Found problems: 15925
2016 CMIMC, 8
Let $r_1$, $r_2$, $\ldots$, $r_{20}$ be the roots of the polynomial $x^{20}-7x^3+1$. If \[\dfrac{1}{r_1^2+1}+\dfrac{1}{r_2^2+1}+\cdots+\dfrac{1}{r_{20}^2+1}\] can be written in the form $\tfrac mn$ where $m$ and $n$ are positive coprime integers, find $m+n$.
2018 Olympic Revenge, 5
Let $p$ a positive prime number and $\mathbb{F}_{p}$ the set of integers $mod \ p$. For $x\in \mathbb{F}_{p}$, define $|x|$ as the cyclic distance of $x$ to $0$, that is, if we represent $x$ as an integer between $0$ and $p-1$, $|x|=x$ if $x<\frac{p}{2}$, and $|x|=p-x$ if $x>\frac{p}{2}$ . Let $f: \mathbb{F}_{p} \rightarrow \mathbb{F}_{p}$ a function such that for every $x,y \in \mathbb{F}_{p}$
\[ |f(x+y)-f(x)-f(y)|<100 \]
Prove that exist $m \in \mathbb{F}_{p}$ such that for every $x \in \mathbb{F}_{p}$
\[ |f(x)-mx|<1000 \]
ABMC Team Rounds, 2021
[u]Round 1[/u]
[b]1.1.[/b] There are $99$ dogs sitting in a long line. Starting with the third dog in the line, if every third dog barks three times, and all the other dogs each bark once, how many barks are there in total?
[b]1.2.[/b] Indigo notices that when she uses her lucky pencil, her test scores are always $66 \frac23 \%$ higher than when she uses normal pencils. What percent lower is her test score when using a normal pencil than her test score when using her lucky pencil?
[b]1.3.[/b] Bill has a farm with deer, sheep, and apple trees. He mostly enjoys looking after his apple trees, but somehow, the deer and sheep always want to eat the trees' leaves, so Bill decides to build a fence around his trees. The $60$ trees are arranged in a $5\times 12$ rectangular array with $5$ feet between each pair of adjacent trees. If the rectangular fence is constructed $6$ feet away from the array of trees, what is the area the fence encompasses in feet squared? (Ignore the width of the trees.)
[u]Round 2[/u]
[b]2.1.[/b] If $x + 3y = 2$, then what is the value of the expression $9^x * 729^y$?
[b]2.2.[/b] Lazy Sheep loves sleeping in, but unfortunately, he has school two days a week. If Lazy Sheep wakes up each day before school's starting time with probability $1/8$ independent of previous days, then the probability that Lazy Sheep wakes up late on at least one school day over a given week is $p/q$ for relatively prime positive integers $p, q$. Find $p + q$.
[b]2.3.[/b] An integer $n$ leaves remainder $1$ when divided by $4$. Find the sum of the possible remainders $n$ leaves when divided by $20$.
[u]Round 3[/u]
[b]3.1. [/b]Jake has a circular knob with three settings that can freely rotate. Each minute, he rotates the knob $120^o$ clockwise or counterclockwise at random. The probability that the knob is back in its original state after $4$ minutes is $p/q$ for relatively prime positive integers $p, q$. Find $p + q$.
[b]3.2.[/b] Given that $3$ not necessarily distinct primes $p, q, r$ satisfy $p+6q +2r = 60$, find the sum of all possible values of $p + q + r$.
[b]3.3.[/b] Dexter's favorite number is the positive integer $x$, If $15x$ has an even number of proper divisors, what is the smallest possible value of $x$? (Note: A proper divisor of a positive integer is a divisor other than itself.)
[u]Round 4[/u]
[b]4.1.[/b] Three circles of radius $1$ are each tangent to the other two circles. A fourth circle is externally tangent to all three circles. The radius of the fourth circle can be expressed as $\frac{a\sqrt{b}-\sqrt{c}}{d}$ for positive integers $a, b, c, d$ where $b$ is not divisible by the square of any prime and $a$ and $d$ are relatively prime. Find $a + b + c + d$.
[b]4.2. [/b]Evaluate $$\frac{\sqrt{15}}{3} \cdot \frac{\sqrt{35}}{5} \cdot \frac{\sqrt{63}}{7}... \cdot \frac{\sqrt{5475}}{73}$$
[b]4.3.[/b] For any positive integer $n$, let $f(n)$ denote the number of digits in its base $10$ representation, and let $g(n)$ denote the number of digits in its base $4$ representation. For how many $n$ is $g(n)$ an integer multiple of $f(n)$?
