Found problems: 15925
MOAA Individual Speed General Rounds, 2021.4
Let $a$, $b$, and $c$ be real numbers such that $0\le a,b,c\le 5$ and $2a + b + c = 10$. Over all possible values of $a$, $b$, and $c$, determine the maximum possible value of $a + 2b + 3c$.
[i]Proposed by Andrew Wen[/i]
1996 China Team Selection Test, 3
Does there exist non-zero complex numbers $a, b, c$ and natural number $h$ such that if integers $k, l, m$ satisfy $|k| + |l| + |m| \geq 1996$, then $|ka + lb + mc| > \frac {1}{h}$ is true?
2015 Princeton University Math Competition, B3
Andrew and Blair are bored in class and decide to play a game. They pick a pair $(a, b)$ with $1 \le a, b \le 100$. Andrew says the next number in the geometric series that begins with $a,b$ and Blair says the next number in the arithmetic series that begins with $a,b$. For how many pairs $(a, b)$ is Andrew's number minus Blair's number a positive perfect square?
2019-2020 Winter SDPC, 3
Find, with proof, all functions $f$ mapping integers to integers with the property that for all integers $m,n$, $$f(m)+f(n)= \max\left(f(m+n),f(m-n)\right).$$
2019 BMT Spring, 3
If $f(x + y) = f(xy)$ for all real numbers $x$ and $y$, and $f(2019) = 17$, what is the value of $f(17)$?
2018 South East Mathematical Olympiad, 1
Assume $c$ is a real number. If there exists $x\in[1,2]$ such that $\max\left\{\left |x+\frac cx\right |, \left |x+\frac cx + 2\right |\right\}\geq 5$, please find the value range of $c$.
2022-23 IOQM India, 14
Let $x,y,z$ be complex numbers such that\\
$\hspace{ 2cm} \frac{x}{y+z}+\frac{y}{z+x}+\frac{z}{x+y}=9$\\
$\hspace{ 2cm} \frac{x^2}{y+z}+\frac{y^2}{z+x}+\frac{z^2}{x+y}=64$\\
$\hspace{ 2cm} \frac{x^3}{y+z}+\frac{y^3}{z+x}+\frac{z^3}{x+y}=488$\\
\\
If $\frac{x}{yz}+\frac{y}{zx}+\frac{z}{xy}=\frac{m}{n}$ where $m,n$ are positive integers with $GCD(m,n)=1$, find $m+n$.
2005 Germany Team Selection Test, 1
Find all monotonically increasing or monotonically decreasing functions $f: \mathbb{R}_+\to\mathbb{R}_+$ which satisfy the equation $f\left(xy\right)\cdot f\left(\frac{f\left(y\right)}{x}\right)=1$ for any two numbers $x$ and $y$ from $\mathbb{R}_+$.
Hereby, $\mathbb{R}_+$ is the set of all positive real numbers.
[i]Note.[/i] A function $f: \mathbb{R}_+\to\mathbb{R}_+$ is called [i]monotonically increasing[/i] if for any two positive numbers $x$ and $y$ such that $x\geq y$, we have $f\left(x\right)\geq f\left(y\right)$.
A function $f: \mathbb{R}_+\to\mathbb{R}_+$ is called [i]monotonically decreasing[/i] if for any two positive numbers $x$ and $y$ such that $x\geq y$, we have $f\left(x\right)\leq f\left(y\right)$.
