Olympiad problems

2001 Tournament Of Towns, 7

The vertices of a triangle have coordinates $(x_1,y_1)$, $(x_2,y_2)$ and $(x_3,y_3)$. For any integers $h$ and $k$, not both 0, both triangles whose vertices have coordinates $(x_1+h,y_1+k),(x_2+h,y_2+k)$ and $(x_3+h,y_3+k)$ has no common interior points with the original triangle. (a) Is it possible for the area of this triangle to be greater than $\tfrac{1}{2}$? (b) What is the maximum area of this triangle?

2009 District Olympiad, 4

Tags: complex number , geometry , absolute value

[b]a)[/b] Let $ z_1,z_2,z_3 $ be three complex numbers of same absolute value, and $ 0=z_1+z_2+z_3. $ Show that these represent the affixes of an equilateral triangle. [b]b)[/b] Find all subsets formed by roots of the same unity that have the property that any three elements of every such, doesn’t represent the vertices of an equilateral triangle.

2009 Ukraine National Mathematical Olympiad, 2

Tags: function , absolute value , algebra

Find all functions $f : \mathbb Z \to \mathbb Z$ such that \[f (n |m|) + f (n(|m| +2)) = 2f (n(|m| +1)) \qquad \forall m,n \in \mathbb Z.\] [b]Note.[/b] $|x|$ denotes the absolute value of the integer $x.$

2007 Federal Competition For Advanced Students, Part 2, 1

Tags: polynomial , pigeonhole principle , absolute value , algebra

Let $ M$ be the set of all polynomials $ P(x)$ with pairwise distinct integer roots, integer coefficients and all absolut values of the coefficients less than $ 2007$. Which is the highest degree among all the polynomials of the set $ M$?

2019 Philippine TST, 3

Tags: quadratic , inequalities , functional inequality , function , absolute value , constant , algebra

Determine all ordered triples $(a, b, c)$ of real numbers such that whenever a function $f : \mathbb{R} \to \mathbb{R}$ satisfies $$|f(x) - f(y)| \le a(x - y)^2 + b(x - y) + c$$ for all real numbers $x$ and $y$, then $f$ must be a constant function.

1987 Romania Team Selection Test, 1

Tags: matrix , absolute value , linear algebra

Let $a,b,c$ be distinct real numbers such that $a+b+c>0$. Let $M$ be the set of $3\times 3$ matrices with the property that each line and each column contain all given numbers $a,b,c$. Find $\{\max \{ \det A \mid A \in M \}$ and the number of matrices which realise the maximum value. [i]Mircea Becheanu[/i]

2014 Harvard-MIT Mathematics Tournament, 23

Tags: probability , expected value , absolute value

Let $S=\{-100,-99,-98,\ldots,99,100\}$. Choose a $50$-element subset $T$ of $S$ at random. Find the expected number of elements of the set $\{|x|:x\in T\}$.

1954 Moscow Mathematical Olympiad, 285

Tags: algebra , quadratic , trinomial , absolute value , root

The absolute values of all roots of the quadratic equation $x^2+Ax+B = 0$ and $x^2+Cx+D = 0$ are less then $1$. Prove that so are absolute values of the roots of the quadratic equation $x^2 + \frac{A + C}{2} x + \frac{B + D}{2} = 0$.

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 \[\displaystyle\sum_{i=1}^n \sin \alpha_i \ge 1\]

2006 Austrian-Polish Competition, 8

Tags: induction , absolute value , algebra

Let $A\subset \{x|0\le x<1\}$ with the following properties: 1. $A$ has at least 4 members. 2. For all pairwise different $a,b,c,d\in A$, $ab+cd\in A$ holds. Prove: $A$ has infinetly many members.

