Olympiad problems

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 $ABC$.

1993 Moldova Team Selection Test, 3

Tags: function , parameterization

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a function defined as the maximum of a finite number of functions $g:\mathbb{R}\rightarrow\mathbb{R}$ of the form $g(x)=C\cdot10^{-|x-d|}$ (with different values of parameters $d{}$ and $C>0$). For real numbers $a<b$ we have $f(a)=f(b)$. Prove that on the segment $[a;b]$ the sum of legnths of segments on which $f$ is increasing is equal to the sum of legnths of segments on which $f$ is decreasing.

1967 IMO Longlists, 28

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 $x.$

1986 ITAMO, 3

Tags: probability , parameterization

Two numbers are randomly selected from interval $I = [0, 1]$. Given $\alpha \in I$, what is the probability that the smaller of the two numbers does not exceed $\alpha$? Is the answer $(100 \alpha)$%, it just seems too easy. :|

2006 Moldova National Olympiad, 10.4

Tags: parameterization , algebra , logarithm

Find all real values of the real parameter $a$ such that the equation \[ 2x^{2}-6ax+4a^{2}-2a-2+\log_{2}(2x^{2}+2x-6ax+4a^{2})= \] \[ =\log_{2}(x^{2}+2x-3ax+2a^{2}+a+1). \] has a unique solution.

2022 German National Olympiad, 1

Tags: system of equations , quadratic equation , parameterization , algebra

Determine all real numbers $a$ for which the system of equations \begin{align*} 3x^2+2y^2+2z^2&=a\\ 4x^2+4y^2+5z^2&=1-a \end{align*} has at least one solution $(x,y,z)$ in the real numbers.

VII Soros Olympiad 2000 - 01, 11.2

Tags: parameterization , algebra , trinomial

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

2007 Today's Calculation Of Integral, 174

Tags: calculus , integration , geometry , parameterization , calculus computations

Let $a$ be a positive number. Assume that the parameterized curve $C: \ x=t+e^{at},\ y=-t+e^{at}\ (-\infty <t< \infty)$ is touched to $x$ axis. (1) Find the value of $a.$ (2) Find the area of the part which is surrounded by two straight lines $y=0, y=x$ and the curve $C.$

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?

1966 IMO Shortlist, 50

Tags: trigonometric equation , parameterization , trigonometry , parabola

Solve the equation $\frac{1}{\sin x}+\frac{1}{\cos x}=\frac 1p$ where $p$ is a real parameter. Discuss for which values of $p$ the equation has at least one real solution and determine the number of solutions in $[0, 2\pi)$ for a given $p.$

2008 Iran MO (3rd Round), 4

Tags: algebra , polynomial , perimeter , analytic geometry , function , parameterization , geometry

=A subset $ S$ of $ \mathbb R^2$ is called an algebraic set if and only if there is a polynomial $ p(x,y)\in\mathbb R[x,y]$ such that \[ S \equal{} \{(x,y)\in\mathbb R^2|p(x,y) \equal{} 0\} \] Are the following subsets of plane an algebraic sets? 1. A square [img]http://i36.tinypic.com/28uiaep.png[/img] 2. A closed half-circle [img]http://i37.tinypic.com/155m155.png[/img]

1963 IMO Shortlist, 4

Tags: parameterization , algebra , system of equations

Find all solutions $x_1, x_2, x_3, x_4, x_5$ of the system \[ x_5+x_2=yx_1 \] \[ x_1+x_3=yx_2 \] \[ x_2+x_4=yx_3 \] \[ x_3+x_5=yx_4 \] \[ x_4+x_1=yx_5 \] where $y$ is a parameter.

1993 All-Russian Olympiad, 1

Tags: modular arithmetic , parameterization , quadratic , number theory

For integers $x$, $y$, and $z$, we have $(x-y)(y-z)(z-x)=x+y+z$. Prove that $27|x+y+z$.

2023 Israel National Olympiad, P4

Tags: algebra , system of equations , parameterization

For each positive integer $n$, find all triples $a,b,c$ of real numbers for which \[\begin{cases}a=b^n+c^n\\ b=c^n+a^n\\ c=a^n+b^n\end{cases}\]

1978 Czech and Slovak Olympiad III A, 2

Tags: inequalities , algebra , parameterization

Determine (at least one) pair of real numbers $k,q$ such that the inequality \[2\left|\sqrt{1-x^2}-kx-q\right|\le\sqrt2-1\] holds for all $x\in[0,1].$

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.

2014 AMC 12/AHSME, 25

Tags: parabola , calculus , derivative , vector , analytic geometry , parameterization , conic

The parabola $P$ has focus $(0,0)$ and goes through the points $(4,3)$ and $(-4,-3)$. For how many points $(x,y)\in P$ with integer coefficients is it true that $|4x+3y|\leq 1000$? $\textbf{(A) }38\qquad \textbf{(B) }40\qquad \textbf{(C) }42\qquad \textbf{(D) }44\qquad \textbf{(E) }46\qquad$

2003 USA Team Selection Test, 4

Tags: function , parameterization , induction , functional equation , strong induction , algebra

Let $\mathbb{N}$ denote the set of positive integers. Find all functions $f: \mathbb{N} \to \mathbb{N}$ such that \[ f(m+n)f(m-n) = f(m^2) \] for $m,n \in \mathbb{N}$.

MathLinks Contest 7th, 4.3

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

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

MathLinks Contest 7th, 5.3

Tags: inequalities , symmetry , function , calculus , derivative , parameterization , logarithm

If $ a\geq b\geq c\geq d > 0$ such that $ abcd\equal{}1$, then prove that \[ \frac 1{1\plus{}a} \plus{} \frac 1{1\plus{}b} \plus{} \frac 1{1\plus{}c} \geq \frac {3}{1\plus{}\sqrt[3]{abc}}.\]

2014 ELMO Shortlist, 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]

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]

1972 IMO Longlists, 2

Tags: parameterization , system of equations , algebra

Find all real values of the parameter $a$ for which the system of equations \[x^4 = yz - x^2 + a,\] \[y^4 = zx - y^2 + a,\] \[z^4 = xy - z^2 + a,\] has at most one real solution.

2009 USAMO, 1

Tags: geometry , analytic geometry , circumcircle , symmetry , parameterization

Given circles $ \omega_1$ and $ \omega_2$ intersecting at points $ X$ and $ Y$, let $ \ell_1$ be a line through the center of $ \omega_1$ intersecting $ \omega_2$ at points $ P$ and $ Q$ and let $ \ell_2$ be a line through the center of $ \omega_2$ intersecting $ \omega_1$ at points $ R$ and $ S$. Prove that if $ P, Q, R$ and $ S$ lie on a circle then the center of this circle lies on line $ XY$.

1986 Traian Lălescu, 2.1

Tags: equation , algebra , parameterization , parametric equation

Find the real values $ m\in\mathbb{R} $ such that all solutions of the equation $$ 1=2mx(2x-1)(2x-2)(2x-3) $$ are real.