Olympiad problems

2013 Romania Team Selection Test, 1

Tags: geometry , 3d geometry , modular arithmetic , symmetry , algebra , polynomial , number theory

Given an integer $n\geq 2,$ let $a_{n},b_{n},c_{n}$ be integer numbers such that \[ \left( \sqrt[3]{2}-1\right) ^{n}=a_{n}+b_{n}\sqrt[3]{2}+c_{n}\sqrt[3]{4}. \] Prove that $c_{n}\equiv 1\pmod{3} $ if and only if $n\equiv 2\pmod{3}.$

2024 Thailand TST, 1

Tags: number theory , polynomial , algebra

Determine all polynomials $P$ with integer coefficients for which there exists an integer $a_n$ such that $P(a_n)=n^n$ for all positive integers $n$.

2007 Putnam, 1

Tags: algebra , polynomial , function

Let $ f$ be a polynomial with positive integer coefficients. Prove that if $ n$ is a positive integer, then $ f(n)$ divides $ f(f(n)\plus{}1)$ if and only if $ n\equal{}1.$

1994 Vietnam Team Selection Test, 3

Tags: polynomial , algebra

Let $P(x)$ be given a polynomial of degree 4, having 4 positive roots. Prove that the equation \[(1-4 \cdot x) \cdot \frac{P(x)}{x^2} + (x^2 + 4 \cdot x - 1) \cdot \frac{P'(x)}{x^2} - P''(x) = 0\] has also 4 positive roots.

1987 Romania Team Selection Test, 10

Tags: modular arithmetic , induction , function , algebra , polynomial , number theory

Let $a,b,c$ be integer numbers such that $(a+b+c) \mid (a^{2}+b^{2}+c^{2})$. Show that there exist infinitely many positive integers $n$ such that $(a+b+c) \mid (a^{n}+b^{n}+c^{n})$. [i]Laurentiu Panaitopol[/i]

2009 Tuymaada Olympiad, 4

Tags: algebra , polynomial , calculus , derivative , floor function , combinatorics

Each of the subsets $ A_1$, $ A_2$, $ \dots,$ $ A_n$ of a 2009-element set $ X$ contains at least 4 elements. The intersection of every two of these subsets contains at most 2 elements. Prove that in $ X$ there is a 24-element subset $ B$ containing neither of the sets $ A_1$, $ A_2$, $ \dots,$ $ A_n$.

2005 IMO Shortlist, 7

Tags: composite number , polynomial , number theory , algebra

Let $P(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\ldots+a_{0}$, where $a_{0},\ldots,a_{n}$ are integers, $a_{n}>0$, $n\geq 2$. Prove that there exists a positive integer $m$ such that $P(m!)$ is a composite number.

1994 Brazil National Olympiad, 4

Tags: function , polynomial , algebra

Let $a, b > 0$ be reals such that \[ a^3=a+1\\ b^6=b+3a \] Show that $a>b$

2008 China Team Selection Test, 3

Tags: gauss , polynomial , geometry , complex number , combinatorial geometry , algebra

Let $ z_{1},z_{2},z_{3}$ be three complex numbers of moduli less than or equal to $ 1$. $ w_{1},w_{2}$ are two roots of the equation $ (z \minus{} z_{1})(z \minus{} z_{2}) \plus{} (z \minus{} z_{2})(z \minus{} z_{3}) \plus{} (z \minus{} z_{3})(z \minus{} z_{1}) \equal{} 0$. Prove that, for $ j \equal{} 1,2,3$, $\min\{|z_{j} \minus{} w_{1}|,|z_{j} \minus{} w_{2}|\}\leq 1$ holds.

2011 Math Prize For Girls Problems, 11

Tags: linear algebra , matrix , algebra , polynomial , trigonometry , mewing

The sequence $a_0$, $a_1$, $a_2$, $\ldots\,$ satisfies the recurrence equation \[ a_n = 2 a_{n-1} - 2 a_{n - 2} + a_{n - 3} \] for every integer $n \ge 3$. If $a_{20} = 1$, $a_{25} = 10$, and $a_{30} = 100$, what is the value of $a_{1331}$?

2013 Romania National Olympiad, 1

Tags: linear algebra , matrix , vector , algebra , polynomial

Given A, non-inverted matrices of order n with real elements, $n\ge 2$ and given ${{A}^{*}}$adjoin matrix A. Prove that $tr({{A}^{*}})\ne -1$ if and only if the matrix ${{I}_{n}}+{{A}^{*}}$ is invertible.

2020 Estonia Team Selection Test, 3

Tags: number theory , polynomial

We say that a set $S$ of integers is [i]rootiful[/i] if, for any positive integer $n$ and any $a_0, a_1, \cdots, a_n \in S$, all integer roots of the polynomial $a_0+a_1x+\cdots+a_nx^n$ are also in $S$. Find all rootiful sets of integers that contain all numbers of the form $2^a - 2^b$ for positive integers $a$ and $b$.

