Found problems: 85335
2001 China Team Selection Test, 2.2
Given distinct positive integers \( g \) and \( h \), let all integer points on the number line \( OX \) be vertices. Define a directed graph \( G \) as follows: for any integer point \( x \), \( x \rightarrow x + g \), \( x \rightarrow x - h \). For integers \( k, l (k < l) \), let \( G[k, l] \) denote the subgraph of \( G \) with vertices limited to the interval \([k, l]\). Find the largest positive integer \( \alpha \) such that for any integer \( r \), the subgraph \( G[r, r + \alpha - 1] \) of \( G \) is acyclic. Clarify the structure of subgraphs \( G[r, r + \alpha - 1] \) and \( G[r, r + \alpha] \) (i.e., how many connected components and what each component is like).
Novosibirsk Oral Geo Oly VII, 2020.2
It is known that four of these sticks can be assembled into a quadrilateral. Is it always true that you can make a triangle out of three of them?
2023 ISL, N7
Let $a,b,c,d$ be positive integers satisfying \[\frac{ab}{a+b}+\frac{cd}{c+d}=\frac{(a+b)(c+d)}{a+b+c+d}.\] Determine all possible values of $a+b+c+d$.
2001 Bundeswettbewerb Mathematik, 4
Prove: For each positive integer is the number of divisors whose decimal representations ends with a 1 or 9 not less than the number of divisors whose decimal representations ends with 3 or 7.
2023 4th Memorial "Aleksandar Blazhevski-Cane", P1
Let $a, b, c, d$ be integers. Prove that for any positive integer $n$, there are at least $\left \lfloor{\frac{n}{4}}\right \rfloor $ positive integers $m \leq n$ such that $m^5 + dm^4 + cm^3 + bm^2 + 2023m + a$ is not a perfect square.
[i]Proposed by Ilir Snopce[/i]
2020 AMC 12/AHSME, 12
Let $\overline{AB}$ be a diameter in a circle of radius $5\sqrt2.$ Let $\overline{CD}$ be a chord in the circle that intersects $\overline{AB}$ at a point $E$ such that $BE=2\sqrt5$ and $\angle AEC = 45^{\circ}.$ What is $CE^2+DE^2?$
$\textbf{(A)}\ 96 \qquad\textbf{(B)}\ 98 \qquad\textbf{(C)}\ 44\sqrt5 \qquad\textbf{(D)}\ 70\sqrt2 \qquad\textbf{(E)}\ 100$
2021 JHMT HS, 10
A pharmaceutical company produces a disease test that has a $95\%$ accuracy rate on individuals who actually have an infection, and a $90\%$ accuracy rate on individuals who do not have an infection. They use their test on a population of mathletes, of which $2\%$ actually have an infection. If a test concludes that a mathlete has an infection, then the probability that the mathlete actually does have an infection is $\tfrac{a}{b},$ where $a$ and $b$ are relatively prime positive integers. Find $a + b.$
1966 Kurschak Competition, 1
Can we arrange $5$ points in space to form a pentagon with equal sides such that the angle between each pair of adjacent edges is $90^o$?
2002 Junior Balkan Team Selection Tests - Moldova, 4
$9$ chess players participate in a chess tournament. According to the regulation, each participant plays a single game with each of the others. At a certain moment of the competition it was found that exactly two participants played the same number of party. To prove that in this case, not a single chess player played any the game, or just one chess player played with everyone else.
2012 National Olympiad First Round, 12
How many subsets of the set $\{1,2,3,4,5,6,7,8,9,10\}$ are there that does not contain 4 consequtive integers?
$ \textbf{(A)}\ 596 \qquad \textbf{(B)}\ 648 \qquad \textbf{(C)}\ 679 \qquad \textbf{(D)}\ 773 \qquad \textbf{(E)}\ 812$
2000 India Regional Mathematical Olympiad, 4
All the $7$ digit numbers containing each of the digits $1,2,3,4,5,6,7$ exactly once , and not divisible by $5$ are arranged in increasing order. Find the $200th$ number in the list.
2018 Danube Mathematical Competition, 2
Prove that there are in finitely many pairs of positive integers $(m, n)$ such that simultaneously $m$ divides $n^2 + 1$ and $n$ divides $m^2 + 1$.
2018 AMC 12/AHSME, 5
What is the sum of all possible values of $k$ for which the polynomials $x^2 - 3x + 2$ and $x^2 - 5x + k$ have a root in common?
