Olympiad problems

1979 IMO Shortlist, 15

Determine all real numbers a for which there exists positive reals $x_{1}, \ldots, x_{5}$ which satisfy the relations $ \sum_{k=1}^{5} kx_{k}=a,$ $ \sum_{k=1}^{5} k^{3}x_{k}=a^{2},$ $ \sum_{k=1}^{5} k^{5}x_{k}=a^{3}.$

2006 Petru Moroșan-Trident, 2

Tags: system of equations , algebra , monotone , equation

Solve in the positive real numbers the following system. $$ \left\{\begin{matrix} x^y=2^3\\y^z=3^4\\z^x=2^4 \end{matrix}\right. $$ [i]Aurel Ene[/i]

2017 Kosovo National Mathematical Olympiad, 2

Tags: algebra , system of equations

Solve the system of equations $x+y+z=\pi$ $\tan x\tan z=2$ $\tan y\tan z=18$

1953 AMC 12/AHSME, 43

Tags: system of equations , percent , algebra

If the price of an article is increased by percent $ p$, then the decrease in percent of sales must not exceed $ d$ in order to yield the same income. The value of $ d$ is: $ \textbf{(A)}\ \frac{1}{1\plus{}p} \qquad\textbf{(B)}\ \frac{1}{1\minus{}p} \qquad\textbf{(C)}\ \frac{p}{1\plus{}p} \qquad\textbf{(D)}\ \frac{p}{p\minus{}1} \qquad\textbf{(E)}\ \frac{1\minus{}p}{1\plus{}p}$

1995 IMO Shortlist, 4

Tags: function , inequalities , optimization , 133 , system of equations

Find all of the positive real numbers like $ x,y,z,$ such that : 1.) $ x \plus{} y \plus{} z \equal{} a \plus{} b \plus{} c$ 2.) $ 4xyz \equal{} a^2x \plus{} b^2y \plus{} c^2z \plus{} abc$ Proposed to Gazeta Matematica in the 80s by VASILE CÎRTOAJE and then by Titu Andreescu to IMO 1995.

1979 Swedish Mathematical Competition, 1

Tags: system , system of equations , algebra

Solve the equations: \[\left\{ \begin{array}{l} x_1 + 2 x_2 + 3 x_3 + \cdots + (n-1) x_{n-1} + n x_n = n \\ 2 x_1 + 3 x_2 + 4 x_3 + \cdots + n x_{n-1} + x_n = n-1 \\ 3 x_1 + 4 x_2 + 5 x_3 + \cdots + x_{n-1} + 2 x_n = n-2 \\ \cdots \cdots \cdots \cdots\cdot\cdots \cdots \cdots \cdots\cdot\cdots \cdots \cdots \cdots\cdot \\ (n-1) x_1 + n x_2 + x_3 + \cdots + (n-3) x_{n-1} + (n-2) x_n = 2 \\ n x_1 + x_2 + 2 x_3 + \cdots + (n-2) x_{n-1} + (n-1) x_n = 1 \end{array} \right. \]

1972 Bulgaria National Olympiad, Problem 2

Tags: algebra , equation , system of equations

Solve the system of equations: $$\begin{cases}\sqrt{\frac{y(t-y)}{t-x}-\frac4x}+\sqrt{\frac{z(t-z)}{t-x}-\frac4x}=\sqrt x\\\sqrt{\frac{z(t-z)}{t-y}-\frac4y}+\sqrt{\frac{x(t-x)}{t-y}-\frac4y}=\sqrt y\\\sqrt{\frac{x(t-x)}{t-z}-\frac4z}+\sqrt{\frac{y(t-y)}{t-z}-\frac4z}=\sqrt z\\x+y+z=2t\end{cases}$$ if the following conditions are satisfied: $0<x<t$, $0<y<t$, $0<z<t$. [i]H. Lesov[/i]

1955 Moscow Mathematical Olympiad, 298

Tags: system of equations , algebra

Find all real solutions of the system $\begin{cases} x^3 + y^3 = 1 \\ x^4 + y^4 = 1 \end{cases}$

2005 Denmark MO - Mohr Contest, 5

Tags: algebra , system , system of equations

For what real numbers $p$ has the system of equations $$\begin{cases} x_1^4+\dfrac{1}{x_1^2}=px_2 \\ \\ x_2^4+\dfrac{1}{x_2^2}=px_3 \\ ... \\ x_{2004}^4+\dfrac{1}{x_{2004}^2}=px_{2005} \\ \\ x_{2005}^4+\dfrac{1}{x_{2005}^2}=px_{1}\end{cases}$$ just one solution $(x_1,x_2,...,x_{2005})$, where $x_1,x_2,...,x_{2005}$ are real numbers?

1990 IMO Longlists, 24

Tags: inequalities , quadratic , system of equations , algebra

Find the real number $t$, such that the following system of equations has a unique real solution $(x, y, z, v)$: \[ \left\{\begin{array}{cc}x+y+z+v=0\\ (xy + yz +zv)+t(xz+xv+yv)=0\end{array}\right. \]

1966 IMO, 5

Tags: algebra , system of equations

Solve the system of equations \[ |a_1-a_2|x_2+|a_1-a_3|x_3+|a_1-a_4|x_4=1 \] \[ |a_2-a_1|x_1+|a_2-a_3|x_3+|a_2-a_4|x_4=1 \] \[ |a_3-a_1|x_1+|a_3-a_2|x_2+|a_3-a_4|x_4=1 \] \[ |a_4-a_1|x_1+|a_4-a_2|x_2+|a_4-a_3|x_3=1 \] where $a_1, a_2, a_3, a_4$ are four different real numbers.

