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

2023 South East Mathematical Olympiad, 7
Tags: inequalities , algebra
The positive integer number $S$ is called a "[i]line number[/i]". if there is a positive integer $n$ and $2n$ positive integers $a_1$, $a_2$,...,$a_n$, $b_1$,$b_2$,...,$b_n$, such that $S = \sum^n_{i=1} a_ib_i$, $\sum^n_{i=1} (a_i^2-b_1^2)=1$, and $\sum^n_{i=1} (a_i+b_i)=2023$, find: (1) The minimum value of [i]line numbers[/i]. (2)The maximum value of [i]line numbers[/i].

2016 CHMMC (Fall), 11
Tags: probability , algebra
Let $a,b \in [0,1], c \in [-1,1]$ be reals chosen independently and uniformly at random. What is the probability that $p(x) = ax^2+bx+c$ has a root in $[0,1]$?

2013 AMC 12/AHSME, 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 $

2006 Federal Competition For Advanced Students, Part 2, 2
Tags: function , functional equation , algebra
Find all monotonous functions $ f: \mathbb{R} \to \mathbb{R}$ that satisfy the following functional equation: \[f(f(x)) \equal{} f( \minus{} f(x)) \equal{} f(x)^2.\]

1982 IMO Longlists, 22
Tags: function , topology , algebra , discrete intermediate value theorem
Let $M$ be the set of real numbers of the form $\frac{m+n}{\sqrt{m^2+n^2}}$, where $m$ and $n$ are positive integers. Prove that for every pair $x \in M, y \in M$ with $x < y$, there exists an element $z \in M$ such that $x < z < y.$

1966 IMO Longlists, 25
Tags: trigonometry , algebra , trigonometric identities , equation
Prove that \[\tan 7 30^{\prime }=\sqrt{6}+\sqrt{2}-\sqrt{3}-2.\]

1994 Balkan MO, 2
Tags: polynomial , calculus , integration , modular arithmetic , algebra
Let $n$ be an integer. Prove that the polynomial $f(x)$ has at most one zero, where \[ f(x) = x^4 - 1994 x^3 + (1993+n)x^2 - 11x + n . \] [i]Greece[/i]

2016 Mathematical Talent Reward Programme, MCQ: P 14
Tags: floor function , greatest integer function , algebra , function
Let $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$. Find $x$ such that $x\lfloor x\lfloor x\lfloor x\rfloor\rfloor \rfloor = 88$ [list=1] [*] $\pi$ [*] 3.14 [*] $\frac{22}{7}$ [*] All of these [/list]

2022 Iran MO (3rd Round), 2
Tags: function , function inequality , number theory , functional equation , algebra
Find all functions $f:\mathbb{N}\to\mathbb{N}$ such that for all $x,y\in\mathbb{N}$: $$0\le y+f(x)-f^{f(y)}(x)\le1$$ that here $$f^n(x)=\underbrace{f(f(\ldots(f}_{n}(x))\ldots)$$

2016 Vietnam National Olympiad, 1
Tags: algebra , functional equation , function
Find all $a\in\mathbb{R}$ such that there is function $f:\mathbb{R}\to\mathbb{R}$ i) $f(1)=2016$ ii) $f(x+y+f(y))=f(x)+ay\quad\forall x,y\in\mathbb{R}$

1992 Romania Team Selection Test, 4
Tags: sequence , combinatorics , algebra , set
Let $A$ be the set of all ordered sequences $(a_1,a_2,...,a_{11})$ of zeros and ones. The elements of $A$ are ordered as follows: The first element is $(0,0,...,0)$, and the $n + 1$−th is obtained from the $n$−th by changing the first component from the right such that the newly obtained sequence was not obtained before. Find the $1992$−th term of the ordered set $A$

1982 All Soviet Union Mathematical Olympiad, 344
Tags: sequence , combinatorics , algebra
Given a sequence of real numbers $a_1, a_2, ... , a_n$. Prove that it is possible to choose some of the numbers providing $3$ conditions: a) not a triple of successive members is chosen, b) at least one of every triple of successive members is chosen, c) the absolute value of chosen numbers sum is not less that one sixth part of the initial numbers' absolute values sum.

2002 IMO Shortlist, 4
Tags: functional equation , algebra
Find all functions $f$ from the reals to the reals such that \[ \left(f(x)+f(z)\right)\left(f(y)+f(t)\right)=f(xy-zt)+f(xt+yz) \] for all real $x,y,z,t$.

2024 ELMO Shortlist, A5
Tags: algebra
Allen and Alan play a game. A nonconstant polynomial $P(x,y)$ with real coefficients and a positive integer $d$ greater than the degree of $P$ are known to both Allen and Alan. Alan thinks of a polynomial $Q(x,y)$ with real coefficients and degree at most $d$ and keeps it secret. Allen can make queries of the form $(s,t)$, where $s$ and $t$ are real numbers such that $P(s,t)\neq0$. Alan must respond with the value $Q(s,t)$. Allen's goal is to determine whether $P$ divides $Q$. Find (in terms of $P$ and $d$) the smallest positive integer, $g$, such that Allen can always achieve this goal making no more than $g$ queries. [i]Linus Tang[/i]

2017 Vietnamese Southern Summer School contest, Problem 1
Tags: inequality , algebra , inequalities
Let $x,y,z$ be the non-negative real numbers satisfying $xy+yz+zx\leq 1$. Prove that: $$1-xy-yz-zx\leq (6-2\sqrt{6})(1-\min\{x,y,z\}).$$

