Found problems: 15925
2009 India IMO Training Camp, 9
Let
$ f(x)\equal{}\sum_{k\equal{}1}^n a_k x^k$ and $ g(x)\equal{}\sum_{k\equal{}1}^n \frac{a_k x^k}{2^k \minus{}1}$ be two polynomials with real coefficients.
Let g(x) have $ 0,2^{n\plus{}1}$ as two of its roots. Prove That $ f(x)$ has a positive root less than $ 2^n$.
2008 Princeton University Math Competition, A1/B3
Given the sequence $1, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 1,...,$ find $n$ such that the sum of the first $n$ terms is $2008$ or $2009$.
1997 Moldova Team Selection Test, 9
Find all $t\in \mathbb Z$ such that: exists a function $f:\mathbb Z^+\to \mathbb Z$ such that:
$f(1997)=1998$
$\forall x,y\in \mathbb Z^+ , \text{gcd}(x,y)=d : f(xy)=f(x)+f(y)+tf(d):P(x,y)$
2004 Iran MO (3rd Round), 12
$\mathbb{N}_{10}$ is generalization of $\mathbb{N}$ that every hypernumber in $\mathbb{N}_{10}$ is something like: $\overline{...a_2a_1a_0}$ with $a_i \in {0,1..9}$
(Notice that $\overline {...000} \in \mathbb{N}_{10}$)
Also we easily have $+,*$ in $\mathbb{N}_{10}$.
first $k$ number of $a*b$= first $k$ nubmer of (first $k$ number of a * first $k$ number of b)
first $k$ number of $a+b$= first $k$ nubmer of (first $k$ number of a + first $k$ number of b)
Fore example $\overline {...999}+ \overline {...0001}= \overline {...000}$
Prove that every monic polynomial in $\mathbb{N}_{10}[x]$ with degree $d$ has at most $d^2$ roots.
1985 IMO Longlists, 70
Let $C$ be a class of functions $f : \mathbb N \to \mathbb N$ that contains the functions $S(x) = x + 1$ and $E(x) = x - [\sqrt x]^2$ for every $x \in \mathbb N$. ($[x]$ is the integer part of $x$.) If $C$ has the property that for every $f, g \in C, f + g, fg, f \circ g \in C$, show that the function $\max(f(x) - g(x), 0)$ is in $C$, for all $f; g \in C$.
2005 Bulgaria National Olympiad, 6
Let $a,b$ and $c$ be positive integers such that $ab$ divides $c(c^{2}-c+1)$ and $a+b$ is divisible by $c^{2}+1$.
Prove that the sets $\{a,b\}$ and $\{c,c^{2}-c+1\}$ coincide.
2019 International Zhautykov OIympiad, 6
We define two types of operation on polynomial of third degree:
a) switch places of the coefficients of polynomial(including zero coefficients), ex:
$ x^3+x^2+3x-2 $ => $ -2x^3+3x^2+x+1$
b) replace the polynomial $P(x)$ with $P(x+1)$
If limitless amount of operations is allowed,
is it possible from $x^3-2$ to get $x^3-3x^2+3x-3$ ?
2015 Mathematical Talent Reward Programme, MCQ: P 9
How many $5 \times 5$ grids are possible such that each element is either 1 or 0 and each row sum and column sum is $4 ?$
[list=1]
[*] 64
[*] 32
[*] 120
[*] 96
[/list]
1965 All Russian Mathematical Olympiad, 056
a) Each of the numbers $x_1,x_2,...,x_n$ can be $1, 0$, or $-1$. What is the minimal possible value of the sum of all products of couples of those numbers.
b) Each absolute value of the numbers $x_1,x_2,...,x_n$ doesn't exceed $1$. What is the minimal possible value of the sum of all products of couples of those numbers.
2022 Kyiv City MO Round 1, Problem 1
Consider $5$ distinct positive integers. Can their mean be
a)Exactly $3$ times larger than their largest common divisor?
b)Exactly $2$ times larger than their largest common divisor?
2022 Bulgarian Spring Math Competition, Problem 10.4
Find the smallest odd prime $p$, such that there exist coprime positive integers $k$ and $\ell$ which satisfy
\[4k-3\ell=12\quad \text{ and }\quad \ell^2+\ell k +k^2\equiv 3\text{ }(\text{mod }p)\]
2023 New Zealand MO, 2
Let $a, b$ and $c$ be positive real numbers such that $a+b+c = abc$. Prove that at least one of $a, b$ or $c$ is greater than $\frac{17}{10}$ .
