Olympiad problems

2007 India IMO Training Camp, 3

Given a finite string $S$ of symbols $X$ and $O$, we denote $\Delta(s)$ as the number of$X'$s in $S$ minus the number of $O'$s (For example, $\Delta(XOOXOOX)=-1$). We call a string $S$ [b]balanced[/b] if every sub-string $T$ of (consecutive symbols) $S$ has the property $-1\leq \Delta(T)\leq 2.$ (Thus $XOOXOOX$ is not balanced, since it contains the sub-string $OOXOO$ whose $\Delta$ value is $-3.$ Find, with proof, the number of balanced strings of length $n$.

MathLinks Contest 7th, 5.1

Tags: polynomial , inequalities , vieta , product of roots , absolute value , algebra

Find all real polynomials $ g(x)$ of degree at most $ n \minus{} 3$, $ n\geq 3$, knowing that all the roots of the polynomial $ f(x) \equal{} x^n \plus{} nx^{n \minus{} 1} \plus{} \frac {n(n \minus{} 1)}2 x^{n \minus{} 2} \plus{} g(x)$ are real.

2006 Iran Team Selection Test, 6

Tags: geometry , perimeter , induction , absolute value , combinatorics

Suppose we have a simple polygon (that is it does not intersect itself, but not necessarily convex). Show that this polygon has a diameter which is completely inside the polygon and the two arcs it creates on the polygon perimeter (the two arcs have 2 vertices in common) both have at least one third of the vertices of the polygon.

1957 Putnam, B1

Tags: linear algebra , matrix , absolute value

Consider the determinant of the matrix $(a_{ij})_{ij}$ with $1\leq i,j \leq 100$ and $a_{ij}=ij.$ Prove that if the absolute value of each of the $100!$ terms in the expansion of this determinant is divided by $101,$ then the remainder is always $1.$

2012 Serbia Team Selection Test, 1

Tags: polynomial , function , inequalities , absolute value , algebra

Let $P(x)$ be a polynomial of degree $2012$ with real coefficients satisfying the condition \[P(a)^3 + P(b)^3 + P(c)^3 \geq 3P(a)P(b)P(c),\] for all real numbers $a,b,c$ such that $a+b+c=0$. Is it possible for $P(x)$ to have exactly $2012$ distinct real roots?

2022 AMC 10, 6

Tags: algebra , absolute value

Which expression is equal to $\left | a-2-\sqrt{(a-1)^2} \right|$ for $a<0$? $\textbf{(A) } 3-2a \qquad \textbf{(B) } 1-a \qquad \textbf{(C) } 1 \qquad \textbf{(D) } a+1 \qquad \textbf{(E) } 3$

2004 AMC 10, 4

Tags: function , absolute value

What is the value of $ x$ if $ |x \minus{} 1| \equal{} |x \minus{} 2|$? $ \textbf{(A)}\ \minus{}\!\frac {1}{2}\qquad \textbf{(B)}\ \frac {1}{2}\qquad \textbf{(C)}\ 1\qquad \textbf{(D)}\ \frac {3}{2}\qquad \textbf{(E)}\ 2$

2012 Today's Calculation Of Integral, 801

Tags: calculus , integration , function , absolute value , calculus computations

Answer the following questions: (1) Let $f(x)$ be a function such that $f''(x)$ is continuous and $f'(a)=f'(b)=0$ for some $a<b$. Prove that $f(b)-f(a)=\int_a^b \left(\frac{a+b}{2}-x\right)f''(x)dx$. (2) Consider the running a car on straight road. After a car which is at standstill at a traffic light started at time 0, it stopped again at the next traffic light apart a distance $L$ at time $T$. During the period, prove that there is an instant　for which the absolute value of the acceleration of the car is more than or equal to $\frac{4L}{T^2}.$

2015 Romania National Olympiad, 1

Tags: complex number , absolute value , algebra

Find all triplets $ (a,b,c) $ of nonzero complex numbers having the same absolute value and which verify the equality: $$ \frac{a}{b} +\frac{b}{c}+\frac{c}{a} =-1 $$

