Found problems: 85335
2000 Irish Math Olympiad, 4
Show that in each set of ten consecutive integers there is one that is coprime with each of the other integers. (For example, in the set $ \{ 114,115,...,123 \}$ there are two such numbers: $ 119$ and $ 121.)$
1996 May Olympiad, 5
In an electronic game of questions and answers, for each correct answer the player adds $5$ points on the screen, for each incorrect answer $2$ points are subtracted and when the player does not answer, no score is added or subtracted. Each game has $30$ questions. Francisco played $5$ games and in all of them he obtained the same number of points, greater than zero, but the number of correct answers, errors and unanswered questions in each game was different. Give all the possible scores that Francisco could obtain.
2024 Thailand Mathematical Olympiad, 3
Let $c$ be a positive real number. Find all functions $f:\mathbb{R}^+\to\mathbb{R}^+$ that satisfy $$x^2f(xf(y))f(x)f(y)=c$$ for all positive reals $x$ and $y$.
2003 Portugal MO, 1
The planet Caramelo is a cube with a $1$ km edge. This planet is going to be wrapped with foam anti-gluttons in order to prevent the presence of greedy ships less than $500$ meters from the planet. What the minimum volume of foam that must surround the planet?
2007 Today's Calculation Of Integral, 249
Determine the sign of $ \int_{\frac{1}{2}}^2 \frac{\ln t}{1\plus{}t^n}\ dt\ (n\equal{}1, 2, \cdots)$.
2006 Dutch Mathematical Olympiad, 2
Given is a acute angled triangle $ABC$. The lengths of the altitudes from $A, B$ and $C$ are successively $h_A, h_B$ and $h_C$. Inside the triangle is a point $P$. The distance from $P$ to $BC$ is $1/3 h_A$ and the distance from $P$ to $AC$ is $1/4 h_B$. Express the distance from $P$ to $AB$ in terms of $h_C$.
2020-2021 Fall SDPC, 3
For some fixed positive integer $n>2$, suppose $x_1$, $x_2$, $x_3$, $\ldots$ is a nonconstant sequence of real numbers such that $x_i=x_j$ if $i \equiv j \pmod{n}$. Let $f(i)=x_i + x_i x_{i+1} + \dots + x_i x_{i+1} \dots x_{i+n-1}$. Given that $$f(1)=f(2)=f(3)=\cdots$$ find all possible values of the product $x_1 x_2 \ldots x_n$.
2019 Jozsef Wildt International Math Competition, W. 19
Let $\{F_n\}_{n\in\mathbb{Z}}$ and $\{L_n\}_{n\in\mathbb{Z}}$ denote the Fibonacci and Lucas numbers, respectively, given by $$F_{n+1} = F_n + F_{n-1}\ \text{and}\ L_{n+1} = L_n + L_{n-1}\ \text{for all}\ n \geq 1$$with $F_0 = 0$, $F_1 = 1$, $L_0 = 2$, and $L_1 = 1$. Prove that for integers $n \geq 1$ and $j \geq 0$
[list=1]
[*]$\sum \limits_{k=1}^n F_{k\pm j}L_{k\mp j}=F_{2n+1}-1+\begin{cases}
0, & \text{if}\ n\ \text{is even}\\
\left(-1\right)^{\pm j}F_{\pm 2j}, & \text{if}\ n\ \text{is odd}
\end{cases}$
[*] $\sum \limits_{k=1}^nF_{k+j}F_{k-j}L_{k+j}L_{k-j}=\frac{F_{4n+2}-1-nL_{4j}}{5}$
[/list]
1958 AMC 12/AHSME, 40
Given $ a_0 \equal{} 1$, $ a_1 \equal{} 3$, and the general relation $ a_n^2 \minus{} a_{n \minus{} 1}a_{n \plus{} 1} \equal{} (\minus{}1)^n$ for $ n \ge 1$. Then $ a_3$ equals:
$ \textbf{(A)}\ \frac{13}{27}\qquad
\textbf{(B)}\ 33\qquad
\textbf{(C)}\ 21\qquad
\textbf{(D)}\ 10\qquad
\textbf{(E)}\ \minus{}17$
1993 Poland - First Round, 1
Prove that the system of equations
$
\begin{cases}
\ a^2 - b = c^2 \\
\ b^2 - a = d^2 \\
\end{cases}
$
has no integer solutions $a, b, c, d$.
2008 China Girls Math Olympiad, 2
Let $ \varphi(x) \equal{} ax^3 \plus{} bx^2 \plus{} cx \plus{} d$ be a polynomial with real coefficients. Given that $ \varphi(x)$ has three positive real roots and that $ \varphi(0) < 0$, prove that
\[ 2b^3 \plus{} 9a^2d \minus{} 7abc \leq 0.
\]
2005 AMC 12/AHSME, 3
Brianna is using part of the money she earned on her weekend job to buy several equally-priced CDs. She used one fifth of her money to buy one third of the CDs. What fraction of her money will she have left after she buys all the CDs?
