Found problems: 85335
2004 National High School Mathematics League, 3
For integer $n\geq4$, find the smallest integer $f(n)$, such that for any positive integer $m$, in any subset with $f(n)$ elements of the set $\{m, m+1, \cdots, m+n-1\}$ there are at least three elements that are relatively prime .
2016 Regional Olympiad of Mexico Northeast, 2
Let $ABC$ be a triangle with $AB = AC$ with centroid $G$. Let $M$ and $N$ be the midpoints of $AB$ and $AC$ respectively and $O$ be the circumcenter of triangle $BCN$ . Prove that $MBOG$ is a cyclic quadrilateral .
1985 Austrian-Polish Competition, 8
A convex $n$-gon $A_0A_1\dots A_{n-1}$ has been partitioned into $n-2$ triangles by certain diagonals not intersecting inside the $n$-gon. Prove that these triangles can be labeled $\triangle_1,\triangle_2,\dots,\triangle_{n-2}$ in such a way that $A_i$ is a vertex of $\triangle_i$, for $i=1,2,\dots,n-2$. Find the number of all such labellings.
2015 AMC 12/AHSME, 23
Let $S$ be a square of side length $1$. Two points are chosen independently at random on the sides of $S$. The probability that the straight-line distance between the points is at least $\tfrac12$ is $\tfrac{a-b\pi}c$, where $a$, $b$, and $c$ are positive integers and $\gcd(a,b,c)=1$. What is $a+b+c$?
$\textbf{(A) }59\qquad\textbf{(B) }60\qquad\textbf{(C) }61\qquad\textbf{(D) }62\qquad\textbf{(E) }63$
1995 National High School Mathematics League, 3
If a person A is taller or heavier than another peoson B, then we note that A is [i]not worse than[/i] B. In 100 persons, if someone is [i]not worse than[/i] other 99 people, we call him [i]excellent boy[/i]. What's the maximum value of the number of [i]excellent boys[/i]?
$\text{(A)}1\qquad\text{(B)}2\qquad\text{(C)}50\qquad\text{(D)}100$
2003 Junior Tuymaada Olympiad, 3
In the acute triangle $ ABC $, the point $ I $ is the center of the inscribed the circle, the point $ O $ is the center of the circumscribed circle and the point $ I_a $ is the center the excircle tangent to the side $ BC $ and the extensions of the sides $ AB $ and $ AC $. Point $ A'$ is symmetric to vertex $ A $ with respect to the line $ BC $. Prove that $ \angle IOI_a = \angle IA'I_a $.
1941 Putnam, B5
A car is being driven so that its wheels, all of radius $a$ feet, have an angular velocity of $\omega$ radians per second.
A particle is thrown off from the tire of one of these wheels, where it is supposed that $a \omega^{2} >g$. Neglecting the resistance of the air, show that the maximum height above the roadway which the particle can reach is
$$\frac{(a \omega+g \omega^{-1})^{2}}{2g}.$$
1974 AMC 12/AHSME, 19
In the adjoining figure $ABCD$ is a square and $CMN$ is an equilateral triangle. If the area of $ABCD$ is one square inch, then the area of $CMN$ in square inches is
[asy]
draw((0,0)--(1,0)--(1,1)--(0,1)--cycle);
draw((.82,0)--(1,1)--(0,.76)--cycle);
label("A", (0,0), S);
label("B", (1,0), S);
label("C", (1,1), N);
label("D", (0,1), N);
label("M", (0,.76), W);
label("N", (.82,0), S);
[/asy]
$ \textbf{(A)}\ 2\sqrt{3}-3 \qquad\textbf{(B)}\ 1-\frac{\sqrt{3}}{3} \qquad\textbf{(C)}\ \frac{\sqrt{3}}{4} \qquad\textbf{(D)}\ \frac{\sqrt{2}}{3} \qquad\textbf{(E)}\ 4-2\sqrt{3} $
2018 AIME Problems, 8
A frog is positioned at the origin in the coordinate plane. From the point $(x,y)$, the frog can jump to any of the points $(x+1, y), (x+2, y), (x, y+1),$ or $(x, y+2)$. Find the number of distinct sequences of jumps in which the frog begins at $(0,0)$ and ends at $(4,4)$.
IV Soros Olympiad 1997 - 98 (Russia), 9.3
Several machines were working in the workshop. After reconstruction, the number of machines decreased, and the percentage by which the number of machines decreased turned out to be equal to the number of remaining machines. What was the smallest number of machines that could have been in the workshop before the reconstruction?
2014 PUMaC Combinatorics A, 6
Let $f(n)$ be the number of points of intersection of diagonals of a $n$-dimensional hypercube that is not the vertex of the cube. For example, $f(3) = 7$ because the intersection points of a cube’s diagonals are at the centers of each face and the center of the cube. Find $f(5)$.
2021 AMC 10 Spring, 22
Ang, Ben, and Jasmin each have $5$ blocks, colored red, blue, yellow, white, and green; and there are $5$ empty boxes. Each of the people randomly and independently of the other two people places one of their blocks into each box. The probability that at least one box receives $3$ blocks all of the same color is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m + n ?$
$\textbf{(A)} ~47 \qquad\textbf{(B)} ~94 \qquad\textbf{(C)} ~227 \qquad\textbf{(D)} ~471 \qquad\textbf{(E)} ~542$
2003 Croatia National Olympiad, Problem 2
For every integer $n>2$, prove the equality
$$\left\lfloor\frac{n(n+1)}{4n-2}\right\rfloor=\left\lfloor\frac{n+1}4\right\rfloor.$$
2012 Online Math Open Problems, 33
You are playing a game in which you have $3$ envelopes, each containing a uniformly random amount of money between $0$ and $1000$ dollars. (That is, for any real $0 \leq a < b \leq 1000$, the probability that the amount of money in a given envelope is between $a$ and $b$ is $\frac{b-a}{1000}$.) At any step, you take an envelope and look at its contents. You may choose either to keep the envelope, at which point you finish, or discard it and repeat the process with one less envelope. If you play to optimize your expected winnings, your expected winnings will be $E$. What is $\lfloor E\rfloor,$ the greatest integer less than or equal to $E$?
