Olympiad problems

1996 Estonia National Olympiad, 1

Tags: algebra

A fisherman, who was sailing in a rowing boat against the current of the river, had a hat falling from the bow of the boat into the water. After half an hour, the fisherman noticed the loss of his cap and immediately turned back. Find the speed of the river if the fisherman caught up with the cap at a distance of $a$ km from the place where it fell into the water (the speed of the river and the movement of the boat relative to the water is considered constant).

1964 Vietnam National Olympiad, 2

Tags: algebra , parametric equation , quadratic

Draw the graph of the functions $y = | x^2 - 1 |$ and $y = x + | x^2 -1 |$. Find the number of roots of the equation $x + | x^2 - 1 | = k$, where $k$ is a real constant.

2008 China Northern MO, 2

Tags: algebra , pascal triangle

The given triangular number table is as follows: [img]https://cdn.artofproblemsolving.com/attachments/a/0/123b7511850047f3cc51494f107703f2757085.png[/img] Among them, the numbers in the first row are $1, 2, 3, ..., 98, 99, 100$. Starting from the second row, each number is equal to the sum of the left and right numbers in the row above it. Find the value of $M$.

2019 India PRMO, 7

Tags: algebra , decimal representation , number theory

Let $s(n)$ denote the sum of digits of a positive integer $n$ in base $10$. If $s(m)=20$ and $s(33m)=120$, what is the value of $s(3m)$?

2022 Switzerland Team Selection Test, 7

Tags: algebra , real coefficients , polynomial

Let $n$ be a positive integer. Find all polynomials $P$ with real coefficients such that $$P(x^2+x-n^2)=P(x)^2+P(x)$$ for all real numbers $x$.

2015 Dutch BxMO/EGMO TST, 5

Tags: functional equation , algebra

Find all functions $f : R \to R$ satisfying $(x^2 + y^2)f(xy) = f(x)f(y)f(x^2 + y^2)$ for all real numbers $x$ and $y$.

VII Soros Olympiad 2000 - 01, 8.7

Tags: algebra , polynomial

In the expression $(x + 100) (x + 99) ... (x-99) (x-100)$, the brackets were expanded and similar terms were given. The expression $x^{201} + ...+ ax^2 + bx + c$ turned out. Find the numbers $a$ and $c$.

2017 Mathematical Talent Reward Programme, MCQ: P 6

Tags: algebra , polynomial

Let $p(x)$ be a polynomial of degree 4 with leading coefficients 1. Suppose $p(1)=1$, $p(2)=2$, $p(3)=3$, $p(4)=4$. Then $p(5)=$ [list=1] [*] 5 [*] $\frac{25}{6}$ [*] 29 [*] 35 [/list]

2024 Ukraine National Mathematical Olympiad, Problem 2

Tags: algebra , number theory , power of 2

You are given a positive integer $n$. Find the smallest positive integer $k$, for which there exist integers $a_1, a_2, \ldots, a_k$, for which the following equality holds: $$2^{a_1} + 2^{a_2} + \ldots + 2^{a_k} = 2^n - n + k$$ [i]Proposed by Mykhailo Shtandenko[/i]

2003 IMC, 3

Tags: polynomial , algebra , linear algebra , matrix

Let $A\in\mathbb{R}^{n\times n}$ such that $3A^3=A^2+A+I$. Show that the sequence $A^k$ converges to an idempotent matrix. (idempotent: $B^2=B$)

1989 IMO Longlists, 90

Tags: algebra , induction

Find the set of all $ a \in \mathbb{R}$ for which there is no infinite sequene $ (x_n)_{n \geq 0} \subset \mathbb{R}$ satisfying $ x_0 \equal{} a,$ and for $ n \equal{} 0,1, \ldots$ we have \[ x_{n\plus{}1} \equal{} \frac{x_n \plus{} \alpha}{\beta x_n \plus{} 1}\] where $ \alpha \beta > 0.$

1984 IMO Longlists, 44

Tags: geometry , pythagorean theorem , system of equations , algebra

Let $a,b,c$ be positive numbers with $\sqrt{a}+\sqrt{b}+\sqrt{c}= \frac{\sqrt{3}}{2}$ Prove that the system of equations \[\sqrt{y-a}+\sqrt{z-a}=1\] \[\sqrt{z-b}+\sqrt{x-b}=1\] \[\sqrt{x-c}+\sqrt{y-c}=1\] has exactly one solution $(x,y,z)$ in real numbers. It was proposed by Poland. Have fun! :lol:

2009 District Olympiad, 2

Tags: algebra , complex number , absolute value

Find the complex numbers $ z_1,z_2,z_3 $ of same absolute value having the property that: $$ 1=z_1z_2z_3=z_1+z_2+z_3. $$

2021 Polish Junior MO Finals, 1

Tags: algebra , positive integer , square root

Positive integers $a$, $b$ an $n$ satisfy \[ \frac{a}{b}=\frac{a^2+n^2}{b^2+n^2}. \] Prove that $\sqrt{ab}$ is an integer.