PS. You should use hide for answers. Rounds 5-8 have been posted [url=https://artofproblemsolving.com/community/c3h2784571p24468619]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2004 Nicolae Coculescu, 4
Let be a function satisfying [url=http://mathworld.wolfram.com/CauchyFunctionalEquation.html]Cauchy's functional equation,[/url] and having the property that it's monotonic on a real interval. Prove that this function is globally monotonic.
[i]Florian Dumitrel[/i]
1981 Poland - Second Round, 3
Prove that there is no continuous function $ f: \mathbb{R} \to \mathbb{R} $ satisfying the condition $ f(f(x)) = - x $ for every $ x $.
STEMS 2023 Math Cat A, 4
Alice has $n > 1$ one variable quadratic polynomials written on paper she keeps secret from Bob. On each move, Bob announces a real number and Alice tells him the value of one of her polynomials at this number. Prove that there exists a constant $C > 0$ such that after $Cn^5$ questions, Bob can determine one of Alice’s polynomials.
[i]Proposed by Rohan Goyal and Anant Mudgal[/i]
2002 Austrian-Polish Competition, 5
Let $A$ be the set $\{2,7,11,13\}$. A polynomial $f$ with integer coefficients possesses the following property: for each integer $n$ there exists $p \in A$ such that $p|f(n)$. Prove that there exists $p \in A$ such that $p|f(n)$ for all integers $n$.
2016 Israel National Olympiad, 1
Nina and Meir are walking on a $3$ km path towards grandma's house. They start walking at the same time from the same point. Meir's speed is $4$ km/h and Nina's speed is $3$ km/h.
Along the path there are several benches. Whenever Nina or Meir reaches a bench, they sit on it for some time and eat a cookie. Nina always takes $t$ minutes to eat a cookie, and Meir always takes $2t$ minutes to eat a cookie, where $t$ is a positive integer.
It turns out that Nina and Meir reached grandma's house at the same time. How many benches were there? Find all of the options.
2021 Final Mathematical Cup, 1
Let $N$ is the set of all positive integers. Determine all mappings $f: N-\{1\} \to N$ such that for every $n \ne m$ the following equation is true $$f(n)f(m)=f\left((nm)^{2021}\right)$$
2006 Estonia Team Selection Test, 1
Let $k$ be any fixed positive integer. Let's look at integer pairs $(a, b)$, for which the quadratic equations $x^2 - 2ax + b = 0$ and $y^2 + 2ay + b = 0$ are real solutions (not necessarily different), which can be denoted by $x_1, x_2$ and $y_1, y_2$, respectively, in such an order that the equation $x_1 y_1 - x_2 y_2 = 4k$.
a) Find the largest possible value of the second component $b$ of such a pair of numbers ($a, b)$.
b) Find the sum of the other components of all such pairs of numbers.
2010 IFYM, Sozopol, 8
Solve this equation with $x \in R$:
$x^3-3x=\sqrt{x+2}$
2020 China Team Selection Test, 5
Let $a_1,a_2,\cdots,a_n$ be a permutation of $1,2,\cdots,n$. Among all possible permutations, find the minimum of $$\sum_{i=1}^n \min \{ a_i,2i-1 \}.$$
2014 Contests, 4
Find all functions $f$ defined on all real numbers and taking real values such that \[f(f(y)) + f(x - y) = f(xf(y) - x),\] for all real numbers $x, y.$
2012 Belarus Team Selection Test, 4
Given $0 < a < b < c$ prove that $$ a^{20}b^{12} + b^{20}c^{12 }+ c^{20}a^{12} <b^{20}a^{12}+ a^{20}c^{12} + c^{20}b^{12} $$
(I. Voronovich)
2019 239 Open Mathematical Olympiad, 6
Find all functions $f : (0, +\infty) \to \mathbb{R}$ satisfying the following conditions:
$(i)$ $f(x) + f(\frac{1}{x}) = 1$ for all $x> 0$;
$(ii)$ $f(xy + x + y) = f(x)f(y)$ for all $x, y> 0$.
2012 China Second Round Olympiad, 9
Given a function $f(x)=a\sin x-\frac{1}{2}\cos 2x+a-\frac{3}{a}+\frac{1}{2}$, where $a\in\mathbb{R}, a\ne 0$.
[b](1)[/b] If for any $x\in\mathbb{R}$, inequality $f(x)\le 0$ holds, find all possible value of $a$.
[b](2)[/b] If $a\ge 2$, and there exists $x\in\mathbb{R}$, such that $f(x)\le 0$. Find all possible value of $a$.
