Found problems: 15925
2025 CMIMC Algebra/NT, 1
Four runners are preparing to begin a $1$-mile race from the same starting line. When the race starts, runners Alice, Bob, and Charlie all travel at constant speeds of $8$ mph, $4$ mph, and $2$ mph, respectively. The fourth runner, Dave, is initially half as slow as Charlie, but Dave has a superpower where he suddenly doubles his running speed every time a runner finishes the race. How many hours does it take for Dave to finish the race?
2011 Hanoi Open Mathematics Competitions, 3
What is the largest integer less than to $\sqrt[3]{(2011)^3 + 3 \times (2011)^2 + 4 \times 2011+ 5}$ ?
(A) $2010$, (B) $2011$, (C) $2012$, (D) $2013$, (E) None of the above.
2016 Junior Regional Olympiad - FBH, 2
If $$w=\sqrt{1+\sqrt{-3+2\sqrt{3}}}-\sqrt{1-\sqrt{-3+2\sqrt{3}}}$$ prove that $w=\sqrt{3}-1$
2015 Balkan MO Shortlist, A1
If ${a, b}$ and $c$ are positive real numbers, prove that
\begin{align*}
a ^ 3b ^ 6 + b ^ 3c ^ 6 + c ^ 3a ^ 6 + 3a ^ 3b ^ 3c ^ 3 &\ge{ abc \left (a ^ 3b ^ 3 + b ^ 3c ^ 3 + c ^ 3a ^ 3 \right) + a ^ 2b ^ 2c ^ 2 \left (a ^ 3 + b ^ 3 + c ^ 3 \right)}.
\end{align*}
[i](Montenegro).[/i]
2005 India National Olympiad, 6
Find all functions $f : \mathbb{R} \longrightarrow \mathbb{R}$ such that \[ f(x^2 + yf(z)) = xf(x) + zf(y) , \] for all $x, y, z \in \mathbb{R}$.
2016 KOSOVO TST, 4
$f:R->R$ such that :
$f(1)=1$ and for any $x\in R$
i) $f(x+5)\geq f(x)+5$
ii)$f(x+1)\leq f(x)+1$
If $g(x)=f(x)+1-x$ find g(2016)
1978 Chisinau City MO, 154
What's more $\sqrt[4]{7}+\sqrt[4]{11}$ or $2\sqrt{\frac{\sqrt{7}+\sqrt{11}}{2}}$ ?
1963 Vietnam National Olympiad, 3
Solve the equation $ \sin^3x \cos 3x \plus{} \cos^3x \sin 3x \equal{} \frac{3}{8}$.
2016 Saudi Arabia IMO TST, 2
Find all functions $f : R \to R$ satisfying the conditions:
1. $f (x + 1) \ge f (x) + 1$ for all $x \in R$
2. $f (x y) \ge f (x)f (y)$ for all $x, y \in R$
1978 All Soviet Union Mathematical Olympiad, 264
Given $0 < a \le x_1\le x_2\le ... \le x_n \le b$. Prove that $$(x_1+x_2+...+x_n)\left ( \frac{1}{x_1}+ \frac{1}{x_2}+...+ \frac{1}{x_n}\right)\le \frac{(a+b)^2}{4ab}n^2$$
2017 Ukraine Team Selection Test, 1
Find the smallest constant $C > 0$ for which the following statement holds: among any five positive real numbers $a_1,a_2,a_3,a_4,a_5$ (not necessarily distinct), one can always choose distinct subscripts $i,j,k,l$ such that
\[ \left| \frac{a_i}{a_j} - \frac {a_k}{a_l} \right| \le C. \]
2003 Junior Balkan Team Selection Tests - Moldova, 6
The real numbers x and у satisfy the equations
$$\begin{cases} \sqrt{3x}\left(1+\dfrac{1}{x+y}\right)=2 \\ \\ \sqrt{7y}\left(1-\dfrac{1}{x+y}\right)=4\sqrt2 \end{cases}$$
Find the numerical value of the ratio $y/x$.
2019 Malaysia National Olympiad, 6
It is known that $2018(2019^{39}+2019^{37}+...+2019)+1$ is prime. How many positive factors does $2019^{41}+1$ have?
