Found problems: 85335
2020 Princeton University Math Competition, 15
Suppose that f is a function $f : R_{\ge 0} \to R$ so that for all $x, y \in R_{\ge 0}$ (nonnegative reals) we have that $$f(x)+f(y) = f(x+y+xy)+f(x)f(y).$$ Given that $f\left(\frac{3}{5} \right) = \frac12$ and$ f(1) = 3$, determine
$$\lfloor \log_2 (-f(10^{2021} - 1)) \rfloor.$$
2021 Durer Math Competition Finals, 13
The trapezoid $ABCD$ satisfies $AB \parallel CD$, $AB = 70$, $AD = 32$ and $BC = 49$. We also know that $\angle ABC = 3 \angle ADC$. How long is the base $CD$?
2021 New Zealand MO, 1
Let $ABCD$ be a convex quadrilateral such that $AB + BC = 2021$ and $AD = CD$. We are also given that $\angle ABC = \angle CDA = 90^o$. Determine the length of the diagonal $BD$.
2017 Junior Balkan Team Selection Tests - Romania, 4
Let $a, b, c, d$ be non-negative real numbers satisfying $a + b + c + d = 3$. Prove that
$$\frac{a}{1 + 2b^3} + \frac{b}{1 + 2c^3} +\frac{c}{1 + 2d^3} +\frac{d}{1 + 2a^3} \ge \frac{a^2 + b^2 + c^2 + d^2}{3}$$
When does the equality hold?
2012 AMC 10, 14
Two equilateral triangles are contained in a square whose side length is $2\sqrt3$. The bases of these triangles are the opposite sides of the square, and their intersection is a rhombus. What is the area of the rhombus?
$ \textbf{(A)}\ \frac{3}{2}\qquad\textbf{(B)}\ \sqrt3\qquad\textbf{(C)}\ 2\sqrt2-1\qquad\textbf{(D)}\ 8\sqrt3-12\qquad\textbf{(E)}\ \frac{4\sqrt3}{3}$
1954 Poland - Second Round, 2
Prove that among ten consecutive natural numbers there is always at least one, and at most four, numbers that are not divisible by any of the numbers $ 2 $, $ 3 $, $ 5 $, $ 7 $.
1982 Bulgaria National Olympiad, Problem 5
Find all values of parameters $a,b$ for which the polynomial
$$x^4+(2a+1)x^3+(a-1)^2x^2+bx+4$$can be written as a product of two monic quadratic polynomials $\Phi(x)$ and $\Psi(x)$, such that the equation $\Psi(x)=0$ has two distinct roots $\alpha,\beta$ which satisfy $\Phi(\alpha)=\beta$ and $\Phi(\beta)=\alpha$.
2022 CMIMC, 1.5
Grant is standing at the beginning of a hallway with infinitely many lockers, numbered, $1, 2, 3, \ldots$ All of the lockers are initially closed. Initially, he has some set $S = \{1, 2, 3, \ldots\}$.
Every step, for each element $s$ of $S$, Grant goes through the hallway and opens each locker divisible by $s$ that is closed, and closes each locker divisible by $s$ that is open. Once he does this for all $s$, he then replaces $S$ with the set of labels of the currently open lockers, and then closes every door again.
After $2022$ steps, $S$ has $n$ integers that divide ${10}^{2022}$. Find $n$.
[i]Proposed by Oliver Hayman[/i]
2010 Morocco TST, 2
Find the integer represented by $\left[ \sum_{n=1}^{10^9} n^{-2/3} \right] $. Here $[x]$ denotes the greatest integer less than or equal to $x.$
1996 Tournament Of Towns, (508) 1
Can one paint four points in the plane red and another four points black so that any three points of the same colour are vertices of a parallelogram whose fourth vertex is a point of the other colour?
(NB Vassiliev)
2023 Ukraine National Mathematical Olympiad, 11.8
There are $2024$ cities in a country, every two of which are bidirectionally connected by exactly one of three modes of transportation - rail, air, or road. A tourist has arrived in this country and has the entire transportation scheme. He chooses a travel ticket for one of the modes of transportation and the city from which he starts his trip. He wants to visit as many cities as possible, but using only the ticket for the specified type of transportation. What is the largest $k$ for which the tourist will always be able to visit at least $k$ cities? During the route, he can return to the cities he has already visited.
[i]Proposed by Bogdan Rublov[/i]
2020-21 IOQM India, 30
Find the number of pairs $(a,b)$ of natural nunbers such that $b$ is a 3-digit number, $a+1$ divides $b-1$ and $b$ divides $a^{2} + a + 2$.
2024 Middle European Mathematical Olympiad, 7
Define [i]glueing[/i] of positive integers as writing their base ten representations one after another and
interpreting the result as the base ten representation of a single positive integer.
Find all positive integers $k$ for which there exists an integer $N_k$ with the following property: for all $n \ge N_k$, we can glue the numbers $1,2,\dots,n$ in some order so that the result is a number divisible by $k$.
[i]Remark[/i]. The base ten representation of a positive integer never starts with zero.
[i]Example[/i]. Glueing $15, 14, 7$ in this order makes $15147$.
