Found problems: 118
Let $n$ be a positive integer and let $p$ be a prime number. Prove that if $a$, $b$, $c$ are integers (not necessarily positive) satisfying the equations \[ a^n + pb = b^n + pc = c^n + pa\] then $a = b = c$.
[i]Proposed by Angelo Di Pasquale, Australia[/i]
Given a polynomial $P$, assume that $L = \{z \in \mathbb{C}: |P(z)| = 1\}$ is a Jordan curve. Show that the zeros of $P'$ are in the interior of $L$.
Let $w$ and $z$ be complex numbers such that $|w| = 1$ and $|z| = 10$. Let $\theta = \arg\left(\tfrac{w-z}{z}\right)$. The maximum possible value of $\tan^2 \theta$ can be written as $\tfrac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$. (Note that $\arg(w)$, for $w \neq 0$, denotes the measure of the angle that the ray from $0$ to $w$ makes with the positive real axis in the complex plane.
What is the greatest real root of the equation $x^3-x^2-x-\frac 13 = 0$?
$
\textbf{(A)}\ \dfrac{\sqrt {3} - \sqrt{2}}{2}
\qquad\textbf{(B)}\ \dfrac{\sqrt [3]{3} - \sqrt[3]{2}}{2}
\qquad\textbf{(C)}\ \dfrac 1{\sqrt[3] {3} - 1}
\qquad\textbf{(D)}\ \dfrac 1{\sqrt[3] {4} - 1}
\qquad\textbf{(E)}\ \text{None of above}
$
We consider a point $P$ in a plane $p$ and a point $Q \not\in p$. Determine all the points $R$ from $p$ for which \[ \frac{QP+PR}{QR} \] is maximum.
Ethan Song and Bryan Guo are playing an unfair game of rock-paper-scissors. In any game, Ethan has a 2/5 chance to win, 2/5 chance to tie, and 1/5 chance to lose. How many games is Ethan expected to win before losing?
[i]2022 CCA Math Bonanza Lightning Round 4.3[/i]
Suppose $P(x)$ is a polynomial with real coefficients, satisfying the condition $P(\cos \theta+\sin \theta)=P(\cos \theta-\sin \theta)$, for every real $\theta$. Prove that $P(x)$ can be expressed in the form$$P(x)=a_0+a_1(1-x^2)^2+a_2(1-x^2)^4+\dots+a_n(1-x^2)^{2n}$$for some real numbers $a_0, a_1, \dots, a_n$ and non-negative integer $n$.
[i]Proposed by C.R. Pranesacher[/i]
We consider a point $P$ in a plane $p$ and a point $Q \not\in p$. Determine all the points $R$ from $p$ for which \[ \frac{QP+PR}{QR} \] is maximum.
Let $n$ be a positive integer and let $p$ be a prime number. Prove that if $a$, $b$, $c$ are integers (not necessarily positive) satisfying the equations \[ a^n + pb = b^n + pc = c^n + pa\] then $a = b = c$.
[i]Proposed by Angelo Di Pasquale, Australia[/i]
Determine the least real number $M$ such that the inequality \[|ab(a^{2}-b^{2})+bc(b^{2}-c^{2})+ca(c^{2}-a^{2})| \leq M(a^{2}+b^{2}+c^{2})^{2}\] holds for all real numbers $a$, $b$ and $c$.
Find all polynomials $f$ that satisfy the equation
\[\frac{f(3x)}{f(x)} = \frac{729 (x-3)}{x-243}\]
for infinitely many real values of $x$.
To celebrate the 20th Thailand Mathematical Olympiad (TMO), Ratchasima Witthayalai School put up flags around the Thao Suranari Monument so that
[list=i]
[*] Each flag is painted in exactly one color, and at least $2$ distinct colors are used.
[*] The number of flags are odd.
[*] Every flags are on a regular polygon such that each vertex has one flag.
[*] Every flags with the same color are on a regular polygon.
[/list]
Prove that there are at least $3$ colors with the same amount of flags.
