Found problems: 85335
2020-IMOC, N2
Find all positive integers $N$ such that the following holds: There exist pairwise coprime positive integers $a,b,c$ with
$$\frac1a+\frac1b+\frac1c=\frac N{a+b+c}.$$
Indonesia Regional MO OSP SMA - geometry, 2013.5
Given an acute triangle $ABC$. The longest line of altitude is the one from vertex $A$ perpendicular to $BC$, and it's length is equal to the length of the median of vertex $B$. Prove that $\angle ABC \le 60^o$
2007 Indonesia Juniors, day 2
p1. Four kite-shaped shapes as shown below ($a > b$, $a$ and $b$ are natural numbers less than $10$) arranged in such a way so that it forms a square with holes in the middle. The square hole in the middle has a perimeter of $16$ units of length. What is the possible perimeter of the outermost square formed if it is also known that $a$ and $b$ are numbers coprime?
[img]https://cdn.artofproblemsolving.com/attachments/4/1/fa95f5f557aa0ca5afb9584d5cee74743dcb10.png[/img]
p2. If $a = 3^p$, $b = 3^q$, $c = 3^r$, and $d = 3^s$ and if $p, q, r$, and $s$ are natural numbers, what is the smallest value of $p\cdot q\cdot r\cdot s$ that satisfies $a^2 + b^3 + c^5 = d^7$
3. Ucok intends to compose a key code (password) consisting of 8 numbers and meet the following conditions:
i. The numbers used are $1, 2, 3, 4, 5, 6, 7, 8$, and $9$.
ii. The first number used is at least $1$, the second number is at least $2$, third digit-at least $3$, and so on.
iii. The same number can be used multiple times.
a) How many different passwords can Ucok compose?
b) How many different passwords can Ucok make, if provision (iii) is replaced with: no numbers may be used more than once.
p 4. For any integer $a, b$, and $c$ applies $a\times (b + c) = (a\times b) + (a\times c)$.
a) Look for examples that show that $a + (b\times c)\ne (a + b)\times (a + c)$.
b) Is it always true that $a + (b\times c) = (a + b)\times(a + c)$? Justify your answer.
p5. The results of a survey of $N$ people with the question whether they maintain dogs, birds, or cats at home are as follows: $50$ people keep birds, $61$ people don't have dogs, $13$ people don't keep a cat, and there are at least $74$ people who keep the most a little two kinds of animals in the house. What is the maximum value and minimum of possible value of $N$ ?
2002 Tuymaada Olympiad, 2
Let $a,b,c,d$ be positive real numbers such that $abcd=1$. Prove that
\[ \frac{1+ab}{1+a} + \frac{1+bc}{1+b} + \frac{1+cd}{1+c} + \frac{1+da}{1+d} \geq 4 . \]
[i]Proposed by A. Khrabrov[/i]
2017 China Team Selection Test, 4
Given a circle with radius 1 and 2 points C, D given on it. Given a constant l with $0<l\le 2$. Moving chord of the circle AB=l and ABCD is a non-degenerated convex quadrilateral. AC and BD intersects at P. Find the loci of the circumcenters of triangles ABP and BCP.
2020 AMC 10, 15
A positive integer divisor of $12!$ is chosen at random. The probability that the divisor chosen is a perfect square can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$?
$\textbf{(A)}\ 3\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 12\qquad\textbf{(D)}\ 18\qquad\textbf{(E)}\ 23$
Ukrainian TYM Qualifying - geometry, 2017.2
Points $P, Q, R$ were marked on the sides $BC, CA, AB$, respectively. Let $a$ be tangent at point $A$ to the circumcircle of triangle $AQR$, $b$ be tangent at point $B$ to the circumcircle of the triangle BPR, $c$ be tangent at point $C$ to the circumscribed circle triangle $CPQ$. Let $X$ be the point of intersection of the lines $b$ and $c, Y$ be the point the intersection of lines $c$ and $a, Z$ is the point of intersection of lines $a$ and $b$. Prove that the lines $AX, BY, CZ$ intersect at one point if and only if the lines $AP, BQ, CR$ intersect at one point.