1993 Nordic, 3

Tags: digit , positive integer , system of equations , number theory

Find all solutions of the system of equations $\begin{cases} s(x) + s(y) = x \\ x + y + s(z) = z \\ s(x) + s(y) + s(z) = y - 4 \end{cases}$ where $x, y$, and $z$ are positive integers, and $s(x), s(y)$, and $s(z)$ are the numbers of digits in the decimal representations of $x, y$, and $z$, respectively.

1961 IMO Shortlist, 1

Tags: system of equations , algebra

Solve the system of equations: \[ x+y+z=a \] \[ x^2+y^2+z^2=b^2 \] \[ xy=z^2 \] where $a$ and $b$ are constants. Give the conditions that $a$ and $b$ must satisfy so that $x,y,z$ are distinct positive numbers.

2022 New Zealand MO, 3

Tags: algebra , system of equations

Find all real numbers$ x$ and $y$ such that $$x^2 + y^2 = 2$$ $$\frac{x^2}{2 - y}+\frac{y^2}{2 - x}= 2.$$

2010 Junior Balkan Team Selection Tests - Romania, 3

Tags: algebra , system of equations

Let $n \ne 0$ be a natural number and integers $x_1, x_2, ...., x_n, y_1, y_2, ...., y_n$ with the properties: a) $x_1 + x_2 + .... + x_n = y_1 + y_2 + .... + y_n = 0,$ b) $x_1 ^ 2 + y_1 ^ 2 = x_2 ^ 2 + y_2 ^ 2 = .... = x_n ^ 2 + y_n ^ 2$. Show that $n$ is even.

1983 Austrian-Polish Competition, 5

Tags: algebra , parameter , system of equations

Let $a_1 < a_2 < a_3 < a_4$ be given positive numbers. Find all real values of parameter $c$ for which the system $$\begin{cases} x_1 + x_2 + x_3 + x_4 = 1 \\ a_1x_1 + a_2 x_2 + a_3x_3 + a_4 x_4 = c \\ a_1^2x_1 + a_2^2 x_2 + a_3^2x_3 + a_4^2 x_4 = c^2\end{cases}$$ has a solution in nonnegative $(x_1,x_2,x_3,x_4)$ real numbers.

1994 India Regional Mathematical Olympiad, 4

Tags: quadratic , complex number , algebra , system of equations

Solve the system of equations for real $x$ and $y$: \begin{eqnarray*} 5x \left( 1 + \frac{1}{x^2 + y^2}\right) &=& 12 \\ 5y \left( 1 - \frac{1}{x^2+y^2} \right) &=& 4 . \end{eqnarray*}

1965 IMO, 4

Tags: system of equations , algebra

Find all sets of four real numbers $x_1, x_2, x_3, x_4$ such that the sum of any one and the product of the other three is equal to 2.

2011 AMC 10, 12

Tags: algebra , system of equations

The players on a basketball team made some three-point shots, some two-point shots, and some one-point free throws. They scored as many points with two-point shots as with three-point shots. Their number of successful free throws was one more than their number of successful two-point shots. The team's total score was 61 points. How many free throws did they make? $\textbf{(A)}\,13 \qquad\textbf{(B)}\,14 \qquad\textbf{(C)}\,15 \qquad\textbf{(D)}\,16 \qquad\textbf{(E)}\,17$

1984 USAMO, 4

Tags: algebra , combinatorics , system of equations

A difficult mathematical competition consisted of a Part I and a Part II with a combined total of $28$ problems. Each contestant solved $7$ problems altogether. For each pair of problems, there were exactly two contestants who solved both of them. Prove that there was a contestant who, in Part I, solved either no problems or at least four problems.

II Soros Olympiad 1995 - 96 (Russia), 11.3

Tags: trigonometry , algebra , system of equations

Solve the system of equations $$\begin{cases} \sin \frac{\pi}{2}xy =z \\ \sin \frac{\pi}{2}yz =x \\ \sin \frac{\pi}{2}zx =y \end{cases} \,\,\, ?$$

2016 Balkan MO Shortlist, A5

Tags: inequalities , algebra , maximum , system of equations , minimum

Let $a, b,c$ and $d$ be real numbers such that $a + b + c + d = 2$ and $ab + bc + cd + da + ac + bd = 0$. Find the minimum value and the maximum value of the product $abcd$.

1967 IMO Shortlist, 5

Tags: linear algebra , matrix , algebra , system of equations , polynomial

Solve the system of equations: $ \begin{matrix} x^2 + x - 1 = y \\ y^2 + y - 1 = z \\ z^2 + z - 1 = x. \end{matrix} $

2018 German National Olympiad, 1

Tags: system of equations , algebra

Find all real numbers $x,y,z$ satisfying the following system of equations: \begin{align*} xy+z&=-30\\ yz+x &= 30\\ zx+y &=-18 \end{align*}

1983 IMO Longlists, 54

Tags: algebra , system of equations

Find all solutions of the following system of $n$ equations in $n$ variables: \[\begin{array}{c}\ x_1|x_1| - (x_1 - a)|x_1 - a| = x_2|x_2|,x_2|x_2| - (x_2 - a)|x_2 - a| = x_3|x_3|,\ \vdots \ x_n|x_n| - (x_n - a)|x_n - a| = x_1|x_1|\end{array}\] where $a$ is a given number.