1978 Austrian-Polish Competition, 1
Determine all functions $f:(0;\infty)\to \mathbb{R}$ that satisfy
$$f(x+y)=f(x^2+y^2)\quad \forall x,y\in (0;\infty)$$
1999 German National Olympiad, 5
Consider the following inequality for real numbers $x,y,z$: $|x-y|+|y-z|+|z-x| \le a \sqrt{x^2 +y^2 +z^2}$ .
(a) Prove that the inequality is valid for $a = 2\sqrt2$
(b) Assuming that $x,y,z$ are nonnegative, show that the inequality is also valid for $a = 2$.
1986 Kurschak Competition, 3
A and B plays the following game: they choose randomly $k$ integers from $\{1,2,\dots,100\}$; if their sum is even, A wins, else B wins. For what values of $k$ does A and B have the same chance of winning?
2016 Kyrgyzstan National Olympiad, 5
Given two monic polynomials $P(x)$ and $Q(x)$ with degrees 2016.
$P(x)=Q(x)$ has no real root. [b]Prove that P(x)=Q(x+1) has at least one real root.[/b]
2016 Regional Olympiad of Mexico West, 1
Indra has a bag for bringing flowers for her grandmother.
The first day she brings $n$ flowers. From the second day Indra tries to bring three times plus one with respect to the number of flowers of the previous day. However, if this number is greater or equal to $40$, Indra substracts multiples of $40$ until the remainder is less than this number, since her bag cannot containt so many flowers. For which value of $n$ Indra will bring $30$ flowers the day $2016$?
2008 Mongolia Team Selection Test, 3
Given positive integers $ m,n > 1$. Prove that the equation
$ (x \plus{} 1)^n \plus{} (x \plus{} 2)^n \plus{} ... \plus{} (x \plus{} m)^n \equal{} (y \plus{} 1)^{2n} \plus{} (y \plus{} 2)^{2n} \plus{} ... \plus{} (y \plus{} m)^{2n}$ has finitely number of solutions $ x,y \in N$
2006 Iran MO (3rd Round), 2
$n$ is a natural number that $\frac{x^{n}+1}{x+1}$ is irreducible over $\mathbb Z_{2}[x]$. Consider a vector in $\mathbb Z_{2}^{n}$ that it has odd number of $1$'s (as entries) and at least one of its entries are $0$. Prove that these vector and its translations are a basis for $\mathbb Z_{2}^{n}$
2012 Romania National Olympiad, 4
For any non-empty numerical numbers $A$ and $B$, denote
$$A + B = \{a + b | a \in A, b \in B\} $$
a) Determine the largest natural number not $p$ with the property:
[i] there exists[/i] $A,B \subset N$ [i]such that[/i] $card \, A = card\, B = p$ [i]and [/i] $A+B = \{0, 1, 2,..., 2012\}$
b) Determine the smallest natural number $n$ with the property:
[i] there exists[/i] $A,B \subset N$ [i]such that[/i] $card \, A = card\, B $ [i]and [/i] $A+B =\{0, 1, 2,..., 2012\}$
1995 APMO, 1
Determine all sequences of real numbers $a_1$, $a_2$, $\ldots$, $a_{1995}$ which satisfy:
\[ 2\sqrt{a_n - (n - 1)} \geq a_{n+1} - (n - 1), \ \mbox{for} \ n = 1, 2, \ldots 1994, \] and \[ 2\sqrt{a_{1995} - 1994} \geq a_1 + 1. \]
IV Soros Olympiad 1997 - 98 (Russia), 9.2
Find all values of the parameter $a$ for which there exist exactly two integer values of $x$ that satisfy the inequality $$x^2+5\sqrt2 x+a<0.$$
2022 Belarusian National Olympiad, 10.3
Through the point $F(0,\frac{1}{4})$ of the coordinate plane two perpendicular lines pass, that intersect parabola $y=x^2$ at points $A,B,C,D$ ($A_x<B_x<C_x<D_x$) The difference of projections of segments $AD$ and $BC$ onto the $Ox$ line is $m$
Find the area of $ABCD$
2022 Bangladesh Mathematical Olympiad, 1
Find all solutions for real $x$, $$\lfloor x\rfloor^3 -7 \lfloor x+\frac{1}{3} \rfloor=-13.$$
2021 BMT, 6
Three distinct integers are chosen uniformly at random from the set
$$\{2021, 2022, 2023, 2024, 2025, 2026, 2027, 2028, 2029, 2030\}.$$
Compute the probability that their arithmetic mean is an integer.