2006 National Olympiad First Round, 12
In how many different ways can the set $\{1,2,\dots, 2006\}$ be divided into three non-empty sets such that no set contains two successive numbers?
$
\textbf{(A)}\ 3^{2006}-3\cdot 2^{2006}+1
\qquad\textbf{(B)}\ 2^{2005}-2
\qquad\textbf{(C)}\ 3^{2004}
\qquad\textbf{(D)}\ 3^{2005}-1
\qquad\textbf{(E)}\ \text{None of above}
$
2011 Singapore Senior Math Olympiad, 2
Determine if there is a set $S$ of 2011 positive integers so that for every pair $m,n$ of distinct elements of $S$, $|m-n|=(m,n)$. Here $(m,n)$ denotes the greatest common divisor of $m$ and $n$.
2002 India Regional Mathematical Olympiad, 4
Suppose the integers $1,2,\ldots 10$ are split into two disjoint collections $a_1,a_2, \ldots a_5$ and $b_1 , \ldots b_5$ such that $a_1 <a _2 < a_3 <a_4 <a _5 , b_1 < b_2 < b_3 < b_4 < b_5$
(i) Show that the larger number in any pair $\{ a_j, b_j \}$ , $1 \leq j \leq 5$ is at least $6$.
(ii) Show that $\sum_{i=1} ^{5} | a_i - b_i|$ = 25 for every such partition.
2018 Turkey Team Selection Test, 7
For integers $a, b$, call the lattice point with coordinates $(a,b)$ [b]basic[/b] if $gcd(a,b)=1$. A graph takes the basic points as vertices and the edges are drawn in such way: There is an edge between $(a_1,b_1)$ and $(a_2,b_2)$ if and only if $2a_1=2a_2\in \{b_1-b_2, b_2-b_1\}$ or $2b_1=2b_2\in\{a_1-a_2, a_2-a_1\}$. Some of the edges will be erased, such that the remaining graph is a forest. At least how many edges must be erased to obtain this forest? At least how many trees exist in such a forest?
2007 Germany Team Selection Test, 1
We define a sequence $ \left(a_{1},a_{2},a_{3},\ldots \right)$ by
\[ a_{n} \equal{} \frac {1}{n}\left(\left\lfloor\frac {n}{1}\right\rfloor \plus{} \left\lfloor\frac {n}{2}\right\rfloor \plus{} \cdots \plus{} \left\lfloor\frac {n}{n}\right\rfloor\right),
\] where $\lfloor x\rfloor$ denotes the integer part of $x$.
[b]a)[/b] Prove that $a_{n+1}>a_n$ infinitely often.
[b]b)[/b] Prove that $a_{n+1}<a_n$ infinitely often.
[i]Proposed by Johan Meyer, South Africa[/i]
Champions Tournament Seniors - geometry, 2018.3
The vertex $F$ of the parallelogram $ACEF$ lies on the side $BC$ of parallelogram $ABCD$. It is known that $AC = AD$ and $AE = 2CD$. Prove that $\angle CDE = \angle BEF$.
2021 MIG, 14
The notation $\lfloor n \rfloor$ denotes the greatest integer less than or equal to $n$. Evaluate $\lfloor 2.1 \lfloor {-}4.3 \rfloor \rfloor$.
$\textbf{(A) }{-}11\qquad\textbf{(B) }{-}10\qquad\textbf{(C) }{-}9\qquad\textbf{(D) }{-}8\qquad\textbf{(E) }{-}4$
2011 Argentina Team Selection Test, 1
Each number from the set $\{1,2,3,4,5,6,7,8\}$ is either colored red or blue, following these rules:
a) The number $4$ is colored red, and there is at least one number colored blue.
b) If two numbers $x, y$ have different colors and $x + y \leq 8$, then the number $x + y$ is colored blue.
c) If two numbers $x, y$ have different colors and $x \cdot y \leq 8$, then the number $x \cdot y$ is colored red.
