Found problems: 744
2024 Canadian Mathematical Olympiad Qualification, 6
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.
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 $x + 1000y$.
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 } $
2009 Tuymaada Olympiad, 1
Three real numbers are given. Fractional part of the product of every two of them is $ 1\over 2$. Prove that these numbers are irrational.
[i]Proposed by A. Golovanov[/i]
1956 Czech and Slovak Olympiad III A, 1
Find all $x,y\in\left(0,\frac{\pi}{2}\right)$ such that
\begin{align*}
\frac{\cos x}{\cos y}&=2\cos^2 y, \\
\frac{\sin x}{\sin y}&=2\sin^2 y.
\end{align*}
2000 District Olympiad (Hunedoara), 1
[b]a)[/b] Solve the system
$$ \left\{\begin{matrix} 3^y-4^x=11\\ \log_4{x} +\log_3 y =3/2\end{matrix}\right. $$
[b]b)[/b] Solve the equation $ \quad 9^{\log_5 (x-2)} -5^{\log_9 (x+2)} = 4. $
2007 South africa National Olympiad, 2
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.
1988 IMO Longlists, 62
Let $x = p, y = q, z = r, w = s$ be the unique solution of the system of linear equations \[ x + a_i \cdot y + a^2_i \cdot z + a^3_i \cdot w = a^4_i, i = 1,2,3,4. \] Express the solutions of the following system in terms of $p,q,r$ and $s:$ \[ x + a^2_i \cdot y + a^4_i \cdot z + a^6_i \cdot w = a^8_i, i = 1,2,3,4. \] Assume the uniquness of the solution.
1999 Moldova Team Selection Test, 2
Let $a,b,c$ be positive numbers. Prove that a triangle with sides $a,b,c$ exists if and only if the system of equations
$$\begin{cases}\dfrac{y}{z}+\dfrac{z}{y}=\dfrac{a}{x} \\ \\ \dfrac{z}{x}+\dfrac{x}{z}=\dfrac{b}{y} \\ \\ \dfrac{x}{y}+\dfrac{y}{x}=\dfrac{c}{z}\end{cases}$$ has a real solution.
1993 IMO Shortlist, 4
Solve the following system of equations, in which $a$ is a given number satisfying $|a| > 1$:
$\begin{matrix} x_{1}^2 = ax_2 + 1 \\ x_{2}^2 = ax_3 + 1 \\ \ldots \\ x_{999}^2 = ax_{1000} + 1 \\ x_{1000}^2 = ax_1 + 1 \\ \end{matrix}$
1967 IMO Shortlist, 4
In what case does the system of equations
$\begin{matrix} x + y + mz = a \\ x + my + z = b \\ mx + y + z = c \end{matrix}$
have a solution? Find conditions under which the unique solution of the above system is an arithmetic progression.
1979 IMO Shortlist, 18
Let $m$ positive integers $a_1, \dots , a_m$ be given. Prove that there exist fewer than $2^m$ positive integers $b_1, \dots , b_n$ such that all sums of distinct $b_k$’s are distinct and all $a_i \ (i \leq m)$ occur among them.
1949-56 Chisinau City MO, 40
Solve the system of equations:
$$\begin{cases} \log_{2} x + \log_{4} y + \log_{4} z =2 \\ \log_{3} y + \log_{9} z + \log_{9} x =2 \\
\log_{4} z + \log_{16} x + \log_{16} y =2\end{cases}$$
2008 India Regional Mathematical Olympiad, 2
Solve the system of equation
$$x+y+z=2;$$$$(x+y)(y+z)+(y+z)(z+x)+(z+x)(x+y)=1;$$$$x^2(y+z)+y^2(z+x)+z^2(x+y)=-6.$$
1926 Eotvos Mathematical Competition, 1
Prove that, if $a$ and $b$ are given integers, the system of equatìons
$$x + y + 2z + 2t = a$$
$$2x - 2y + z- t = b$$
has a solution in integers $x, y,z,t$.
1992 All Soviet Union Mathematical Olympiad, 564
Find all real $x, y$ such that $$\begin{cases}(1 + x)(1 + x^2)(1 + x^4) = 1+ y^7 \\
(1 + y)(1 + y^2)(1 + y^4) = 1+ x^7 \end{cases}$$
II Soros Olympiad 1995 - 96 (Russia), 10.4
Solve the system of equations
$$\begin{cases} x^2+ [y]=10
\\ y^2+[x]=13
\end{cases}$$
($[x]$ is the integer part of $x$, $[x]$ is equal to the largest integer not exceeding $x$. For example, $[3,33] = 3$, $[2] = 2$, $[- 3.01] = -4$).
1945 Moscow Mathematical Olympiad, 097
The system $\begin{cases} x^2 - y^2 = 0 \\
(x - a)^2 + y^2 = 1 \end{cases}$ generally has four solutions. For which $a$ the number of solutions of the system is equal to three or two?
2017 VJIMC, 3
Let $n \ge 2$ be an integer. Consider the system of equations
\begin{align} x_1+\frac{2}{x_2}=x_2+\frac{2}{x_3}=\dots=x_n+\frac{2}{x_1} \end{align}
1. Prove that $(1)$ has infinitely many real solutions $(x_1,\dotsc,x_n)$ such that the numbers $x_1,\dotsc,x_n$ are distinct.
2. Prove that every solution of $(1)$, such that the numbers $x_1,\dotsc,x_n$ are not all equal, satisfies $\vert x_1x_2\cdots x_n\vert=2^{n/2}$.
2011 Romanian Masters In Mathematics, 2
Determine all positive integers $n$ for which there exists a polynomial $f(x)$ with real coefficients, with the following properties:
(1) for each integer $k$, the number $f(k)$ is an integer if and only if $k$ is not divisible by $n$;
(2) the degree of $f$ is less than $n$.
[i](Hungary) Géza Kós[/i]
2018 Iran MO (1st Round), 14
For how many integers $k$ does the following system of equations has a solution other than $a=b=c=0$ in the set of real numbers? \begin{align*} \begin{cases} a^2+b^2=kc(a+b),\\ b^2+c^2 = ka(b+c),\\ c^2+a^2=kb(c+a).\end{cases}\end{align*}
2012 AMC 10, 22
The sum of the first $m$ positive odd integers is $212$ more than the sum of the first $n$ positive even integers. What is the sum of all possible values of $n$?
$ \textbf{(A)}\ 255
\qquad\textbf{(B)}\ 256
\qquad\textbf{(C)}\ 257
\qquad\textbf{(D)}\ 258
\qquad\textbf{(E)}\ 259
$
2001 Putnam, 2
Find all pairs of real numbers $(x,y)$ satisfying the system of equations:
\begin{align*}\frac{1}{x} + \frac{1}{2y} &= (x^2+3y^2)(3x^2+y^2)\\
\frac{1}{x} - \frac{1}{2y} &= 2(y^4-x^4)\end{align*}
2014 Ukraine Team Selection Test, 9
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 $x, y, z$.
1994 China Team Selection Test, 2
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.