Olympiad problems

1961 AMC 12/AHSME, 26

For a given arithmetic series the sum of the first $50$ terms is $200$, and the sum of the next $50$ terms is $2700$. The first term in the series is: ${{ \textbf{(A)}\ -1221 \qquad\textbf{(B)}\ -21.5 \qquad\textbf{(C)}\ -20.5 \qquad\textbf{(D)}\ 3 }\qquad\textbf{(E)}\ 3.5 } $

2016 Germany National Olympiad (4th Round), 1

Tags: algebra , system of equations

Find all real pairs $(a,b)$ that solve the system of equation \begin{align*} a^2+b^2 &= 25, \\ 3(a+b)-ab &= 15. \end{align*} [i](German MO 2016 - Problem 1)[/i]

2013 India PRMO, 11

Tags: algebra , system of equations

Three real numbers $x, y, z$ are such that $x^2 + 6y = -17, y^2 + 4z = 1$ and $z^2 + 2x = 2$. What is the value of $x^2 + y^2 + z^2$?

2008 Czech-Polish-Slovak Match, 1

Tags: system of equations , algebra

Determine all triples $(x, y, z)$ of positive real numbers which satisfies the following system of equations \[2x^3=2y(x^2+1)-(z^2+1), \] \[ 2y^4=3z(y^2+1)-2(x^2+1), \] \[ 2z^5=4x(z^2+1)-3(y^2+1).\]

III Soros Olympiad 1996 - 97 (Russia), 10.4

Tags: number theory , system of equations

Find natural $a, b, c, d$ that satisfy the system $$\begin{cases} ab+cd=34 \\ ac-bd=19 \end{cases}$$

2010 Bosnia And Herzegovina - Regional Olympiad, 1

Tags: algebra , system of equations

Find all real numbers $(x,y)$ satisfying the following: $$x+\frac{3x-y}{x^2+y^2}=3$$ $$y-\frac{x+3y}{x^2+y^2}=0$$

2024 Canadian Mathematical Olympiad Qualification, 6

Tags: algebra , system of equations

For certain real constants $ p, q, r$, we are given a system of equations $$\begin{cases} a^2 + b + c = p \\ a + b^2 + c = q \\ a + b + c^2 = r \end{cases}$$ What is the maximum number of solutions of real triplets $(a, b, c)$ across all possible $p, q, r$? Give an example of the $p$, $q$, $r$ that achieves this maximum.

2006 Regional Competition For Advanced Students, 2

Tags: algebra , system of equations

Let $ n>1$ be a positive integer an $ a$ a real number. Determine all real solutions $ (x_1,x_2,\dots,x_n)$ to following system of equations: $ x_1\plus{}ax_2\equal{}0$ $ x_2\plus{}a^2x_3\equal{}0$ … $ x_k\plus{}a^kx_{k\plus{}1}\equal{}0$ … $ x_n\plus{}a^nx_1\equal{}0$

2007 South africa National Olympiad, 2

Tags: polynomial , system of equations , absolute value , algebra

Consider the equation $ x^4 \equal{} ax^3 \plus{} bx^2 \plus{} cx \plus{} 2007$, where $ a,b,c$ are real numbers. Determine the largest value of $ b$ for which this equation has exactly three distinct solutions, all of which are integers.

2007 ITest, 46

Tags: algebra , floor function , system of equations

Let $(x,y,z)$ be an ordered triplet of real numbers that satisfies the following system of equations: \begin{align*}x+y^2+z^4&=0,\\y+z^2+x^4&=0,\\z+x^2+y^4&=0.\end{align*} If $m$ is the minimum possible value of $\lfloor x^3+y^3+z^3\rfloor$, find the modulo $2007$ residue of $m$.

2015 Lusophon Mathematical Olympiad, 4

Tags: algebra , system of equations

Let $a$ be a real number, such that $a\ne 0, a\ne 1, a\ne -1$ and $m,n,p,q$ be natural numbers . Prove that if $a^m+a^n=a^p+a^q$ and $a^{3m}+a^{3n}=a^{3p}+a^{3q}$ , then $m \cdot n = p \cdot q$.

1966 IMO, 1

Tags: algebra , combinatorics , system of equations

In a mathematical contest, three problems, $A,B,C$ were posed. Among the participants ther were 25 students who solved at least one problem each. Of all the contestants who did not solve problem $A$, the number who solved $B$ was twice the number who solved $C$. The number of students who solved only problem $A$ was one more than the number of students who solved $A$ and at least one other problem. Of all students who solved just one problem, half did not solve problem $A$. How many students solved only problem $B$?