Find all possible ways the numbers from this set can be colored.
2000 All-Russian Olympiad Regional Round, 8.8
There are 2000 cities in the country. Every city is connected by non-stop two-way airlines with some other cities, and for each city, the number of airlines originating from it is a factor of two. (i.e. $1$, $2$, $4$, $8$, $...$). For each city $A$, the statistician calculated the number routes with no more than one transfer connecting $A$ with other cities, and then summed up the results for all $2000$ cities. He got $100,000$. Prove that the statistician was wrong.
2012 Grigore Moisil Intercounty, 3
$ \lim_{n\to\infty } \frac{1}{n}\sum_{i,j=1}^n \frac{i+j}{i^2+j^2} $
2014 Junior Balkan MO, 1
Find all triples of primes $(p,q,r)$ satisfying $3p^{4}-5q^{4}-4r^{2}=26$.
2007 Today's Calculation Of Integral, 196
Calculate
\[\frac{\int_{0}^{\pi}e^{-x}\sin^{n}x\ dx}{\int_{0}^{\pi}e^{x}\sin^{n}x \ dx}\ (n=1,\ 2,\ \cdots). \]
2010 Germany Team Selection Test, 3
On a $999\times 999$ board a [i]limp rook[/i] can move in the following way: From any square it can move to any of its adjacent squares, i.e. a square having a common side with it, and every move must be a turn, i.e. the directions of any two consecutive moves must be perpendicular. A [i]non-intersecting route[/i] of the limp rook consists of a sequence of pairwise different squares that the limp rook can visit in that order by an admissible sequence of moves. Such a non-intersecting route is called [i]cyclic[/i], if the limp rook can, after reaching the last square of the route, move directly to the first square of the route and start over.
How many squares does the longest possible cyclic, non-intersecting route of a limp rook visit?
[i]Proposed by Nikolay Beluhov, Bulgaria[/i]
2024 CAPS Match, 2
For a positive integer $n$, an $n$-configuration is a family of sets $\left\langle A_{i,j}\right\rangle_{1\le i,j\le n}.$ An $n$-configuration is called [i]sweet[/i] if for every pair of indices $(i, j)$ with $1\le i\le n -1$ and $1\le j\le n$ we have $A_{i,j}\subseteq A_{i+1,j}$ and $A_{j,i}\subseteq A_{j,i+1}.$ Let $f(n, k)$ denote the number of sweet $n$-configurations such that $A_{n,n}\subseteq \{1, 2,\ldots , k\}$. Determine which number is larger: $f\left(2024, 2024^2\right)$ or $f\left(2024^2, 2024\right).$
2010 Purple Comet Problems, 22
Ten distinct points are placed on a circle. All ten of the points are paired so that the line segments connecting the pairs do not intersect. In how many different ways can this pairing be done?
