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

2019 Saudi Arabia Pre-TST + Training Tests, 1.3
Tags: functional , algebra
Find all functions $f : R^+ \to R^+$ such that $f(3 (f (xy))^2 + (xy)^2) = (xf (y) + yf (x))^2$ for any $x, y > 0$.

1977 Bulgaria National Olympiad, Problem 5
Tags: algebra , polynomial , function
Let $Q(x)$ be a non-zero polynomial and $k$ be a natural number. Prove that the polynomial $P(x) = (x-1)^kQ(x)$ has at least $k+1$ non-zero coefficients.

2013 NZMOC Camp Selection Problems, 3
Tags: reciprocal , algebra , number theory
Prove that for any positive integer $n > 2$ we can find $n$ distinct positive integers, the sum of whose reciprocals is equal to $1$.

2017 Estonia Team Selection Test, 2
Tags: algebra
Find the smallest constant $C > 0$ for which the following statement holds: among any five positive real numbers $a_1,a_2,a_3,a_4,a_5$ (not necessarily distinct), one can always choose distinct subscripts $i,j,k,l$ such that \[ \left| \frac{a_i}{a_j} - \frac {a_k}{a_l} \right| \le C. \]

2002 China Team Selection Test, 3
Tags: polynomial , algebra
Let \[ f(x_1,x_2,x_3) = -2 \cdot (x_1^3+x_2^3+x_3^3) + 3 \cdot (x_1^2(x_2+x_3) + x_2^2 \cdot (x_1+x_3) + x_3^2 \cdot ( x_1+x_2 ) - 12x_1x_2x_3. \] For any reals $r,s,t$, we denote \[ g(r,s,t)=\max_{t\leq x_3\leq t+2} |f(r,r+2,x_3)+s|. \] Find the minimum value of $g(r,s,t)$.

1982 Vietnam National Olympiad, 1
Tags: polynomial , calculus , integration , trigonometry , vieta , algebra
Determine a quadric polynomial with intergral coefficients whose roots are $\cos 72^{\circ}$ and $\cos 144^{\circ}.$

2006 Putnam, A2
Tags: factorial , algebra , polynomial , function , probability , logarithm
Alice and Bob play a game in which they take turns removing stones from a heap that initially has $n$ stones. The number of stones removed at each turn must be one less than a prime number. The winner is the player who takes the last stone. Alice plays first. Prove that there are infinitely many such $n$ such that Bob has a winning strategy. (For example, if $n=17,$ then Alice might take $6$ leaving $11;$ then Bob might take $1$ leaving $10;$ then Alice can take the remaining stones to win.)

Oliforum Contest I 2008, 1
Tags: sequence , number theory , algebra , recurrence relation
Consider the sequence of integer such that: $ a_1 = 2$ $ a_2 = 5$ $ a_{n + 1} = (2 - n^2)a_n + (2 + n^2)a_{n - 1}, \forall n\ge 2$ Find all triplies $ (x,y,z) \in \mathbb{N}^3$ such that $ a_xa_y = a_z$.

2015 Dutch IMO TST, 2
Tags: algebra , polynomial
Determine all polynomials P(x) with real coefficients such that [(x + 1)P(x − 1) − (x − 1)P(x)] is a constant polynomial.

2010 India IMO Training Camp, 9
Tags: geometry , rectangle , inequalities , algebra
Let $A=(a_{jk})$ be a $10\times 10$ array of positive real numbers such that the sum of numbers in row as well as in each column is $1$. Show that there exists $j<k$ and $l<m$ such that \[a_{jl}a_{km}+a_{jm}a_{kl}\ge \frac{1}{50}\]

2022 ELMO Revenge, 5
Tags: algebra
Let $f(x)=x+3x^{\frac 23}, g(x)=x+x^{\frac 13}$. Call a sequence $\{a_i\}_{i\ge 0}$ satisfactory if for all $i\ge 1, a_i\in \{f(a_{i-1}), g(a_{i-1})\}$. Find all pairs of real numbers $(x,y)$ such that there exist satisfactory sequences $(a_i)_{i\ge 0}, (b_i)_{i\ge 0}$ and positive integers $m$ and $n$, such that $a_0 =x$, $b_0 = y$, and $$|a_m-b_n|<1$$

2016 BMT Spring, 6
Tags: algebra , calculus
Amy is traveling on the $xy$-plane in a spaceship where her motion is described by the following equation $xe^y = ye^x$. Given that her $x$-component of velocity is a constant $3$ mph , the magnitude of her velocity as she approaches $(1,-1)$ can be expressed as $\sqrt{\frac{a + be^4}{ c + de^2}}$ . Find $\frac{ac}{bd}$ . (You may assume that the initial conditions do allow her to approach $(1,-1)$)

2017 Hong Kong TST, 6
Tags: algebra
Given infinite sequences $a_1,a_2,a_3,\cdots$ and $b_1,b_2,b_3,\cdots$ of real numbers satisfying $\displaystyle a_{n+1}+b_{n+1}=\frac{a_n+b_n}{2}$ and $\displaystyle a_{n+1}b_{n+1}=\sqrt{a_nb_n}$ for all $n\geq1$. Suppose $b_{2016}=1$ and $a_1>0$. Find all possible values of $a_1$