1988 IMO Longlists, 55
Suppose $\alpha_i > 0, \beta_i > 0$ for $1 \leq i \leq n, n > 1$ and that \[ \sum^n_{i=1} \alpha_i = \sum^n_{i=1} \beta_i = \pi. \] Prove that \[ \sum^n_{i=1} \frac{\cos(\beta_i)}{\sin(\alpha_i)} \leq \sum^n_{i=1} \cot(\alpha_i). \]
2020 Romanian Master of Mathematics, 2
Let $N \geq 2$ be an integer, and let $\mathbf a$ $= (a_1, \ldots, a_N)$ and $\mathbf b$ $= (b_1, \ldots b_N)$ be sequences of non-negative integers. For each integer $i \not \in \{1, \ldots, N\}$, let $a_i = a_k$ and $b_i = b_k$, where $k \in \{1, \ldots, N\}$ is the integer such that $i-k$ is divisible by $n$. We say $\mathbf a$ is $\mathbf b$-[i]harmonic[/i] if each $a_i$ equals the following arithmetic mean: \[a_i = \frac{1}{2b_i+1} \sum_{s=-b_i}^{b_i} a_{i+s}.\]
Suppose that neither $\mathbf a $ nor $\mathbf b$ is a constant sequence, and that both $\mathbf a$ is $\mathbf b$-[i]harmonic[/i] and $\mathbf b$ is $\mathbf a$-[i]harmonic[/i].
Prove that at least $N+1$ of the numbers $a_1, \ldots, a_N,b_1, \ldots, b_N$ are zero.
2004 District Olympiad, 1
We say that the real numbers $a$ and $b$ have property $P$ if: $a^2+b \in Q$ and $b^2 + a \in Q$.Prove that:
a) The numbers $a= \frac{1+\sqrt2}{2}$ and $b= \frac{1-\sqrt2}{2}$ are irrational and have property $P$
b) If $a, b$ have property $P$ and $a+b \in Q -\{1\}$, then $a$ and $b$ are rational numbers
c) If $a, b$ have property $P$ and $\frac{a}{b} \in Q$, then $a$ and $b$ are rational numbers.
1966 IMO Shortlist, 10
How many real solutions are there to the equation $x = 1964 \sin x - 189$ ?
1996 Estonia National Olympiad, 1
Prove that for any positive numbers $x,y$ it holds that $x^xy^y \ge x^yy^x$.
2010 N.N. Mihăileanu Individual, 1
Let be two real reducible quadratic polynomials $ P,Q $ in one variable. Prove that if $ P-Q $ is irreducible, then $ P+Q $ is reducible.
2005 AIME Problems, 6
Let $P$ be the product of the nonreal roots of $x^4-4x^3+6x^2-4x=2005$. Find $\lfloor P\rfloor$.
2023-IMOC, A1
Find all functions $f:\mathbb{N} \rightarrow \mathbb{N}$ such that for all positive integers $n$, there exists an unique positive integer $k$, satisfying $f^k(n)\leq n+k+1$.
2010 CHMMC Fall, 14
A $4$-dimensional hypercube of edge length $1$ is constructed in $4$-space with its edges parallel to
the coordinate axes and one vertex at the origin. The coordinates of its sixteen vertices are given
by $(a, b, c, d)$, where each of $a, b, c,$ and $d$ is either $0$ or $1$. The $3$-dimensional hyperplane given
by $x + y + z + w = 2$ intersects the hypercube at $6$ of its vertices. Compute the $3$-dimensional
volume of the solid formed by the intersection.
2022 Irish Math Olympiad, 9
9. Let [i]k[/i] be a positive integer and let $x_0, x_1, x_2, \cdots$ be an infinite sequence defined by the relationship
$$x_0 = 0$$
$$x_1 = 1$$
$$x_{n+1} = kx_n +x_{n-1}$$
For all [i]n[/i] $\ge$ 1
(a) For the special case [i]k[/i] = 1, prove that $x_{n-1}x_{n+1}$ is never a perfect square for [i]n[/i] $\ge$ 2
(b) For the general case of integers [i]k[/i] $\ge$ 1, prove that $x_{n-1}x_{n+1}$ is never a perfect square for [i]n[/i] $\ge$ 2
1965 Leningrad Math Olympiad, grade 6
[b]6.1 [/b] The bindery had 92 sheets of white paper and $135$ sheets of colored paper. It took a sheet of white paper to bind each book. and a sheet of colored paper. After binding several books of white Paper turned out to be half as much as the colored one. How many books were bound?
