Found problems: 85335
2017 AIME Problems, 2
When each of 702, 787, and 855 is divided by the positive integer $m$, the remainder is always the positive integer $r$. When each of 412, 722, and 815 is divided by the positive integer $n$, the remainder is always the positive integer $s \neq r$. Fine $m+n+r+s$.
2015 Romania Team Selection Test, 4
Let $k$ be a positive integer congruent to $1$ modulo $4$ which is not a perfect square and let $a=\frac{1+\sqrt{k}}{2}$.
Show that $\{\left \lfloor{a^2n}\right \rfloor-\left \lfloor{a\left \lfloor{an}\right \rfloor}\right \rfloor : n \in \mathbb{N}_{>0}\}=\{1 , 2 , \ldots ,\left \lfloor{a}\right \rfloor\}$.
1983 IMO Longlists, 61
Let $a$ and $b$ be integers. Is it possible to find integers $p$ and $q$ such that the integers $p+na$ and $q +nb$ have no common prime factor no matter how the integer $n$ is chosen ?
2019 Brazil Team Selection Test, 1
Determine all pairs $(n, k)$ of distinct positive integers such that there exists a positive integer $s$ for which the number of divisors of $sn$ and of $sk$ are equal.
2008 HMNT, 2
What is the units digit of $7^{2009}$?
2018 CMIMC Combinatorics, 9
Compute the number of rearrangements $a_1, a_2, \dots, a_{2018}$ of the sequence $1, 2, \dots, 2018$ such that $a_k > k$ for $\textit{exactly}$ one value of $k$.
2012 National Olympiad First Round, 9
The chord $[CD]$ of the circle with diameter $[AB]$ is perpendicular to $[AB]$. Let $M$ and $N$ be the midpoints of $[BC]$ and $[AD]$, respectively. If $|BC|=6$ and $|AD|=2\sqrt{3}$, then $|MN|=?$
$ \textbf{(A)}\ 4 \qquad \textbf{(B)}\ 3 \sqrt 2 \qquad \textbf{(C)}\ \sqrt{21} \qquad \textbf{(D)}\ 5 \qquad \textbf{(E)}\ \text{None}$
2021-IMOC, C1
The numbers $1,2,\cdots,2021$ are arranged in a circle. For any $1 \le i \le 2021$, if $i,i+1,i+2$ are three consecutive numbers in some order such that $i+1$ is not in the middle, then $i$ is said to be a good number. Indices are taken mod $2021$. What is the maximum possible number of good numbers?
[i]CSJL[/i]
2006 Romania National Olympiad, 2
Let $n$ be a positive integer. Prove that there exists an integer $k$, $k\geq 2$, and numbers $a_i \in \{ -1, 1 \}$, such that \[ n = \sum_{1\leq i < j \leq k } a_ia_j . \]
2004 Bulgaria National Olympiad, 3
A group consist of n tourists. Among every 3 of them there are 2 which are not familiar. For every partition of the tourists in 2 buses you can find 2 tourists that are in the same bus and are familiar with each other. Prove that is a tourist familiar to at most $\displaystyle \frac 2{5}n$ tourists.
2005 AMC 12/AHSME, 6
In $ \triangle ABC$, we have $ AC \equal{} BC \equal{} 7$ and $ AB \equal{} 2$. Suppose that $ D$ is a point on line $ AB$ such that $ B$ lies between $ A$ and $ D$ and $ CD \equal{} 8$. What is $ BD$?
$ \textbf{(A)}\ 3\qquad
\textbf{(B)}\ 2 \sqrt {3}\qquad
\textbf{(C)}\ 4\qquad
\textbf{(D)}\ 5\qquad
\textbf{(E)}\ 4 \sqrt {2}$
1970 AMC 12/AHSME, 24
An equilateral triangle and a regular hexagon have equal perimeters. If the area of the triangle is $2$, then the area of the hexagon is
$\textbf{(A) }2\qquad\textbf{(B) }3\qquad\textbf{(C) }4\qquad\textbf{(D) }6\qquad \textbf{(E) }12$
2018 Saudi Arabia BMO TST, 4
Let $ABC$ be an acute, non isosceles with $I$ is its incenter. Denote $D, E$ as tangent points of $(I)$ on $AB,AC$, respectively. The median segments respect to vertex $A$ of triangles $ABE$ and $ACD$ meet$ (I)$ at$ P,Q,$ respectively. Take points $M, N$ on the line $DE$ such that $AM \parallel BE$ and $AN \parallel C D$ respectively.
a) Prove that $A$ lies on the radical axis of $(MIP)$ and $(NIQ)$.
b) Suppose that the orthocenter $H$ of triangle $ABC$ lies on $(I)$. Prove that there exists a line which is tangent to three circles of center $A, B, C$ and all pass through $H$.
