Olympiad problems

1984 AMC 12/AHSME, 30

For any complex number $w = a + bi$, $|w|$ is defined to be the real number $\sqrt{a^2 + b^2}$. If $w = \cos{40^\circ} + i\sin{40^\circ}$, then \[ |w + 2w^2 + 3w^3 + \cdots + 9w^9|^{-1} \] equals $\textbf{(A)}\ \frac{1}{9}\sin{40^\circ} \qquad \textbf{(B)}\ \frac{2}{9}\sin{20^\circ} \qquad \textbf{(C)}\ \frac{1}{9}\cos{40^\circ} \qquad \textbf{(D)}\ \frac{1}{18}\cos{20^\circ} \qquad \textbf{(E)}\text{ none of these}$

2008 Argentina National Olympiad, 2

Tags: abstract algebra , absolute value , algebra

In every cell of a $ 60 \times 60$ board is written a real number, whose absolute value is less or equal than $ 1$. The sum of all numbers on the board equals $ 600$. Prove that there is a $ 12 \times 12$ square in the board such that the absolute value of the sum of all numbers on it is less or equal than $ 24$.

2003 Putnam, 3

Tags: algebra , domain , absolute value , function , trigonometry , calculus

Find the minimum value of \[|\sin{x} + \cos{x} + \tan{x} + \cot{x} + \sec{x} + \csc{x}|\] for real numbers $x$.

2010 Romanian Master of Mathematics, 2

Tags: algebra , cauchy inequality , absolute value , inequalities , function

For each positive integer $n$, find the largest real number $C_n$ with the following property. Given any $n$ real-valued functions $f_1(x), f_2(x), \cdots, f_n(x)$ defined on the closed interval $0 \le x \le 1$, one can find numbers $x_1, x_2, \cdots x_n$, such that $0 \le x_i \le 1$ satisfying \[|f_1(x_1)+f_2(x_2)+\cdots f_n(x_n)-x_1x_2\cdots x_n| \ge C_n\] [i]Marko Radovanović, Serbia[/i]

2010 Junior Balkan Team Selection Tests - Romania, 2

Tags: sum , absolute value , algebra , inequalities

Let $a_1, a_2, ..., a_n$ real numbers such that $a_1 + a_2 + ... + a_n = 0$ and $| a_1 | + | a_2 | + ... + | a_n | = 1$. Show that: $| a _ 1 + 2 a _ 2 + ... + n a _ n | \le \frac {n-1} {2}$.

2009 IMS, 4

Tags: limit , algebra , polynomial , absolute value , vector , induction , ratio

In this infinite tree, degree of each vertex is equal to 3. A real number $ \lambda$ is given. We want to assign a real number to each node in such a way that for each node sum of numbers assigned to its neighbors is equal to $ \lambda$ times of the number assigned to this node. Find all $ \lambda$ for which this is possible.

2020 AMC 10, 5

Tags: absolute value

What is the sum of all real numbers $x$ for which $|x^2-12x+34|=2?$ $\textbf{(A) } 12 \qquad \textbf{(B) } 15 \qquad \textbf{(C) } 18 \qquad \textbf{(D) } 21 \qquad \textbf{(E) } 25$

1961 Putnam, B4

Tags: absolute value , inequalities

Let $x_1 , x_2 ,\ldots, x_n$ be real numbers in $[0,1].$ Determine the maximum value of the sum of the $\frac{n(n-1)}{2}$ terms: $$\sum_{i<j}|x_i -x_j |.$$

2014 Math Prize For Girls Problems, 20

Tags: absolute value , ceiling function , trigonometry

How many complex numbers $z$ such that $\left| z \right| < 30$ satisfy the equation \[ e^z = \frac{z - 1}{z + 1} \, ? \]

1975 Czech and Slovak Olympiad III A, 4

Tags: analytic geometry , parameterization , linear algebra , absolute value , line , algebra

Determine all real values of parameter $p$ such that the equation \[|x-2|+|y-3|+y=p\] is an equation of a ray in the plane $xy.$

2008 China Team Selection Test, 2

Tags: modular arithmetic , polynomial , absolute value , algebra

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.

2006 Iran Team Selection Test, 6

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

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.

