Olympiad problems

1968 IMO Shortlist, 19

We are given a fixed point on the circle of radius $1$, and going from this point along the circumference in the positive direction on curved distances $0, 1, 2, \ldots $ from it we obtain points with abscisas $n = 0, 1, 2, .\ldots$ respectively. How many points among them should we take to ensure that some two of them are less than the distance $\frac 15$ apart ?

2021 JHMT HS, 9

Tags: algebra , logarithm

Let $a$ and $b$ be positive real numbers such that $\log_{43}{a} = \log_{47} (3a + 4b) = \log_{2021}b^2$. Then, the value of $\tfrac{b^2}{a^2}$ can be written as $m + \sqrt{n}$, where $m$ and $n$ are integers. Find $m + n$.

2017 239 Open Mathematical Olympiad, 4

Tags: polynomial , algebra

A polynomial $f(x)$ with integer coefficients is given. We define $d(a,k)=|f^k(a)-a|.$ It is known that for each integer $a$ and natural number $k$, $d(a,k)$ is positive. Prove that for all such $a,k$, $$d(a,k) \geq \frac{k}{3}.$$ ($f^k(x)=f(f^{k-1}(x)), f^0(x)=x.$)

2012 Bundeswettbewerb Mathematik, 1

Tags: number theory , algebra

given a positive integer $n$. the set $\{ 1,2,..,2n \}$ is partitioned into $a_1<a_2<...<a_n $ and $b_1>b_2>...>b_n$. find the value of : $ \sum_{i=1}^{n}|a_i - b_i| $

2008 AMC 10, 15

Tags: algebra , difference of squares , special factorizations

How many right triangles have integer leg lengths $ a$ and $ b$ and a hypotenuse of length $ b\plus{}1$, where $ b<100$? $ \textbf{(A)}\ 6 \qquad \textbf{(B)}\ 7 \qquad \textbf{(C)}\ 8 \qquad \textbf{(D)}\ 9 \qquad \textbf{(E)}\ 10$

2017 South Africa National Olympiad, 6

Tags: algebra , polynomial , integer polynomial , number theory

Determine all pairs $(P, d)$ of a polynomial $P$ with integer coefficients and an integer $d$ such that the equation $P(x) - P(y) = d$ has infinitely many solutions in integers $x$ and $y$ with $x \neq y$.

2006 China Team Selection Test, 1

Tags: polynomial , induction , modular arithmetic , pigeonhole principle , algebra

Let $k$ be an odd number that is greater than or equal to $3$. Prove that there exists a $k^{th}$-degree integer-valued polynomial with non-integer-coefficients that has the following properties: (1) $f(0)=0$ and $f(1)=1$; and. (2) There exist infinitely many positive integers $n$ so that if the following equation: \[ n= f(x_1)+\cdots+f(x_s), \] has integer solutions $x_1, x_2, \dots, x_s$, then $s \geq 2^k-1$.

1984 IMO Longlists, 45

Tags: algebra

Let $X$ be an arbitrary nonempty set contained in the plane and let sets $A_1, A_2,\cdots,A_m$ and $B_1, B_2,\cdots, B_n$ be its images under parallel translations. Let us suppose that $A_1\cup A_2 \cup \cdots\cup A_m \subset B_1 \cup B_2 \cup\cdots\cup B_n$ and that the sets $A_1, A_2,\cdots,A_m$ are disjoint. Prove that $m \le n$.

1990 IMO Longlists, 35

Tags: calculus , derivative , geometric series , algebra

Prove that if $|x| < 1$, then \[ \frac{x}{(1-x)^2}+\frac{x^2}{(1+x^2)^2} + \frac{x^3}{(1-x^3)^2}+\cdots=\frac{x}{1-x}+\frac{2x^2}{1+x^2}+\frac{3x^3}{1-x^3}+\cdots\]

2007 Germany Team Selection Test, 2

Tags: absolute value , algebra

Determine the sum of absolute values for the complex roots of $ 20 x^8 \plus{} 7i x^7 \minus{}7ix \plus{} 20.$

2014 International Zhautykov Olympiad, 1

Tags: polynomial , calculus , integration , algebra

Does there exist a polynomial $P(x)$ with integral coefficients such that $P(1+\sqrt 3) = 2+\sqrt 3$ and $P(3+\sqrt 5) = 3+\sqrt 5 $? [i]Proposed by Alexander S. Golovanov, Russia[/i]

2016 Korea Junior Math Olympiad, 7

Tags: algebra , inequalities

positive integers $a_1, a_2, . . . , a_9$ satisfying $a_1+a_2+ . . . +a_9 =90$ find maximum of $$\frac{1^{a_1} \cdot 2^{a_2} \cdot . . . \cdot 9^{a_9}}{a_1! \cdot a_2! \cdot . . . \cdot a_9!}$$ [hide=mention] I was really shocked because there are no inequality problems at KJMO and the test difficulty even more lower...[/hide]

2021 South East Mathematical Olympiad, 7

Tags: inequalities , algebra

Let $a,b,c$ be pairwise distinct positive real, Prove that$$\dfrac{ab+bc+ca}{(a+b)(b+c)(c+a)}<\dfrac17(\dfrac{1}{|a-b|}+\dfrac{1}{|b-c|}+\dfrac{1}{|c-a|}).$$

2009 Hanoi Open Mathematics Competitions, 6

Tags: algebra , system of equations

Suppose that $4$ real numbers $a, b,c,d$ satisfy the conditions $\begin{cases} a^2 + b^2 = 4\\ c^2 + d^2 = 4 \\ ac + bd = 2 \end{cases}$ Find the set of all possible values the number $M = ab + cd$ can take.

