Found problems: 15925
2024 Malaysia IMONST 2, 4
For all $n \geq 1$, define $a_{n}$ to be the fraction $\frac{k}{2^n}$ such that $a_{n}$ is the closest to $\frac{1}{3}$ over all integer values of $k$.
Prove that the sequence $a_{1}, a_{2}, \cdots $satisfies the equation $2a_{i+2} = a_{i+1} + a_{i}$ for all $i \geq 1$.
2019 Romania Team Selection Test, 1
Let be a natural number $ n\ge 3. $ Find
$$ \inf_{\stackrel{ x_1,x_2,\ldots ,x_n\in\mathbb{R}_{>0}}{1=P\left( x_1,x_2,\ldots ,x_n\right)}}\sum_{i=1}^n\left( \frac{1}{x_i} -x_i \right) , $$
where $ P\left( x_1,x_2,\ldots ,x_n\right) :=\sum_{i=1}^n \frac{1}{x_i+n-1} , $ and find in which circumstances this infimum is attained.
2011 IMO Shortlist, 2
Determine all sequences $(x_1,x_2,\ldots,x_{2011})$ of positive integers, such that for every positive integer $n$ there exists an integer $a$ with \[\sum^{2011}_{j=1} j x^n_j = a^{n+1} + 1\]
[i]Proposed by Warut Suksompong, Thailand[/i]
2006 Estonia National Olympiad, 1
Calculate the sum $$\frac{1}{1+2^{-2006}}+...+ \frac{1}{1+2^{-1}}+ \frac{1}{1+2^{0}}+ \frac{1}{1+2^{1}}+...+ \frac{1}{1+2^{2006}}$$
2011 USA TSTST, 1
Find all real-valued functions $f$ defined on pairs of real numbers, having the following property: for all real numbers $a, b, c$, the median of $f(a,b), f(b,c), f(c,a)$ equals the median of $a, b, c$.
(The [i]median[/i] of three real numbers, not necessarily distinct, is the number that is in the middle when the three numbers are arranged in nondecreasing order.)
2015 BMT Spring, 7
Evaluate $\sum_{k=0}^{37}(-1)^k\binom{75}{2k}$.
2024 Chile TST Ibero., 4
Prove that if \( a \), \( b \), and \( c \) are positive real numbers, then the following inequality holds:
\[
\frac{a + 3c}{a + b} + \frac{c + 3a}{b + c} + \frac{4b}{c + a} \geq 6.
\]
2020 Vietnam National Olympiad, 2
a)Let$a,b,c\in\mathbb{R}$ and $a^2+b^2+c^2=1$.Prove that:
$|a-b|+|b-c|+|c-a|\le2\sqrt{2}$
b) Let $a_1,a_2,..a_{2019}\in\mathbb{R}$ and $\sum_{i=1}^{2019}a_i^2=1$.Find the maximum of:
$S=|a_1-a_2|+|a_2-a_3|+...+|a_{2019}-a_1|$
V Soros Olympiad 1998 - 99 (Russia), 9.1
Place parentheses in the expression $$2:2 -3:3 - 4: 4-5:5$$ so that the result is a number greater than $39$.
1978 IMO Longlists, 29
Given a nonconstant function $f : \mathbb{R}^+ \longrightarrow\mathbb{R}$ such that $f(xy) = f(x)f(y)$ for any $x, y > 0$, find functions $c, s : \mathbb{R}^+ \longrightarrow \mathbb{R}$ that satisfy $c\left(\frac{x}{y}\right) = c(x)c(y)-s(x)s(y)$ for all $x, y > 0$ and $c(x)+s(x) = f(x)$ for all $x > 0$.
2021 Canadian Mathematical Olympiad Qualification, 7
If $A, B$ and $C$ are real angles such that
$$\cos (B-C)+\cos (C-A)+\cos (A-B)=-3/2,$$
find
$$\cos (A)+\cos (B)+\cos (C)$$
2021 Durer Math Competition Finals, 1
In Sixcountry there are $ 12$ months, but each month consists of $6$ weeks. The month are named the same way we do, from January to December, but in each month the weeks have different lengths. In the $k$-th month the weeks consist of $6^{k-1}$ days. What is the number of days of the spring (March, April, May together)?
