This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 134

2004 Bundeswettbewerb Mathematik, 2
Tags: geometry , perimeter , geometric transformation , reflection , parameterization , function , integration
Let $k$ be a positive integer. In a circle with radius $1$, finitely many chords are drawn. You know that every diameter of the circle intersects at most $k$ of these chords. Prove that the sum of the lengths of all these chords is less than $k \cdot \pi$.

2007 Bulgarian Autumn Math Competition, Problem 12.1
Tags: trigonometric equation , parameter , parameterization , algebra
Determine the values of the real parameter $a$, such that the equation \[\sin 2x\sin 4x-\sin x\sin 3x=a\] has a unique solution in the interval $[0,\pi)$.

2003 Flanders Math Olympiad, 4
Tags: analytic geometry , parameterization , number theory , least common multiple , ratio , calculus , integration
Consider all points with integer coordinates in the carthesian plane. If one draws a circle with M(0,0) and a well-chose radius r, the circles goes through some of those points. (like circle with $r=2\sqrt2$ goes through 4 points) Prove that $\forall n\in \mathbb{N}, \exists r$ so that the circle with midpoint 0,0 and radius $r$ goes through at least $n$ points.

1988 Spain Mathematical Olympiad, 6
Tags: diophantine equation , parameterization , number theory
For all integral values of parameter $t$, find all integral solutions $(x,y)$ of the equation $$ y^2 = x^4-22x^3+43x^2+858x+t^2+10452(t+39)$$ .

2000 Spain Mathematical Olympiad, 1
Tags: polynomial , parameterization , quadratic , algebra
Consider the polynomials \[P(x) = x^4 + ax^3 + bx^2 + cx + 1 \quad \text{and} \quad Q(x) = x^4 + cx^3 + bx^2 + ax + 1.\] Find the conditions on the parameters $a, b, $c with $a\neq c$ for which $P(x)$ and $Q(x)$ have two common roots and, in such cases, solve the equations $P(x) = 0$ and $Q(x) = 0.$

2024 Moldova EGMO TST, 1
Tags: parabola , algebra , parameterization
Let $P$ be the set of all parabolas with the equation of the form $$y=(a-1)x^2-2(a+2)x+a+1$$ where $a$ is a real parameter and $a\neq1$. Prove that there exists an unique point $M$ such that all parabolas in $P$ pass through $M$.

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]

2012 Gulf Math Olympiad, 2
Tags: inequalities , parameterization , algebra
Prove that if $a, b, c$ are positive real numbers, then the least possible value of \[6a^3 + 9b^3 + 32c^3 + \frac{1}{4abc}\] is $6$. For which values of $a, b$ and $c$ is equality attained?

2022 Israel TST, 1
Tags: algebra , system of equations , parameterization
Let $n>1$ be an integer. Find all $r\in \mathbb{R}$ so that the system of equations in real variables $x_1, x_2, \dots, x_n$: \begin{align*} &(r\cdot x_1-x_2)(r\cdot x_1-x_3)\dots (r\cdot x_1-x_n)=\\ =&(r\cdot x_2-x_1)(r\cdot x_2-x_3)\dots (r\cdot x_2-x_n)=\\ &\qquad \qquad \qquad \qquad \vdots \\ =&(r\cdot x_n-x_1)(r\cdot x_n-x_2)\dots (r\cdot x_n-x_{n-1}) \end{align*} has a solution where the numbers $x_1, x_2, \dots, x_n$ are pairwise distinct.

2000 Slovenia National Olympiad, Problem 2
Tags: parameterization , absolute value
Find all real numbers $a$ for which the following equation has a unique real solution: $$|x-1|+|x-2|+\ldots+|x-99|=a.$$

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?

2010 Today's Calculation Of Integral, 607
Tags: calculus , integration , analytic geometry , geometry , function , parameterization , logarithm
On the coordinate plane, Let $C$ be the graph of $y=(\ln x)^2\ (x>0)$ and for $\alpha >0$, denote $L(\alpha)$ be the tangent line of $C$ at the point $(\alpha ,\ (\ln \alpha)^2).$ (1) Draw the graph. (2) Let $n(\alpha)$ be the number of the intersection points of $C$ and $L(\alpha)$. Find $n(\alpha)$. (3) For $0<\alpha <1$, let $S(\alpha)$ be the area of the region bounded by $C,\ L(\alpha)$ and the $x$-axis. Find $S(\alpha)$. 2010 Tokyo Institute of Technology entrance exam, Second Exam.

1966 IMO Shortlist, 31
Tags: function , parameterization , equation , algebra
Solve the equation $|x^2 -1|+ |x^2 - 4| = mx$ as a function of the parameter $m$. Which pairs $(x,m)$ of integers satisfy this equation?

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]

1991 Arnold's Trivium, 20
Tags: calculus , derivative , parameterization , function , inequalities
Find the derivative of the solution of the equation $\ddot{x} =x + A\dot{x}^2$, with initial conditions $x(0) = 1$, $\dot{x}(0) = 0$, with respect to the parameter $A$ for $A = 0$.

1966 IMO Shortlist, 18
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 $p.$

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 $\frac{32}{225}$.

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 $x, y\in\mathbb R.$

2006 JBMO ShortLists, 4
Tags: parameterization , number theory
Determine the biggest possible value of $ m$ for which the equation $ 2005x \plus{} 2007y \equal{} m$ has unique solution in natural numbers.

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 $a$?

1970 Czech and Slovak Olympiad III A, 3
Tags: parameterization , algebra , root , inequalities
Let $p>0$ be a given parameter. Determine all real $x$ such that \[\frac{1}{\,x+\sqrt{p-x^2\,}\,}+\frac{1}{\,x-\sqrt{p-x^2\,}\,}\ge\frac{1}{\,p\,}.\]

2024 Polish MO Finals, 4
Tags: system of equations , parameterization , algebra
Do there exist real numbers $a,b,c$ such that the system of equations \begin{align*} x+y+z&=a\\ x^2+y^2+z^2&=b\\ x^4+y^4+z^4&=c \end{align*} has infinitely many real solutions $(x,y,z)$?

PEN H Problems, 4
Tags: diophantine equation , ellipse , parameterization , greatest common divisor , conic
Find all pairs $(x, y)$ of positive rational numbers such that $x^{2}+3y^{2}=1$.

1991 Arnold's Trivium, 28
Tags: parameterization
Sketch the phase portrait and investigate its variation under variation of the small complex parameter $\epsilon$: \[\dot{z}=\epsilon z-(1+i)z|z|^2+\overline{z}^4\]

2002 Mongolian Mathematical Olympiad, Problem 3
Tags: parameterization
Find all positive integer $n$ for which there exist real number $a_1,a_2,\ldots,a_n$ such that $$\{a_j-a_i|1\le i<j\le n\}=\left\{1,2,\ldots,\frac{n(n-1)}2\right\}.$$