This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

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Found problems: 15925

2021 Princeton University Math Competition, B2
Tags: algebra
Kris is asked to compute $\log_{10} (x^y)$, where $y$ is a positive integer and $x$ is a positive real number. However, they misread this as $(\log_{10} x)^y$ , and compute this value. Despite the reading error, Kris still got the right answer. Given that $x > 10^{1.5}$ , determine the largest possible value of $y$.

2020 HK IMO Preliminary Selection Contest, 2
Tags: integer , algebra
Let $x$, $y$, $z$ be positive integers satisfying $x<y<z$ and $x+xy+xyz=37$. Find the greatest possible value of $x+y+z$.

2017 Harvard-MIT Mathematics Tournament, 1
Tags: algebra , polynomial
Let $Q(x)=a_0+a_1x+\dots+a_nx^n$ be a polynomial with integer coefficients, and $0\le a_i<3$ for all $0\le i\le n$. Given that $Q(\sqrt{3})=20+17\sqrt{3}$, compute $Q(2)$.

1974 Polish MO Finals, 5
Tags: binomial coefficient , algebra , arithmetic sequence
Prove that for any natural numbers $n,r$ with $r + 3 \le n $the binomial coefficients $n \choose r$, $n \choose r+1$, $n \choose r+2 $, $n \choose r+3 $ cannot be successive terms of an arithmetic progression.

2020 MMATHS, I12
Tags: trigonometry , algebra , polynomial
Let $p(x)$ be the monic cubic polynomial with roots $\sin^2(1^{\circ})$, $\sin^2(3^{\circ})$, and $\sin^2(9^{\circ})$. Suppose that $p\left(\frac{1}{4}\right)=\frac{\sin(a^{\circ})}{n\sin(b^{\circ})}$, where $0 <a,b \le 90$ and $a,b,n$ are positive integers. What is $a+b+n$? [i]Proposed by Andrew Yuan[/i]

2021 Saint Petersburg Mathematical Olympiad, 1
Tags: algebra , system of equations
Solve the following system of equations $$\sin^2{x} + \cos^2{y} = y^4. $$ $$\sin^2{y} + \cos^2{x} = x^2. $$ [i]A. Khrabov[/i]

2006 Cezar Ivănescu, 1
Tags: monotone , equation , quadratic formula , cristinel mortici , algebra
Solve the equation [b]a)[/b] $ \log_2^2 +(x-1)\log_2 x =6-2x $ in $ \mathbb{R} . $ [b]b)[/b] $ 2^{x+1}+3^{x+1} +2^{1/x^2}+3^{1/x^2}=18 $ in $ (0,\infty ) . $ [i]Cristinel Mortici[/i]

2000 Moldova National Olympiad, Problem 6
Tags: algebra
Find all real values of the parameter $a$ for which the system \begin{align*} &1+\left(4x^2-12x+9\right)^2+2^{y+2}=a\\ &\log_3\left(x^2-3x+\frac{117}4\right)+32=a+\log_3(2y+3) \end{align*}has a unique real solution. Solve the system for these values of $a$.

1985 Tournament Of Towns, (102) 6
Tags: sequence , sum , integer part , recurrence relation , algebra
The numerical sequence $x_1 , x_2 ,.. $ satisfies $x_1 = \frac12$ and $x_{k+1} =x^2_k+x_k$ for all natural integers $k$ . Find the integer part of the sum $\frac{1}{x_1+1}+\frac{1}{x_2+1}+...+\frac{1}{x_{100}+1}$ {A. Andjans, Riga)

1993 Baltic Way, 7
Tags: algebra
Solve the system of equations in integers: \[\begin{cases}z^x=y^{2x}\\ 2^z=4^x\\ x+y+z=20.\end{cases}\]

2016 BMT Spring, 12
Tags: algebra
What is the number of nondecreasing positive integer sequences of length $7$ whose last term is at most $9$?

2016 Postal Coaching, 5
Tags: algebra , polynomial
A real polynomial of odd degree has all positive coefficients. Prove that there is a (possibly trivial) permutation of the coefficients such that the resulting polynomial has exactly one real zero.

2019 Denmark MO - Mohr Contest, 2
Tags: algebra , polynomial
Two distinct numbers a and b satisfy that the two equations $x^{2019} + ax + 2b = 0$ and $x^{2019}+ bx + 2a = 0$ have a common solution. Determine all possible values of $a + b$.

