Found problems: 15925
2005 Chile National Olympiad, 4
Let $f(a)$ be the largest integer less than or equal to the fourth root of " $a$". Calculate $$f(1)+f(2)+...+f(2005).$$
2002 Federal Competition For Advanced Students, Part 2, 1
Find all polynomials $P(x)$ of the smallest possible degree with the following properties:
(i) The leading coefficient is $200$;
(ii) The coefficient at the smallest non-vanishing power is $2$;
(iii) The sum of all the coefficients is $4$;
(iv) $P(-1) = 0, P(2) = 6, P(3) = 8$.
1966 IMO Longlists, 18
Solve the equation $\frac{1}{\sin x}+\frac{1}{\cos x}=\frac{1}{p}, $ where $p$ is a real parameter. Investigate for which values of $p$ solutions exist and how many solutions exist.
(Of course, the last question ''how many solutions exist'' should be understood as ''how many solutions exists modulo $2\pi $''.)
2018 Cono Sur Olympiad, 3
Define the product $P_n=1! \cdot 2!\cdot 3!\cdots (n-1)!\cdot n!$
a) Find all positive integers $m$, such that $\frac {P_{2020}}{m!}$ is a perfect square.
b) Prove that there are infinite many value(s) of $n$, such that $\frac {P_{n}}{m!}$ is a perfect square, for at least two positive integers $m$.
1914 Eotvos Mathematical Competition, 2
Suppose that
$$-1 \le ax^2 + bx + c \le 1 \ \ for \ \ -1 \le x \le 1 , $$
where a, b, c are real numbers. Prove that
$$-4 \le 2ax + b \le 4 \ \ for \ \ -1 \le x \le 1 , $$
1968 All Soviet Union Mathematical Olympiad, 095
What is greater, $31^{11}$ or $17^{14}$ ?
1989 Brazil National Olympiad, 3
A function $f$, defined for the set of integers, is such that $f(x)=x-10$ if $x>100$ and $f(x)=f(f(x+11))$ if $x \leq 100$.
Determine, justifying your answer, the set of all possible values for $f$.
2018 BMT Spring, Tie 3
Find $$\sum^{k=672}_{k=0} { 2018\choose {3k+2}} \,\, (mod \, 3)$$
2004 Switzerland Team Selection Test, 7
The real numbers $a,b,c,d$ satisfy the equations:
$$\begin{cases} a =\sqrt{45-\sqrt{21-a}} \\ b =\sqrt{45+\sqrt{21-b}}\\ c =\sqrt{45-\sqrt{21+c}}\ \\ d=\sqrt{45+\sqrt{21+d}} \end {cases}$$
Prove that $abcd = 2004$.
2006 China Second Round Olympiad, 2
Let $x,y$ be real numbers. Define a sequence $\{a_n \}$ through the recursive formula
\[ a_0=x,a_1=y,a_{n+1}=\frac{a_na_{n-1}+1}{a_n+a_{n-1}},\]
Find $a_n$.
1995 Belarus National Olympiad, Problem 6
Let $p$ and $q$ be distinct positive integers. Prove that at least one of the equations $x^2+px+q=0$ and $x^2+qx+p=0$ has a real root.
2017 CHMMC (Fall), 1
Let $a, b$ be the roots of the quadratic polynomial $Q(x) = x^2 + x + 1$, and let $u, v$ be the roots of the quadratic polynomial $R(x) = 2x^2 + 7x + 1$.
Suppose $P$ is a cubic polynomial which satis es the equations
$$\begin{cases}
P(au) = Q(u)R(a) \\
P(bu) = Q(u)R(b) \\
P(av) = Q(v)R(a) \\
P(bv) = Q(v)R(b)
\end{cases}$$
If $M$ and$ N$ are the coeffcients of $x^2$ and $x$ respectively in $P(x)$, what is the value of $M+ N$?
