Found problems: 252
2007 South africa National Olympiad, 2
Consider the equation $ x^4 \equal{} ax^3 \plus{} bx^2 \plus{} cx \plus{} 2007$, where $ a,b,c$ are real numbers. Determine the largest value of $ b$ for which this equation has exactly three distinct solutions, all of which are integers.
1982 AMC 12/AHSME, 11
How many integers with four different digits are there between $1,000$ and $9,999$ such that the absolute value of the difference between the first digit and the last digit is $2$?
$\textbf {(A) } 672 \qquad \textbf {(B) } 784 \qquad \textbf {(C) } 840 \qquad \textbf {(D) } 896 \qquad \textbf {(E) } 1008$
2013 Philippine MO, 3
3. Let n be a positive integer. The numbers 1, 2, 3,....., 2n are randomly assigned to 2n distinct points on a circle. To each chord joining two of these points, a value is assigned equal to the absolute value of the difference between the assigned numbers at its endpoints.
Show that one can choose n pairwise non-intersecting chords such that the sum of the values assigned to them is $n^2$ .
2019 Canada National Olympiad, 4
Prove that for $n>1$ and real numbers $a_0,a_1,\dots, a_n,k$ with $a_1=a_{n-1}=0$,
\[|a_0|-|a_n|\leq \sum_{i=0}^{n-2}|a_i-ka_{i+1}-a_{i+2}|.\]
1985 Miklós Schweitzer, 5
Let $F(x,y)$ and $G(x,y)$ be relatively prime homogeneous polynomials of degree at least one having integer coefficients. Prove that there exists a number $c$ depending only on the degrees and the maximum of the absolute values of the coefficients of $F$ and $G$ such that $F(x,y)\neq G(x,y)$ for any integers $x$ and $y$ that are relatively prime and satisfy $\max \{ |x|,|y|\} > c$. [K. Gyory]
1989 Balkan MO, 2
Let $\overline{a_{n}a_{n-1}\ldots a_{1}a_{0}}$ be the decimal representation of a prime positive integer such that $n>1$ and $a_{n}>1$. Prove that the polynomial $P(x)=a_{n}x^{n}+\ldots +a_{1}x+a_{0}$ cannot be written as a product of two non-constant integer polynomials.
2006 Australia National Olympiad, 4
There are $n$ points on a circle, such that each line segment connecting two points is either red or blue.
$P_iP_j$ is red if and only if $P_{i+1} P_{j+1}$ is blue, for all distinct $i, j$ in $\left\{1, 2, ..., n\right\}$.
(a) For which values of $n$ is this possible?
(b) Show that one can get from any point on the circle to any other point, by doing a maximum of 3 steps, where one step is moving from a point to another point through a red segment connecting these points.
2014 Vietnam National Olympiad, 3
Find all sets of not necessary distinct 2014 rationals such that:if we remove an arbitrary number in the set, we can divide remaining 2013 numbers into three sets such that each set has exactly 671 elements and the product of all elements in each set are the same.
2024 Indonesia TST, A
Given real numbers $x,y,z$ which satisfies
$$|x+y+z|+|xy+yz+zx|+|xyz| \le 1$$
Show that $max\{ |x|,|y|,|z|\} \le 1$.
2020-21 IOQM India, 5
Find the number of integer solutions to $||x| - 2020| < 5$.
2002 France Team Selection Test, 3
Let $n$ be a positive integer and let $(a_1,a_2,\ldots ,a_{2n})$ be a permutation of $1,2,\ldots ,2n$ such that the numbers $|a_{i+1}-a_i|$ are pairwise distinct for $i=1,\ldots ,2n-1$.
Prove that $\{a_2,a_4,\ldots ,a_{2n}\}=\{1,2,\ldots ,n\}$ if and only if $a_1-a_{2n}=n$.
2008 China Team Selection Test, 2
Prove that for all $ n\geq 2,$ there exists $ n$-degree polynomial $ f(x) \equal{} x^n \plus{} a_{1}x^{n \minus{} 1} \plus{} \cdots \plus{} a_{n}$ such that
(1) $ a_{1},a_{2},\cdots, a_{n}$ all are unequal to $ 0$;
(2) $ f(x)$ can't be factorized into the product of two polynomials having integer coefficients and positive degrees;
(3) for any integers $ x, |f(x)|$ isn't prime numbers.
2008 Harvard-MIT Mathematics Tournament, 3
Determine all real numbers $ a$ such that the inequality $ |x^2 \plus{} 2ax \plus{} 3a|\le2$ has exactly one solution in $ x$.