Farmer Tim is lost in the densely-forested Cartesian plane. Starting from the origin he walks a sinusoidal path in search of home; that is, after $t$ minutes he is at position $(t,\sin t)$.
Five minutes after he sets out, Alex enters the forest at the origin and sets out in search of Tim. He walks in such a way that after he has been in the forest for $m$ minutes, his position is $(m,\cos t)$.
What is the greatest distance between Alex and Farmer Tim while they are walking in these paths?
Let $x$ be a positive real number. Define
\[
A = \sum_{k=0}^{\infty} \frac{x^{3k}}{(3k)!}, \quad
B = \sum_{k=0}^{\infty} \frac{x^{3k+1}}{(3k+1)!}, \quad\text{and}\quad
C = \sum_{k=0}^{\infty} \frac{x^{3k+2}}{(3k+2)!}.
\] Given that $A^3+B^3+C^3 + 8ABC = 2014$, compute $ABC$.
[i]Proposed by Evan Chen[/i]
Let $a,b,c \in \left[ \frac 12, 1 \right]$. Prove that \[ 2 \leq \frac{ a+b}{1+c} + \frac{ b+c}{1+a} + \frac{ c+a}{1+b} \leq 3 . \]
[i]selected by Mircea Lascu[/i]
Let $a_1, a_2, \dots$ be an arithmetic sequence and $b_1, b_2, \dots$ be a geometric sequence. Suppose that $a_1 b_1 = 20$, $a_2 b_2 = 19$, and $a_3 b_3 = 14$. Find the greatest possible value of $a_4 b_4$.
Let $ x$ and $ y$ positive real numbers such that $ (1\plus{}x)(1\plus{}y)\equal{}2$. Show that $ xy\plus{}\frac{1}{xy}\geq\ 6$
Let $P(x)$ and $Q(x)$ be non-constant integer-coefficient polynomials such that for any integer $x\in \mathbb Z$, there exists integer $y\in \mathbb Z$ such that $P(x)=Q(y)$. Prove that the degree of $Q$ divides the degree of $P$.
Use calculus to find the behaviour of the function
\[y=e^x\sin{x} \ \ \ \ \ \ \ -\infty <x< +\infty\]
and sketch the graph of the function for $-2\pi \le x \le 2\pi$. Show clearly the locations of the maxima, minima and points of inflection in your graph.
Let $a$ and $b$ be positive integers, and let $A$ and $B$ be finite sets of integers satisfying
(i) $A$ and $B$ are disjoint;
(ii) if an integer $i$ belongs to either to $A$ or to $B$, then either $i+a$ belongs to $A$ or $i-b$ belongs to $B$.
Prove that $a\left\lvert A \right\rvert = b \left\lvert B \right\rvert$. (Here $\left\lvert X \right\rvert$ denotes the number of elements in the set $X$.)
For each integer $\, n \geq 2, \,$ determine, with proof, which of the two positive real numbers $\, a \,$ and $\, b \,$ satisfying \[ a^n = a + 1, \hspace{.3in} b^{2n} = b + 3a \] is larger.
Let $ S \equal{} \{2^0,2^1,2^2,\ldots,2^{10}\}$. Consider all possible positive differences of pairs of elements of $ S$. Let $ N$ be the sum of all of these differences. Find the remainder when $ N$ is divided by $ 1000$.
Find all real-valued continuously differentiable functions $f$ on the real line such that for all $x$, \[ \left( f(x) \right)^2 = \displaystyle\int_0^x \left[ \left( f(t) \right)^2 + \left( f'(t) \right)^2 \right] \, \mathrm{d}t + 1990. \]
Across all $x \in \mathbb{R}$, find the maximum value of the expression $$\sin x + \sin 3x + \sin 5x.$$
For what a from the interval $[0,\pi]$ do there exist $a$ and $b$ that are not simultaneously equal to zero, for which the inequality
$$a \cos x + b \cos 2x \le 0$$ is satisfied for all $x$ belonging to the segment $[a, \pi]$?