Found problems: 85335
2006 Hanoi Open Mathematics Competitions, 6
On the circle of radius $30$ cm are given $2$ points A,B with $AB = 16$ cm and $C$ is a midpoint of $AB$. What is the perpendicular distance from $C$ to the circle?
1966 IMO Shortlist, 54
We take $100$ consecutive natural numbers $a_{1},$ $a_{2},$ $...,$ $a_{100}.$ Determine the last two digits of the number $a_{1}^{8}+a_{2}^{8}+...+a_{100}^{8}.$
2020 AMC 12/AHSME, 9
How many solutions does the equation $\tan{(2x)} = \cos{(\tfrac{x}{2})}$ have on the interval $[0, 2\pi]?$
$\textbf{(A) } 1 \qquad \textbf{(B) } 2 \qquad \textbf{(C) } 3 \qquad \textbf{(D) } 4 \qquad \textbf{(E) } 5$
2012 CIIM, Problem 4
Let $f(x) = \frac{\sin(x)}{x}$ Find $$ \lim_{T\to\infty}\frac{1}{T}\int_0^T\sqrt{1+f'(x)^2}dx.$$
2009 Middle European Mathematical Olympiad, 4
Determine all integers $ k\ge 2$ such that for all pairs $ (m$, $ n)$ of different positive integers not greater than $ k$, the number $ n^{n\minus{}1}\minus{}m^{m\minus{}1}$ is not divisible by $ k$.
2016 CMIMC, 3
Let $ABC$ be a triangle. The angle bisector of $\angle B$ intersects $AC$ at point $P$, while the angle bisector of $\angle C$ intersects $AB$ at a point $Q$. Suppose the area of $\triangle ABP$ is 27, the area of $\triangle ACQ$ is 32, and the area of $\triangle ABC$ is $72$. The length of $\overline{BC}$ can be written in the form $m\sqrt n$ where $m$ and $n$ are positive integers with $n$ as small as possible. What is $m+n$?
2018 PUMaC Live Round, Misc. 3
Suppose $x,y\in\mathbb{Z}$ satisfy
$$y^4+4y^3+28y+8x^3+6y^2+32x+1=(x^2-y^2)(x^2+y^2+24).$$
Find the sum of all possible values of $|xy|$.
2025 AIME, 7
The twelve letters $A$,$B$,$C$,$D$,$E$,$F$,$G$,$H$,$I$,$J$,$K$, and $L$ are randomly grouped into six pairs of letters. The two letters in each pair are placed next to each other in alphabetical order to form six two-letter words, and then those six words are listed alphabetically. For example, a possible result is $AB$, $CJ$, $DG$, $EK$, $FL$, $HI$. The probability that the last word listed contains $G$ is $\frac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
2000 IMO Shortlist, 1
Determine all positive integers $ n\geq 2$ that satisfy the following condition: for all $ a$ and $ b$ relatively prime to $ n$ we have \[a \equiv b \pmod n\qquad\text{if and only if}\qquad ab\equiv 1 \pmod n.\]
2019 IFYM, Sozopol, 2
Does there exist a strictly increasing function $f:\mathbb{N}\rightarrow \mathbb{N}$, such that for $\forall$ $n\in \mathbb{N}$:
$f(f(f(n)))=n+2f(n)$?
2017 CMIMC Combinatorics, 7
Given a finite set $S \subset \mathbb{R}^3$, define $f(S)$ to be the mininum integer $k$ such that there exist $k$ planes that divide $\mathbb{R}^3$ into a set of regions, where no region contains more than one point in $S$. Suppose that
\[M(n) = \max\{f(S) : |S| = n\} \text{ and } m(n) = \min\{f(S) : |S| = n\}.\]
Evaluate $M(200) \cdot m(200)$.
1976 Bundeswettbewerb Mathematik, 4
In a plane are given $n > 2$ distinct points. Some pairs of these points are connected by segments so that no two of the segments intersect. Prove that there are at most $3n-6$ segments.
2013 Purple Comet Problems, 30
Suppose $x,y$ and $z$ are integers that satisfy the system of equations \[x^2y+y^2z+z^2x=2186\] \[xy^2+yz^2+zx^2=2188.\] Evaluate $x^2+y^2+z^2.$
2018 Caucasus Mathematical Olympiad, 4
Morteza places a function $[0,1]\to [0,1]$ (that is a function with domain [0,1] and values from [0,1]) in each cell of an $n \times n$ board. Pavel wants to place a function $[0,1]\to [0,1]$ to the left of each row and below each column (i.e. to place $2n$ functions in total) so that the following condition holds for any cell in this board:
If $h$ is the function in this cell, $f$ is the function below its column, and $g$ is the function to the left of its row, then $h(x) = f(g(x))$ for all $x \in [0, 1]$.
Prove that Pavel can always fulfil his plan.
Kvant 2023, M2730
On each cell of a $3\times 6$ the board lies one coin. It is known that some two coins lying on adjacent cells are fake. They have the same weigh, but are lighter than the real ones. All the other coins are real. How can one find both counterfeit coins in three weightings on a double-pan balance, without using weights?
[i]Proposed by K. Knop[/i]
2014 Contests, 1
Determine all triples $(a,b,c)$, where $a, b$, and $c$ are positive integers that satisfy
$a \le b \le c$ and $abc = 2(a + b + c)$.
2020 China Team Selection Test, 6
Given a simple, connected graph with $n$ vertices and $m$ edges. Prove that one can find at least $m$ ways separating the set of vertices into two parts, such that the induced subgraphs on both parts are connected.