2014 Hanoi Open Mathematics Competitions, 3
How many $0$'s are there in the sequence $x_1, x_2,..., x_{2014}$ where $x_n =\big[ \frac{n + 1}{\sqrt{2015}}\big] -\big[ \frac{n }{\sqrt{2015}}\big]$ , $n = 1, 2,...,2014$ ?
(A): $1128$, (B): $1129$, (C): $1130$, (D): $1131$, (E) None of the above.
1967 IMO Longlists, 33
In what case does the system of equations
$\begin{matrix} x + y + mz = a \\ x + my + z = b \\ mx + y + z = c \end{matrix}$
have a solution? Find conditions under which the unique solution of the above system is an arithmetic progression.
2014 India PRMO, 4
In a triangle with integer side lengths, one side is three times as long as a second side, and the length of the third side is $17$. What is the greatest possible perimeter of the triangle?
2001 District Olympiad, 2
Two numbers $(z_1,z_2)\in \mathbb{C}^*\times \mathbb{C}^*$ have the property $(P)$ if there is a real number $a\in [-2,2]$ such that $z_1^2-az_1z_2+z_2^2=0$. Prove that if $(z_1,z_2)$ have the property $(P)$, then $(z_1^n,z_2^n)$ satisfy this property, for any positive integer $n$.
[i]Dorin Andrica[/i]
1957 Moscow Mathematical Olympiad, 355
a) A student takes a subway to an Olympiad, pays one ruble and gets his change. Prove that if he takes a tram (street car) on his way home, he will have enough coins to pay the fare without change.
b) A student is going to a club. (S)he takes a tram, pays one ruble and gets the change. Prove that on the way back by a tram (s)he will be able to pay the fare without any need to change.
Note: In $1957$, the price of a subway ticket was $50$ kopeks, that of a tram ticket $30$ kopeks, the denominations of the coins were $1, 2, 3, 5, 10, 15$, and $20$ kopeks. ($1$ rouble = $100$ kopeks.)
2021 Kyiv Mathematical Festival, 4
Find all collections of $63$ integer numbers such that the square of each number is equal to the sum of all other numbers, and not all the numbers are equal. (O. Rudenko)
1984 Tournament Of Towns, (053) O1
The price of $175$ Humpties is more than the price of $125$ Dumpties but less than that of $126$ Dumpties.
Prove that you cannot buy three Humpties and one Dumpty for
(a) $80$ cents.
(b) $1$ dollar.
(S Fomin, Leningrad)
PS. (a) for Juniors , (a),(b) for Seniors
2019 Grand Duchy of Lithuania, 1
Let $x, y, z$ be positive numbers such that $\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = 1$. Prove that
$$\sqrt{x + yz} +\sqrt{y + zx} +\sqrt{z + xy} \ge\sqrt{xyz}+\sqrt{x }+\sqrt{y} +\sqrt{z}$$
1996 Estonia National Olympiad, 2
For which positive $x$ does the expression $x^{1000}+x^{900}+x^{90}+x^6+\frac{1996}{x}$ attain the smallest value?
MathLinks Contest 7th, 1.2
Let $ a,b,c,d$ be four distinct positive integers in arithmetic progression. Prove that $ abcd$ is not a perfect square.
MathLinks Contest 4th, 3.2
Determine all functions $f : R \to R$ such that $f(x) \ge 0$ for all positive reals $x$, $f(0) = 0$ and for all reals $x, y$
$$f(x + y -xy) = f(x) + f(y) - f(xy).$$
2010 Contests, 3
Find all functions $f: \mathbb R \rightarrow \mathbb R$ such that
\[f(x+xy+f(y)) = \left(f(x)+\frac{1}{2}\right) \left(f(y)+\frac{1}{2}\right)\]
holds for all real numbers $x,y$.
2018 Kazakhstan National Olympiad, 3
Is there exist a function $f:\mathbb {N}\to \mathbb {N}$ with for $\forall m,n \in \mathbb {N}$ $$f\left(mf\left(n\right)\right)=f\left(m\right)f\left(m+n\right)+n ?$$