Olympiad problems

2000 Baltic Way, 17

Tags: polynomial , vieta , algebra , quadratic , system of equations

Find all real solutions to the following system of equations: \[\begin{cases} x+y+z+t=5\\xy+yz+zt+tx=4\\xyz+yzt+ztx+txy=3\\xyzt=-1\end{cases}\]

2005 Today's Calculation Of Integral, 64

Tags: trigonometry , integration , algebra , calculus , polynomial , calculus computations

Let $f(t)$ be the cubic polynomial for $t$ such that $\cos 3x=f(\cos x)$ holds for all real number $x$. Evaluate \[\int_0^1 \{f(t)\}^2 \sqrt{1-t^2}dt\]

1997 Romania National Olympiad, 3

Tags: polynomial , function , algebra

Suppose that $a,b,c,d\in\mathbb{R}$ and $f(x)=ax^3+bx^2+cx+d$ such that $f(2)+f(5)<7<f(3)+f(4)$. Prove that there exists $u,v\in\mathbb{R}$ such that $u+v=7 , f(u)+f(v)=7$

2009 Princeton University Math Competition, 8

Tags: geometry , polynomial , quadratic , algebra

The real numbers $x$, $y$, $z$, and $t$ satisfy the following equation: \[2x^2 + 4xy + 3y^2 - 2xz -2 yz + z^2 + 1 = t + \sqrt{y + z - t} \] Find 100 times the maximum possible value for $t$.

1994 All-Russian Olympiad Regional Round, 10.6

Tags: polynomial , algebra , integer polynomial

Find the free coefficient of the polynomial $P(x)$ with integer coefficients, knowing that it is less than $1000$ in absolute value and that $P(19) = P(94) = 1994$.

2014-2015 SDML (High School), 6

Tags: algebra , polynomial , sum of roots

Find the largest integer $k$ such that $$k\leq\sqrt{2}+\sqrt[3]{\frac{3}{2}}+\sqrt[4]{\frac{4}{3}}+\sqrt[5]{\frac{5}{4}}+\cdots+\sqrt[2015]{\frac{2015}{2014}}.$$

Find all positive integers $n$ and sequence of integers $a_0,a_1,\ldots, a_n$ such that the following hold: 1. $a_n\neq 0$; 2. $f(a_{i-1})=a_i$ for all $i=1,\ldots, n$, where $f(x) = a_nx^n+a_{n-1}x^{n-1}+\cdots +a_0$. [i] Proposed by usjl[/i]

1958 AMC 12/AHSME, 41

Tags: algebra , polynomial , vieta

The roots of $ Ax^2 \plus{} Bx \plus{} C \equal{} 0$ are $ r$ and $ s$. For the roots of \[ x^2 \plus{} px \plus{} q \equal{} 0 \] to be $ r^2$ and $ s^2$, $ p$ must equal: $ \textbf{(A)}\ \frac{B^2 \minus{} 4AC}{A^2}\qquad \textbf{(B)}\ \frac{B^2 \minus{} 2AC}{A^2}\qquad \textbf{(C)}\ \frac{2AC \minus{} B^2}{A^2}\qquad \\ \textbf{(D)}\ B^2 \minus{} 2C\qquad \textbf{(E)}\ 2C \minus{} B^2$

2002 Mongolian Mathematical Olympiad, Problem 2

Tags: polynomial , irreducibility , algebra

Prove that for each $n\in\mathbb N$ the polynomial $(x^2+x)^{2^n}+1$ is irreducible over the polynomials with integer coefficients.

2012 Harvard-MIT Mathematics Tournament, 9

Tags: algebra , trigonometry , hmmt , polynomial , function

How many real triples $(a,b,c)$ are there such that the polynomial $p(x)=x^4+ax^3+bx^2+ax+c$ has exactly three distinct roots, which are equal to $\tan y$, $\tan 2y$, and $\tan 3y$ for some real number $y$?

1977 IMO Shortlist, 9

Tags: algebra , polynomial , functional equation

For which positive integers $n$ do there exist two polynomials $f$ and $g$ with integer coefficients of $n$ variables $x_1, x_2, \ldots , x_n$ such that the following equality is satisfied: \[\sum_{i=1}^n x_i f(x_1, x_2, \ldots , x_n) = g(x_1^2, x_2^2, \ldots , x_n^2) \ ? \]

2019 Vietnam National Olympiad, Day 1

Tags: polynomial

For each real coefficient polynomial $f(x)={{a}_{0}}+{{a}_{1}}x+\cdots +{{a}_{n}}{{x}^{n}}$, let $$\Gamma (f(x))=a_{0}^{2}+a_{1}^{2}+\cdots +a_{m}^{2}.$$ Let be given polynomial $P(x)=(x+1)(x+2)\ldots (x+2020).$ Prove that there exists at least $2019$ pairwise distinct polynomials ${{Q}_{k}}(x)$ with $1\le k\le {{2}^{2019}}$ and each of it satisfies two following conditions: i) $\deg {{Q}_{k}}(x)=2020.$ ii) $\Gamma \left( {{Q}_{k}}{{(x)}^{n}} \right)=\Gamma \left( P{{(x)}^{n}} \right)$ for all positive initeger $n$.

