Found problems: 85335
A diabolical combination lock has $n$ dials (each with $c$ possible states), where $n,c>1$. The dials are initially set to states $d_1, d_2, \ldots, d_n$, where $0\le d_i\le c-1$ for each $1\le i\le n$. Unfortunately, the actual states of the dials (the $d_i$'s) are concealed, and the initial settings of the dials are also unknown. On a given turn, one may advance each dial by an integer amount $c_i$ ($0\le c_i\le c-1$), so that every dial is now in a state $d_i '\equiv d_i+c_i \pmod{c}$ with $0\le d_i ' \le c-1$. After each turn, the lock opens if and only if all of the dials are set to the zero state; otherwise, the lock selects a random integer $k$ and cyclically shifts the $d_i$'s by $k$ (so that for every $i$, $d_i$ is replaced by $d_{i-k}$, where indices are taken modulo $n$).
Show that the lock can always be opened, regardless of the choices of the initial configuration and the choices of $k$ (which may vary from turn to turn), if and only if $n$ and $c$ are powers of the same prime.
[i]Bobby Shen.[/i]
On the trapezoid $ABCD$ , side $DA$ is perpendicular to the bases $AB$ and $CD$ . The base $AB$ measures $45$, the base $CD$ measures $20$ and the $BC$ side measures $65$. Let $P$ on the $BC$ side such that $BP$ measures $45$ and $M$ is the midpoint of $DA$. Calculate the measure of the $PM$ segment.
Suppose $P(x)$ is a monic polynomial of degree $2023$ such that $P(k) = k^{2023}P(1-\frac{1}{k})$ for every positive integer $1 \leq k \leq 2023$. Then $P(-1) = \frac{a}{b}$ where $a$ and $b$ are relatively prime integers. Compute the unique integer $0 \leq n < 2027$ such that $bn-a$ is divisible by the prime $2027$.
The year $1978$ was “peculiar” in that the sum of the numbers formed with the first two digits and the last two digits is equal to the number formed with the middle two digits, i.e., $19+78=97$. What was the last previous peculiar year, and when will the next one occur?
Let $a_1 = 2,$ and for $n\ge 1,$ let $a_{n+1} = 2a_n + 1.$ Find the smallest value of an $a_n$ that is not a prime number.
Let $f(x)$ be a function acting on a string of $0$s and $1$s, defined to be the number of substrings of $x$ that have at least one $1$, where a substring is a contiguous sequence of characters in $x$. Let $S$ be the set of binary strings with $24$ ones and $100$ total digits. Compute the maximum possible value of $f(s)$ over all $s\in S$.
For example, $f(110) = 5$ as $\underline{1}10$, $1\underline{1}0$, $\underline{11}0$, $1\underline{10}$, and $\underline{110}$ are all substrings including a $1$. Note that $11\underline{0}$ is not such a substring.
Let $ABCD$ be a cyclic quadrilateral satisfying $AD^2 + BC^2 = AB^2$. The diagonals of $ABCD$ intersect at $E$. Let $P$ be a point on side $\overline{AB}$ satisfying $\angle APD = \angle BPC$. Show that line $PE$ bisects $\overline{CD}$.
[i]Proposed by Ankan Bhattacharya[/i]
Berit has $11$ twenty kroner coins, $14$ ten kroner coins, and $12$ five kroner coins. An exchange machine can exchange three ten kroner coins into one twenty kroner coin and two five kroner coins, and the reverse. It can also exchange two twenty kroner coins into three ten kroner coins and two five kroner coins, and the reverse.
(i) Can Berit get the same number of twenty kroner and ten kroner coins, but no five kroner coins?
(ii) Can she get the same number each of twenty kroner, ten kroner, and five kroner coins?
Distinct points $ A$, $ B$, $ C$, and $ D$ lie on a line, with $ AB\equal{}BC\equal{}CD\equal{}1$. Points $ E$ and $ F$ lie on a second line, parallel to the first, with $ EF\equal{}1$. A triangle with positive area has three of the six points as its vertices. How many possible values are there for the area of the triangle?
$ \textbf{(A)}\ 3 \qquad
\textbf{(B)}\ 4 \qquad
\textbf{(C)}\ 5 \qquad
\textbf{(D)}\ 6 \qquad
\textbf{(E)}\ 7$
Let $n$ be a positive integer number and let $a_1, a_2, \ldots, a_n$ be $n$ positive real numbers. Prove that $f : [0, \infty) \rightarrow \mathbb{R}$, defined by
\[f(x) = \dfrac{a_1 + x}{a_2 + x} + \dfrac{a_2 + x}{a_3 + x} + \cdots + \dfrac{a_{n-1} + x}{a_n + x} + \dfrac{a_n + x}{a_1 + x}, \]
is a decreasing function.
