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

2010 Putnam, B1
Tags: vector , function , parameterization , geometric sequence , limit , inequalities
Is there an infinite sequence of real numbers $a_1,a_2,a_3,\dots$ such that \[a_1^m+a_2^m+a_3^m+\cdots=m\] for every positive integer $m?$

2002 Vietnam Team Selection Test, 2
Tags: diophantine equation , parameterization , derivative , calculus , polynomial , algebra
Find all polynomials $P(x)$ with integer coefficients such that the polynomial \[ Q(x)=(x^2+6x+10) \cdot P^2(x)-1 \] is the square of a polynomial with integer coefficients.

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

2024 Baltic Way, 4
Tags: parameterization , inequalities , algebra
Find the largest real number $\alpha$ such that, for all non-negative real numbers $x$, $y$ and $z$, the following inequality holds: \[ (x+y+z)^3 + \alpha (x^2z + y^2x + z^2y) \geq \alpha (x^2y + y^2z + z^2x). \]

1986 Traian Lălescu, 2.1
Tags: equation , parametric equation , parameterization , algebra
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.

1978 Vietnam National Olympiad, 2
Tags: parameterization , algebra , system of equations
Find all values of the parameter $m$ such that the equations $x^2 = 2^{|x|} + |x| - y - m = 1 - y^2$ have only one root.

2013 Greece Team Selection Test, 2
Tags: parametric equation , analytic geometry , parameterization
For the several values of the parameter $m\in \mathbb{N^{*}}$,find the pairs of integers $(a,b)$ that satisfy the relation $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{[a,m]+[b,m]}{(a+b)m}=\frac{10}{11}$, and,moreover,on the Cartesian plane $Oxy$ the lie in the square $D=\{(x,y):1\leq x\leq 36,1\leq y\leq 36\}$. [i][u]Note:[/u]$[k,l]$ denotes the least common multiple of the positive integers $k,l$.[/i]

1996 Vietnam Team Selection Test, 2
Tags: parameterization , combinatorics , matrix , linear algebra
There are some people in a meeting; each doesn't know at least 56 others, and for any pair, there exist a third one who knows both of them. Can the number of people be 65?

1966 IMO Shortlist, 50
Tags: parabola , parameterization , trigonometry , trigonometric equation
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 $p.$

1971 IMO Longlists, 30
Tags: system of equations , algebra , parameterization
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: parameterization , function , integration , calculus , logarithm , geometry , analytic geometry
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.

2009 AIME Problems, 10
Tags: matrix , parameterization , function , linear algebra
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$.

1966 IMO Longlists, 31
Tags: equation , function , parameterization , 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?

1997 Finnish National High School Mathematics Competition, 1
Tags: quadratic , product of roots , parameterization , logarithm , polynomial , algebra
Determine the real numbers $a$ such that the equation $a 3^x + 3^{-x} = 3$ has exactly one solution $x.$

2008 District Olympiad, 3
Tags: euler , function , parameterization , algebra
For any real $ a$ define $ f_a : \mathbb{R} \rightarrow \mathbb{R}^2$ by the law $ f_a(t) \equal{} \left( \sin(t), \cos(at) \right)$. a) Prove that $ f_{\pi}$ is not periodic. b) Determine the values of the parameter $ a$ for which $ f_a$ is periodic. [b]Remark[/b]. L. Euler proved in $ 1737$ that $ \pi$ is irrational.

2010 Contests, 2
Tags: rearrangement inequality , function , parameterization , induction , algebra , inequalities
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]

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 $$\{a_j-a_i|1\le i<j\le n\}=\left\{1,2,\ldots,\frac{n(n-1)}2\right\}.$$

2018 Belarusian National Olympiad, 11.1
Tags: parameterization , functional equation , algebra
Find all real numbers $a$ for which there exists a function $f$ defined on the set of all real numbers which takes as its values all real numbers exactly once and satisfies the equality $$ f(f(x))=x^2f(x)+ax^2 $$ for all real $x$.

1987 Greece National Olympiad, 3
Tags: parameterization , algebra , inequalities
Solve for real values of parameter $a$, the inequality : $$\sqrt{a+x}+ \sqrt{a-x}>a , \ \ x\in\mathbb{R}$$

2013 Greece Team Selection Test, 2
Tags: parametric equation , parameterization , analytic geometry
For the several values of the parameter $m\in \mathbb{N^{*}}$,find the pairs of integers $(a,b)$ that satisfy the relation $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{[a,m]+[b,m]}{(a+b)m}=\frac{10}{11}$, and,moreover,on the Cartesian plane $Oxy$ the lie in the square $D=\{(x,y):1\leq x\leq 36,1\leq y\leq 36\}$. [i][u]Note:[/u]$[k,l]$ denotes the least common multiple of the positive integers $k,l$.[/i]

2017 District Olympiad, 2
Tags: equation , parameterization , diophantine
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. $

2009 Indonesia TST, 2
Tags: logarithm , parameterization , inequalities , algebra
Find the value of real parameter $ a$ such that $ 2$ is the smallest integer solution of \[ \frac{x\plus{}\log_2 (2^x\minus{}3a)}{1\plus{}\log_2 a} >2.\]

2017 Bosnia And Herzegovina - Regional Olympiad, 1
Tags: parameterization , logarithm , algebra , inequalities
In terms of real parameter $a$ solve inequality: $\log _{a} {x} + \mid a+\log _{a} {x} \mid \cdot \log _{\sqrt{x}} {a} \geq a\log _{x} {a}$ in set of real numbers

2014 Ukraine Team Selection Test, 9
Tags: number theory , diophantine , prime number , diophantine equation , parameterization , system of equations
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 $x, y, z$.

1976 Dutch Mathematical Olympiad, 4
Tags: parameter , equation , parameterization , trinomial , algebra
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 $a$?