1935 Moscow Mathematical Olympiad, 016
How many real solutions does the following system have ?$\begin{cases} x+y=2 \\
xy - z^2 = 1 \end{cases}$
2008 Romania National Olympiad, 1
Find functions $ f: \mathbb{N} \rightarrow \mathbb{N}$, such that $ f(x^2 \plus{} f(y)) \equal{} xf(x) \plus{} y$, for $ x,y \in \mathbb{N}$.
2015 Princeton University Math Competition, B1
Roy is starting a baking company and decides that he will sell cupcakes. He sells $n$ cupcakes for $(n + 20)(n + 15)$ cents. A man walks in and buys $\$10.50$ worth of cupcakes. Roy bakes cupcakes at a rate of $10$ cupcakes an hour. How many minutes will it take Roy to complete the order?
2023 NMTC Junior, P3
Let $a_i (i=1,2,3,4,5,6)$ are reals. The polynomial
$f(x)=a_1+a_2x+a_3x^2+a_4x^3+a_5x^4+a_6a^5+7x^6-4x^7+x^8$ can be factorized into linear factors $x-x_i$ where
$i \in {1,2,3,...,8}$.
Find the possible values of $a_1$.
2006 Balkan MO, 4
Let $m$ be a positive integer and $\{a_n\}_{n\geq 0}$ be a sequence given by $a_0 = a \in \mathbb N$, and \[ a_{n+1} = \begin{cases} \displaystyle \frac{a_n}2 & \textrm { if } a_n \equiv 0 \pmod 2, \\ a_n + m & \textrm{ otherwise. } \end{cases} \]
Find all values of $a$ such that the sequence is periodical (starting from the beginning).
2015 China Western Mathematical Olympiad, 1
Let the integer $n \ge 2$ , and $x_1,x_2,\cdots,x_n $ be real numbers such that $\sum_{k=1}^nx_k$ be integer . $d_k=\underset{m\in {Z}}{\min}\left|x_k-m\right| $, $1\leq k\leq n$ .Find the maximum value of $\sum_{k=1}^nd_k$.
2024 Nigerian MO Round 2, Problem 2
Solve the system of equations:
\[x>y>z\]
\[x+y+z=1\]
\[x^2+y^2+z^2=69\]
\[x^3+y^3+z^3=271\]
[hide=Answer]x=7, y=-2, z=-4[/hide]
2019 ITAMO, 4
Let $\lfloor x \rfloor$ denote the greatest integer less than or equal to $x.$
Let $\lambda \geq 1$ be a real number and $n$ be a positive integer with the property that $\lfloor \lambda^{n+1}\rfloor, \lfloor \lambda^{n+2}\rfloor ,\cdots, \lfloor \lambda^{4n}\rfloor$ are all perfect squares$.$ Prove that $\lfloor \lambda \rfloor$ is a perfect square$.$
2018 Greece Junior Math Olympiad, 1
a) Does there exist a real number $x$ such that $x+\sqrt{3}$ and $x^2+\sqrt{3}$ are both rationals?
b) Does there exist a real number $y$ such that $y+\sqrt{3}$ and $y^3+\sqrt{3}$ are both rationals?
1994 Dutch Mathematical Olympiad, 2
A sequence of integers $ a_1,a_2,a_3,...$ is such that $ a_1\equal{}2, a_2\equal{}3$, and
$ a_{n\plus{}1}\equal{}2a_{n\minus{}1}$ or $ 3a_n\minus{}2a_{n\minus{}1}$ for all $ n \ge 2$.
Prove that no number between $ 1600$ and $ 2000$ can be an element of the sequence.
1993 IMO Shortlist, 3
Prove that \[ \frac{a}{b+2c+3d} +\frac{b}{c+2d+3a} +\frac{c}{d+2a+3b}+ \frac{d}{a+2b+3c} \geq \frac{2}{3} \] for all positive real numbers $a,b,c,d$.
2005 Belarusian National Olympiad, 5
For $0<a,b,c,d<\frac{\pi}{2}$ is true that $$\cos 2a+\cos 2b+ \cos 2c+ \cos 2d= 4 (\sin a \sin b \sin c \sin d -\cos a \cos b \cos c \cos d)$$
Find all possible values of $a+b+c+d$
2016 PAMO, 4
Let $x,y,z$ be positive real numbers such that $xyz=1$. Prove that
$\frac{1}{(x+1)^2+y^2+1}$ $+$ $\frac{1}{(y+1)^2+z^2+1}$ $+$ $\frac{1}{(z+1)^2+x^2+1}$ $\leq$ ${\frac{1}{2}}$.