$ \textbf{(A)}\ \frac{1}{5} \qquad
\textbf{(B)}\ \frac{1}{3} \qquad
\textbf{(C)}\ \frac{2}{5} \qquad
\textbf{(D)}\ \frac{2}{3} \qquad
\textbf{(E)}\ \frac{4}{5}$
Swiss NMO - geometry, 2021.8
Let $\triangle ABC$ be a triangle with $AB =AC$ and $\angle BAC = 20^{\circ}$. Let $D$ be point on the side $AB$ such that $\angle BCD = 70^{\circ}$. Let $E$ be point on the side $AC$ such that $\angle CBE = 60^{\circ}$. Determine the value of angle $\angle CDE$.
2015 Math Prize for Girls Problems, 18
Let $n$ be a positive integer. When the leftmost digit of (the standard base 10 representation of) $n$ is shifted to the rightmost position (the units position), the result is $n/3$. Find the smallest possible value of the sum of the digits of $n$.
2000 Harvard-MIT Mathematics Tournament, 12
Calculate the number of ways of choosing $4$ numbers from the set ${1,2,\cdots ,11}$ such that at least $2$ of the numbers are consecutive.
2022 Adygea Teachers' Geometry Olympiad, 3
The incircle of triangle $ABC$ touches its sides at points $A'$, $B'$, $C'$. $I$ is its center. Straight line $B'I$ intersects segment $A'C'$ at point $P$. Prove that straight line $BP$ passes through the midpoint of $AC$.
2021 Kyiv City MO Round 1, 11.4
For positive real numbers $a, b, c$ with sum $\frac{3}{2}$, find the smallest possible value of the following expression:
$$\frac{a^3}{bc} + \frac{b^3}{ca} + \frac{c^3}{ab} + \frac{1}{abc}$$
[i]Proposed by Serhii Torba[/i]
1990 Romania Team Selection Test, 7
The sequence $ (x_n)_{n \geq 1}$ is defined by:
$ x_1\equal{}1$
$ x_{n\plus{}1}\equal{}\frac{x_n}{n}\plus{}\frac{n}{x_n}$
Prove that $ (x_n)$ increases and $ [x_n^2]\equal{}n$.
2013 NIMO Problems, 14
Let $p$, $q$, and $r$ be primes satisfying \[ pqr = 189999999999999999999999999999999999999999999999999999962.
\] Compute $S(p) + S(q) + S(r) - S(pqr)$, where $S(n)$ denote the sum of the decimals digits of $n$.
[i]Proposed by Evan Chen[/i]
2006 APMO, 3
Let $p\ge5$ be a prime and let $r$ be the number of ways of placing $p$ checkers on a $p\times p$ checkerboard so that not all checkers are in the same row (but they may all be in the same column). Show that $r$ is divisible by $p^5$. Here, we assume that all the checkers are identical.
2019 Thailand TST, 2
A circle $\omega$ with radius $1$ is given. A collection $T$ of triangles is called [i]good[/i], if the following conditions hold:
[list=1]
[*] each triangle from $T$ is inscribed in $\omega$;
[*] no two triangles from $T$ have a common interior point.
[/list]
Determine all positive real numbers $t$ such that, for each positive integer $n$, there exists a good collection of $n$ triangles, each of perimeter greater than $t$.
2001 AMC 8, 1
Casey's shop class is making a golf trophy. He has to paint 300 dimples on a golf ball. If it takes him 2 seconds to paint one dimple, how many minutes will he need to do his job?
$ \text{(A)}\ 4\qquad\text{(B)}\ 6\qquad\text{(C)}\ 8\qquad\text{(D)}\ 10\qquad\text{(E)}\ 12 $
2018 Turkey Team Selection Test, 8
For integers $m\geq 3$, $n$ and $x_1,x_2, \ldots , x_m$ if $x_{i+1}-x_i \equiv x_i-x_{i-1} (mod n) $ for every $2\leq i \leq m-1$, we say that the $m$-tuple $(x_1,x_2,\ldots , x_m)$ is an arithmetic sequence in $(mod n)$. Let $p\geq 5$ be a prime number and $1<a<p-1$ be an integer. Let ${a_1,a_2,\ldots , a_k}$ be the set of all possible remainders when positive powers of $a$ are divided by $p$. Show that if a permutation of ${a_1,a_2,\ldots , a_k}$ is an arithmetic sequence in $(mod p)$, then $k=p-1$.
1999 AMC 12/AHSME, 24
Six points on a circle are given. Four of the chords joining pairs of the six points are selected at random. What is the probability that the four chords are the sides of a convex quadrilateral?
$ \textbf{(A)}\ \frac{1}{15}\qquad
\textbf{(B)}\ \frac{1}{91}\qquad
\textbf{(C)}\ \frac{1}{273}\qquad
\textbf{(D)}\ \frac{1}{455}\qquad
\textbf{(E)}\ \frac{1}{1365}$
2016 JBMO Shortlist, 1
Determine the largest positive integer $n$ that divides $p^6 - 1$ for all primes $p > 7$.