1993 Denmark MO - Mohr Contest, 3
Determine all real solutions $x,y$ to the system of equations
$$\begin{cases} x^2 + y^2 = 1 \\ x^6 + y^6 = \dfrac{7}{16} \end{cases}$$
2004 Thailand Mathematical Olympiad, 6
Let $f(x) = x^6 + x^5 + x^4 + x^3 + x^2 + x + 1$. Find the remainder when $f(x^7)$ is divided by $f(x)$.
1997 Akdeniz University MO, 4
A plane dividing like a chessboard and write a real number each square such that, for a squares' number equal to its up, down ,left and right squares' numbers arithmetic mean. Prove that all number are equal.
2019 MIG, 7
How many positive integers less than or equal to $150$ have exactly three distinct prime factors?
2022 BMT, 3
Suppose we have four real numbers $a,b,c,d$ such that $a$ is nonzero, $a,b,c$ form a geometric sequence, in that order, and $b,c,d$ form an arithmetic sequence, in that order. Compute the smallest possible value of $\frac{d}{a}.$ (A geometric sequence is one where every succeeding term can be written as the previous term multiplied by a constant, and an arithmetic sequence is one where every succeeeding term can be written as the previous term added to a constant.)
III Soros Olympiad 1996 - 97 (Russia), 9.6
Find the common fraction with the smallest positive denominator lying between the fractions $\frac{96}{35}$ and $\frac{97}{36} $.
2014 Austria Beginners' Competition, 3
Let $a, b, c$ and $d$ be real numbers with $a < b < c < d$.
Sort the numbers $x = a \cdot b + c \cdot d, y = b \cdot c + a \cdot d$ and $z = c \cdot a + b \cdot d$ in ascending\order and prove the correctness of your result.
(R. Henner, Vienna)
2010 CHMMC Winter, 10
Compute the number of $10$-bit sequences of $0$’s and $1$’s do not contain $001$ as a subsequence.
2005 National Olympiad First Round, 14
We call a number $10^3 < n < 10^6$ a [i]balanced [/i]number if the sum of its last three digits is equal to the sum of its other digits. What is the sum of all balanced numbers in $\bmod {13}$?
$
\textbf{(A)}\ 0
\qquad\textbf{(B)}\ 5
\qquad\textbf{(C)}\ 7
\qquad\textbf{(D)}\ 11
\qquad\textbf{(E)}\ 12
$
2012 AMC 10, 20
A $3\times3$ square is partitioned into $9$ unit squares. Each unit square is painted either white or black with each color being equally likely, chosen independently and at random. The square is the rotated $90^\circ$ clockwise about its center, and every white square in a position formerly occupied by a black square is painted black. The colors of all other squares are left unchanged. What is the probability that the grid is now entirely black?
$ \textbf{(A)}\ \dfrac{49}{512}
\qquad\textbf{(B)}\ \dfrac{7}{64}
\qquad\textbf{(C)}\ \dfrac{121}{1024}
\qquad\textbf{(D)}\ \dfrac{81}{512}
\qquad\textbf{(E)}\ \dfrac{9}{32}
$
2021 BMT, 20
For some positive integer $n$, $(1 + i) + (1 + i)^2 + (1 + i)^3 + ... + (1 + i)^n = (n^2 - 1)(1 - i)$, where $i = \sqrt{-1}$. Compute the value of $n$.
2011 IFYM, Sozopol, 1
$In$ $triangle$ $ABC$ $bisectors$ $AA_1$, $BB_1$ $and$ $CC_1$ $are$ $drawn$. $Bisectors$ $AA_1$ $and$ $CC_1$ $intersect$ $segments$ $C_1B_1$ $and$ $B_1A_1$ $at$ $points$ $M$ $and$ $N$, $respectively$. $Prove$ $that$ $\angle$$MBB_1$ = $\angle$$NBB_1$.