2017-IMOC, A3

Tags: algebra , system of equations

Solve the following system of equations: $$\begin{cases} x^3+y+z=1\\ x+y^3+z=1\\ x+y+z^3=1\end{cases}$$

1976 Swedish Mathematical Competition, 2

Tags: algebra , system , system of equations

For which real $a$ are there distinct reals $x$, $y$ such that $$\begin{cases} x = a - y^2 \\ y = a - x^2 \,\,\, ? \end {cases}$$

1954 Moscow Mathematical Olympiad, 266

Tags: algebra , system of equations

Find all solutions of the system consisting of $3$ equations: $x \left(1 - \frac{1}{2^n}\right) +y \left(1 - \frac{1}{2^{n+1}}\right) +z \left(1 - \frac{1}{2^{n+2}}\right) = 0$ for $n = 1, 2, 3$.

IV Soros Olympiad 1997 - 98 (Russia), 11.12

Tags: algebra , system of equations , parameter

Find how many different solutions depending on $a$ has the system of equations : $$\begin{cases} x+z=2a \\ y+u+xz=a-3 \\ yz+xu=2a \\ yu=1 \end{cases}$$

2015 District Olympiad, 2

Tags: algebra , exponential equation , system of equations

Solve in $ \mathbb{Z} $ the following system of equations: $$ \left\{\begin{matrix} 5^x-\log_2 (y+3) = 3^y\\ 5^y -\log_2 (x+3)=3^x\end{matrix}\right. . $$

2008 Federal Competition For Advanced Students, P1, 2

Tags: system of equations , algebra

Given $a \in R^{+}$ and an integer $n > 4$ determine all n-tuples ($x_1, ...,x_n$) of positive real numbers that satisfy the following system of equations: $\begin {cases} x_1x_2(3a-2x_3) = a^3\\ x_2x_3(3a-2x_4) = a^3\\ ...\\ x_{n-2}x_{n-1}(3a-2x_n) = a^3\\ x_{n-1}x_n(3a-2x_1) = a^3 \\ x_nx_1(3a-2x_2) = a^3 \end {cases}$ .

1984 IMO Longlists, 44

Tags: geometry , pythagorean theorem , system of equations , algebra

Let $a,b,c$ be positive numbers with $\sqrt{a}+\sqrt{b}+\sqrt{c}= \frac{\sqrt{3}}{2}$ Prove that the system of equations \[\sqrt{y-a}+\sqrt{z-a}=1\] \[\sqrt{z-b}+\sqrt{x-b}=1\] \[\sqrt{x-c}+\sqrt{y-c}=1\] has exactly one solution $(x,y,z)$ in real numbers. It was proposed by Poland. Have fun! :lol:

1950 Poland - Second Round, 1

Tags: algebra , system , system of equations

Solve the system of equations $$\begin{cases} x^2+x+y=8\\ y^2+2xy+z=168\\ z^2+2yz+2xz=12480 \end{cases}$$

2024 Kyiv City MO Round 2, Problem 1

Tags: system of equations , algebra

Solve the following system of equations in real numbers: $$\left\{\begin{array}{l}x^2=y^2+z^2,\\x^{2024}=y^{2024}+z^{2024},\\x^{2025}=y^{2025}+z^{2025}.\end{array}\right.$$ [i]Proposed by Mykhailo Shtandenko, Anton Trygub, Bogdan Rublov[/i]

1936 Moscow Mathematical Olympiad, 027

Tags: algebra , parameter , system of equations

Solve the system $\begin{cases} x+y=a \\ x^5 +y^5 = b^5 \end{cases}$

2010 Saudi Arabia Pre-TST, 4.1

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

Find all triples $(a, b, c)$ of positive integers for which $$\begin{cases} a + bc=2010 \\ b + ca = 250\end{cases}$$

2004 Austrian-Polish Competition, 3

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

Solve the following system of equations in $\mathbb{R}$ where all square roots are non-negative: $ \begin{matrix} a - \sqrt{1-b^2} + \sqrt{1-c^2} = d \\ b - \sqrt{1-c^2} + \sqrt{1-d^2} = a \\ c - \sqrt{1-d^2} + \sqrt{1-a^2} = b \\ d - \sqrt{1-a^2} + \sqrt{1-b^2} = c \\ \end{matrix} $

1989 IberoAmerican, 1

Tags: algebra , system of equations

Determine all triples of real numbers that satisfy the following system of equations: \[x+y-z=-1\\ x^2-y^2+z^2=1\\ -x^3+y^3+z^3=-1\]