Olympiad problems

2020 Silk Road, 3

Tags: algebra , polynomial

A polynomial $ Q (x) = k_n x ^ n + k_ {n-1} x ^ {n-1} + \ldots + k_1 x + k_0 $ with real coefficients is called [i]powerful[/i] if the equality $ | k_0 | = | k_1 | + | k_2 | + \ldots + | k_ {n-1} | + | k_n | $, and [i]non-increasing[/i] , if $ k_0 \geq k_1 \geq \ldots \geq k_ {n-1} \geq k_n $. Let for the polynomial $ P (x) = a_d x ^ d + a_ {d-1} x ^ {d-1} + \ldots + a_1 x + a_0 $ with nonzero real coefficients, where $ a_d> 0 $, the polynomial $ P (x) (x-1) ^ t (x + 1) ^ s $ is [i]powerful[/i] for some non-negative integers $ s $ and $ t $ ($ s + t> 0 $). Prove that at least one of the polynomials $ P (x) $ and $ (- 1) ^ d P (-x) $ is [i]nonincreasing[/i].

2016 BMT Spring, 8

Tags: algebra , calculus

Evaluate the following limit $$\lim_{x\to 0} (1 + 2x + 3x^2 + 4x^3 +...)^{1/x}$$

PEN A Problems, 112

Tags: induction , algebra , relatively prime , vieta , divisibility theory , number theory

Prove that there exist infinitely many pairs $(a, b)$ of relatively prime positive integers such that \[\frac{a^{2}-5}{b}\;\; \text{and}\;\; \frac{b^{2}-5}{a}\] are both positive integers.

2020 Bulgaria Team Selection Test, 5

Tags: inequalities , algebra , functional equation , function , additive function , induction

Given is a function $f:\mathbb{R}\rightarrow \mathbb{R}$ such that $|f(x+y)-f(x)-f(y)|\leq 1$. Prove the existence of an additive function $g:\mathbb{R}\rightarrow \mathbb{R}$ (that is $g(x+y)=g(x)+g(y)$) such that $|f(x)-g(x)|\leq 1$ for any $x \in \mathbb{R}$

2010 Contests, 2

Tags: algebra

Let $\{a_{n}\}$ be a sequence which satisfy $a_{1}=5$ and $a_{n=}\sqrt[n]{a_{n-1}^{n-1}+2^{n-1}+2.3^{n-1}} \qquad \forall n\geq2$ [b](a)[/b] Find the general fomular for $a_{n}$ [b](b)[/b] Prove that $\{a_{n}\}$ is decreasing sequences

2003 Greece JBMO TST, 1

Tags: computational , algebra

If point $M(x,y)$ lies on the line with equation $y=x+2$ and $1<y<3$, calculate the value of $A=\sqrt{y^2-8x}+\sqrt{y^2+2x+5}$

2023 Myanmar IMO Training, 8

Tags: algebra , system of equations

Find all real numbers $a, b, c$ that satisfy $$ 2a - b =a^2b, \qquad 2b-c = b^2 c, \qquad 2c-a= c^2 a.$$

2006 Iran Team Selection Test, 4

Tags: number theory , algebra , divisibility tests , polynomial

Let $n$ be a fixed natural number. Find all $n$ tuples of natural pairwise distinct and coprime numbers like $a_1,a_2,\ldots,a_n$ such that for $1\leq i\leq n$ we have \[ a_1+a_2+\ldots+a_n|a_1^i+a_2^i+\ldots+a_n^i \]

2014 ELMO Shortlist, 3

Tags: reflection , function , algebra , geometry

We say a finite set $S$ of points in the plane is [i]very[/i] if for every point $X$ in $S$, there exists an inversion with center $X$ mapping every point in $S$ other than $X$ to another point in $S$ (possibly the same point). (a) Fix an integer $n$. Prove that if $n \ge 2$, then any line segment $\overline{AB}$ contains a unique very set $S$ of size $n$ such that $A, B \in S$. (b) Find the largest possible size of a very set not contained in any line. (Here, an [i]inversion[/i] with center $O$ and radius $r$ sends every point $P$ other than $O$ to the point $P'$ along ray $OP$ such that $OP\cdot OP' = r^2$.) [i]Proposed by Sammy Luo[/i]

2022 All-Russian Olympiad, 3

Tags: algebra

Initially, a pair of numbers $(1,1)$ is written on the board. If for some $x$ and $y$ one of the pairs $(x, y-1)$ and $(x+y, y+1)$ is written on the board, then you can add the other one. Similarly for $(x, xy)$ and $(\frac {1} {x}, y)$. Prove that for each pair that appears on the board, its first number will be positive.

1963 German National Olympiad, 2

Tags: algebra , trigonometry

For which numbers $x$of the interval $0 < x <\pi$ holds: $$\frac{\tan 2x}{\tan x} -\frac{2 \cot 2x}{\cot x}=1$$

2011 Peru MO (ONEM), 2

Tags: algebra , inequalities , trigonometry

If $\alpha, \beta, \gamma$ are angles whose measures in radians belong to the interval $\left[0, \frac{\pi}{2}\right]$ such that: $$\sin^2 \alpha + \sin^2 \beta + \sin^2 \gamma = 1$$ calculate the minimum possible value of $\cos \alpha + \cos \beta + \cos \gamma$.