2014 ASDAN Math Tournament, 6
Consider a circle of radius $4$ with center $O_1$, a circle of radius $2$ with center $O_2$ that lies on the circumference of circle $O_1$, and a circle of radius $1$ with center $O_3$ that lies on the circumference of circle $O_2$. The centers of the circle are collinear in the order $O_1$, $O_2$, $O_3$. Let $A$ be a point of intersection of circles $O_1$ and $O_2$ and $B$ be a point of intersection of circles $O_2$ and $O_3$ such that $A$ and $B$ lie on the same semicircle of $O_2$. Compute the length of $AB$.
2023 Ukraine National Mathematical Olympiad, 10.1
Find all positive integers $k$, for which the product of some consecutive $k$ positive integers ends with $k$.
[i]Proposed by Oleksiy Masalitin[/i]
2015 Online Math Open Problems, 1
Evaluate \[ \sqrt{\binom82+\binom92+\binom{15}2+\binom{16}2}. \]
[i] Proposed by Evan Chen [/i]
2013 IMO Shortlist, N4
Determine whether there exists an infinite sequence of nonzero digits $a_1 , a_2 , a_3 , \cdots $ and a positive integer $N$ such that for every integer $k > N$, the number $\overline{a_k a_{k-1}\cdots a_1 }$ is a perfect square.
2007 India IMO Training Camp, 2
Find all integer solutions of the equation \[\frac {x^{7} \minus{} 1}{x \minus{} 1} \equal{} y^{5} \minus{} 1.\]
2013 Flanders Math Olympiad, 4
Consider (in the plane) three concentric circles with radii $1, 2$ and $3$ and equilateral triangle $\Delta$ such that on each of the three circles is one vertex of $\Delta$ . Calculate the length of the side of $\Delta$ .
[img]https://1.bp.blogspot.com/-q40dl3TSQqE/Xy1QAcno_9I/AAAAAAAAMR8/11nsSA0syNAaGb3W7weTHsNpBeGQZXnHACLcBGAsYHQ/s0/flanders%2B2013%2Bp4.png[/img]
2012 AMC 12/AHSME, 7
Small lights are hung on a string 6 inches apart in the order red, red, green, green, green, red, red, green, green, green, and so on continuing this pattern of 2 red lights followed by 3 green lights. How many feet separate the third red light and the 21st red light?
[b]Note:[/b] 1 foot is equal to 12 inches.
$\textbf{(A)}\ 18 \qquad\textbf{(B)}\ 18.5 \qquad\textbf{(C)}\ 20 \qquad\textbf{(D)}\ 20.5 \qquad\textbf{(E)}\ 22.5 $
MMPC Part II 1996 - 2019, 2001
[b]p1. [/b] A clock has a long hand for minutes and a short hand for hours. A placement of those hands is [i]natural [/i] if you will see it in a correctly functioning clock. So, having both hands pointing straight up toward $12$ is natural and so is having the long hand pointing toward $6$ and the short hand half-way between $2$ and $3$. A natural placement of the hands is symmetric if you get another natural placement by interchanging the long and short hands. One kind of symmetric natural placement is when the hands are pointed in exactly the same direction.
Are there symmetric natural placements of the hands in which the two hands are not pointed in exactly the same direction? If so, describe one such placement. If not, explain why none are possible.
[b]p2.[/b] Let $\frac{m}{n}$ be a fraction such that when you write out the decimal expansion of $\frac{m}{n}$ , it eventually ends up with the four digits $2001$ repeated over and over and over. Prove that $101$ divides $n$.
[b]p3.[/b] Consider the following two questions:
Question $1$: I am thinking of a number between $0$ and $15$. You get to ask me seven yes-or-no questions, and I am allowed to lie at most once in answering your questions. What seven questions can you ask that will always allow you to determine the number? Note: You need to come up with seven questions that are independent of the answers that are received. In other words, you are not allowed to say, "If the answer to question $1$ is yes, then question $2$ is XXX; but if the answer to question $1$ is no, then question $2$ is YYY."
Question $2$: Consider the set $S$ of all seven-tuples of zeros and ones. What sixteen elements of $S$ can you choose so that every pair of your chosen seven-tuples differ in at least three coordinates?
a. These two questions are closely related. Show that an answer to Question $1$ gives an answer to Question $2$.
b. Answer either Question $1$ or Question $2$.
[b]p4.[/b] You may wish to use the angle addition formulas for the sine and cosine functions:
$\sin (\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta$
$\cos (\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta$
a) Prove the identity $(\sin x)(1 + 2 \cos 2x) = \sin (3x)$.
b) For any positive integer $n$, prove the identity $$(sin x)(1 + 2 \cos 2x + 2\cos 4x + ... +2\cos 2nx) = \sin
((2n +1)x)$$
[b]p5.[/b] Define the set $\Omega$ in the $xy$-plane as the union of the regions bounded by the three geometric figures: triangle $A$ with vertices $(0.5, 1.5)$, $(1.5, 0.5)$ and $(0.5,-0.5)$, triangle $B$ with vertices $(-0.5,1.5)$, $(-1.5,-0.5)$ and $(-0.5, 0.5)$, and rectangle $C$ with corners $(0.5, 1.0)$, $(-0.5, 1.0)$, $(-0.5,-1.0)$, and $(0.5,-1.0)$.
a. Explain how copies of $\Omega$ can be used to cover the $xy$-plane. The copies are obtained by translating $\Omega$ in the $xy$-plane, and copies can intersect only along their edges.
b. We can define a transformation of the plane as follows: map any point $(x, y)$ to $(x + G, x + y + G)$, where $G = 1$ if $y < -2x$, $G = -1$ if $y > -2x$, and $G = 0$ if $y = -2x$. Prove that every point in $\Omega$ is transformed into another point in $\Omega$, and that there are at least two points in $\Omega$ that are transformed into the same point.
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