Found problems: 85335
1975 AMC 12/AHSME, 21
Suppose $ f(x)$ is defined for all real numbers $ x$; $ f(x)>0$ for all $ x$, and $ f(a)f(b)\equal{}f(a\plus{}b)$ for all $ a$ and $ b$. Which of the following statements is true?
$ \text{I. } f(0)\equal{}1$
$ \text{II. } f(\minus{}a)\equal{}1/f(a) \text{ for all } a$
$ \text{III. } f(a) \equal{} \sqrt[3]{f(3a)} \text{ for all } a$
$ \text{IV. } f(b)>f(a) \text{ if } b>a$
$ \textbf{(A)}\ \text{III and IV only} \qquad$
$ \textbf{(B)}\ \text{I, III, and IV only} \qquad$
$ \textbf{(C)}\ \text{I, II, and IV only} \qquad$
$ \textbf{(D)}\ \text{I, II, and III only} \qquad$
$ \textbf{(E)}\ \text{All are true.}$
2019 Saudi Arabia Pre-TST + Training Tests, 3.3
All of the numbers $1, 2,3,...,1000000$ are initially colored black. On each move it is possible to choose the number $x$ (among the colored numbers) and change the color of $x$ and of all of the numbers that are not co-prime with $x$ (black into white, white into black). Is it possible to color all of the numbers white?
2023 LMT Spring, 6
Aidan, Boyan, Calvin, Derek, Ephram, and Fanalex are all chamber musicians at a concert. In each performance, any combination of musicians of them can perform for all the others to watch. What is the minimum number of performances necessary to ensure that each musician watches every other musician play?
Kvant 2020, M2633
There are two round tables with $n{}$ dwarves sitting at each table. Each dwarf has only two friends: his neighbours to the left and to the right. A good wizard wants to seat the dwarves at one round table so that each two neighbours are friends. His magic allows him to make any $2n$ pairs of dwarves into pairs of friends (the dwarves in a pair may be from the same or from different tables). However, he knows that an evil sorcerer will break $n{}$ of those new friendships. For which $n{}$ is the good wizard able to achieve his goal no matter what the evil sorcerer does?
[i]Mikhail Svyatlovskiy[/i]
2022 AMC 8 -, 7
When the World Wide Web first became popular in the $1990$s, download speeds reached a maximum of about $56$ kilobits per second. Approximately how many minutes would the download of a $4.2$-megabyte song have taken at that speed? (Note that there are $8000$ kilobits in a megabyte.)
$\textbf{(A)} ~0.6\qquad\textbf{(B)} ~10\qquad\textbf{(C)} ~1800\qquad\textbf{(D)} ~7200\qquad\textbf{(E)} ~36000\qquad$
2022 Chile Junior Math Olympiad, 2
In a trapezoid $ABCD$ whose parallel sides $AB$ and $CD$ are in ratio $\frac{AB}{CD}=\frac32$, the points $ N$ and $M$ are marked on the sides $BC$ and $AB$ respectively, in such a way that $BN = 3NC$ and $AM = 2MB$ and segments $AN$ and $DM$ are drawn that intersect at point $P$, find the ratio between the areas of triangle $APM$ and trapezoid $ABCD$.
[img]https://cdn.artofproblemsolving.com/attachments/7/8/21d59ca995d638dfcb76f9508e439fd93a5468.png[/img]
2007 Belarusian National Olympiad, 8
Let $(m,n)$ be a pair of positive integers.
(a) Prove that the set of all positive integers can be partitioned into four pairwise disjoint nonempty subsets such that none of them has two numbers with absolute value of their difference equal to either $m$, $n$, or $m+n$.
(b) Find all pairs $(m,n)$ such that the set of all positive integers can not be partitioned into three pairwise disjoint nonempty subsets satisfying the above condition.
2007 IMC, 3
Let $ C$ be a nonempty closed bounded subset of the real line and $ f: C\to C$ be a nondecreasing continuous function. Show that there exists a point $ p\in C$ such that $ f(p) \equal{} p$.
(A set is closed if its complement is a union of open intervals. A function $ g$ is nondecreasing if $ g(x)\le g(y)$ for all $ x\le y$.)
2010 ISI B.Stat Entrance Exam, 1
Let $a_1,a_2,\cdots, a_n$ and $b_1,b_2,\cdots, b_n$ be two permutations of the numbers $1,2,\cdots, n$. Show that
\[\sum_{i=1}^n i(n+1-i) \le \sum_{i=1}^n a_ib_i \le \sum_{i=1}^n i^2\]
2023 VN Math Olympiad For High School Students, Problem 4
Determine whether or not the length of symmedian is not greater than the length of the angle bisector drawn from the same vertex?
2015 British Mathematical Olympiad Round 1, 4
James has a red jar, a blue jar and a pile of $100$ pebbles. Initially both jars are empty. A move consists of moving a pebble from the pile into one of the jars or returning a pebble from one of the jars to the pile. The numbers of pebbles in the red and blue jars determine the state of the game. The followwing conditions must be satisfied:
(a) The red jar may never contain fewer pebbles than the blue jar;
(b) The game may never be returned to a previous state.
What is the maximum number of moves that James can make?
2014 Online Math Open Problems, 30
Let $p = 2^{16}+1$ be an odd prime. Define $H_n = 1+ \frac{1}{2} + \frac{1}{3} + ... + \frac{1}{n}$. Compute the remainder when \[ (p-1)! \sum_{n = 1}^{p-1} H_n \cdot 4^n \cdot \binom{2p-2n}{p-n} \] is divided by $p$.