2016 Hong Kong TST, 2
Determine all positive integers $n$ for which there exist pairwise distinct positive real numbers $a_1, a_2, \cdots, a_n$ satisfying $\displaystyle \left\{a_i+\frac{(-1)^i}{a_i}\mid 1\leq i \leq n\right\}=\{a_i\mid 1\leq i \leq n\}$
2010 Contests, 4
Determine whether there exists a polynomial $f(x_1, x_2)$ with two variables, with integer coefficients, and two points $A=(a_1, a_2)$ and $B=(b_1, b_2)$ in the plane, satisfying the following conditions:
(i) $A$ is an integer point (i.e $a_1$ and $a_2$ are integers);
(ii) $|a_1-b_1|+|a_2-b_2|=2010$;
(iii) $f(n_1, n_2)>f(a_1, a_2)$ for all integer points $(n_1, n_2)$ in the plane other than $A$;
(iv) $f(x_1, x_2)>f(b_1, b_2)$ for all integer points $(x_1, x_2)$ in the plane other than $B$.
[i]Massimo Gobbino, Italy[/i]
2020 Saint Petersburg Mathematical Olympiad, 6.
The points $(1,1),(2,3),(4,5)$ and $(999,111)$ are marked in the coordinate system. We continue to mark points in the following way :
[list]
[*]If points $(a,b)$ are marked then $(b,a)$ and $(a-b,a+b)$ can be marked
[*]If points $(a,b)$ and $(c,d)$ are marked then so can be $(ad+bc, 4ac-4bd)$.
[/list]
Can we, after some finite number of these steps, mark a point belonging to the line $y=2x$.
2003 Turkey Team Selection Test, 4
Find the least
a. positive real number
b. positive integer
$t$ such that the equation $(x^2+y^2)^2 + 2tx(x^2 + y^2) = t^2y^2$ has a solution where $x,y$ are positive integers.
1984 IMO Longlists, 31
Let $f_1(x) = x^3+a_1x^2+b_1x+c_1 = 0$ be an equation with three positive roots $\alpha>\beta>\gamma > 0$. From the equation $f_1(x) = 0$, one constructs the equation $f_2(x) = x^3 +a_2x^2 +b_2x+c_2 = x(x+b_1)^2 -(a_1x+c_1)^2 = 0$. Continuing this process, we get equations $f_3,\cdots, f_n$. Prove that
\[\lim_{n\to\infty}\sqrt[2^{n-1}]{-a_n} = \alpha\]
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)
2018 Junior Balkan Team Selection Tests - Moldova, 2
Let $x$,$y$ be positive real numbers such that $\frac{1}{1+x+x^2}+\frac{1}{1+y+y^2}+\frac{1}{1+x+y}=1$.Prove that $xy=1.$
1992 Swedish Mathematical Competition, 3
Solve:
$$\begin{cases} 2x_1 - 5x_2 + 3x_3 \ge 0 \\
2x_2 - 5x_3 + 3x4 \ge 0 \\
...\\
2x_{23} - 5x_{24} + 3x_{25} \ge 0\\
2x_{24} - 5x_{25} + 3x_1 \ge 0\\
2x_{25} - 5x_1 + 3x_2 \ge 0 \end{cases}$$
2011 Graduate School Of Mathematical Sciences, The Master Cource, The University Of Tokyo, 1
Let $A=\left(
\begin{array}{ccc}
1 & 1& 0 \\
0 & 1& 0 \\
0 &0 & 2
\end{array}
\right),\ B=\left(
\begin{array}{ccc}
a & 1& 0 \\
b & 2& c \\
0 &0 & a+1
\end{array}
\right)\ (a,\ b,\ c\in{\mathbb{C}}).$
(1) Find the condition for $a,\ b,\ c$ such that ${\text{rank} (AB-BA})\leq 1.$
(2) Under the condition of (1), find the condition for $a,\ b,\ c$ such that $B$ is diagonalizable.
2018 Polish Junior MO Finals, 4
Real numbers $a, b, c$ are not equal $0$ and are solution of the system:
$\begin{cases} a^2 + a = b^2 \\ b^2 + b = c^2 \\ c^2 +c = a^2 \end{cases}$
Prove that $(a - b)(b - c)(c - a) = 1$.
2010 Moldova Team Selection Test, 2
Prove that for any real number $ x$ the following inequality is true:
$ \max\{|\sin x|, |\sin(x\plus{}2010)|\}>\dfrac1{\sqrt{17}}$
1990 Canada National Olympiad, 5
The function $f : \mathbb N \to \mathbb R$ satisfies $f(1) = 1, f(2) = 2$ and \[f (n+2) = f(n+2 - f(n+1) ) + f(n+1 - f(n) ).\] Show that $0 \leq f(n+1) - f(n) \leq 1$. Find all $n$ for which $f(n) = 1025$.