Found problems: 15925
2004 Peru MO (ONEM), 3
Let $x,y,z$ be positive real numbers, less than $\pi$, such that:
$$\cos x + \cos y + \cos z = 0$$
$$\cos 2x + \cos 2 y + \cos 2z = 0$$
$$\cos 3x + \cos 3y + \cos 3z = 0$$
Find all the values that $\sin x + \sin y + \sin z$ can take.
2022 Thailand Mathematical Olympiad, 2
Define a function $f:\mathbb{N}\times \mathbb{N}\to\{-1,1\}$ such that
$$f(m,n)=\begin{cases} 1 &\text{if }m,n\text{ have the same parity, and} \\ -1 &\text{if }m,n\text{ have different parity}\end{cases}$$
for every positive integers $m,n$. Determine the minimum possible value of
$$\sum_{1\leq i<j\leq 2565} ijf(x_i,x_j)$$
across all permutations $x_1,x_2,x_3,\dots,x_{2565}$ of $1,2,\dots,2565$.
1966 IMO Longlists, 31
Solve the equation $|x^2 -1|+ |x^2 - 4| = mx$ as a function of the parameter $m$. Which pairs $(x,m)$ of integers satisfy this equation?
2015 South Africa National Olympiad, 2
Determine all pairs of real numbers $a$ and $x$ that satisfy the simultaneous equations $$5x^3 + ax^2 + 8 = 0$$ and $$5x^3 + 8x^2 + a = 0.$$
2018 Brazil Undergrad MO, 5
Consider the set $A = \left\{\frac{j}{4}+\frac{100}{j}|j=1,2,3,..\right\} $ What is the smallest number that belongs to the $ A $ set?
2019 India PRMO, 14
Let $\mathcal{R}$ denote the circular region in the $xy-$plane bounded by the circle $x^2+y^2=36$. The lines $x=4$ and $y=3$ divide $\mathcal{R}$ into four regions $\mathcal{R}_i ~ , ~i=1,2,3,4$. If $\mid \mathcal{R}_i \mid$ denotes the area of the region $\mathcal{R}_i$ and if $\mid \mathcal{R}_1 \mid >$ $\mid \mathcal{R}_2 \mid >$ $\mid \mathcal{R}_3 \mid > $ $\mid \mathcal{R}_4 \mid $, determine $\mid \mathcal{R}_1 \mid $ $-$ $\mid \mathcal{R}_2 \mid $ $-$ $\mid \mathcal{R}_3 \mid $ $+$ $\mid \mathcal{R}_4 \mid $.
1994 Vietnam National Olympiad, 1
Find all real solutions to
\[x^{3}+3x-3+\ln{(x^{2}-x+1)}=y,\]
\[y^{3}+3y-3+\ln{(y^{2}-y+1)}=z,\]
\[z^{3}+3z-3+\ln{(z^{2}-z+1)}=x.\]
2008 Grigore Moisil Intercounty, 1
Find all monotonic functions $ f:\mathbb{R}\longrightarrow\mathbb{R} $ with the property that
$$ (f(\sin x))^2-3f(x)=-2, $$
for any real numbers $ x. $
[i]Dorin Andrica[/i] and [i]Mihai Piticari[/i]
1989 IMO Longlists, 46
Let S be the point of intersection of the two lines $ l_1 : 7x \minus{} 5y \plus{} 8 \equal{} 0$ and $ l_2 : 3x \plus{} 4y \minus{} 13 \equal{} 0.$ Let $ P \equal{} (3, 7), Q \equal{} (11, 13),$ and let $ A$ and $ B$ be points on the line $ PQ$ such that $ P$ is between $ A$ and $ Q,$ and $ B$ is between $ P$ and $ Q,$ and such that \[ \frac{PA}{AQ} \equal{} \frac{PB}{BQ} \equal{} \frac{2}{3}.\] Without finding the coordinates of $ B$ find the equations of the lines $ SA$ and $ SB.$
2010 Puerto Rico Team Selection Test, 4
Find the largest possible value in the real numbers of the term $$\frac{3x^2 + 16xy + 15y^2}{x^2 + y^2}$$ with $x^2 + y^2 \ne 0$.
