Olympiad problems

2018 Purple Comet Problems, 13

Suppose $x$ and $y$ are nonzero real numbers simultaneously satisfying the equations $x + \frac{2018}{y}= 1000$ and $ \frac{9}{x}+ y = 1$. Find the maximum possible value of $x + 1000y$.

2017 Istmo Centroamericano MO, 4

Tags: diophantine , diophantine equation , inequalities , number theory , system of equations

Suppose that $a$ and $ b$ are distinct positive integers satisfying $20a + 17b = p$ and $17a + 20b = q$ for certain primes $p$ and $ q$. Determine the minimum value of $p + q$.

1991 Austrian-Polish Competition, 8

Tags: system of equations , number theory

Consider the system of congruences $$\begin{cases} xy \equiv - 1 \,\, (mod z) \\ yz \equiv 1 \, \, (mod x) \\zx \equiv 1 \, \, (mod y)\end {cases}$$ Find the number of triples $(x,y, z) $ of distinct positive integers satisfying this system such that one of the numbers $x,y, z$ equals $19$.

II Soros Olympiad 1995 - 96 (Russia), 10.4

Tags: algebra , system of equations

Solve system of equations $$\begin{cases} x+\dfrac{x+y}{x^2+y^2}=1 \\ x+\dfrac{x-y}{x^2+y^2}=2 \end{cases}$$

1967 IMO Longlists, 5

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

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

2011 Indonesia TST, 1

Tags: system of equations , algebra

Find all $4$-tuple of real numbers $(x, y, z, w)$ that satisfy the following system of equations: $$x^2 + y^2 + z^2 + w^2 = 4$$ $$\frac{1}{x^2} +\frac{1}{y^2} +\frac{1}{z^2 }+\frac{1}{w^2} = 5 -\frac{1}{(xyzw)^2}$$

1942 Eotvos Mathematical Competition, 2

Tags: system , system of equations , diophantine , number theory

Let $a, b, c $and $d$ be integers such that for all integers m and n, there exist integers $x$ and $y$ such that $ax + by = m$, and $cx + dy = n$. Prove that $ad - bc = \pm 1$.

2014 Finnish National High School Mathematics, 1

Tags: system of equations , algebra

Determine the value of the expression $x^2 + y^2 + z^2$, if $x + y + z = 13$ , $xyz= 72$ and $\frac1x + \frac1y + \frac1z = \frac34$.

2016 Dutch BxMO TST, 2

Tags: system of equations , algebra

Determine all triples (x, y, z) of non-negative real numbers that satisfy the following system of equations $\begin{cases} x^2 - y = (z - 1)^2\\ y^2 - z = (x - 1)^2 \\ z^2 - x = (y -1)^2 \end{cases}$.

2012 Singapore Junior Math Olympiad, 4

Tags: algebra , system of equations

Determine the values of the positive integer $n$ for which the following system of equations has a solution in positive integers $x_1, x_2,...,, x_n$. Find all solutions for each such $n$. $$\begin{cases} x_1 + x_2 +...+ x_n = 16 \\ \\ \dfrac{1}{x_1} + \dfrac{1}{x_2} +...+ \dfrac{1}{x_n} = 1\end{cases}$$

2011 Morocco National Olympiad, 1

Tags: system of equations , algebra , linear algebra

Solve the following equation in $\mathbb{R}^+$ : \[\left\{\begin{matrix} \frac{1}{x}+\frac{1}{y}+\frac{1}{z}=2010\\ x+y+z=\frac{3}{670} \end{matrix}\right.\]

1992 Poland - First Round, 2

Tags: system of equations , algebra

Given is a natural number $n \geq 3$. Solve the system of equations: $\[ \begin{cases} \tan (x_1) + 3 \cot (x_1) &= 2 \tan (x_2) \\ \tan (x_2) + 3 \cot (x_2) &= 2 \tan (x_3) \\ & \dots \\ \tan (x_n) + 3 \cot (x_n) &= 2 \tan (x_1) \\ \end{cases} \]$

2021 AMC 10 Fall, 14

Tags: system of equations , absolute value , algebra

How many ordered pairs $(x,y)$ of real numbers satisfy the following system of equations? \begin{align*} x^2+3y&=9\\ (|x|+|y|-4)^2&=1\\ \end{align*} $\textbf{(A)}\: 1\qquad\textbf{(B)} \: 2\qquad\textbf{(C)} \: 3\qquad\textbf{(D)} \: 5\qquad\textbf{(E)} \: 7$

2014 Saudi Arabia IMO TST, 3

Tags: number theory , induction , system of equations , algebra

Show that it is possible to write a $n \times n$ array of non-negative numbers (not necessarily distinct) such that the sums of entries on each row and each column are pairwise distinct perfect squares.