2004 Iran MO (3rd Round), 15
This problem is easy but nobody solved it.
point $A$ moves in a line with speed $v$ and $B$ moves also with speed $v'$ that at every time the direction of move of $B$ goes from $A$.We know $v \geq v'$.If we know the point of beginning of path of $A$, then $B$ must be where at first that $B$ can catch $A$.
2001 IMC, 4
Let $A=(a_{k,l})_{k,l=1,...,n}$ be a complex $n \times n$ matrix such that for each $m \in \{1,2,...,n\}$ and $1 \leq j_{1} <...<j_{m}$ the determinant of the matrix $(a_{j_{k},j_{l}})_{k,l=1,...,n}$ is zero. Prove that $A^{n}=0$ and that there exists a permutation $\sigma \in S_{n}$ such that the matrix $(a_{\sigma(k),\sigma(l)})_{k,l=1,...,n}$ has all of its nonzero elements above the diagonal.
1998 Belarus Team Selection Test, 2
Let $ p$ be a prime number and $ f$ an integer polynomial of degree $ d$ such that $ f(0) = 0,f(1) = 1$ and $ f(n)$ is congruent to $ 0$ or $ 1$ modulo $ p$ for every integer $ n$. Prove that $ d\geq p - 1$.
2018 AMC 12/AHSME, 19
Let $A$ be the set of positive integers that have no prime factors other than $2$, $3$, or $5$. The infinite sum $$\frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \frac{1}{6} + \frac{1}{8} + \frac{1}{9} + \frac{1}{10} + \frac{1}{12} + \frac{1}{15} + \frac{1}{16} + \frac{1}{18} + \frac{1}{20} + \cdots$$ of the reciprocals of the elements of $A$ can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$?
$\textbf{(A)} \text{ 16} \qquad \textbf{(B)} \text{ 17} \qquad \textbf{(C)} \text{ 19} \qquad \textbf{(D)} \text{ 23} \qquad \textbf{(E)} \text{ 36}$
2017 Mexico National Olympiad, 2
A set of $n$ positive integers is said to be [i]balanced[/i] if for each integer $k$ with $1 \leq k \leq n$, the average of any $k$ numbers in the set is an integer. Find the maximum possible sum of the elements of a balanced set, all of whose elements are less than or equal to $2017$.
2015 Stars Of Mathematics, 3
Let $n$ be a positive integer and let $a_1,a_2,...,a_n$ be non-zero positive integers.Prove that $$\sum_{k=1}^n\frac{\sqrt{a_k}}{1+a_1+a_2+...+a_k}<\sum_{k=1}^{n^2}\frac{1}{k}.$$
2022 Korea Winter Program Practice Test, 3
Let $n\ge 3$ be a positive integer. Amy wrote all the integers from $1$ to $n^2$ on the $n\times n$ grid, so that each cell contains exactly one number. For $i=1,2,\cdots ,n^2-1$, the cell containing $i$ shares a common side with the cell containing $i+1$. Each turn, Bred can choose one cell, and check what number is written. Bred wants to know where $1$ is written by less than $3n$ turns. Determine whether $n$ such that Bred can always achieve his goal is infinite.
2019 Jozsef Wildt International Math Competition, W. 35
Calculate$$\lim \limits_{n \to \infty}\frac{n!\left(1+\frac{1}{n}\right)^{n^2+n}}{n^{n+\frac{1}{2}}}$$
2023 AMC 8, 3
[i]Wind chill[/i] is a measure of how cold people feel when exposed to wind outside. A good estimate for wind chill can be found using this calculation: $$(\text{wind chill}) = (\text{air temperature}) - 0.7 \times (\text{wind speed}),$$ where temperature is measured in degrees Fahrenheit $(^{\circ}\text{F})$ and and the wind speed is measured in miles per hour (mph). Suppose the air temperature is $36^{\circ}\text{F} $ and the wind speed is $18$ mph. Which of the following is closest to the approximate wind chill?
$\textbf{(A)}~18\qquad\textbf{(B)}~23\qquad\textbf{(C)}~28\qquad\textbf{(D)}~32\qquad\textbf{(E)}~35$
2014 Tuymaada Olympiad, 6
Radius of the circle $\omega_A$ with centre at vertex $A$ of a triangle $\triangle{ABC}$ is equal to the radius of the excircle tangent to $BC$. The circles $\omega_B$ and $\omega_C$ are defined similarly. Prove that if two of these circles are tangent then every two of them are tangent to each other.
[i](L. Emelyanov)[/i]
2020 Online Math Open Problems, 15
Let $ABC$ be a triangle with $AB = 20$ and $AC = 22$. Suppose its incircle touches $\overline{BC}$, $\overline{CA}$, and $\overline{AB}$ at $D$, $E$, and $F$ respectively, and $P$ is the foot of the perpendicular from $D$ to $\overline{EF}$. If $\angle BPC = 90^{\circ}$, then compute $BC^2$.
[i]Proposed by Ankan Bhattacharya[/i]
2012 IMAR Test, 4
Design a planar finite non-empty set $S$ satisfying the following two conditions:
(a) every line meets $S$ in at most four points; and
(b) every $2$-colouring of $S$ - that is, each point of $S$ is coloured one of two colours - yields (at least) three monochromatic collinear points.