1989 Greece National Olympiad, 1
Let $a,b,c,d x,y,z, w$ be real numbers such that $$\begin{matrix}
ax -by-c z-dw =0\\
b x +a y -d z +cw=0\\
c x+ d y +a z -b w=0\\
dx-c y+bz+aw=0
\end{matrix}$$
prove that $$a=b=c=d=0, \ \ or \ \ x=y=z=w=0$$
The Golden Digits 2024, P1
Determine all functions $f:\mathbb{R}_+\to\mathbb{R}_+$ which satisfy \[f\left(\frac{y}{f(x)}\right)+x=f(xy)+f(f(x)),\]for any positive real numbers $x$ and $y$.
[i]Proposed by Pavel Ciurea[/i]
1980 Swedish Mathematical Competition, 1
Show that $\log_{10} 2$ is irrational.
2014 Contests, 3
Let $n \ge 2$ be a positive integer, and write in a digit form \[\frac{1}{n}=0.a_1a_2\dots.\] Suppose that $n = a_1 + a_2 + \cdots$. Determine all possible values of $n$.
2022 China Second Round, 3
Let $a_1,a_2,\cdots ,a_{100}$ be non-negative integers such that
$(1)$ There are positive integers$ k\leq 100$ such that $a_1\leq a_2\leq \cdots\leq a_{k}$
and $a_i=0$ $(i>k);$
$(2)$ $ a_1+a_2+a_3+\cdots +a_{100}=100;$
$(3)$ $ a_1+2a_2+3a_3+\cdots +100a_{100}=2022.$
Find the minimum of $ a_1+2^2a_2+3^2a_3+\cdots +100^2a_{100}.$
2003 South africa National Olympiad, 5
Prove that the sum of the squares of two consecutive positive integers cannot be equal to a sum of the fourth powers of two consecutive positive integers.
2015 China National Olympiad, 3
Let $a_1,a_2,...$ be a sequence of non-negative integers such that for any $m,n$ \[ \sum_{i=1}^{2m} a_{in} \leq m.\] Show that there exist $k,d$ such that \[ \sum_{i=1}^{2k} a_{id} = k-2014.\]
2012 CHMMC Spring, 4
The expression below has six empty boxes. Each box is to be fi lled in with a number from $1$ to $6$, where all six numbers are used exactly once, and then the expression is evaluated. What is the maximum possible final result that can be achieved?
$$\dfrac{\frac{\square}{\square}+\frac{\square}{\square}}{\frac{\square}{\square}}$$
2011 Dutch BxMO TST, 3
Find all triples $(x, y, z)$ of real numbers that satisfy $x^2 + y^2 + z^2 + 1 = xy + yz + zx +|x - 2y + z|$.
2023 Ecuador NMO (OMEC), 3
We define a sequence of numbers $a_n$ such that $a_0=1$ and for all $n\ge0$:
\[2a_{n+1} ^3 + 2a_n ^3 = 3 a_{n +1} ^2 a_n + 3a_{n+1}a_n^2\]
Find the sum of all $a_{2023}$'s possible values.
2010 IFYM, Sozopol, 8
Solve this equation with $x \in R$:
$x^3-3x=\sqrt{x+2}$
2010 ISI B.Math Entrance Exam, 7
We are given $a,b,c \in \mathbb{R}$ and a polynomial $f(x)=x^3+ax^2+bx+c$ such that all roots (real or complex) of $f(x)$ have same absolute value. Show that $a=0$ iff $b=0$.
2017 Taiwan TST Round 2, 4
Find all integer $c\in\{0,1,...,2016\}$ such that the number of $f:\mathbb{Z}\rightarrow\{0,1,...,2016\}$ which satisfy the following condition is minimal:\\
(1) $f$ has periodic $2017$\\
(2) $f(f(x)+f(y)+1)-f(f(x)+f(y))\equiv c\pmod{2017}$\\
Proposed by William Chao
2011 ELMO Shortlist, 3
Let $N$ be a positive integer. Define a sequence $a_0,a_1,\ldots$ by $a_0=0$, $a_1=1$, and $a_{n+1}+a_{n-1}=a_n(2-1/N)$ for $n\ge1$. Prove that $a_n<\sqrt{N+1}$ for all $n$.
[i]Evan O'Dorney.[/i]
1987 Greece National Olympiad, 3
Solve for real values of parameter $a$, the inequality : $$\sqrt{a+x}+ \sqrt{a-x}>a , \ \ x\in\mathbb{R}$$