2019 LIMIT Category B, Problem 6
Let $f(x)=a_0+a_1|x|+a_2|x|^2+a_3|x|^3$, where $a_0,a_1,a_2,a_3$ are constant. Then
$\textbf{(A)}~f(x)\text{ is differentiable at }x=0\text{ if whatever be }a_0,a_1,a_2,a_3$
$\textbf{(B)}~f(x)\text{ is not differentiable at }x=0\text{ if whatever be }a_0,a_1,a_2,a_3$
$\textbf{(C)}~f(x)\text{ is differentiable at }x=0\text{ only if }a_1=0$
$\textbf{(D)}~f(x)\text{ is differentiable at }x=0\text{ only if }a_1=0,a_3=0$
2009 South africa National Olympiad, 5
A game is played on a board with an infinite row of holes labelled $0, 1, 2, \dots$. Initially, $2009$ pebbles are put into hole $1$; the other holes are left empty. Now steps are performed according to the following scheme:
(i) At each step, two pebbles are removed from one of the holes (if possible), and one pebble is put into each of the neighbouring holes.
(ii) No pebbles are ever removed from hole $0$.
(iii) The game ends if there is no hole with a positive label that contains at least two pebbles.
Show that the game always terminates, and that the number of pebbles in hole $0$ at the end of the game is independent of the specific sequence of steps. Determine this number.
2009 Math Prize For Girls Problems, 1
How many ordered pairs of integers $ (x, y)$ are there such that
\[ 0 < \left\vert xy \right\vert < 36?\]
2024 Ukraine National Mathematical Olympiad, Problem 2
There is a table with $n > 2$ cells in the first row, $n-1$ cells in the second row is a cell, $n-2$ in the third row, $\ldots$, $1$ cell in the $n$-th row. The cells are arranged as shown below.
[img]https://i.ibb.co/0Z1CR0c/UMO24-8-2.png[/img]
In each cell of the top row Petryk writes a number from $1$ to $n$, so that each number is written exactly once. For each other cell, if the cells directly above it contains numbers $a, b$, it contains number $|a-b|$. What is the largest number that can be written in a single cell of the bottom row?
[i]Proposed by Bogdan Rublov[/i]
2012 AIME Problems, 12
For a positive integer $p$, define the positive integer $n$ to be $p$-safe if $n$ differs in absolute value by more than $2$ from all multiples of $p$. For example, the set of $10$-safe numbers is $\{3, 4, 5, 6, 7, 13, 14, 15, 16, 17,23, \ldots \}$. Find the number of positive integers less than or equal to $10,000$ which are simultaneously $7$-safe, $11$-safe, and $13$-safe.·
1959 AMC 12/AHSME, 25
The symbol $|a|$ means $+a$ if $a$ is greater than or equal to zero, and $-a$ if $a$ is less than or equal to zero; the symbol $<$ means "less than"; the symbol $>$ means "greater than."
The set of values $x$ satisfying the inequality $|3-x|<4$ consists of all $x$ such that:
$ \textbf{(A)}\ x^2<49 \qquad\textbf{(B)}\ x^2>1 \qquad\textbf{(C)}\ 1<x^2<49\qquad\textbf{(D)}\ -1<x<7\qquad\textbf{(E)}\ -7<x<1 $
2016 District Olympiad, 2
Let $ a,b,c\in\mathbb{C}^* $ pairwise distinct, having the same absolute value, and satisfying:
$$ a^2+b^2+c^2-ab-bc-ca=0. $$
Prove that $ a,b,c $ represents the affixes of the vertices of a right or equilateral triangle.
2009 Today's Calculation Of Integral, 423
Let $ f(x)\equal{}x^2\plus{}3$ and $ y\equal{}g(x)$ be the equation of the line with the slope $ a$, which pass through the point $ (0,\ f(0))$ .
Find the maximum and minimum values of $ I(a)\equal{}3\int_{\minus{}1}^1 |f(x)\minus{}g(x)|\ dx$.
2013 AMC 12/AHSME, 25
Let $f : \mathbb{C} \to \mathbb{C} $ be defined by $ f(z) = z^2 + iz + 1 $. How many complex numbers $z $ are there such that $ \text{Im}(z) > 0 $ and both the real and the imaginary parts of $f(z)$ are integers with absolute value at most $ 10 $?
${ \textbf{(A)} \ 399 \qquad \textbf{(B)} \ 401 \qquad \textbf{(C)} \ 413 \qquad \textbf{(D}} \ 431 \qquad \textbf{(E)} \ 441 $
PEN O Problems, 9
Let $n$ be an integer, and let $X$ be a set of $n+2$ integers each of absolute value at most $n$. Show that there exist three distinct numbers $a, b, c \in X$ such that $c=a+b$.
2004 AIME Problems, 2
Set $A$ consists of $m$ consecutive integers whose sum is $2m$, and set $B$ consists of $2m$ consecutive integers whose sum is $m$. The absolute value of the difference between the greatest element of $A$ and the greatest element of $B$ is $99$. Find $m$.
2010 Tuymaada Olympiad, 3
Let $f(x) = ax^2+bx+c$ be a quadratic trinomial with $a$,$b$,$c$ reals such that any quadratic trinomial obtained by a permutation of $f$'s coefficients has an integer root (including $f$ itself).
Show that $f(1)=0$.