2008 IMO Shortlist, 5
Tags: inequalities , algebra
Let $ a$, $ b$, $ c$, $ d$ be positive real numbers such that $ abcd \equal{} 1$ and $ a \plus{} b \plus{} c \plus{} d > \dfrac{a}{b} \plus{} \dfrac{b}{c} \plus{} \dfrac{c}{d} \plus{} \dfrac{d}{a}$. Prove that \[ a \plus{} b \plus{} c \plus{} d < \dfrac{b}{a} \plus{} \dfrac{c}{b} \plus{} \dfrac{d}{c} \plus{} \dfrac{a}{d}\] [i]Proposed by Pavel Novotný, Slovakia[/i]

2018 Serbia JBMO TST, 2
Tags: inequalities , algebra
Show that for $a,b,c > 0$ the following inequality holds: $\frac{\sqrt{ab}}{a+b+2c}+\frac{\sqrt{bc}}{b+c+2a}+\frac{\sqrt{ca}}{c+a+2b} \le \frac {3}{4}$.

1990 Czech and Slovak Olympiad III A, 1
Tags: algebra , recurrence relation , periodic , sequence
Let $(a_n)_{n\ge1}$ be a sequence given by \begin{align*} a_1 &= 1, \\ a_{2^k+j} &= -a_j\text{ for any } k\ge0,1\le j\le 2^k. \end{align*} Show that the sequence is not periodic.

1987 Traian Lălescu, 2.1
Tags: abstract algebra , ring theory , extension rings , superrings , algebra , polynomial
Any polynom, with coefficients in a given division ring, that is irreducible over it, is also irreducible over a given extension skew ring of it that's finite. Prove that the ring and its extension coincide.

1987 IMO, 3
Tags: algebra , polynomial , number theory , prime number
Let $n\ge2$ be an integer. Prove that if $k^2+k+n$ is prime for all integers $k$ such that $0\le k\le\sqrt{n\over3}$, then $k^2+k+n$ is prime for all integers $k$ such that $0\le k\le n-2$.

2014 BMT Spring, 9
Tags: algebra
Suppose $a_1, a_2, ...$ and $b_1, b_2,...$ are sequences satisfying $a_n + b_n = 7$, $a_n = 2b_{n-1} - a_{n-1}$, and $b_n = 2a_{n-1} - b_{n-1}$, for all $n$. If $a_1 = 2$, find $(a_{2014})^2 - (b_{2014})^2$. .

2015 British Mathematical Olympiad Round 1, 1
Tags: algebra
On Thursday 1st January 2015, Anna buys one book and one shelf. For the next two years she buys one book every day and one shelf on alternate Thursdays, so she next buys a shelf on 15th January. On how many days in the period Thursday 1st January 2015 until (and including) Saturday 31st December 2016 is it possible for Anna to put all her books on all her shelves, so that there is an equal number of books on each shelf?

2017 Costa Rica - Final Round, A1
Tags: polynomial , algebra
Let $P (x)$ be a polynomial of degree $2n$, such that $P (k) =\frac{k}{k + 1}$ for $k = 0,...,2n$. Determine $P (2n + 1)$.

2016 Middle European Mathematical Olympiad, 1
Tags: system of equations , quadratic , algebra
Find all triples $(a, b, c)$ of real numbers such that $$ a^2 + ab + c = 0, $$ $$b^2 + bc + a = 0, $$ $$c^2 + ca + b = 0.$$

1989 Greece Junior Math Olympiad, 4
Tags: algebra , simplification
Simplify i) $1+\frac{2a+\dfrac{2}{a}}{a+\dfrac{1}{a}}$ ii) $\frac{3b+\dfrac{3}{b}+\dfrac{3}{b^2}}{b+\dfrac{1}{b}+\dfrac{1}{b^2}}$ iii) $\frac{\left(\dfrac{1}{a^2}+\dfrac{1}{b^2}+\dfrac{1}{ab}\right)a^6b^2-a^6-a^5b}{a^4b}$

2016 Serbia Additional Team Selection Test, 1
Tags: polynomial , algebra
Let $P_0(x)=x^3-4x$. Sequence of polynomials is defined as following:\\ $P_{n+1}=P_n(1+x)P_n(1-x)-1$.\\ Prove that $x^{2016}|P_{2016}(x)$.

2022 Belarusian National Olympiad, 10.7
Tags: algebra , polynomial
Find all positive integers $a$ for which there exists a polynomial $p(x)$ with integer coefficients such that $p(\sqrt{2}+1)=2-\sqrt{2}$ and $p(\sqrt{2}+2)=a$