Olympiad problems

VII Soros Olympiad 2000 - 01, 11.2

For all valid values of $a, b$, and $c$, solve the equation $$\frac{a (x-b) (x-c) }{(a-b) (a-c)} + \frac{b (x-c) (x-a)}{(b-c) (b-a)} +\frac{c (x-a) (x-b) }{(c-a ) (c-b)} = x^2$$

2013 Kosovo National Mathematical Olympiad, 4

Tags: parameterization

Find all value of parameter $a$ such that equations $x^2-ax+1=0$ and $x^2-x+a=0$ have at least one same solution. For this value $a$ find same solution of this equations(real or imaginary).

2012 Graduate School Of Mathematical Sciences, The Master Course, Kyoto University, 4

Tags: parameterization , calculus , integration , logarithm , real analysis

Let $\alpha,\ \beta$ be real numbers. Find the ranges of $\alpha,\ \beta$ such that the improper integral $\int_1^{\infty} \frac{x^{\alpha}\ln x}{(1+x)^{\beta}}$ converges.

2007 Indonesia TST, 2

Tags: parameterization , algebra

Solve the equation \[ x\plus{}a^3\equal{}\sqrt[3]{a\minus{}x}\] where $ a$ is a real parameter.

2013 USAJMO, 6

Tags: parameterization , inequalities

Find all real numbers $x,y,z\geq 1$ satisfying \[\min(\sqrt{x+xyz},\sqrt{y+xyz},\sqrt{z+xyz})=\sqrt{x-1}+\sqrt{y-1}+\sqrt{z-1}.\]

2014 Ukraine Team Selection Test, 9

Tags: number theory , prime number , parameterization , diophantine , system of equations , diophantine equation

Let $m, n$ be odd prime numbers. Find all pairs of integers numbers $a, b$ for which the system of equations: $x^m+y^m+z^m=a$, $x^n+y^n+z^n=b$ has many solutions in integers $x, y, z$.

2017 District Olympiad, 2

Tags: diophantine , parameterization , equation

Let $ E(x,y)=\frac{x}{y} +\frac{x+1}{y+1} +\frac{x+2}{y+2} . $ [b]a)[/b] Solve in $ \mathbb{N}^2 $ the equation $ E(x,y)=3. $ [b]b)[/b] Show that there are infinitely many natural numbers $ n $ such that the equation $ E(x,y)=n $ has at least one solution in $ \mathbb{N}^2. $

2010 Contests, 2

Tags: function , parameterization , induction , inequalities , rearrangement inequality , algebra

Let $n$ be a positive integer number and let $a_1, a_2, \ldots, a_n$ be $n$ positive real numbers. Prove that $f : [0, \infty) \rightarrow \mathbb{R}$, defined by \[f(x) = \dfrac{a_1 + x}{a_2 + x} + \dfrac{a_2 + x}{a_3 + x} + \cdots + \dfrac{a_{n-1} + x}{a_n + x} + \dfrac{a_n + x}{a_1 + x}, \] is a decreasing function. [i]Dan Marinescu et al.[/i]

1967 IMO Shortlist, 4

Tags: parameterization , calculus , trigonometry , algebra , trigonometric identities

Find values of the parameter $u$ for which the expression \[y = \frac{ \tan(x-u) + \tan(x) + \tan(x+u)}{ \tan(x-u)\tan(x)\tan(x+u)}\] does not depend on $x.$

2013 ELMO Shortlist, 1

Tags: parameterization , number theory , easy problem.

Find all ordered triples of non-negative integers $(a,b,c)$ such that $a^2+2b+c$, $b^2+2c+a$, and $c^2+2a+b$ are all perfect squares. [i]Proposed by Matthew Babbitt[/i]

MathLinks Contest 7th, 4.3

Tags: inequalities , function , parameterization , polynomial , vieta , cauchy inequality , algebra

Let $ a,b,c$ be positive real numbers such that $ ab\plus{}bc\plus{}ca\equal{}3$. Prove that \[ \frac 1{1\plus{}a^2(b\plus{}c)} \plus{} \frac 1{1\plus{}b^2(c\plus{}a)} \plus{} \frac 1 {1\plus{}c^2(a\plus{}b) } \leq \frac 3 {1\plus{}2abc} .\]

2014 Contests, 3

Tags: parameterization , inequalities

Let $a,b,c,d,e,f$ be positive real numbers. Given that $def+de+ef+fd=4$, show that \[ ((a+b)de+(b+c)ef+(c+a)fd)^2 \geq\ 12(abde+bcef+cafd). \][i]Proposed by Allen Liu[/i]

2007 Kurschak Competition, 1

Tags: parameterization , combinatorics

We have placed $n>3$ cards around a circle, facing downwards. In one step we may perform the following operation with three consecutive cards. Calling the one on the center $B$, the two on the ends $A$ and $C$, we put card $C$ in the place of $A$, then move $A$ and $B$ to the places originally occupied by $B$ and $C$, respectively. Meanwhile, we flip the cards $A$ and $B$. Using a number of these steps, is it possible to move each card to its original place, but facing upwards?

