Found problems: 15925
2014 HMNT, 10
Let $z$ be a complex number and k a positive integer such that $z^k$ is a positive real number other than $1$. Let $f(n)$ denote the real part of the complex number $z^n$. Assume the parabola $p(n) = an^2 +bn+c$ intersects $f(n)$ four times, at $n = 0, 1, 2, 3$. Assuming the smallest possible value of $k$, find the largest possible value of $a$.
2021 Serbia Team Selection Test, P6
Let $S=\{1,2, \ldots ,10^{10}\}$. Find all functions $f:S \rightarrow S$, such that $$f(x+1)=f(f(x))+1 \pmod {10^{10}}$$ for each $x \in S$ (assume $f(10^{10}+1)=f(1)$).
2010 Saudi Arabia BMO TST, 3
Let $a > 0$ be a real number and let $f : R \to R$ be a function satisfying $$f(x_1) + f(x_2) \ge a f(x_1 + x_2), \forall x_1 ,x_2 \in R.$$ Prove that $$f(x_1) + f(x_2) +(x_3) \ge \frac{3a^2}{a+2} f(x_1+ x_2 + x_3), \forall x_1 ,x_2,x_3 \in R$$.
2016 Israel National Olympiad, 1
Nina and Meir are walking on a $3$ km path towards grandma's house. They start walking at the same time from the same point. Meir's speed is $4$ km/h and Nina's speed is $3$ km/h.
Along the path there are several benches. Whenever Nina or Meir reaches a bench, they sit on it for some time and eat a cookie. Nina always takes $t$ minutes to eat a cookie, and Meir always takes $2t$ minutes to eat a cookie, where $t$ is a positive integer.
It turns out that Nina and Meir reached grandma's house at the same time. How many benches were there? Find all of the options.
2001 Saint Petersburg Mathematical Olympiad, 11.1
Do there exist distinct numbers $x,y,z$ from $[0,\dfrac{\pi}{2}]$, such that six number $\sin x$, $\sin y$,$\sin z$, $\cos x$, $\cos y$, $\cos z$ could be partitioned into 3 pairs with equal sums?
[I]Proposed by A. Golovanov[/i]
1978 Chisinau City MO, 154
What's more $\sqrt[4]{7}+\sqrt[4]{11}$ or $2\sqrt{\frac{\sqrt{7}+\sqrt{11}}{2}}$ ?
2010 Brazil Team Selection Test, 4
Let $f$ be any function that maps the set of real numbers into the set of real numbers. Prove that there exist real numbers $x$ and $y$ such that \[f\left(x-f(y)\right)>yf(x)+x\]
[i]Proposed by Igor Voronovich, Belarus[/i]
1998 All-Russian Olympiad Regional Round, 11.1
Two identical decks have 36 cards each. One deck is shuffled and put on top of the second. For each card of the top deck, we count the number of cards between it and the corresponding card of the bottom deck. What is the sum of these numbers?
Sorry if this has been posted before but I would like to know if I solved it correctly. Thanks!
2007 Singapore MO Open, 4
find all functions $f:\mathbb{N}\rightarrow\mathbb{N}$ st
$f(f(m)+f(n))=m+n \,\forall m,n\in\mathbb{N}$
related:
https://artofproblemsolving.com/community/c6h381298
1989 Canada National Olympiad, 3
Define $ \{ a_n \}_{n\equal{}1}$ as follows: $ a_1 \equal{} 1989^{1989}; \ a_n, n > 1,$ is the sum of the digits of $ a_{n\minus{}1}$. What is the value of $ a_5$?
2013 Brazil Team Selection Test, 4
Find all functions $f:\mathbb{R} \rightarrow \mathbb{R}$ that satisfy the conditions
\[f(1+xy)-f(x+y)=f(x)f(y) \quad \text{for all } x,y \in \mathbb{R},\]
and $f(-1) \neq 0$.
2011 NIMO Problems, 14
In circle $\theta_1$ with radius $1$, circles $\phi_1, \phi_2, \dots, \phi_8$, with equal radii, are drawn such that for $1 \le i \le 8$, $\phi_i$ is tangent to $\omega_1$, $\phi_{i-1}$, and $\phi_{i+1}$, where $\phi_0 = \phi_8$ and $\phi_1 = \phi_9$. There exists a circle $\omega_2$ such that $\omega_1 \neq \omega_2$ and $\omega_2$ is tangent to $\phi_i$ for $1 \le i \le 8$. The radius of $\omega_2$ can be expressed in the form $a - b\sqrt{c} -d\sqrt{e - \sqrt{f}} + g \sqrt{h - j \sqrt{k}}$ such that $a, b, \dots, k$ are positive integers and the numbers $e, f, k, \gcd(h, j)$ are squarefree. What is $a+b+c+d+e+f+g+h+j+k$.