2005 IMC, 4

Tags: polynomial , algebra

4) find all polynom with coeffs a permutation of $[1,...,n]$ and all roots rational

2019 USA TSTST, 7

Tags: greatest common divisor , number theory , functional equation , algebra

Let $f: \mathbb Z\to \{1, 2, \dots, 10^{100}\}$ be a function satisfying $$\gcd(f(x), f(y)) = \gcd(f(x), x-y)$$ for all integers $x$ and $y$. Show that there exist positive integers $m$ and $n$ such that $f(x) = \gcd(m+x, n)$ for all integers $x$. [i]Ankan Bhattacharya[/i]

VMEO IV 2015, 10.1

Tags: recurrence relation , algebra , sequence , recurrence , limit of sequence

Where $n$ is a positive integer, the sequence $a_n$ is determined by the formula $$a_{n+1}=\frac{1}{a_1 + a_2 +... + a_n} -\sqrt2, \,a_1 = 1.$$ Find the limit of the sequence $S_n$ defined by $S_n=a_1 + a_2 +... + a_n$.

2015 Vietnam Team selection test, Problem 6

Tags: algebra

Find the smallest positive interger number $n$ such that there exists $n$ real numbers $a_1,a_2,\ldots,a_n$ satisfied three conditions as follow: a. $a_1+a_2+\cdots+a_n>0$; b. $a_1^3+a_2^3+\cdots+a_n^3<0$; c. $a_1^5+a_2^5+\cdots+a_n^5>0$.

1971 Spain Mathematical Olympiad, 6

Tags: algebra , inequalities

The velocities of a submerged and surfaced submarine are, respectively, $v$ and $kv$. It is situated at a point $P$ at $30$ miles from the center $O$ of a circle of $60$ mile radius. The surveillance of an enemy squadron forces him to navigate submerged while inside the circle. Discuss, according to the values of $k$, the fastest path to move to the opposite end of the diameter that passes through $P$ . (Consider the case particular $k =\sqrt5$.)

2017 Thailand TSTST, 4

Tags: functional equation , function , algebra

Find all function $f:\mathbb{N}^*\rightarrow \mathbb{N}^*$ that satisfy: $(f(1))^3+(f(2))^3+...+(f(n))^3=(f(1)+f(2)+...+f(n))^2$

2001 Cuba MO, 6

Tags: inequalities , algebra

The roots of the equation $ax^2 - 4bx + 4c = 0$ with $ a > 0$ belong to interval $[2, 3]$. Prove that: a) $a \le b \le c < a + b.$ b) $\frac{a}{a+c} + \frac{b}{b+a} > \frac{c}{b+c} .$

2009 Stanford Mathematics Tournament, 10

Tags: algebra

Evaluate $\sum_{n=2009}^{\infty} \frac{ {n \choose 2009}}{2^n}$

2020 Jozsef Wildt International Math Competition, W26

Tags: sequence , recurrence relation , algebra

Let $P_n$ denote the $n$-th Pell number defined by $P_{n+1}=2P_n+P_{n-1}$, $P_0=0$, $P_1=1$. Furthermore, let $T_n$ denote the $n$-th triangular number, that is $T_n=\binom{n+1}2$. Show that $$\sum_{n=0}^\infty4T_n\cdot\frac{P_n}{3^{n+2}}=P_3+P_4$$ [i]Proposed by Ángel Plaza[/i]

2004 China Team Selection Test, 3

Tags: algebra

Find all positive integer $ n$ satisfying the following condition: There exist positive integers $ m$, $ a_1$, $ a_2$, $ \cdots$, $ a_{m\minus{}1}$, such that $ \displaystyle n \equal{} \sum_{i\equal{}1}^{m\minus{}1} a_i(m\minus{}a_i)$, where $ a_1$, $ a_2$, $ \cdots$, $ a_{m\minus{}1}$ may not distinct and $ 1 \leq a_i \leq m\minus{}1$.

2023 Iran MO (3rd Round), 1

Tags: algebra , polynomial

Find all integers $n > 4$ st for every two subsets $A,B$ of $\{0,1,....,n-1\}$ , there exists a polynomial $f$ with integer coefficients st either $f(A) = B$ or $f(B) = A$ where the equations are considered mod n. We say two subsets are equal mod n if they produce the same set of reminders mod n. and the set $f(X)$ is the set of reminders of $f(x)$ where $x \in X$ mod n.

2005 Romania National Olympiad, 4

Tags: algebra , inequalities

a) Prove that for all positive reals $u,v,x,y$ the following inequality takes place: \[ \frac ux + \frac vy \geq \frac {4(uy+vx)}{(x+y)^2} . \] b) Let $a,b,c,d>0$. Prove that \[ \frac a{b+2c+d} + \frac b{c+2d+a} + \frac c{d+2a+b} + \frac d{a+2b+c} \geq 1.\] [i]Traian Tămâian[/i]

2021 China Team Selection Test, 5

Tags: algebra , function

Determine all $ f:R\rightarrow R $ such that $$ f(xf(y)+y^3)=yf(x)+f(y)^3 $$