2015 AMC 12/AHSME, 18

Tags: quadratic , function , polynomial , vieta , sfft , algebra

The zeroes of the function $f(x)=x^2-ax+2a$ are integers. What is the sum of all possible values of $a$? $\textbf{(A) }7\qquad\textbf{(B) }8\qquad\textbf{(C) }16\qquad\textbf{(D) }17\qquad\textbf{(E) }18$

Kvant 2022, M2714

Tags: number theory , polynomial , divisibility

Let $f{}$ and $g{}$ be polynomials with integers coefficients. The leading coefficient of $g{}$ is equal to 1. It is known that for infinitely many natural numbers $n{}$ the number $f(n)$ is divisible by $g(n)$ . Prove that $f(n)$ is divisible by $g(n)$ for all positive integers $n{}$ such that $g(n)\neq 0$. [i]From the folklore[/i]

India EGMO 2024 TST, 3

Tags: function , algebra , polynomial

Find all functions $f: \mathbb{N} \mapsto \mathbb{N}$ so that for any positive integer $n$ and finite sequence of positive integers $a_0, \dots, a_n$, whenever the polynomial $a_0+a_1x+\dots+a_nx^n$ has at least one integer root, so does \[f(a_0)+f(a_1)x+\dots+f(a_n)x^n.\] [i]Proposed by Sutanay Bhattacharya[/i]

2020-IMOC, A6

Tags: algebra , polynomial

$\definecolor{A}{RGB}{255,0,0}\color{A}\fbox{A6.}$ Let $ P (x)$ be a polynomial with real coefficients such that $\deg P \ge 3$ is an odd integer. Let $f : \mathbb{R}\rightarrow\mathbb{Z}$ be a function such that $$\definecolor{A}{RGB}{0,0,200}\color{A}\forall_{x\in\mathbb{R}}\ f(P(x)) = P(f(x)).$$ $\definecolor{A}{RGB}{255,150,0}\color{A}\fbox{(a)}$ Prove that the range of $f$ is finite. $\definecolor{A}{RGB}{255,150,0}\color{A}\fbox{(b)}$ Show that for any positive integer $n$, there exist $P$, $f$ that satisfies the above condition and also that the range of $f$ has cardinality $n$. [i]Proposed by [/i][b][color=#419DAB]ltf0501[/color][/b]. [color=#3D9186]#1735[/color]

2019 Iran Team Selection Test, 4

Tags: algebra , polynomial

Let $1<t<2$ be a real number. Prove that for all sufficiently large positive integers like $d$, there is a monic polynomial $P(x)$ of degree $d$, such that all of its coefficients are either $+1$ or $-1$ and $$\left|P(t)-2019\right| <1.$$ [i]Proposed by Navid Safaei[/i]

2010 USA Team Selection Test, 8

Tags: induction , algebra , polynomial , combinatorics

Let $m,n$ be positive integers with $m \geq n$, and let $S$ be the set of all $n$-term sequences of positive integers $(a_1, a_2, \ldots a_n)$ such that $a_1 + a_2 + \cdots + a_n = m$. Show that \[\sum_S 1^{a_1} 2^{a_2} \cdots n^{a_n} = {n \choose n} n^m - {n \choose n-1} (n-1)^m + \cdots + (-1)^{n-2} {n \choose 2} 2^m + (-1)^{n-1} {n \choose 1}.\]

2010 Indonesia TST, 1

Tags: geometry , trigonometry , algebra , polynomial , induction , function , ratio

Is there a triangle with angles in ratio of $ 1: 2: 4$ and the length of its sides are integers with at least one of them is a prime number? [i]Nanang Susyanto, Jogjakarta[/i]

2010 Romania Team Selection Test, 2

Tags: ceiling function , algebra , polynomial , number theory

(a) Given a positive integer $k$, prove that there do not exist two distinct integers in the open interval $(k^2, (k + 1)^2)$ whose product is a perfect square. (b) Given an integer $n > 2$, prove that there exist $n$ distinct integers in the open interval $(k^n, (k + 1)^n)$ whose product is the $n$-th power of an integer, for all but a finite number of positive integers $k$. [i]AMM Magazine[/i]

2006 Pre-Preparation Course Examination, 4

Tags: algebra , polynomial , geometry

Find a 3rd degree polynomial whose roots are $r_a$, $r_b$ and $r_c$ where $r_a$ is the radius of the outer inscribed circle of $ABC$ with respect to $A$.

1996 Romania National Olympiad, 2

Tags: polynomial , algebra

Find all polynomials $p_n(x)=a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0$ ($n\geq 2$) with real and non-zero coeficients s.t. $p_n(x)-p_1(x)p_2(x)...p_{n-1}(x)$ be a constant polynomial. ;)

2019 Dutch IMO TST, 1

Tags: trinomial , quadratic trinomial , algebra , polynomial

Let $P(x)$ be a quadratic polynomial with two distinct real roots. For all real numbers $a$ and $b$ satisfying $|a|,|b| \ge 2017$, we have $P(a^2+b^2) \ge P(2ab)$. Show that at least one of the roots of $P$ is negative.

1970 Miklós Schweitzer, 12

Tags: probability , parameterization , induction , algebra , polynomial , probability and stats

Let $ \vartheta_1,...,\vartheta_n$ be independent, uniformly distributed, random variables in the unit interval $ [0,1]$. Define \[ h(x)\equal{} \frac1n \# \{k: \; \vartheta_k<x\ \}.\] Prove that the probability that there is an $ x_0 \in (0,1)$ such that $ h(x_0)\equal{}x_0$, is equal to $ 1\minus{} \frac1n.$ [i]G. Tusnady[/i]

1984 USAMO, 1

Tags: algebra , polynomial

The product of two of the four roots of the quartic equation $x^4 - 18x^3 + kx^2+200x-1984=0$ is $-32$. Determine the value of $k$.