Found problems: 15925
2024 Dutch IMO TST, 3
Let $a,b,c$ be real numbers such that $0 \le a \le b \le c$ and $a+b+c=1$. Show that
\[ab\sqrt{b-a}+bc\sqrt{c-b}+ac\sqrt{c-a}<\frac{1}{4}.\]
2012 Princeton University Math Competition, A1
Compute the smallest positive integer $a$ for which $$\sqrt{a +\sqrt{a +...}} - \frac{1}{a +\frac{1}{a+...}}> 7$$
2014 Contests, A2
Let $x,y$ and $z$ be positive real numbers such that $xy+yz+xz=3xyz$. Prove that \[ x^2y+y^2z+z^2x \ge 2(x+y+z)-3 \] and determine when equality holds.
[i]UK - David Monk[/i]
2017 Dutch IMO TST, 2
let $a_1,a_2,...a_n$ a sequence of real numbers such that $a_1+....+a_n=0$.
define $b_i=a_1+a_2+....a_i$ for all $1 \leq i \leq n$ .suppose $b_i(a_{j+1}-a_{i+1}) \geq 0$ for all $1 \leq i \leq j \leq n-1$.
Show that $$\max_{1 \leq l \leq n} |a_l| \geq \max_{1 \leq m \leq n} |b_m|$$
2021 AMC 10 Fall, 14
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*}
$\textbf{(A)}\: 1\qquad\textbf{(B)} \: 2\qquad\textbf{(C)} \: 3\qquad\textbf{(D)} \: 5\qquad\textbf{(E)} \: 7$
1999 Israel Grosman Mathematical Olympiad, 4
Consider a polynomial $f(x) = x^4 +ax^3 +bx^2 +cx+d$ with integer coefficients.
Prove that if $f(x)$ has exactly one real root, then it can be factored into nonconstant polynomials with rational coefficients
1990 Mexico National Olympiad, 4
Find $0/1 + 1/1 + 0/2 + 1/2 + 2/2 + 0/3 + 1/3 + 2/3 + 3/3 + 0/4 + 1/4 + 2/4 + 3/4 + 4/4 + 0/5 + 1/5 + 2/5 + 3/5 + 4/5 + 5/5 + 0/6 + 1/6 + 2/6 + 3/6 + 4/6 + 5/6 + 6/6$
2014 Contests, 1
Let $a_0 < a_1 < a_2 < \dots$ be an infinite sequence of positive integers. Prove that there exists a unique integer $n\geq 1$ such that
\[a_n < \frac{a_0+a_1+a_2+\cdots+a_n}{n} \leq a_{n+1}.\]
[i]Proposed by Gerhard Wöginger, Austria.[/i]
2021 CMIMC, 2.1
Find the unique 3 digit number $N=\underline{A}$ $\underline{B}$ $\underline{C},$ whose digits $(A, B, C)$ are all nonzero, with the property that the product $P=\underline{A}$ $\underline{B}$ $\underline{C}$ $\times$ $\underline{A}$ $\underline{B}$ $\times$ $\underline{A}$ is divisible by $1000$.
[i]Proposed by Kyle Lee[/i]
2002 AIME Problems, 15
Circles $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ intersect at two points, one of which is $(9,6),$ and the product of the radii is $68.$ The x-axis and the line $y=mx$, where $m>0,$ are tangent to both circles. It is given that $m$ can be written in the form $a\sqrt{b}/c,$ where $a,$ $b,$ and $c$ are positive integers, $b$ is not divisible by the square of any prime, and $a$ and $c$ are relatively prime. Find $a+b+c.$
2022 OMpD, 2
We say that a sextuple of positive real numbers $(a_1, a_2, a_3, b_1, b_2, b_3)$ is $\textit{phika}$ if $a_1 + a_2 + a_3 = b_1 + b_2 + b_3 = 1$.
(a) Prove that there exists a $\textit{phika}$ sextuple $(a_1, a_2, a_3, b_1, b_2, b_3)$ such that:
$$a_1(\sqrt{b_1} + a_2) + a_2(\sqrt{b_2} + a_3) + a_3(\sqrt{b_3} + a_1) > 1 - \frac{1}{2022^{2022}}$$
(b) Prove that for every $\textit{phika}$ sextuple $(a_1, a_2, a_3, b_1, b_2, b_3)$, we have:
$$a_1(\sqrt{b_1} + a_2) + a_2(\sqrt{b_2} + a_3) + a_3(\sqrt{b_3} + a_1) < 1$$
2014 Online Math Open Problems, 27
Let $p = 2^{16}+1$ be a prime, and let $S$ be the set of positive integers not divisible by $p$.
Let $f: S \to \{0, 1, 2, ..., p-1\}$ be a function satisfying
\[ f(x)f(y) \equiv f(xy)+f(xy^{p-2}) \pmod{p} \quad\text{and}\quad f(x+p) = f(x) \]
for all $x,y \in S$.
