Found problems: 15925
2024 China Western Mathematical Olympiad, 8
Given a positive integer $n \geq 2$. Let $a_{ij}$ $(1 \leq i,j \leq n)$ be $n^2$ non-negative reals and their sum is $1$. For $1\leq i \leq n$, define $R_i=max_{1\leq k \leq n}(a_{ik})$. For $1\leq j \leq n$, define $C_j=min_{1\leq k \leq n}(a_{kj})$
Find the maximum value of $C_1C_2 \cdots C_n(R_1+R_2+ \cdots +R_n)$
2011 Benelux, 3
If $k$ is an integer, let $\mathrm{c}(k)$ denote the largest cube that is less than or equal to $k$. Find all positive integers $p$ for which the following sequence is bounded:
$a_0 = p$ and $a_{n+1} = 3a_n-2\mathrm{c}(a_n)$ for $n \geqslant 0$.
2021 Balkan MO Shortlist, A6
Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that
$$f(xy) = f(x)f(y) + f(f(x + y))$$
holds for all $x, y \in \mathbb{R}$.
1981 Poland - Second Round, 1
Prove that for any real numbers $ x_1, x_2, \ldots, x_{1981} $, $ y_1, y_2, \ldots, y_{1981} $ such that $ \sum_{j=1}^{1981} x_j = 0 $, $ \sum_{j=1}^{1981} y_j = 0 $ the inequality occurs
$$
\sqrt{\sum_{j=1}^{1981} (x_j^2+y_j^2)} \leq \frac{1}{\sqrt{2}} \sum_{j=1}^{1981} \sqrt{x_j^2+y_j^2}.$$
1978 Romania Team Selection Test, 8
For any set $ A $ we say that two functions $ f,g:A\longrightarrow A $ are [i]similar,[/i] if there exists a bijection $ h:A\longrightarrow A $ such that $ f\circ h=h\circ g. $
[b]a)[/b] If $ A $ has three elements, construct a finite, arbitrary number functions, having as domain and codomain $ A, $ that are two by two similar, and every other function with the same domain and codomain as the ones determined is similar to, at least, one of them.
[b]b)[/b] For $ A=\mathbb{R} , $ show that the functions $ \sin $ and $ -\sin $ are similar.
2015 Indonesia MO, 4
Let function pair $f,g : \mathbb{R^+} \rightarrow \mathbb{R^+}$ satisfies
\[
f(g(x)y + f(x)) = (y+2015)f(x)
\]
for every $x,y \in \mathbb{R^+} $
a. Prove that $f(x) = 2015g(x)$ for every $x \in \mathbb{R^+}$
b. Give an example of function pair $(f,g)$ that satisfies the statement above and $f(x), g(x) \geq 1$ for every $x \in \mathbb{R^+}$
2001 Dutch Mathematical Olympiad, 2
The function f has the following properties :
$f(x + y) = f(x) + f(y) + xy$ for all real $x$ and $y$
$f(4) = 10$
Calculate $f(2001)$.
2005 Junior Balkan Team Selection Tests - Moldova, 8
The families of second degree functions $f_m, g_m: R\to R, $ are considered , $f_m (x) = (m^2 + 1) x^2 + 3mx + m^2 - 1$, $g_m (x) = m^2x^2 + mx - 1$, where $m$ is a real nonzero parameter.
Show that, for any function $h$ of the second degree with the property that $g_m (x) \le h (x) \le f_m (x)$ for any real $x$, there exists $\lambda \in [0, 1]$ which verifies the condition $h (x) = \lambda f_m (x) + (1- \lambda) g_m (x)$, whatever real $x$ is.
2021 Harvard-MIT Mathematics Tournament., 7
Suppose that $x$, $y$, and $z$ are complex numbers of equal magnitude that satisfy
\[x+y+z = -\frac{\sqrt{3}}{2}-i\sqrt{5}\]
and
\[xyz=\sqrt{3} + i\sqrt{5}.\]
If $x=x_1+ix_2, y=y_1+iy_2,$ and $z=z_1+iz_2$ for real $x_1,x_2,y_1,y_2,z_1$ and $z_2$ then
\[(x_1x_2+y_1y_2+z_1z_2)^2\]
can be written as $\tfrac{a}{b}$ for relatively prime positive integers $a$ and $b$. Compute $100a+b.$
2015 Saudi Arabia GMO TST, 1
Find all functions $f : R \to R$ satisfying the following conditions
(a) $f(1) = 1$,
(b) $f(x + y) = f(x) + f(y)$, $\forall (x,y) \in R^2$
(c) $f\left(\frac{1}{x}\right) =\frac{ f(x)}{x^2 }$, $\forall x \in R -\{0\}$
Trần Nam Dũng
2007 Nicolae Coculescu, 1
Let be two real numbers $ x,y, $ and a natural number $ n_0 $ such that $ \{ n_0x \} = \{ n_0y \} $ and $ \{ (n_0+1)x \} = \{ (n_0+1)y \} ,$ where $ \{\} $ denotes the fractional part. Show that $ \{ nx \} =\{ ny \} , $ for any natural number $ n. $
[i]Ovidiu Pop[/i]
1968 Czech and Slovak Olympiad III A, 1
Let $a_1,\ldots,a_n\ (n>2)$ be real numbers with at most one zero. Solve the system
\begin{align*}
x_1x_2 &= a_1, \\
x_2x_3 &= a_2, \\
&\ \vdots \\
x_{n-1}x_n &= a_{n-1}, \\
x_nx_1 &\ge a_n.