[i]Proposed by Eugene Chen
[/i]
1987 IberoAmerican, 1
Find the function $f(x)$ such that
\[f(x)^2f\left(\frac{1-x}{x+1}\right) =64x \]
for $x\not=0,x\not=1,x\not=-1$.
1987 Czech and Slovak Olympiad III A, 3
Let $f:(0,\infty)\to(0,\infty)$ be a function satisfying $f\bigl(xf(y)\bigr)+f\bigl(yf(x)\bigr)=2xy$ for all $x,y>0$. Show that $f(x) = x$ for all positive $x$.
2015 Taiwan TST Round 3, 2
Consider all polynomials $P(x)$ with real coefficients that have the following property: for any two real numbers $x$ and $y$ one has \[|y^2-P(x)|\le 2|x|\quad\text{if and only if}\quad |x^2-P(y)|\le 2|y|.\] Determine all possible values of $P(0)$.
[i]Proposed by Belgium[/i]
1975 Bulgaria National Olympiad, Problem 3
Let $f(x)=a_0x^3+a_1x^2+a_2x+a_3$ be a polynomial with real coefficients ($a_0\ne0$) such that $|f(x)|\le1$ for every $x\in[-1,1]$. Prove that
(a) there exist a constant $c$ (one and the same for all polynomials with the given property), for which
(b) $|a_0|\le4$.
[i]V. Petkov[/i]
2009 Peru IMO TST, 1
Show that there are infinitely many triples $(x, y, z)$ of real numbers such that $$\displaystyle{x^2+y = y^2+z= z^2 + x}$$ and $x\ne y\ne z \ne x.$
1994 India National Olympiad, 6
Find all real-valued functions $f$ on the reals such that $f(-x) = -f(x)$, $f(x+1) = f(x) + 1$ for all $x$, and $f\left(\dfrac{1}{x}\right) = \dfrac{f(x)}{x^2}$ for $x \not = 0$.
2002 Estonia National Olympiad, 1
Peeter, Juri, Kati and Mari are standing at the entrance of a dark tunnel. They have one torch and none of them dares to be in the tunnel without it, but the tunnel is so narrow that at most two people can move together. It takes $1$ minute for Peeter, $2$ minutes for Juri, $5$ for Kati and $10$ for Mari to pass the tunnel. Find the minimum time in which they can all pass through the tunnel.
2013 Poland - Second Round, 4
Solve equation
$(x^4 + 3y^2)\sqrt{|x + 2| + |y|}=4|xy^2|$
in real numbers $x$, $y$.
2002 Regional Competition For Advanced Students, 4
Let $a_0, a_1, ..., a_{2002}$ be real numbers.
a) Show that the smallest of the values $a_k (1-a_{2002-k})$ ($0 \le k \le 2002$) the following applies:
it is smaller or equal to $1/4$.
b) Does this statement always apply to the smallest of the values $a_k (1-a_{2003-k})$ ($1 \le k \le 2002$) ?
c) Show for positive real numbers $a_0, a_1, ..., a_{2002}$ :
the smallest of the values $a_k (1-a_{2003-k})$ ($1 \le k \le 2002$) is less than or equal to $1/4$.
1997 Polish MO Finals, 2
Find all real solutions to: \begin{eqnarray*} 3(x^2 + y^2 + z^2) &=& 1 \\ x^2y^2 + y^2z^2 + z^2x^2 &=& xyz(x + y + z)^3. \end{eqnarray*}
VII Soros Olympiad 2000 - 01, 8.7
In the expression $(x + 100) (x + 99) ... (x-99) (x-100)$, the brackets were expanded and similar terms were given. The expression $x^{201} + ...+ ax^2 + bx + c$ turned out. Find the numbers $a$ and $c$.
2015 Thailand TSTST, 1
Let $a,b,c$ be a real numbers such that this equations:
$a^2x + b^2y + c^2z = 1$
$xy + yz + xz = 1$
have only one solution $(x, y, z)$ in real numbers. Prove that $a, b, c$ are sides of the triangle
1997 Cono Sur Olympiad, 1
We have $98$ cards, in each one we will write one of the numbers: $1, 2, 3, 4,...., 97, 98$.
We can order the $98$ cards, in a sequence such that two consecutive numbers $X$ and $Y$ and the number $X - Y$ is greater than $48$, determine how and how many ways we can make this sequence!!