1979 IMO Longlists, 44
Tags: diophantine equation , algebra , system of equations , cauchy-schwarz inequality
Determine all real numbers a for which there exists positive reals $x_{1}, \ldots, x_{5}$ which satisfy the relations $ \sum_{k=1}^{5} kx_{k}=a,$ $ \sum_{k=1}^{5} k^{3}x_{k}=a^{2},$ $ \sum_{k=1}^{5} k^{5}x_{k}=a^{3}.$

2024 Malaysia IMONST 2, 4
Tags: algebra
For all $n \geq 1$, define $a_{n}$ to be the fraction $\frac{k}{2^n}$ such that $a_{n}$ is the closest to $\frac{1}{3}$ over all integer values of $k$. Prove that the sequence $a_{1}, a_{2}, \cdots $satisfies the equation $2a_{i+2} = a_{i+1} + a_{i}$ for all $i \geq 1$.

2015 IMO Shortlist, A6
Tags: algebra , polynomial
Let $n$ be a fixed integer with $n \ge 2$. We say that two polynomials $P$ and $Q$ with real coefficients are [i]block-similar[/i] if for each $i \in \{1, 2, \ldots, n\}$ the sequences \begin{eqnarray*} P(2015i), P(2015i - 1), \ldots, P(2015i - 2014) & \text{and}\\ Q(2015i), Q(2015i - 1), \ldots, Q(2015i - 2014) \end{eqnarray*} are permutations of each other. (a) Prove that there exist distinct block-similar polynomials of degree $n + 1$. (b) Prove that there do not exist distinct block-similar polynomials of degree $n$. [i]Proposed by David Arthur, Canada[/i]

2016 CHMMC (Fall), 10
Tags: number theory , algebra
For a positive integer $n$, let $p(n)$ denote the number of prime divisors of $n$, counting multiplicity (i.e. $p(12)=3$). A sequence $a_n$ is defined such that $a_0 = 2$ and for $n > 0$, $a_n = 8^{p(a_{n-1})} + 2$. Compute $$\sum_{n=0}^{\infty} \frac{a_n}{2^n}$$

2017 Saudi Arabia IMO TST, 2
Tags: algebra , functional equation
Find all $f:\mathbb{R}\to\mathbb{R}$ satisfying: $$f(xf(y)-y)+f(xy-x)+f(x+y)=2xy,\quad\forall x,y\in\mathbb{R}.$$

1984 Iran MO (2nd round), 1
Tags: function , floor function , domain , limit , algebra
Let $f$ and $g$ be two functions such that \[f(x)=\frac{1}{\lfloor | x | \rfloor}, \quad g(x)=\frac{1}{|\lfloor x \rfloor |}.\] Find the domains of $f$ and $g$ and then prove that \[\lim_{x \to -1^+} f(x)= \lim_{x \to 1^- } g(x).\]

2017 Iran Team Selection Test, 6
Tags: algebra , number theory
Let $k>1$ be an integer. The sequence $a_1,a_2, \cdots$ is defined as: $a_1=1, a_2=k$ and for all $n>1$ we have: $a_{n+1}-(k+1)a_n+a_{n-1}=0$ Find all positive integers $n$ such that $a_n$ is a power of $k$. [i]Proposed by Amirhossein Pooya[/i]

2019 VJIMC, 1
Tags: discrete math , number theory , algebra
Let $\{a_n \}_{n=0}^{\infty}$ be a sequence given recrusively such that $a_0=1$ and $$a_{n+1}=\frac{7a_n+\sqrt{45a_n^2-36}}{2}$$ for $n\geq 0$ Show that : a) $a_n$ is a positive integer. b) $a_n a_{n+1}-1$ is a square of an integer. [i]Proposed by Stefan Gyurki (Matej Bel University, Banska Bystrica).[/i]

2016 HMNT, 10
Tags: algebra
Determine the largest integer $n$ such that there exist monic quadratic polynomials $p_1(x)$, $p_2(x)$, $p_3(x)$ with integer coefficients so that for all integers $ i \in [1, n]$ there exists some $j \in [1, 3]$ and $m \in Z$ such that $p_j (m) = i$.

1991 Vietnam National Olympiad, 1
Tags: function , algebra
Find all functions $f: \mathbb{R}\to\mathbb{R}$ satisfying: $\frac{f(xy)+f(xz)}{2} - f(x)f(yz) \geq \frac{1}{4}$ for all $x,y,z \in \mathbb{R}$

1994 Baltic Way, 4
Tags: algebra
Is there an integer $n$ such that $\sqrt{n-1}+\sqrt{n+1}$ is a rational number?

1977 USAMO, 1
Tags: algebra , polynomial , modular arithmetic
Determine all pairs of positive integers $ (m,n)$ such that $ (1\plus{}x^n\plus{}x^{2n}\plus{}\cdots\plus{}x^{mn})$ is divisible by $ (1\plus{}x\plus{}x^2\plus{}\cdots\plus{}x^{m})$.