[i]Author: Alex Zhu[/i]
1957 AMC 12/AHSME, 14
If $ y \equal{} \sqrt{x^2 \minus{} 2x \plus{} 1} \plus{} \sqrt{x^2 \plus{} 2x \plus{} 1}$, then $ y$ is:
$ \textbf{(A)}\ 2x\qquad
\textbf{(B)}\ 2(x \plus{} 1)\qquad
\textbf{(C)}\ 0\qquad
\textbf{(D)}\ |x \minus{} 1| \plus{} |x \plus{} 1|\qquad
\textbf{(E)}\ \text{none of these}$
KoMaL A Problems 2020/2021, A. 785
Let $k\ge t\ge 2$ positive integers. For integers $n\ge k$ let $p_n$ be the probability that if we choose $k$ from the first $n$ positive integers randomly, any $t$ of the $k$ chosen integers have greatest common divisor $1$. Let qn be the probability that if we choose $k-t+1$ from the first $n$ positive integers the product is not divisible by a perfect $t^{th}$ power that is greater then $1$.
Prove that sequences $p_n$ and $q_n$ converge to the same value.
JBMO Geometry Collection, 2016
A trapezoid $ABCD$ ($AB || CF$,$AB > CD$) is circumscribed.The incircle of the triangle $ABC$ touches the lines $AB$ and $AC$ at the points $M$ and $N$,respectively.Prove that the incenter of the trapezoid $ABCD$ lies on the line $MN$.
1974 IMO Longlists, 8
Let $x, y, z$ be real numbers each of whose absolute value is different from $\frac{1}{\sqrt 3}$ such that $x + y + z = xyz$. Prove that
\[\frac{3x - x^3}{1-3x^2} + \frac{3y - y^3}{1-3y^2} + \frac{3z -z^3}{1-3z^2} = \frac{3x - x^3}{1-3x^2} \cdot \frac{3y - y^3}{1-3y^2} \cdot \frac{3z - z^3}{1-3z^2}\]
2010 Contests, 2
Prove that for any real number $ x$ the following inequality is true:
$ \max\{|\sin x|, |\sin(x\plus{}2010)|\}>\dfrac1{\sqrt{17}}$
1954 Moscow Mathematical Olympiad, 270
Consider $\vartriangle ABC$ and a point $S$ inside it. Let $A_1, B_1, C_1$ be the intersection points of $AS, BS, CS$ with $BC, AC, AB$, respectively. Prove that at least in one of the resulting quadrilaterals $AB_1SC_1, C_1SA_1B, A_1SB_1C$ both angles at either $C_1$ and $B_1$, or $C_1$ and $A_1$, or $A_1$ and $B_1$ are not acute.
2016 Harvard-MIT Mathematics Tournament, 16
Determine the number of integers $2 \le n \le 2016$ such that $n^n-1$ is divisible by $2$, $3$, $5$, $7$.
2019 HMIC, 2
Annie has a permutation $(a_1, a_2, \dots ,a_{2019})$ of $S=\{1,2,\dots,2019\}$, and Yannick wants to guess her permutation. With each guess Yannick gives Annie an $n$-tuple $(y_1, y_2, \dots, y_{2019})$ of integers in $S$, and then Annie gives the number of indices $i\in S$ such that $a_i=y_i$.
(a) Show that Yannick can always guess Annie's permutation with at most $1200000$ guesses.
(b) Show that Yannick can always guess Annie's permutation with at most $24000$ guesses.
[i]Yannick Yao[/i]
2019 Latvia Baltic Way TST, 8
A $20 \times 20$ rectangular grid has been given. It is known that one of the grid's unit squares contains a hidden treasure. To find the treasure, we have been given an opportunity to order several scientific studies at the same time, results of which will be known only after some time. For each study we must choose one $1 \times 4$ rectangle, and the study will tell whether the rectangle contains the treasure. The $1 \times 4$ rectangle can be either horizontal or vertical, and it can extend over a side of the $20 \times 20$ grid, coming back in at the opposite side (you can think of the $20 \times 20$ grid as a torus - the opposite sides are connected).
What is the minimal amount of studies that have to ordered for us to precisely determine the unit square containing the treasure?
2022 Kyiv City MO Round 1, Problem 5
Find the smallest integer $n$ for which it's possible to cut a square into $2n$ squares of two sizes: $n$ squares of one size, and $n$ squares of another size.
[i](Proposed by Bogdan Rublov)[/i]
2011 China National Olympiad, 1
Let $a_1,a_2,\ldots,a_n$ are real numbers, prove that;
\[\sum_{i=1}^na_i^2-\sum_{i=1}^n a_i a_{i+1} \le \left\lfloor \frac{n}{2}\right\rfloor(M-m)^2.\]
where $a_{n+1}=a_1,M=\max_{1\le i\le n} a_i,m=\min_{1\le i\le n} a_i$.