$
\textbf{(A) }3 \qquad
\textbf{(B) }4 \qquad
\textbf{(C) }5 \qquad
\textbf{(D) }6 \qquad
\textbf{(E) }10 \qquad
$
2012 IFYM, Sozopol, 6
Calculate the sum
$1+\frac{\binom{2}{1}}{8}+\frac{\binom{4}{2}}{8^2}+\frac{\binom{6}{3}}{8^3}+...+\frac{\binom{2n}{n}}{8^n}+...$
2020 Canada National Olympiad, 3
There are finite many coins in David’s purse. The values of these coins are pair wisely distinct positive integers. Is that possible to make such a purse, such that David has exactly $2020$ different ways to select the coins in his purse and the sum of these selected coins is $2020$?
2023 239 Open Mathematical Olympiad, 4
There are a million numbered chairs at a large round table. The Sultan has seated a million wise men on them. Each of them sees the thousand people following him in clockwise order. Each of them was given a cap of black or white color, and they must simultaneously write down on their own piece of paper a guess about the color of their cap. Those who do not guess will be executed. The wise men had the opportunity to agree on a strategy before the test. What is the largest number of survivors that they can guarantee?
May Olympiad L1 - geometry, 1997.2
In the rectangle $ABCD, M, N, P$ and $Q$ are the midpoints of the sides. If the area of the shaded triangle is $1$, calculate the area of the rectangle $ABCD$.
[img]https://2.bp.blogspot.com/-9iyKT7WP5fc/XNYuXirLXSI/AAAAAAAAKK4/10nQuSAYypoFBWGS0cZ5j4vn_hkYr8rcwCK4BGAYYCw/s400/may3.gif[/img]
1980 Poland - Second Round, 2
Prove that for any real numbers $ x_1, x_2, x_3, \ldots, x_n $ the inequality is true
$$ x_1x_2x_3\ldots x_n \leq \frac{x_1^2}{2} + \frac{x_2^4}{4} + \frac{x_3^8}{8} + \ldots + \frac{x_n^{2^ n}}{2^n} + \frac{1}{2^n}$$
2004 AMC 8, 2
How many different four-digit numbers can be formed by rearranging the four digits in $2004$?
$\textbf{(A)}\ 4\qquad
\textbf{(B)}\ 6\qquad
\textbf{(C)}\ 16\qquad
\textbf{(D)}\ 24\qquad
\textbf{(E)}\ 81$
2016 AMC 8, 14
Karl's car uses a gallon of gas every $35$ miles, and his gas tank holds $14$ gallons when it is full. One day, Karl started with a full tank of gas, drove $350$ miles, bought $8$ gallons of gas, and continued driving to his destination. When he arrived, his gas tank was half full. How many miles did Karl drive that day?
$\textbf{(A)}\mbox{ }525\qquad\textbf{(B)}\mbox{ }560\qquad\textbf{(C)}\mbox{ }595\qquad\textbf{(D)}\mbox{ }665\qquad\textbf{(E)}\mbox{ }735$
2016 Saudi Arabia GMO TST, 1
Let $ABC$ be an acute, non-isosceles triangle which is inscribed in a circle $(O)$. A point $I$ belongs to the segment $BC$. Denote by $H$ and $K$ the projections of $I$ on $AB$ and $AC$, respectively. Suppose that the line $HK$ intersects $(O)$ at $M, N$ ($H$ is between $M, K$ and $K$ is between $H, N$). Prove the following assertions:
a) If $A$ is the center of the circle $(IMN)$, then $BC$ is tangent to $(IMN)$.
b) If $I$ is the midpoint of $BC$, then $BC$ is equal to $4$ times of the distance between the centers of two circles $(ABK)$ and $(ACH)$.
2014 Putnam, 3
Let $a_0=5/2$ and $a_k=a_{k-1}^2-2$ for $k\ge 1.$ Compute \[\prod_{k=0}^{\infty}\left(1-\frac1{a_k}\right)\] in closed form.
2019 Saudi Arabia Pre-TST + Training Tests, 1.2
Determine all arithmetic sequences $a_1, a_2,...$ for which there exists integer $N > 1$ such that for any positive integer $k$ the following divisibility holds $a_1a_2 ...a_k | a_{N+1}a_{N+2}...a_{N+k}$ .
2015 Princeton University Math Competition, A5/B7
Alice has an orange $\text{3-by-3-by-3}$ cube, which is comprised of $27$ distinguishable, $\text{1-by-1-by-1}$ cubes. Each small cube was initially orange, but Alice painted $10$ of the small cubes completely black. In how many ways could she have chosen $10$ of these smaller cubes to paint black such that every one of the $27$ $\text{3-by-1-by-1}$ sub-blocks of the $\text{3-by-3-by-3}$ cube contains at least one small black cube?
2012 Turkey MO (2nd round), 3
Find all non-decreasing functions from real numbers to itself such that for all real numbers $x,y$ $f(f(x^2)+y+f(y))=x^2+2f(y)$ holds.