2024 CAPS Match, 5

Tags: algebra , functional equation , parameterization

Let $\alpha\neq0$ be a real number. Determine all functions $f:\mathbb R\to\mathbb R$ such that \[f\left(x^2+y^2\right)=f(x-y)f(x+y)+\alpha yf(y)\] holds for all $x, y\in\mathbb R.$

2007 Indonesia TST, 2

Tags: parameterization , algebra

Solve the equation \[ x\plus{}a^3\equal{}\sqrt[3]{a\minus{}x}\] where $ a$ is a real parameter.

2009 Italy TST, 2

Tags: parameterization , geometry

$ABC$ is a triangle in the plane. Find the locus of point $P$ for which $PA,PB,PC$ form a triangle whose area is equal to one third of the area of triangle $ABC$.

1975 Czech and Slovak Olympiad III A, 4

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

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 $xy.$

1956 Moscow Mathematical Olympiad, 332

Tags: algebra , system of equations , parameterization

Prove that the system of equations $\begin{cases} x_1 - x_2 = a \\ x_3 - x_4 = b \\ x_1 + x_2 + x_3 + x_4 = 1\end{cases}$ has at least one solution in positive numbers ($x_1 ,x_2 ,x_3 ,x_4>0$) if and only if $|a| + |b| < 1$.

1957 Czech and Slovak Olympiad III A, 1

Tags: equation , parametric equation , algebra , parameterization

Find all real numbers $p$ such that the equation $$\sqrt{x^2-5p^2}=px-1$$ has a root $x=3$. Then, solve the equation for the determined values of $p$.

2007 Bulgarian Autumn Math Competition, Problem 8.1

Tags: algebra , system of equations , parameter , parameterization

Determine all real $a$, such that the solutions to the system of equations $\begin{cases} \frac{3x-5}{3}+\frac{3x+5}{4}\geq \frac{x}{7}-\frac{1}{15}\\ (2x-a)^3+(2x+a)(1-4x^2)+16x^2a-6x^2a+a^3\leq 2a^2+a \end{cases}$ form an interval with length $\frac{32}{225}$.

2009 AIME Problems, 10

Tags: function , parameterization , linear algebra , matrix

The Annual Interplanetary Mathematics Examination (AIME) is written by a committee of five Martians, five Venusians, and five Earthlings. At meetings, committee members sit at a round table with chairs numbered from $ 1$ to $ 15$ in clockwise order. Committee rules state that a Martian must occupy chair $ 1$ and an Earthling must occupy chair $ 15$. Furthermore, no Earthling can sit immediately to the left of a Martian, no Martian can sit immediately to the left of a Venusian, and no Venusian can sit immediately to the left of an Earthling. The number of possible seating arrangements for the committee is $ N\cdot (5!)^3$. Find $ N$.

1971 IMO Longlists, 30

Tags: parameterization , system of equations , algebra

Prove that the system of equations \[2yz+x-y-z=a,\\ 2xz-x+y-z=a,\\ 2xy-x-y+z=a, \] $a$ being a parameter, cannot have five distinct solutions. For what values of $a$ does this system have four distinct integer solutions?

1976 Dutch Mathematical Olympiad, 4

Tags: algebra , parameter , parameterization , equation , trinomial

For $a,b, x \in R$ holds: $x^2 - (2a^2 + 4)x + a^2 + 2a + b = 0$. For which $b$ does this equation have at least one root between $0$ and $1$ for all $a$?

1978 Miklós Schweitzer, 10

Tags: parameterization , limit , probability and stats

Let $ Y_n$ be a binomial random variable with parameters $ n$ and $ p$. Assume that a certain set $ H$ of positive integers has a density and that this density is equal to $ d$. Prove the following statements: (a) $ \lim _{n \rightarrow \infty}P(Y_n\in H)\equal{}d$ if $ H$ is an arithmetic progression. (b) The previous limit relation is not valid for arbitrary $ H$. (c) If $ H$ is such that $ P(Y_n \in H)$ is convergent, then the limit must be equal to $ d$. [i]L. Posa[/i]

1995 Canada National Olympiad, 5

Tags: parameterization , trigonometry , algebra

$u$ is a real parameter such that $0<u<1$. For $0\le x \le u$, $f(x)=0$. For $u\le x \le n$, $f(x)=1-\left(\sqrt{ux}+\sqrt{(1-u)(1-x)}\right)^2$. The sequence $\{u_n\}$ is define recursively as follows: $u_1=f(1)$ and $u_n=f(u_{n-1})$ $\forall n\in \mathbb{N}, n\neq 1$. Show that there exists a positive integer $k$ for which $u_k=0$.