2016 Junior Balkan Team Selection Tests - Moldova, 5

Tags: system , algebra , system of equations

Real numbers $a$ and $b$ satisfy the system of equations $$\begin{cases} a^3-a^2+a-5=0 \\ b^3-2b^2+2b+4=0 \end{cases}$$ Find the numerical value of the sum $a+ b$.

2008 ITest, 8

Tags: system of equations , algebra

The math team at Jupiter Falls Middle School meets twice a month during the Summer, and the math team coach, Mr. Fischer, prepares some Olympics-themed problems for his students. One of the problems Joshua and Alexis work on boils down to a system of equations: \begin{align*}2x+3y+3z&=8,\\3x+2y+3z&=808,\\3x+3y+2z&=80808.\end{align*} Their goal is not to find a solution $(x,y,z)$ to the system, but instead to compute the sum of the variables. Find the value of $x+y+z$.

1979 Romania Team Selection Tests, 3.

Tags: simultaneous equation , algebra , system of equations , inequalities

Let $a,b,c\in \mathbb{R}$ with $a^2+b^2+c^2=1$ and $\lambda\in \mathbb{R}_{>0}\setminus\{1\}$. Then for each solution $(x,y,z)$ of the system of equations: \[ \begin{cases} x-\lambda y=a,\\ y-\lambda z=b,\\ z-\lambda x=c. \end{cases} \] we have $\displaystyle x^2+y^2+z^2\leqslant \frac1{(\lambda-1)^2}$. [i]Radu Gologan[/i]

1994 China Team Selection Test, 2

Tags: arithmetic sequence , function , system of equations , combinatorics , algebra , ratio

An $n$ by $n$ grid, where every square contains a number, is called an $n$-code if the numbers in every row and column form an arithmetic progression. If it is sufficient to know the numbers in certain squares of an $n$-code to obtain the numbers in the entire grid, call these squares a key. [b]a.) [/b]Find the smallest $s \in \mathbb{N}$ such that any $s$ squares in an $n-$code $(n \geq 4)$ form a key. [b]b.)[/b] Find the smallest $t \in \mathbb{N}$ such that any $t$ squares along the diagonals of an $n$-code $(n \geq 4)$ form a key.

2013 Kosovo National Mathematical Olympiad, 3

Tags: analytic geometry , system of equations

Prove that solution of equation $y=x^2+ax+b$ and $x=y^2+cy+d$ it belong a circle.

2022 Azerbaijan National Mathematical Olympiad, 4

Tags: system of equations , system , algebra

Find all quadruplets $(x_1, x_2, x_3, x_4)$ of real numbers such that the next six equalities apply: $$\begin{cases} x_1 + x_2 = x^2_3 + x^2_4 + 6x_3x_4\\ x_1 + x_3 = x^2_2 + x^2_4 + 6x_2x_4\\ x_1 + x_4 = x^2_2 + x^2_3 + 6x_2x_3\\ x_2 + x_3 = x^2_1 + x^2_4 + 6x_1x_4\\ x_2 + x_4 = x^2_1 + x^2_3 + 6x_1x_3 \\ x_3 + x_4 = x^2_1 + x^2_2 + 6x_1x_2 \end{cases}$$

2013 Costa Rica - Final Round, 1

Tags: system of equations , algebra

Determine and justify all solutions $(x,y, z)$ of the system of equations: $x^2 = y + z$ $y^2 = x + z$ $z^2 = x + y$

1989 Greece National Olympiad, 1

Tags: algebra , system of equations

Let $a,b,c,d x,y,z, w$ be real numbers such that $$\begin{matrix} ax -by-c z-dw =0\\ b x +a y -d z +cw=0\\ c x+ d y +a z -b w=0\\ dx-c y+bz+aw=0 \end{matrix}$$ prove that $$a=b=c=d=0, \ \ or \ \ x=y=z=w=0$$

III Soros Olympiad 1996 - 97 (Russia), 11.3

Tags: algebra , system of equations

Find the greatest $a$ for which there is $b$ such that the system $$\begin{cases} y=x^4+a \\ x=\dfrac{1}{y^4}+b \end{cases}$$ has exactly two solutions.

2010 Albania National Olympiad, 4

Tags: algorithm , system of equations , number theory , algebra

The sequence of Fibonnaci's numbers if defined from the two first digits $f_1=f_2=1$ and the formula $f_{n+2}=f_{n+1}+f_n$, $\forall n \in N$. [b](a)[/b] Prove that $f_{2010} $ is divisible by $10$. [b](b)[/b] Is $f_{1005}$ divisible by $4$? Albanian National Mathematical Olympiad 2010---12 GRADE Question 4.

2009 Greece JBMO TST, 4

Tags: algebra , minimum , sum , system of equations

Find positive real numbers $x,y,z$ that are solutions of the system $x+y+z=xy+yz+zx$ and $xyz=1$ , and have the smallest possible sum.