. Given a positive intger $k$, find the smallest integer $m(k) > 1$ such that there exist $\Gamma \in F$ for which $\Gamma (x) = m(k)$ has exactly $k$ distinct integer roots.
2010 Brazil Team Selection Test, 4
Find all functions $f$ from the set of real numbers into the set of real numbers which satisfy for all $x$, $y$ the identity \[ f\left(xf(x+y)\right) = f\left(yf(x)\right) +x^2\]
[i]Proposed by Japan[/i]
2006 China Team Selection Test, 3
$k$ and $n$ are positive integers that are greater than $1$. $N$ is the set of positive integers. $A_1, A_2, \cdots A_k$ are pairwise not-intersecting subsets of $N$ and $A_1 \cup A_2 \cup \cdots \cup A_k = N$.
Prove that for some $i \in \{ 1,2,\cdots,k \}$, there exsits infinity many non-factorable n-th degree polynomials so that coefficients of one polynomial are pairwise distinct and all the coeficients are in $A_i$.
2002 All-Russian Olympiad Regional Round, 8.7
''Moskvich'' and ''Zaporozhets'' drove past the observer on the highway and the Niva moving towards them. It is known that when the Moskvich caught up with the observer, it was equidistant from the Zaporozhets and the Niva, and when the Niva caught up with the observer, it was equal. but removed from ''Moskvich'' and ''Zaporozhets''. Prove that ''Zaporozhets'' at the moment of passing by the observer was equidistant from the Niva and ''Moskvich''.
2011 Postal Coaching, 3
Let $P (x)$ be a polynomial with integer coefficients. Given that for some integer $a$ and some positive integer $n$, where
\[\underbrace{P(P(\ldots P}_{\text{n times}}(a)\ldots)) = a,\]
is it true that $P (P (a)) = a$?
2016 Singapore Junior Math Olympiad, 2
Let $a_1,a_2,...,a_9$ be a sequence of numbers satisfying $0 < p \le a_i \le q$ for each $i = 1,2,..., 9$.
Prove that $\frac{a_1}{a_9}+\frac{a_2}{a_8}+...+\frac{a_9}{a_1} \le 1 + \frac{4(p^2+q^2)}{pq}$
1999 China Second Round Olympiad, 2
Let $a$,$b$,$c$ be real numbers.
Let $z_{1}$,$z_{2}$,$z_{3}$ be complex numbers such that $|z_{k}|=1$ $(k=1,2,3)$ $~$ and $~$ $\frac{z_{1}}{z_{2}}+\frac{z_{2}}{z_{3}}+\frac{z_{3}}{z_{1}}=1$
Find $|az_{1}+bz_{2}+cz_{3}|$.
2023 Indonesia TST, 3
Find all positive integers $n \geqslant 2$ for which there exist $n$ real numbers $a_1<\cdots<a_n$ and a real number $r>0$ such that the $\tfrac{1}{2}n(n-1)$ differences $a_j-a_i$ for $1 \leqslant i<j \leqslant n$ are equal, in some order, to the numbers $r^1,r^2,\ldots,r^{\frac{1}{2}n(n-1)}$.
1975 Chisinau City MO, 97
Find the smallest value of the expression $(x-1) (x -2) (x -3) (x - 4) + 10$.