[i]Proposed by Yang Liu[/i]
2014 VTRMC, Problem 4
Suppose we are given a $19\times19$ chessboard (a table with $19^2$ squares) and remove the central square. Is it possible to tile the remaining $19^2-1=360$ squares with $4\times1$ and $1\times4$ rectangles? (So that each of the $360$ squares is covered by exactly one rectangle.) Justify your answer.
2024 HMNT, 21
Two points are chosen independently and uniformly at random from the interior of the $X$-pentomino shown below. Compute the probability that the line segment between these two points lies entirely within the $X$-pentomino.
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2003 Paraguay Mathematical Olympiad, 3
Today the age of Pedro is written and then the age of Luisa, obtaining a number of four digits that is a perfect square. If the same is done in $33$ years from now, there would be a perfect square of four digits . Find the current ages of Pedro and Luisa.
2022 HMNT, 7
Compute the number of ordered pairs of positive integers $(a, b)$ satisfying the equation
$$gcd(a, b) \cdot a + b^2 = 10000.$$
2021 Iran Team Selection Test, 5
Point $X$ is chosen inside the non trapezoid quadrilateral $ABCD$ such that $\angle AXD +\angle BXC=180$.
Suppose the angle bisector of $\angle ABX$ meets the $D$-altitude of triangle $ADX$ in $K$, and the angle bisector of $\angle DCX$ meets the $A$-altitude of triangle $ADX$ in $L$.We know $BK \perp CX$ and $CL \perp BX$. If the circumcenter of $ADX$ is on the line $KL$ prove that $KL \perp AD$.
Proposed by [i]Alireza Dadgarnia[/i]
2016 Azerbaijan Team Selection Test, 1
Determine all positive integers $M$ such that the sequence $a_0, a_1, a_2, \cdots$ defined by \[ a_0 = M + \frac{1}{2} \qquad \textrm{and} \qquad a_{k+1} = a_k\lfloor a_k \rfloor \quad \textrm{for} \, k = 0, 1, 2, \cdots \] contains at least one integer term.
2010 Dutch BxMO TST, 4
The two circles $\Gamma_1$ and $\Gamma_2$ intersect at $P$ and $Q$. The common tangent that's on the same side as $P$, intersects the circles at $A$ and $B$,respectively. Let $C$ be the second intersection with $\Gamma_2$ of the tangent to $\Gamma_1$ at $P$, and let $D$ be the second intersection with $\Gamma_1$ of the tangent to $\Gamma_2$ at $Q$. Let $E$ be the intersection of $AP$ and $BC$, and let $F$ be the intersection of $BP$ and $AD$. Let $M$ be the image of $P$ under point reflection with respect to the midpoint of $AB$. Prove that $AMBEQF$ is a cyclic hexagon.
Russian TST 2017, P1
A planar country has an odd number of cities separated by pairwise distinct distances. Some of these cities are connected by direct two-way flights. Each city is directly connected to exactly two ther cities, and the latter are located farthest from it. Prove that, using these flights, one may go from any city to any other city
2017 Dutch IMO TST, 4
Find all functions $f : \mathbb{R} \rightarrow \mathbb{R}$ such that
$$(y + 1)f(x) + f(xf(y) + f(x + y))= y$$
for all $x, y \in \mathbb{R}$.
1982 Vietnam National Olympiad, 1
Find all positive integers $x, y, z$ such that $2^x + 2^y + 2^z = 2336$.
2022 IMAR Test, 1
Find all pairs of primes $p, q<2023$ such that $p \mid q^2+8$ and $q \mid p^2+8$.
2021 Indonesia TST, N
For every positive integer $n$, let $p(n)$ denote the number of sets $\{x_1, x_2, \dots, x_k\}$ of integers with $x_1 > x_2 > \dots > x_k > 0$ and $n = x_1 + x_3 + x_5 + \dots$ (the right hand side here means the sum of all odd-indexed elements). As an example, $p(6) = 11$ because all satisfying sets are as follows: $$\{6\}, \{6, 5\}, \{6, 4\}, \{6, 3\}, \{6, 2\}, \{6, 1\}, \{5, 4, 1\}, \{5, 3, 1\}, \{5, 2, 1\}, \{4, 3, 2\}, \{4, 3, 2, 1\}.$$ Show that $p(n)$ equals to the number of partitions of $n$ for every positive integer $n$.
2013 Vietnam National Olympiad, 3
Let $ABC$ be a triangle such that $ABC$ isn't a isosceles triangle. $(I)$ is incircle of triangle touches $BC,CA,AB$ at $D,E,F$ respectively. The line through $E$ perpendicular to $BI$ cuts $(I)$ again at $K$. The line through $F$ perpendicular to $CI$ cuts $(I)$ again at $L$.$J$ is midpoint of $KL$.
[b]a)[/b] Prove that $D,I,J$ collinear.
[b]b)[/b] $B,C$ are fixed points,$A$ is moved point such that $\frac{AB}{AC}=k$ with $k$ is constant.$IE,IF$ cut $(I)$ again at $M,N$ respectively.$MN$ cuts $IB,IC$ at $P,Q$ respectively. Prove that bisector perpendicular of $PQ$ through a fixed point.