[asy]
import graph; size(12cm);
real labelscalefactor = 0.5;
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps);
pen dotstyle = black;
draw((2.46,0.12)--(3.05,-0.69));
draw((2.46,1.12)--(4,-1));
draw((5.54,0.12)--(4.95,-0.69));
draw((3.05,1.93)--(5.54,1.12));
draw((4.95,1.93)--(4,2.24));
draw((8.05,1.93)--(7.46,1.12));
draw((7.46,0.12)--(8.05,-0.69));
draw((9,2.24)--(9,-1));
draw((9.95,-0.69)--(9.95,1.93));
draw((10.54,1.12)--(10.54,0.12));
draw((15.54,1.12)--(15.54,0.12));
draw((14.95,-0.69)--(12.46,0.12));
draw((13.05,-0.69)--(14,-1));
draw((12.46,1.12)--(14.95,1.93));
draw((14,2.24)--(13.05,1.93));
label("1",(-1.08,2.03),SE*labelscalefactor);
label("2",(-0.3,1.7),SE*labelscalefactor);
label("3",(0.05,1.15),SE*labelscalefactor);
label("4",(0.00,0.38),SE*labelscalefactor);
label("5",(-0.33,-0.12),SE*labelscalefactor);
label("6",(-1.08,-0.4),SE*labelscalefactor);
label("7",(-1.83,-0.19),SE*labelscalefactor);
label("8",(-2.32,0.48),SE*labelscalefactor);
label("9",(-2.3,1.21),SE*labelscalefactor);
label("10",(-1.86,1.75),SE*labelscalefactor);
dot((-1,-1),dotstyle);
dot((-0.05,-0.69),dotstyle);
dot((0.54,0.12),dotstyle);
dot((0.54,1.12),dotstyle);
dot((-0.05,1.93),dotstyle);
dot((-1,2.24),dotstyle);
dot((-1.95,1.93),dotstyle);
dot((-2.54,1.12),dotstyle);
dot((-2.54,0.12),dotstyle);
dot((-1.95,-0.69),dotstyle);
dot((4,-1),dotstyle);
dot((4.95,-0.69),dotstyle);
dot((5.54,0.12),dotstyle);
dot((5.54,1.12),dotstyle);
dot((4.95,1.93),dotstyle);
dot((4,2.24),dotstyle);
dot((3.05,1.93),dotstyle);
dot((2.46,1.12),dotstyle);
dot((2.46,0.12),dotstyle);
dot((3.05,-0.69),dotstyle);
dot((9,-1),dotstyle);
dot((9.95,-0.69),dotstyle);
dot((10.54,0.12),dotstyle);
dot((10.54,1.12),dotstyle);
dot((9.95,1.93),dotstyle);
dot((9,2.24),dotstyle);
dot((8.05,1.93),dotstyle);
dot((7.46,1.12),dotstyle);
dot((7.46,0.12),dotstyle);
dot((8.05,-0.69),dotstyle);
dot((14,-1),dotstyle);
dot((14.95,-0.69),dotstyle);
dot((15.54,0.12),dotstyle);
dot((15.54,1.12),dotstyle);
dot((14.95,1.93),dotstyle);
dot((14,2.24),dotstyle);
dot((13.05,1.93),dotstyle);
dot((12.46,1.12),dotstyle);
dot((12.46,0.12),dotstyle);
dot((13.05,-0.69),dotstyle);[/asy]
2021 MMATHS, 9
Suppose that $P(x)$ is a monic cubic polynomial with integer roots, and suppose that $\frac{P(a)}{a}$ is an integer for exactly $6$ integer values of $a$. Suppose furthermore that exactly one of the distinct numbers $\frac{P(1) + P(-1)}{2}$ and $\frac{P(1) - P(-1)}{2}$ is a perfect square. Given that $P(0) > 0$, find the second-smallest possible value of $P(0).$
[i]Proposed by Andrew Wu[/i]
V Soros Olympiad 1998 - 99 (Russia), 10.3
It is known that $\sin 3x = 3 \sin x - 4 \sin^3x$. It is also easy to prove that $\sin nx$ for odd $n$ can be represented as a polynomial of degree $n$ of $\sin x$. Let $\sin 1999x = P(\sin x)$, where $P(t)$ is a polynomial of the $1999$th degree of $t$. Solve the equation $$P \left(\cos \frac{x}{1999}\right) = \frac12 .$$
1964 AMC 12/AHSME, 20
The sum of the numerical coefficients of all the terms in the expansion of $(x-2y)^{18}$ is:
$ \textbf{(A)}\ 0\qquad\textbf{(B)}\ 1\qquad\textbf{(C)}\ 19\qquad\textbf{(D)}\ -1\qquad\textbf{(E)}\ -19 $