\end{align*}
2004 Balkan MO, 1
The sequence $\{a_n\}_{n\geq 0}$ of real numbers satisfies the relation:
\[ a_{m+n} + a_{m-n} - m + n -1 = \frac12 (a_{2m} + a_{2n}) \]
for all non-negative integers $m$ and $n$, $m \ge n$. If $a_1 = 3$ find $a_{2004}$.
2004 All-Russian Olympiad Regional Round, 10.1
The sum of positive numbers $a, b, c$ is equal to $\pi/2$. Prove that
$$\cos a + \cos b + \cos c > \sin a + \sin b + \sin c.$$
2018 Bundeswettbewerb Mathematik, 2
Find all real numbers $x$ satisfying the equation
\[\left\lfloor \frac{20}{x+18}\right\rfloor+\left\lfloor \frac{x+18}{20}\right\rfloor=1.\]
2020 Iran Team Selection Test, 4
Given a function $g:[0,1] \to \mathbb{R}$ satisfying the property that for every non empty dissection of the trivial $[0,1]$ to subsets $A,B$ we have either $\exists x \in A; g(x) \in B$ or $\exists x \in B; g(x) \in A$ and we have furthermore $g(x)>x$ for $x \in [0,1]$. Prove that there exist infinite $x \in [0,1]$ with $g(x)=1$.
[i]Proposed by Ali Zamani [/i]
2001 All-Russian Olympiad, 2
In a magic square $n \times n$ composed from the numbers $1,2,\cdots,n^2$, the centers of any two squares are joined by a vector going from the smaller number to the bigger one. Prove that the sum of all these vectors is zero. (A magic square is a square matrix such that the sums of entries in all its rows and columns are equal.)
2006 Bulgaria Team Selection Test, 1
Find all sequences of positive integers $\{a_n\}_{n=1}^{\infty}$, for which $a_4=4$ and
\[\frac{1}{a_1a_2a_3}+\frac{1}{a_2a_3a_4}+\cdots+\frac{1}{a_na_{n+1}a_{n+2}}=\frac{(n+3)a_n}{4a_{n+1}a_{n+2}}\]
for all natural $n \geq 2$.
[i]Peter Boyvalenkov[/i]
2006 Germany Team Selection Test, 2
Four real numbers $ p$, $ q$, $ r$, $ s$ satisfy $ p+q+r+s = 9$ and $ p^{2}+q^{2}+r^{2}+s^{2}= 21$. Prove that there exists a permutation $ \left(a,b,c,d\right)$ of $ \left(p,q,r,s\right)$ such that $ ab-cd \geq 2$.
2023 Spain Mathematical Olympiad, 4
Let $x_1\leq x_2\leq x_3\leq x_4$ be real numbers. Prove that there exist polynomials of degree two $P(x)$ and $Q(x)$ with real coefficients such that $x_1$, $x_2$, $x_3$ and $x_4$ are the roots of $P(Q(x))$ if and only if $x_1+x_4=x_2+x_3$.
2011 India IMO Training Camp, 2
Let the real numbers $a,b,c,d$ satisfy the relations $a+b+c+d=6$ and $a^2+b^2+c^2+d^2=12.$ Prove that
\[36 \leq 4 \left(a^3+b^3+c^3+d^3\right) - \left(a^4+b^4+c^4+d^4 \right) \leq 48.\]
[i]Proposed by Nazar Serdyuk, Ukraine[/i]
1993 Tournament Of Towns, (391) 3
Each of the numbers $1, 2, 3,... 25$ is arranged in a $5$ by $5$ table. In each row they appear in increasing order (left to right). Find the maximal and minimal possible sum of the numbers in the third column.
(Folklore)
2007 Silk Road, 4
The set of polynomials $f_1, f_2, \ldots, f_n$ with real coefficients is called [i]special [/i], if for any different $i,j,k \in \{ 1,2, \ldots, n\}$ polynomial $\dfrac{2}{3}f_i + f_j + f_k$ has no real roots, but for any different $p,q,r,s \in \{ 1,2, \ldots, n\}$ of a polynomial $f_p + f_q + f_r + f_s$ there is a real root.
a) Give an example of a [i]special [/i] set of four polynomials whose sum is not a zero polynomial.
b) Is there a [i]special [/i] set of five polynomials?
2021 Portugal MO, 3
All sequences of $k$ elements $(a_1,a_2,...,a_k)$ are considered, where each $a_i$ belongs to the set $\{1,2,... ,2021\}$. What is the sum of the smallest elements of all these sequences?
2002 Denmark MO - Mohr Contest, 5
Homer Grog has written the numbers $1, 3, 4, 5, 7, 9, 11, 13, 15,17$, one number on each note. He arranges the bills in a circle and tries to get the largest sum $S$ of the numbers of three consecutive bills to be the least possible. What is the smallest value $S$ can assume?