2020 CIIM, 6
Tags: real analysis , algebra , combinatorics
For a set $A$, we define $A + A = \{a + b: a, b \in A \}$. Determine whether there exists a set $A$ of positive integers such that $$\sum_{a \in A} \frac{1}{a} = +\infty \quad \text{and} \quad \lim_{n \rightarrow +\infty} \frac{|(A+A) \cap \{1,2,\cdots,n \}|}{n}=0.$$ [hide=Note]Google translated from [url=http://ciim.uan.edu.co/ciim-2020-pruebas-virtuales/pruebas-virtuales]http://ciim.uan.edu.co/ciim-2020-pruebas-virtuales/pruebas-virtuales[/url][/hide]

2020 Taiwan TST Round 3, 3
Tags: algebra , functional equation , arithmetic sequence
Let $\mathbb Z$ be the set of integers. We consider functions $f :\mathbb Z\to\mathbb Z$ satisfying \[f\left(f(x+y)+y\right)=f\left(f(x)+y\right)\] for all integers $x$ and $y$. For such a function, we say that an integer $v$ is [i]f-rare[/i] if the set \[X_v=\{x\in\mathbb Z:f(x)=v\}\] is finite and nonempty. (a) Prove that there exists such a function $f$ for which there is an $f$-rare integer. (b) Prove that no such function $f$ can have more than one $f$-rare integer. [i]Netherlands[/i]

2017 Romanian Master of Mathematics Shortlist, A1
Tags: algebra , binary operation , set , cardinality
A set $A$ is endowed with a binary operation $*$ satisfying the following four conditions: (1) If $a, b, c$ are elements of $A$, then $a * (b * c) = (a * b) * c$ , (2) If $a, b, c$ are elements of $A$ such that $a * c = b *c$, then $a = b$ , (3) There exists an element $e$ of $A$ such that $a * e = a$ for all $a$ in $A$, and (4) If a and b are distinct elements of $A-\{e\}$, then $a^3 * b = b^3 * a^2$, where $x^k = x * x^{k-1}$ for all integers $k \ge 2$ and all $x$ in $A$. Determine the largest cardinality $A$ may have. proposed by Bojan Basic, Serbia

1995 Czech and Slovak Match, 6
Tags: algebra , number theory , prime number
Find all triples $(x; y; p)$ of two non-negative integers $x, y$ and a prime number p such that $ p^x-y^p=1 $

1998 Belarusian National Olympiad, 8
Tags: algebra , functional equation , functional
a) Prove that for no real a such that $0 \le a <1$ there exists a function defined on the set of all positive numbers and taking values in the same set, satisfying for all positive $x$ the equality $$f\left(f(x)+\frac{1}{f(x)}\right)=x+a \,\,\,\,\,\,\, (*) $$ b) Prove that for any $a>1$ there are infinitely many functions defined on the set of all positive numbers, with values in the same set, satisfying ($*$) for all positive x.

2020 Poland - Second Round, 6.
Tags: algebra
Let $(a_0,a_1,a_2,...)$ and $(b_0,b_1,b_2,...)$ be such sequences of non-negative real numbers, that for every integer $i\geqslant 1$ holds $a_i^2\leqslant a_{i-1}a_{i+1}$ and $b_i^2\leqslant b_{i-1}b_{i+1}$. Define sequence $c_0,c_1,c_2,...$ as $$c_0=a_0b_0, \; c_n=\sum_{i=0}^{n} {{n}\choose{i}} a_ib_{n-i}.$$ Prove that for every integer $k\geqslant 1$ holds $c_{k}^2\leqslant c_{k-1}c_{k+1}$.

2023 Regional Olympiad of Mexico West, 3
Tags: algebra , floor function , inequalities
Let $x>1$ be a real number that is not an integer. Denote $\{x\}$ as its decimal part and $\lfloor x\rfloor$ the floor function. Prove that $$ \left(\frac{x+\{x\}}{\lfloor x\rfloor}-\frac{\lfloor x\rfloor}{x+\{x\}}\right)+\left(\frac{x+\lfloor x\rfloor}{\{x\}}-\frac{\{x\}}{x+\lfloor x\rfloor}\right)>\frac{16}{3}$$

1999 Rioplatense Mathematical Olympiad, Level 3, 4
Tags: inequalities , algebra
Prove the following inequality: $$ \frac{1}{\sqrt[3]{1^2}+\sqrt[3]{1 \cdot 2}+\sqrt[3]{2^2} }+\frac{1}{\sqrt[3]{3^2}+\sqrt[3]{3 \cdot 4}+\sqrt[3]{4^2} }+...+ \frac{1}{\sqrt[3]{999^2}+\sqrt[3]{999 \cdot 1000}+\sqrt[3]{1000^2} }> \frac{9}{2}$$ (The member on the left has 500 fractions.)

2023 Romania National Olympiad, 1
Tags: exponential , exponential equation , algebra
Solve the following equation for real values of $x$: \[ 2 \left( 5^x + 6^x - 3^x \right) = 7^x + 9^x. \]

2010 ELMO Shortlist, 5
Tags: algebra
Given a prime $p$, let $d(a,b)$ be the number of integers $c$ such that $1 \leq c < p$, and the remainders when $ac$ and $bc$ are divided by $p$ are both at most $\frac{p}{3}$. Determine the maximum value of \[\sqrt{\sum_{a=1}^{p-1}\sum_{b=1}^{p-1}d(a,b)(x_a + 1)(x_b + 1)} - \sqrt{\sum_{a=1}^{p-1}\sum_{b=1}^{p-1}d(a,b)x_ax_b}\] over all $(p-1)$-tuples $(x_1,x_2,\ldots,x_{p-1})$ of real numbers. [i]Brian Hamrick.[/i]