1980 Vietnam National Olympiad, 1

Tags: trigonometry , vector , absolute value , inequalities

Let $\alpha_{1}, \alpha_{2}, \cdots , \alpha_{n}$ be numbers in the interval $[0, 2\pi]$ such that the number $\displaystyle\sum_{i=1}^n (1 + \cos \alpha_{i})$ is an odd integer. Prove that \[\displaystyle\sum_{i=1}^n \sin \alpha_i \ge 1\]

2003 Putnam, 4

Tags: quadratic , function , calculus , derivative , inequalities , absolute value

Suppose that $a, b, c, A, B, C$ are real numbers, $a \not= 0$ and $A \not= 0$, such that \[|ax^2+ bx + c| \le |Ax^2+ Bx + C|\] for all real numbers $x$. Show that \[|b^2- 4ac| \le |B^2- 4AC|\]

2014 Putnam, 2

Tags: function , integration , real analysis , absolute value

Suppose that $f$ is a function on the interval $[1,3]$ such that $-1\le f(x)\le 1$ for all $x$ and $\displaystyle \int_1^3f(x)\,dx=0.$ How large can $\displaystyle\int_1^3\frac{f(x)}x\,dx$ be?

2021 OMpD, 3

Tags: inequalities , algebra , n-variable inequality , absolute value

Let $a$ and $b$ be positive real numbers, with $a < b$ and let $n$ be a positive integer. Prove that for all real numbers $x_1, x_2, \ldots , x_n \in [a, b]$: $$ |x_1 - x_2| + |x_2 - x_3| + \cdots + |x_{n-1} - x_n| + |x_n - x_1| \leq \frac{2(b - a)}{b + a}(x_1 + x_2 + \cdots + x_n)$$ And determine for what values of $n$ and $x_1, x_2, \ldots , x_n$ the equality holds.

1956 Czech and Slovak Olympiad III A, 3

Tags: system of equations , absolute value , algebra

Find all real pairs $x,y$ such that \begin{align*} x-|y+1|&=1, \\ x^2+y&=10. \end{align*}

2013 Bogdan Stan, 2

Tags: function , calculus , integration , absolute value

Let be a sequence of continuous functions $ \left( f_n \right)_{n\ge 1} :[0,1]\longrightarrow\mathbb{R} $ satisfying the following properties: $ \text{a) } $ for any natural $ n $ and $ x\in [1/n,1] ,$ it follows $ \left| f_n(x) \right|\leqslant 1/n. $ $ \text{b) } $ for any natural $ n, $ it follows $ \int_0^1 f_n^2(t)dt\leqslant 1. $ Then, $\lim_{n\to 0} \int_0^1\left| f_n(t) \right| dt=0 $ [i]Cristinel Mortici[/i]

2010 Belarus Team Selection Test, 7.1

Tags: number theory , minimum value , minimum , absolute value

Find the smallest value of the expression $|3 \cdot 5^m - 11 \cdot 13^n|$ for all $m,n \in N$. (Folklore)

2011 Putnam, B1

Tags: integration , inequalities , floor function , absolute value , putnam analysis

Let $h$ and $k$ be positive integers. Prove that for every $\varepsilon >0,$ there are positive integers $m$ and $n$ such that \[\varepsilon < \left|h\sqrt{m}-k\sqrt{n}\right|<2\varepsilon.\]

1968 AMC 12/AHSME, 27

Tags: absolute value

Let $S_n=1-2+3-4+\cdots +(-1)^{n-1}n,\ n=1, 2, \cdots$. Then $S_{17}+S_{33}+S_{50}$ equals: $\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ -1 \qquad\textbf{(E)}\ -2$

2009 Croatia Team Selection Test, 1

Tags: polynomial , absolute value , algebra

Determine the lowest positive integer n such that following statement is true: If polynomial with integer coefficients gets value 2 for n different integers, then it can't take value 4 for any integer.

2007 Germany Team Selection Test, 2

Tags: absolute value , algebra

Determine the sum of absolute values for the complex roots of $ 20 x^8 \plus{} 7i x^7 \minus{}7ix \plus{} 20.$

2015 China National Olympiad, 3