[b]6.2[/b] Prove that if you multiply all the integers from $1$ to $1965$, you get the number, the last whose non-zero digit is even.
[b]6.3[/b] The front tires of a car wear out after $25,000$ kilometers, and the rear tires after $15,000$ kilometers of travel. When should you swap tires so that they wear out at the same time?
[b]6.4[/b] A rectangle $19$ cm $\times 65$ cm is divided by straight lines parallel to its sides into squares with side 1 cm. How many parts will this rectangle be divided into if you also draw a diagonal in it?
[b]6.5[/b] Find the dividend, divisor and quotient in the example:
[center][img]https://cdn.artofproblemsolving.com/attachments/2/e/de053e7e11e712305a89d3b9e78ac0901dc775.png[/img]
[/center]
[b]6.6[/b] Odd numbers from $1$ to $49$ are written out in table form
$$\,\,\,1\,\,\,\,\,\, 3\,\,\,\,\,\, 5\,\,\,\,\,\, 7\,\,\,\,\,\, 9$$
$$11\,\,\, 13\,\,\, 15\,\,\, 17\,\,\, 19$$
$$21\,\,\, 23\,\,\, 25\,\,\, 27\,\,\, 29$$
$$31\,\,\, 33\,\,\, 35\,\,\, 37\,\,\, 39$$
$$41\,\,\, 43\,\,\, 45\,\,\, 47\,\,\, 49$$
$5$ numbers are selected, any two of which are not on the same line or in one column. What is their sum?
PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3988081_1965_leningrad_math_olympiad]here[/url].
2003 India Regional Mathematical Olympiad, 4
Find the number of ordered triples $(x,y,z)$ of non-negative integers satisfying
(i) $x \leq y \leq z$
(ii) $x + y + z \leq 100.$
2014 Korea Junior Math Olympiad, 2
Let there be $2n$ positive reals $a_1,a_2,...,a_{2n}$. Let $s = a_1 + a_3 +...+ a_{2n-1}$, $t = a_2 + a_4 + ... + a_{2n}$, and
$x_k = a_k + a_{k+1} + ... + a_{k+n-1}$ (indices are taken modulo $2n$). Prove that
$$\frac{s}{x_1}+\frac{t}{x_2}+\frac{s}{x_3}+\frac{t}{x_4}+...+\frac{s}{x_{2n-1}}+\frac{t}{x_{2n}}>\frac{2n^2}{n+1}$$
2014 District Olympiad, 4
Let $n\geq2$ be a positive integer. Determine all possible values of the sum
\[ S=\left\lfloor x_{2}-x_{1}\right\rfloor +\left\lfloor x_{3}-x_{2}\right\rfloor+...+\left\lfloor x_{n}-x_{n-1}\right\rfloor \]
where $x_i\in \mathbb{R}$ satisfying $\lfloor{x_i}\rfloor=i$ for $i=1,2,\ldots n$.
V Soros Olympiad 1998 - 99 (Russia), 10.8
It is known that for all $x$ such that $|x| < 1$, the following inequality holds $$ax^2+bx+c\le \frac{1}{\sqrt{1-x^2}}$$Find the greatest value of $a + 2c$.
1998 USAMTS Problems, 2
There are infinitely many ordered pairs $(m,n)$ of positive integers for which the sum
\[ m + ( m + 1) + ( m + 2) +... + ( n - 1 )+n\]
is equal to the product $mn$. The four pairs with the smallest values of $m$ are $(1, 1), (3, 6), (15, 35),$ and $(85, 204)$. Find three more $(m, n)$ pairs.
2003 Bosnia and Herzegovina Team Selection Test, 3
Prove that for every positive integer $n$ holds:
$(n-1)^n+2n^n \leq (n+1)^{n} \leq 2(n-1)^n+2n^{n}$