1935 Eotvos Mathematical Competition, 1
Let $n$ be a positive integer. Prove that
$$\frac{a_1}{b_1}+ \frac{a_2}{b_2}+ ...+\frac{a_n}{b_n} \ge n $$
where $(b_1, b_2, ..., b_n)$ is any permutation of the positive real numbers $a_1, a_2, ..., a_n$.
2021 Stanford Mathematics Tournament, R6
[b]p21[/b]. If $f = \cos(\sin (x))$. Calculate the sum $\sum^{2021}_{n=0} f'' (n \pi)$.
[b]p22.[/b] Find all real values of $A$ that minimize the difference between the local maximum and local minimum of $f(x) = \left(3x^2 - 4\right)\left(x - A + \frac{1}{A}\right)$.
[b]p23.[/b] Bessie is playing a game. She labels a square with vertices labeled $A, B, C, D$ in clockwise order. There are $7$ possible moves: she can rotate her square $90$ degrees about the center, $180$ degrees about the center, $270$ degrees about the center; or she can flip across diagonal $AC$, flip across diagonal $BD$, flip the square horizontally (flip the square so that vertices A and B are switched and vertices $C$ and $D$ are switched), or flip the square vertically (vertices $B$ and $C$ are switched, vertices $A$ and $D$ are switched). In how many ways can Bessie arrive back at the original square for the first time in $3$ moves?
[b]p24.[/b] A positive integer is called [i]happy [/i] if the sum of its digits equals the two-digit integer formed by its two leftmost digits. Find the number of $5$-digit happy integers.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
1986 USAMO, 3
What is the smallest integer $n$, greater than one, for which the root-mean-square of the first $n$ positive integers is an integer?
$\mathbf{Note.}$ The root-mean-square of $n$ numbers $a_1, a_2, \cdots, a_n$ is defined to be
\[\left[\frac{a_1^2 + a_2^2 + \cdots + a_n^2}n\right]^{1/2}\]
2000 India National Olympiad, 3
If $a,b,c,x$ are real numbers such that $abc \not= 0$ and \[ \frac{xb + (1-x)c}{a} = \frac{xc + (1-x)a}{b} = \frac{xa + (1-x) b }{c}, \] then prove that $a = b = c$.
1994 French Mathematical Olympiad, Problem 5
Assume $f:\mathbb N_0\to\mathbb N_0$ is a function such that $f(1)>0$ and, for any nonnegative integers $m$ and $n$,
$$f\left(m^2+n^2\right)=f(m)^2+f(n)^2.$$(a) Calculate $f(k)$ for $0\le k\le12$.
(b) Calculate $f(n)$ for any natural number $n$.
2010 Belarus Team Selection Test, 5.1
The following expression $x^{30} + *x^{29} +...+ *x+8 = 0$ is written on a blackboard. Two players $A$ and $B$ play the following game. $A$ starts the game. He replaces all the asterisks by the natural numbers from $1$ to $30$ (using each of them exactly once). Then player $B$ replace some of" $+$ "by ” $-$ "(by his own choice). The goal of $A$ is to get the equation having a real root greater than $10$, while the goal of $B$ is to get the equation having a real root less that or equal to $10$. If both of the players achieve their goals or nobody of them achieves his goal, then the result of the game is a draw. Otherwise, the player achieving his goal is a winner.
Who of the players wins if both of them play to win?
(I.Bliznets)
2017 Mathematical Talent Reward Programme, SAQ: P 4
An irreducible polynomial is a not-constant polynomial that cannot be factored into product of two non-constant polynomials. Consider the following statements :-
[b]Statement 1 :[/b] $p(x)$ be any monic irreducible polynomial with integer coefficients and degree $\geq 4$. Then $p(n)$ is a prime for at least one natural number $n$
[b]Statement 2 :[/b] $n^2+1$ is prime for infinitely many values of natural number $n$
Show that if [b]Statement 1[/b] is true then [b]Statement 2[/b] is also true
2022 Switzerland Team Selection Test, 5
Let $a, b, c, \lambda$ be positive real numbers with $\lambda \geq 1/4$. Show that $$\frac{a}{\sqrt{b^2+\lambda bc+c^2}}+\frac{b}{\sqrt{c^2+\lambda ca+a^2}}+\frac{c}{\sqrt{a^2+\lambda ab+b^2}} \geq \frac{3}{\sqrt{\lambda +2}}.$$
2004 Korea Junior Math Olympiad, 5
Show that there exists no function $f:\mathbb {R}\rightarrow \mathbb {R}$ that satisfies $f(f(x))-x^2+x+3=0$ for arbitrary real variable $x$.
(Same as KMO 2004 P1)