Today's calculation of integrals, 852

Tags: calculus , integration , trigonometry , polynomial , absolute value , calculus computations , algebra

Let $f(x)$ be a polynomial. Prove that if $\int_0^1 f(x)g_n(x)\ dx=0\ (n=0,\ 1,\ 2,\ \cdots)$, then all coefficients of $f(x)$ are 0 for each case as follows. (1) $g_n(x)=(1+x)^n$ (2) $g_n(x)=\sin n\pi x$ (3) $g_n(x)=e^{nx}$

2007 IMAR Test, 3

Tags: absolute value , number theory

Prove that $ N\geq 2n \minus{} 2$ integers, of absolute value not higher than $ n > 2$, and of absolute value of their sum $ S$ less than $ n \minus{} 1,$ there exist some of sum $ 0.$ Show that for $ |S| \equal{} n \minus{} 1$ this is not anymore true, and neither for $ N \equal{} 2n \minus{} 3$ (when even for $ |S| \equal{} 1$ this is not anymore true).

1975 Czech and Slovak Olympiad III A, 4

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

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 $xy.$

1970 Miklós Schweitzer, 4

Tags: modular arithmetic , quadratic , function , absolute value , number theory

If $ c$ is a positive integer and $ p$ is an odd prime, what is the smallest residue (in absolute value) of \[ \sum_{n=0}^{\frac{p-1}{2}} \binom{2n}{n}c^n \;(\textrm{mod}\;p\ ) \ ?\] J. Suranyi

1975 Putnam, A2

Tags: analytic geometry , algebra , absolute value

Describe the region $R$ consisting of the points $(a,b)$ of the cartesian plane for which both (possibly complex) roots of the polynomial $z^2+az+b$ have absolute value smaller than $1$.

2016 AMC 12/AHSME, 3

Tags: absolute value

Let $x=-2016$. What is the value of $\left| \ \bigl \lvert { \ \lvert x\rvert -x }\bigr\rvert -|x|{\frac{}{}}^{}_{}\right|-x$? $\textbf{(A)}\ -2016\qquad\textbf{(B)}\ 0\qquad\textbf{(C)}\ 2016\qquad\textbf{(D)}\ 4032\qquad\textbf{(E)}\ 6048$

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 $1.$

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.

1979 AMC 12/AHSME, 27

Tags: probability , quadratic , search , absolute value

An ordered pair $( b , c )$ of integers, each of which has absolute value less than or equal to five, is chosen at random, with each such ordered pair having an equal likelihood of being chosen. What is the probability that the equation $x^ 2 + bx + c = 0$ will [i]not[/i] have distinct positive real roots? $\textbf{(A) }\frac{106}{121}\qquad\textbf{(B) }\frac{108}{121}\qquad\textbf{(C) }\frac{110}{121}\qquad\textbf{(D) }\frac{112}{121}\qquad\textbf{(E) }\text{none of these}$

1956 Moscow Mathematical Olympiad, 342

Tags: sequence , absolute value , algebra

Given three numbers $x, y, z$ denote the absolute values of the differences of each pair by $x_1,y_1, z_1$. From $x_1, y_1, z_1$ form in the same fashion the numbers $x_2, y_2, z_2$, etc. It is known that $x_n = x,y_n = y, z_n = z$ for some $n$. Find $y$ and $z$ if $x = 1$.

2023 Ecuador NMO (OMEC), 1

Tags: absolute value , algebra

Find all reals $(a, b, c)$ such that $$\begin{cases}a^2+b^2+c^2=1\\ |a+b|=\sqrt{2}\end{cases}$$

2003 Putnam, 3

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

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

2010 Romanian Masters In Mathematics, 2

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

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]

2012 Poland - Second Round, 3

Tags: least common multiple , vector , absolute value , 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.

1958 AMC 12/AHSME, 39

Tags: quadratic , function , absolute value

We may say concerning the solution of \[ |x|^2 \plus{} |x| \minus{} 6 \equal{} 0 \] that: $ \textbf{(A)}\ \text{there is only one root}\qquad \textbf{(B)}\ \text{the sum of the roots is }{\plus{}1}\qquad \textbf{(C)}\ \text{the sum of the roots is }{0}\qquad \\ \textbf{(D)}\ \text{the product of the roots is }{\plus{}4}\qquad \textbf{(E)}\ \text{the product of the roots is }{\minus{}6}$