[i]Dan Marinescu et al.[/i]
A point $P$ and a segment $AB$ with length $20$ are randomly drawn on a plane. Suppose that the probability that a randomly selected line passing through $P$ intersects segment $AB$ is $\tfrac{1}{2}$. Next, randomly choose point $Q$ on segment $AB$. What is the probability with respect to choosing $Q$ that a circle centered at $Q$ passing through $P$ contains both $A$ and $B$ in its interior?
The inscribed circle of the triangle $ABC{}$ touches the sides $BC,CA$ and $AB{}$ at $A',B'$ and $C'{}$ respectively. Let $I_a$ be the $A$-excenter of $ABC{}.$ Prove that $I_aA'$ is perpendicular to the line determined by the circumcenters of $I_aBC'$ and $I_aCB'.$
a) Suppose that a sequence of numbers $x_1,x_2,x_3,...$ satisfies the inequality $x_n-2x_{n+1}+x_{n+2} \le 0$ for any $n$ . Moreover $x_o=1,x_{20}=9,x_{200}=6$. What is the maximal value of $x_{2009}$ can be?
b) Suppose that a sequence of numbers $x_1,x_2,x_3,...$ satisfies the inequality $2x_n-3x_{n+1}+x_{n+2} \le 0$ for any $n$. Moreover $x_o=1,x_1=2,x_3=1$. Can $x_{2009}$ be greater then $0,678$ ?
Some cities in country are connected with oneway road. It is known that every closed cyclic route, that don`t break traffic laws, consists of even roads. Prove that king of city can place military bases in some cities such that there are not roads between these cities, but for every city without base we can go from city with base by no more than $1$ road.
[hide=PS]I think it should be one more condition, like there is cycle that connect all cities [/hide]
Let $T = \{1,2,...,10\}$. Find the number of bijective functions $f : T\to T$ that satis es the following for all $x \in T$:
$f(f(x)) = x$
$|f(x) - x| \ge 2$
Let $S$ be a set of $n^2 + 1$ closed intervals ($n$ a positive integer). Prove that at least one of the following assertions holds:
[b](i)[/b] There exists a subset $S'$ of $n+1$ intervals from $S$ such that the intersection of the intervals in $S'$ is nonempty.
[b](ii)[/b] There exists a subset $S''$ of $n + 1$ intervals from $S$ such that any two of the intervals in $S''$ are disjoint.
Find the number of positive integers $n$ less than $10000$ such that there are more $4$’s in the digits of $n + 1$ than in the digits of $n$.
Find all functions $f : \mathbb{N} \rightarrow \mathbb{N}$ such that for any two positive integers $m,n$ we have :
$$f(n)+1400m^2|n^2+f(f(m))$$
Find the number of functions $f$ that map the set $\{1,2, 3,4\}$ into itself such that the range of the function $f(x)$ is the same as the range of the function $f(f(x))$.
Let $f_1 (x) = x^2 - 3$ and $f_n (x) = f_1(f_{n-1} (x))$ for $n \ge 2$. Let $m_n$ be the smallest positive root of $f_n$, and $M_n$ be the largest positive root of $f_n$. If $x$ is the least number such that $M_n \le m_n \cdot x$ for all $n \ge 1$, compute $x^2$.
Given that $A_k=\frac{k(k-1)}2\cos\frac{k(k-1)\pi}2,$ find $|A_{19}+A_{20}+\cdots+A_{98}|.$
Let $f(x)$ and $g(x)$ be polynomials with non-negative integer coefficients, and let m be the largest coefficient of $f.$ Suppose that there exist natural numbers $a < b$ such that $f(a) = g(a)$ and $f(b) = g(b)$. Show that if $b > m,$ then $f = g.$
Let $O$ be an interior point of acute triangle $ABC$. Let $A_1$ lie on $BC$ with $OA_1$ perpendicular to $BC$. Define $B_1$ on $CA$ and $C_1$ on $AB$ similarly. Prove that $O$ is the circumcenter of $ABC$ if and only if the perimeter of $A_1B_1C_1$ is not less than any one of the perimeters of $AB_1C_1, BC_1A_1$, and $CA_1B_1$.
In $\vartriangle ABC$, the points $D, E$ and $F$ lie on $AB, BC$ and $CA$ respectively. The line segments $AE, BF$ and $CD$ meet at the point $G$. Suppose that the area of each of $\vartriangle BGD, \vartriangle ECG$ and $\vartriangle GFA$ is $1$ cm$^2$. Prove that the area of each of $\vartriangle BEG, \vartriangle GCF$ and $\vartriangle ADG$ is also $1$ cm$^2$.
[img]https://cdn.artofproblemsolving.com/attachments/e/7/ec090135bd2e47a9681d767bb984797d87218c.png[/img]
Let equilateral triangle $ABC$ with side length $6$ be inscribed in a circle and let $P$ be on arc $AC$ such that $AP \cdot P C = 10$. Find the length of $BP$.