2010 AMC 12/AHSME, 21

Tags: algebra , vieta , analytic geometry , polynomial

The graph of $ y \equal{} x^6 \minus{} 10x^5 \plus{} 29x^4 \minus{} 4x^3 \plus{} ax^2$ lies above the line $ y \equal{} bx \plus{} c$ except at three values of $ x$, where the graph and the line intersect. What is the largest of those values? $ \textbf{(A)}\ 4 \qquad \textbf{(B)}\ 5 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 7 \qquad \textbf{(E)}\ 8$

2023 Iran Team Selection Test, 5

Tags: function , algebra

Suppose that $n\ge2$ and $a_1,a_2,...,a_n$ are natural numbers that $ (a_1,a_2,...,a_n)=1$. Find all strictly increasing function $f: \mathbb{Z} \to \mathbb{R} $ that: $$ \forall x_1,x_2,...,x_n \in \mathbb{Z} : f(\sum_{i=1}^{n} {x_ia_i}) = \sum_{i=1}^{n} {f(x_ia_i})$$ [i]Proposed by Navid Safaei and Ali Mirzaei [/i]

2022 Estonia Team Selection Test, 5

Tags: combinatorics , algebra

For each integer $n\ge 1,$ compute the smallest possible value of \[\sum_{k=1}^{n}\left\lfloor\frac{a_k}{k}\right\rfloor\] over all permutations $(a_1,\dots,a_n)$ of $\{1,\dots,n\}.$ [i]Proposed by Shahjalal Shohag, Bangladesh[/i]

2015 Balkan MO Shortlist, A4

Tags: algebra , function , functional equation

Find all functions $f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ such that $$ (x+y)f(2yf(x)+f(y))=x^{3}f(yf(x)), \ \ \ \forall x,y\in \mathbb{R}^{+}.$$ (Albania)

2019 ELMO Shortlist, A1

Tags: inequalities , algebra , inequality

Let $a$, $b$, $c$ be positive reals such that $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=1$. Show that $$a^abc+b^bca+c^cab\ge 27bc+27ca+27ab.$$ [i]Proposed by Milan Haiman[/i]

2022 Durer Math Competition (First Round), 5

Tags: algebra , sequence

Let $a_1 \le a_2 \le ... \le a_n$ be real numbers for which $$\sum_{i=1}^{n} a_i^{2k+1} = 0$$ holds for all integers $0 \le k < n$. Show that in this case, $a_i = -a_{n+1-i}$ holds for all $1 \le i \le n$.

2009 Kosovo National Mathematical Olympiad, 3

Tags: inequalities , algebra

Let $a,b$ and $c$ be the sides of a triangle, prove that $\frac {a}{b+c}+\frac {b}{c+a}+\frac {c}{a+b}<2$.

1949 Moscow Mathematical Olympiad, 162

Tags: algebra , geometric progression , geometric sequence , combinatorics

Given a set of $4n$ positive numbers such that any distinct choice of ordered foursomes of these numbers constitutes a geometric progression. Prove that at least $4$ numbers of the set are identical.

1950 Poland - Second Round, 1

Tags: algebra , system , system of equations

Solve the system of equations $$\begin{cases} x^2+x+y=8\\ y^2+2xy+z=168\\ z^2+2yz+2xz=12480 \end{cases}$$

2016 Korea Winter Program Practice Test, 3

Tags: functional equation , algebra

Determine all the functions $f : \mathbb{R}\rightarrow\mathbb{R}$ that satisfies the following. $f(xf(y)+yf(z)+zf(x))=yf(x)+zf(y)+xf(z)$

1987 Brazil National Olympiad, 1

Tags: number theory , algebra , coprime , prime number , polynomial

$p(x_1, x_2, ... , x_n)$ is a polynomial with integer coefficients. For each positive integer $r, k(r)$ is the number of $n$-tuples $(a_1, a_2,... , a_n)$ such that $0 \le a_i \le r-1 $ and $p(a_1, a_2, ... , a_n)$ is prime to $r$. Show that if $u$ and $v$ are coprime then $k(u\cdot v) = k(u)\cdot k(v)$, and if p is prime then $k(p^s) = p^{n(s-1)} k(p)$.