Let $N$ be the product of all possible nonzero values of $f(81)$.
Find the remainder when when $N$ is divided by $p$.
[i]Proposed by Yang Liu and Ryan Alweiss[/i]
2019 China Girls Math Olympiad, 5
Let $p$ be a prime number such that $p\mid (2^{2019}-1) .$ The sequence $a_1,a_2,...,a_n$ satisfies the following conditions: $a_0=2, a_1=1 ,a_{n+1}=a_n+\frac{p^2-1}{4}a_{n-1}$ $(n\geq 1).$ Prove that $p\nmid (a_n+1),$ for any $n\geq 0.$
2014 Korea National Olympiad, 4
Prove that there exists a function $f : \mathbb{N} \rightarrow \mathbb{N}$ that satisfies the following
(1) $\{f(n) : n\in\mathbb{N}\}$ is a finite set; and
(2) For nonzero integers $x_1, x_2, \ldots, x_{1000}$ that satisfy $f(\left|x_1\right|)=f(\left|x_2\right|)=\cdots=f(\left|x_{1000}\right|)$, then $x_1+2x_2+2^2x_3+2^3x_4+2^4x_5+\cdots+2^{999}x_{1000}\ne 0$.
1973 IMO Shortlist, 17
$G$ is a set of non-constant functions $f$. Each $f$ is defined on the real line and has the form $f(x)=ax+b$ for some real $a,b$. If $f$ and $g$ are in $G$, then so is $fg$, where $fg$ is defined by $fg(x)=f(g(x))$. If $f$ is in $G$, then so is the inverse $f^{-1}$. If $f(x)=ax+b$, then $f^{-1}(x)= \frac{x-b}{a}$. Every $f$ in $G$ has a fixed point (in other words we can find $x_f$ such that $f(x_f)=x_f$. Prove that all the functions in $G$ have a common fixed point.
1999 Ukraine Team Selection Test, 4
If $n \in N$ and $0 < x <\frac{\pi}{2n}$, prove the inequality $\frac{\sin 2x}{\sin x}+\frac{\sin 3x}{\sin 2x} +...+\frac{\sin (n+1)x}{\sin nx} < 2\frac{\cos x}{\sin^2 x}$.
.
1981 AMC 12/AHSME, 8
For all positive numbers $x,y,z$ the product $(x+y+z)^{-1}(x^{-1}+y^{-1}+z^{-1})(xy+yz+xz)^{-1}[(xy)^{-1}+(yz)^{-1}+(xz)^{-1}]$ equals
$\text{(A)}\ x^{-2}y^{-2}z^{-2} \qquad \text{(B)}\ x^{-2}+y^{-2}+z^{-2} \qquad \text{(C)}\ (x+y+z)^{-1}$
$\text{(D)}\ \frac{1}{xyz} \qquad \text{(E)}\ \frac{1}{xy+yz+xz}$
2023 Puerto Rico Team Selection Test, 3
Let $p(x)$ be a polynomial of degree $2022$ such that:
$$p(k) =\frac{1}{k+1}\,\,\, \text{for }\,\,\, k = 0, 1, . . . , 2022$$
Find $p(2023)$.
1975 Swedish Mathematical Competition, 2
Is there a positive integer $n$ such that the fractional part of
\[
\left(3+\sqrt{5}\right)^n >0.99 ?
\]
2005 Poland - Second Round, 1
The polynomial $W(x)=x^2+ax+b$ with integer coefficients has the following property: for every prime number $p$ there is an integer $k$ such that both $W(k)$ and $W(k+1)$ are divisible by $p$. Show that there is an integer $m$ such that $W(m)=W(m+1)=0$.
1992 Nordic, 2
Let $n > 1$ be an integer and let $a_1, a_2,... , a_n$ be $n$ different integers. Show that the polynomial
$f(x) = (x -a_1)(x - a_2)\cdot ... \cdot (x -a_n) - 1$ is not divisible by any polynomial with integer coefficients
and of degree greater than zero but less than $n$ and such that the highest power of $x$ has coefficient $1$.
2004 IMO Shortlist, 3
Does there exist a function $s\colon \mathbb{Q} \rightarrow \{-1,1\}$ such that if $x$ and $y$ are distinct rational numbers satisfying ${xy=1}$ or ${x+y\in \{0,1\}}$, then ${s(x)s(y)=-1}$? Justify your answer.
[i]Proposed by Dan Brown, Canada[/i]
2001 APMO, 4
A point in the plane with a cartesian coordinate system is called a [i]mixed point[/i] if one of its coordinates is rational and the other one is irrational. Find all polynomials with real coefficients such that their graphs do not contain any mixed point.
1986 Polish MO Finals, 4
Find all $n$ such that there is a real polynomial $f(x)$ of degree $n$ such that $f(x) \ge f'(x)$ for all real $x$.
2001 China Team Selection Test, 3
$$F(x)=x^{6}+15x^{5}+85x^{4}+225x^{3}+274x^{2}+120x+1$$