Olympiad problems

2019 AIME Problems, 6

In a Martian civilization, all logarithms whose bases are not specified are assumed to be base $b$, for some fixed $b \geq 2$. A Martian student writes down \begin{align*}3 \log(\sqrt{x}\log x) &= 56\\\log_{\log (x)}(x) &= 54 \end{align*} and finds that this system of equations has a single real number solution $x > 1$. Find $b$.

2018 Purple Comet Problems, 9

Tags: system of equations , algebra

For some $k > 0$ the lines $50x + ky = 1240$ and $ky = 8x + 544$ intersect at right angles at the point $(m,n)$. Find $m + n$.

2016 Uzbekistan National Olympiad, 5

Tags: system of equations , algebra

Solve following system equations: \[\left\{ \begin{array}{c} 3x+4y=26\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ \sqrt{x^2+y^2-4x+2y+5}+\sqrt{x^2+y^2-20x-10y+125}=10\ \end{array} \right.\ \ \]

1966 AMC 12/AHSME, 39

Tags: algebra , system of equations , geometric sequence

In base $R_1$ the expanded fraction $F_1$ becomes $0.373737...$, and the expanded fraction $F_2$ becomes $0.737373...$. In base $R_2$ fraction $F_1$, when expanded, becomes $0.252525...$, while fraction $F_2$ becomes $0.525252...$. The sum of $R_1$ and $R_2$, each written in base ten is: $\text{(A)}\ 24 \qquad \text{(B)}\ 22\qquad \text{(C)}\ 21\qquad \text{(D)}\ 20\qquad \text{(E)}\ 19$

2020 Vietnam National Olympiad, 5

Tags: algebra , system of equations

Let a system of equations: $\left\{\begin{matrix}x-ay=yz\\y-az=zx\\z-ax=xy\end{matrix}\right.$ a)Find (x,y,z) if a=0 b)Prove that: the system have 5 distinct roots $\forall$a>1,a$\in\mathbb{R}.$

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 $\frac{32}{225}$.

1982 Polish MO Finals, 3

Tags: algebra , system of equations , parameter

Find all pairs of positive numbers $(x,y)$ which satisfy the system of equations $$\begin{cases} x^2 +y^2 = a^2 +b^2 \\ x^3 +y^3 = a^3 +b^3 \end{cases}$$ where $a$ and $b$ are given positive numbers.

2003 Czech And Slovak Olympiad III A, 1

Tags: system of equations , algebra

Solve the following system in the set of real numbers: $x^2 -xy+y^2 = 7$, $x^2y+xy^2 = -2$.

2012 Czech And Slovak Olympiad IIIA, 6

Tags: algebra , system of equations

In the set of real numbers solve the system of equations $x^4+y^2+4=5yz$ $y^4+z^2+4=5zx$ $z^4+x^2+4=5xy$

2009 JBMO Shortlist, 3

Tags: algebra , system of equations

Find all values of the real parameter $a$, for which the system $(|x| + |y| - 2)^2 = 1$ $y = ax + 5$ has exactly three solutions

1967 IMO Longlists, 46

Tags: algebra , trigonometric equation , trigonometry , system of equations

If $x,y,z$ are real numbers satisfying relations \[x+y+z = 1 \quad \textrm{and} \quad \arctan x + \arctan y + \arctan z = \frac{\pi}{4},\] prove that $x^{2n+1} + y^{2n+1} + z^{2n+1} = 1$ holds for all positive integers $n$.

2022 German National Olympiad, 1

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

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.

1976 Czech and Slovak Olympiad III A, 4

Tags: algebra , system of equations , linear algebra , linear equation

Determine all solutions of the linear system of equations \begin{align*} &x_1& &-x_2& &-x_3& &-\cdots& &-x_n& &= 2a, \\ -&x_1& &+3x_2& &-x_3& &-\cdots& &-x_n& &= 4a, \\ -&x_1& &-x_2& &+7x_3& &-\cdots& &-x_n& &= 8a, \\ &&&&&&&&&&&\vdots \\ -&x_1& &-x_2& &-x_3& &-\cdots& &+\left(2^n-1\right)x_n& &= 2^na, \end{align*} with unknowns $x_1,\ldots,x_n$ and a real parameter $a.$

2010 Contests, 1

Tags: algebra , pigeonhole principle , system of equations

Find all triples $(a,b,c)$ of positive real numbers satisfying the system of equations \[ a\sqrt{b}-c \&= a,\qquad b\sqrt{c}-a \&= b,\qquad c\sqrt{a}-b \&= c. \]

2003 Junior Balkan Team Selection Tests - Moldova, 6

Tags: system of equations , system , algebra

The real numbers x and у satisfy the equations $$\begin{cases} \sqrt{3x}\left(1+\dfrac{1}{x+y}\right)=2 \\ \\ \sqrt{7y}\left(1-\dfrac{1}{x+y}\right)=4\sqrt2 \end{cases}$$ Find the numerical value of the ratio $y/x$.

2018 Polish Junior MO Finals, 4

Tags: algebra , system of equations

Real numbers $a, b, c$ are not equal $0$ and are solution of the system: $\begin{cases} a^2 + a = b^2 \\ b^2 + b = c^2 \\ c^2 +c = a^2 \end{cases}$ Prove that $(a - b)(b - c)(c - a) = 1$.

1994 Vietnam National Olympiad, 1

Tags: algebra , system of equations , logarithm

Find all real solutions to \[x^{3}+3x-3+\ln{(x^{2}-x+1)}=y,\] \[y^{3}+3y-3+\ln{(y^{2}-y+1)}=z,\] \[z^{3}+3z-3+\ln{(z^{2}-z+1)}=x.\]

2008 JBMO Shortlist, 4

Tags: algebra , system of equations

Find all triples $(x,y,z)$ of real numbers that satisfy the system $\begin{cases} x + y + z = 2008 \\ x^2 + y^2 + z^2 = 6024^2 \\ \frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{1}{2008} \end{cases}$

2024 AMC 12/AHSME, 13

Tags: system of equations

There are real numbers $x,y,h$ and $k$ that satisfy the system of equations $$x^2 + y^2 - 6x - 8y = h$$ $$x^2 + y^2 - 10x + 4y = k$$ What is the minimum possible value of $h+k$? $ \textbf{(A) }-54 \qquad \textbf{(B) }-46 \qquad \textbf{(C) }-34 \qquad \textbf{(D) }-16 \qquad \textbf{(E) }16 \qquad $