2004 Unirea, 3

Tags: polynomial , integration , real analysis , trigonometry , algebra

Hello, I've been trying to solve this for a while now, but no success! I mean, I have an expression for this but not a closed one. I derived something in terms of Tchebychev Polynomials : cos(nx) = P_n(cos(x)). Any help is appreciated. Compute the following primitive: \[ \int \frac{x\sin\left(2004 x\right)}{\tan x}\ dx\]

2025 Vietnam Team Selection Test, 6

Tags: algebra , polynomial

For each prime $p$ of the form $4k+3$ with $k \in \mathbb{Z}^+$, consider the polynomial $$Q(x)=px^{2p} - x^{2p-1} + p^2x^{\frac{3p+1}{2}} - px^{p+1} +2(p^2+1)x^p -px^{p-1}+ p^2 x^{\frac{p-1}{2}} -x + p.$$ Determine all ordered pairs of polynomials $f, g$ with integer coefficients such that $Q(x)=f(x)g(x)$.

1991 IMO Shortlist, 23

Tags: algebra , function , functional equation , polynomial

Let $ f$ and $ g$ be two integer-valued functions defined on the set of all integers such that [i](a)[/i] $ f(m \plus{} f(f(n))) \equal{} \minus{}f(f(m\plus{} 1) \minus{} n$ for all integers $ m$ and $ n;$ [i](b)[/i] $ g$ is a polynomial function with integer coefficients and g(n) = $ g(f(n))$ $ \forall n \in \mathbb{Z}.$

1988 Poland - Second Round, 1

Tags: algebra , polynomial

Let $ f(x) $ be a polynomial, $ n $ - a natural number. Prove that if $ f(x^{n}) $ is divisible by $ x-1 $, then it is also divisible by $ x^{n-1} + x^{n-2} + \ldots + x + $1.

2003 National Olympiad First Round, 8

Tags: polynomial , algebra

Let $P$ be a polynomial such that $(x-4)P(2x) = 4(x-1)P(x)$, for every real $x$. If $P(0) \neq 0$, what is the degree of $P$? $ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ \text{None of the preceding} $

2003 India National Olympiad, 1

Tags: algebra , polynomial , trigonometry , geometry , angle bisector

Let $P$ be an interior point of an acute-angled triangle $ABC$. The line $BP$ meets the line $AC$ at $E$, and the line $CP$ meets the line $AB$ at $F$. The lines $AP$ and $EF$ intersect each other at $D$. Let $K$ be the foot of the perpendicular from the point $D$ to the line $BC$. Show that the line $KD$ bisects the angle $\angle EKF$.

1995 All-Russian Olympiad, 3

Tags: polynomial , symmetry , algebra , quadratic , complex number , function

Can the equation $f(g(h(x))) = 0$, where $f$, $g$, $h$ are quadratic polynomials, have the solutions $1, 2, 3, 4, 5, 6, 7, 8$? [i]S. Tokarev[/i]

2015 Saudi Arabia Pre-TST, 1.2

Tags: polynomial , algebra

How many polynomials $P$ of integer coefficients and degree at most $4$ satisfy $0 \le P(x) < 72$ for all $x\in \{0, 1, 2, 3, 4\}$? Harvard-MIT Mathematics Tournament 2011

1998 Harvard-MIT Mathematics Tournament, 4

Tags: derivative , polynomial , calculus , function , algebra , trigonometry

Find the range of $ f(A)=\frac{\sin A(3\cos^{2}A+\cos^{4}A+3\sin^{2}A+\sin^{2}A\cos^{2}A)}{\tan A (\sec A-\sin A\tan A)} $ if $A\neq \dfrac{n\pi}{2}$.

2022 ISI Entrance Examination, 7

Tags: polynomial , limit

Let $$P(x)=1+2 x+7 x^{2}+13 x^{3}~,\qquad x \in \mathbb{R} .$$ Calculate for all $x \in \mathbb{R},$ $$\lim _{n \rightarrow \infty}\left(P\left(\frac{x}{n}\right)\right)^{n}$$

2019 Singapore MO Open, 4

Tags: number theory , prime number , polynomial , algebra

Let $p \equiv 2 \pmod 3$ be a prime, $k$ a positive integer and $P(x) = 3x^{\frac{2p-1}{3}}+3x^{\frac{p+1}{3}}+x+1$. For any integer $n$, let $R(n)$ denote the remainder when $n$ is divided by $p$ and let $S = \{0,1,\cdots,p-1\}$. At each step, you can either (a) replaced every element $i$ of $S$ with $R(P(i))$ or (b) replaced every element $i$ of $S$ with $R(i^k)$. Determine all $k$ such that there exists a finite sequence of steps that reduces $S$ to $\{0\}$. [i]Proposed by fattypiggy123[/i]

1997 Romania Team Selection Test, 4

Tags: polynomial , modular arithmetic , number theory , algebra , greatest common divisor

Let $n\ge 2$ be an integer and let $P(X)=X^n+a_{n-1}X^{n-1}+\ldots +a_1X+1$ be a polynomial with positive integer coefficients. Suppose that $a_k=a_{n-k}$ for all $k\in 1,2,\ldots,n-1$. Prove that there exist infinitely many pairs of positive integers $x,y$ such that $x|P(y)$ and $y|P(x)$. [i]Remus Nicoara[/i]

2006 Moldova Team Selection Test, 3

Tags: algebra , polynomial , inequalities

Let $a,b,c$ be sides of the triangle. Prove that \[ a^2\left(\frac{b}{c}-1\right)+b^2\left(\frac{c}{a}-1\right)+c^2\left(\frac{a}{b}-1\right)\geq 0 . \]