2012 Serbia Team Selection Test, 1

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

2012 Poland - Second Round, 3

Tags: least common multiple , absolute value , vector , number theory

Let $m,n\in\mathbb{Z_{+}}$ be such numbers that set $\{1,2,\ldots,n\}$ contains exactly $m$ different prime numbers. Prove that if we choose any $m+1$ different numbers from $\{1,2,\ldots,n\}$ then we can find number from $m+1$ choosen numbers, which divide product of other $m$ numbers.

2009 Harvard-MIT Mathematics Tournament, 1

Tags: absolute value

How many ways can the integers from $-7$ to $7$ inclusive be arranged in a sequence such that the absolute value of the numbers in the sequence does not decrease?

1999 Romania Team Selection Test, 11

Tags: polynomial , absolute value , inequalities , algebra

Let $a,n$ be integer numbers, $p$ a prime number such that $p>|a|+1$. Prove that the polynomial $f(x)=x^n+ax+p$ cannot be represented as a product of two integer polynomials. [i]Laurentiu Panaitopol[/i]

2014 Taiwan TST Round 1, 1

Tags: absolute value , polynomial , inequalities , algebra

Let $f(x) = x^n + a_{n-2} x^{n-2} + a_{n-3}x^{n-3} + \dots + a_1x + a_0$ be a polynomial with real coefficients $(n \ge 2)$. Suppose all roots of $f$ are real. Prove that the absolute value of each root is at most $\sqrt{\frac{2(1-n)}n a_{n-2}}$.

1990 AMC 8, 7

Tags: absolute value

When three different numbers from the set $ \{-3,-2,-1, 4, 5\} $ are multiplied, the largest possible product is $ \text{(A)}\ 10\qquad\text{(B)}\ 20\qquad\text{(C)}\ 30\qquad\text{(D)}\ 40\qquad\text{(E)}\ 60 $

PEN C Problems, 6

Tags: algebra , quadratic residue , absolute value , polynomial

Let $a, b, c$ be integers and let $p$ be an odd prime with \[p \not\vert a \;\; \text{and}\;\; p \not\vert b^{2}-4ac.\] Show that \[\sum_{k=1}^{p}\left( \frac{ak^{2}+bk+c}{p}\right) =-\left( \frac{a}{p}\right).\]

2007 National Olympiad First Round, 18

Tags: absolute value

How many integers $n$ are there such that $n^3+8$ has at most $3$ positive divisors? $ \textbf{(A)}\ 4 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 1 \qquad\textbf{(E)}\ \text{None of the above} $

2004 National Olympiad First Round, 35

Tags: absolute value

We are placing $n$ integers whose sum is $94$ over a circle such that each number is equal to the absolute value of the difference of (clockwise) next two numbers. What is the largest $n$ that makes such placing possible? $ \textbf{(A)}\ 188 \qquad\textbf{(B)}\ 186 \qquad\textbf{(C)}\ 141 \qquad\textbf{(D)}\ 100 \qquad\textbf{(E)}\ 47 $

2022 Iran MO (3rd Round), 3

Tags: algebra , complex number , absolute value

Prove that for natural number $n$ it's possible to find complex numbers $\omega_1,\omega_2,\cdots,\omega_n$ on the unit circle that $$\left\lvert\sum_{j=1}^{n}\omega_j\right\rvert=\left\lvert\sum_{j=1}^{n}\omega_j^2\right\rvert=n-1$$ iff $n=2$ occurs.

2012 Math Prize For Girls Problems, 16

Tags: geometry , conic , ellipse , absolute value , analytic geometry

Say that a complex number $z$ is [i]three-presentable[/i] if there is a complex number $w$ of absolute value $3$ such that $z = w - \frac{1}{w}$. Let $T$ be the set of all three-presentable complex numbers. The set $T$ forms a closed curve in the complex plane. What is the area inside $T$?

2023 Junior Balkan Team Selection Tests - Romania, P4

Tags: algebra , absolute value

Let be $a$ be positive real number. Prove that there are no real numbers $b$ and $c$, with $b < c$, so that for any distinct numbers $x, y \in (b, c)$ we have $|\frac{x+y} {x-y}| \leq a$.

1993 Poland - First Round, 2

Tags: function , absolute value

The sequence of functions $f_0,f_1,f_2,...$ is given by the conditions: $f_0(x) = |x|$ for all $x \in R$ $f_{n+1}(x) = |f_n(x)-2|$ for $n=0,1,2,...$ and all $x \in R$. For each positive integer $n$, solve the equation $f_n(x)=1$.