1963 AMC 12/AHSME, 17

Tags: algebra , function , domain

The expression $\dfrac{\dfrac{a}{a+y}+\dfrac{y}{a-y}}{\dfrac{y}{a+y}-\dfrac{a}{a-y}}$, a real, $a\neq 0$, has the value $-1$ for: $\textbf{(A)}\ \text{all but two real values of }y \qquad \textbf{(B)}\ \text{only two real values of }y \qquad$ $\textbf{(C)}\ \text{all real values of }y \qquad \textbf{(D)}\ \text{only one real value of }y \qquad \textbf{(E)}\ \text{no real values of }y$

2017-IMOC, A1

Tags: inequalities , algebra

Prove that for all $a,b>0$ with $a+b=2$, we have $$\left(a^n+1\right)\left(b^n+1\right)\ge4$$ for all $n\in\mathbb N_{\ge2}$.

2016 South East Mathematical Olympiad, 5

Tags: complex number , inequalities , algebra , natural logarithm , china southeast mo

Let a constant $\alpha$ as $0<\alpha<1$, prove that: $(1)$ There exist a constant $C(\alpha)$ which is only depend on $\alpha$ such that for every $x\ge 0$, $\ln(1+x)\le C(\alpha)x^\alpha$. $(2)$ For every two complex numbers $z_1,z_2$, $|\ln|\frac{z_1}{z_2}||\le C(\alpha)\left(|\frac{z_1-z_2}{z_2}|^\alpha+|\frac{z_2-z_1}{z_1}|^\alpha\right)$.

2013 AMC 10, 5

Tags: algebra

Tom, Dorothy, and Sammy went on a vacation and agreed to split the costs evenly. During their trip Tom paid $\$105$, Dorothy paid $\$125$, and Sammy paid $\$175$. In order to share the costs equally, Tom gave Sammy $t$ dollars, and Dorothy gave Sammy $d$ dollars. What is $t-d$? $ \textbf{(A)}\ 15\qquad\textbf{(B)}\ 20\qquad\textbf{(C)}\ 25\qquad\textbf{(D)}\ 30\qquad\textbf{(E)}\ 35 $

2012 IFYM, Sozopol, 6

Tags: algebra

If $a$, $b$, and $c$ are positive numbers, determine the least possible value of the following expression: $\frac{1}{\frac{a}{b}+\frac{b}{c}+\frac{c}{a}}-\frac{2}{\frac{a}{c}+\frac{c}{b}+\frac{b}{a}}$.

2016 Japan Mathematical Olympiad Preliminary, 6

Tags: number theory , combinatorics , algebra

Integers $1 \le n \le 200$ are written on a blackboard just one by one. We surrounded just $100$ integers with circle. We call a square of the sum of surrounded integers minus the sum of not surrounded integers $score$ of this situation. Calculate the average score in all ways.

2023 China Girls Math Olympiad, 3

Tags: algebra , inequality , inequalities

Let $a,b,c,d \in [0,1] .$ Prove that$$\frac{1}{1+a+b}+\frac{1}{1+b+c}+\frac{1}{1+c+d}+\frac{1}{1+d+a}\leq \frac{4}{1+2\sqrt[4]{abcd}}$$

2020 Princeton University Math Competition, A1

Tags: algebra

Let $a_1, . . . , a_{2020}$ be a sequence of real numbers such that $a_1 = 2^{-2019}$, and $a^2_{n-1}a_n = a_n-a_{n-1}$. Prove that $a_{2020} <\frac{1}{2^{2019} -1}$

1949-56 Chisinau City MO, 48

Tags: trigonometry , algebra

Calculate $\sin^3 a + \cos^3 a$ if you know that $\sin a+ \cos a = m$.

2012 Indonesia TST, 1

Tags: polynomial , trigonometry , algebra

Suppose $P(x,y)$ is a homogenous non-constant polynomial with real coefficients such that $P(\sin t, \cos t) = 1$ for all real $t$. Prove that $P(x,y) = (x^2+y^2)^k$ for some positive integer $k$. (A polynomial $A(x,y)$ with real coefficients and having a degree of $n$ is homogenous if it is the sum of $a_ix^iy^{n-i}$ for some real number $a_i$, for all integer $0 \le i \le n$.)

1995 Czech and Slovak Match, 2

Tags: functional equation in z , algebra

Find all pairs of functions $f ,g : Z \rightarrow Z $ that satisfy $f (g(x)+y) = g( f (y)+x) $ for all integers $ x,y$ and such that $g(x) = g(y)$ only if $x = y$.