Found problems: 5923
Find the largest prime number less than $2008$ that is a divisor of some integer in the infinite sequence \[\left\lfloor\dfrac{2008}1\right\rfloor,\,\,\,\,\,\,\,\,\,\left\lfloor\dfrac{2008^2}2\right\rfloor,\,\,\,\,\,\,\,\,\,\left\lfloor\dfrac{2008^3}3\right\rfloor,\,\,\,\,\,\,\,\,\,\left\lfloor\dfrac{2008^4}4\right\rfloor,\,\,\,\,\,\,\,\,\,\ldots.\]
Let $a_1,a_2,a_3,\ldots$ be an infinite sequence of positive integers such that $a_{n+2m}$ divides $a_{n}+a_{n+m}$ for all positive integers $n$ and $m.$ Prove that this sequence is eventually periodic, i.e. there exist positive integers $N$ and $d$ such that $a_n=a_{n+d}$ for all $n>N.$
Let $P_0(x)=x^3-4x$. Sequence of polynomials is defined as following:\\
$P_{n+1}=P_n(1+x)P_n(1-x)-1$.\\
Prove that $x^{2016}|P_{2016}(x)$.
The sequence of real numbers $a_0,a_1,a_2,\ldots$ is defined recursively by \[a_0=-1,\qquad\sum_{k=0}^n\dfrac{a_{n-k}}{k+1}=0\quad\text{for}\quad n\geq 1.\]Show that $ a_{n} > 0$ for all $ n\geq 1$.
[i]Proposed by Mariusz Skalba, Poland[/i]
Let $a_1,\dots,a_n$ be a non increasing sequence of positive real numbers. Prove that \[\sqrt{a_1^2+a_2^2+\cdots+a_n^2}\le a_1+\frac{a_2}{\sqrt{2}+1}+\cdots+\frac{a_n}{\sqrt{n}+\sqrt{n-1}}.\] When does equality hold?
Define a sequence by $s_0 = 1$ and for $d \geq 1$, $s_d = s_{d-1} + X_d$, where $X_d$ is chosen uniformly at random from the set $\{1, 2, \dots, d\}$.
What is the probability that the sequence $s_0, s_1, s_2, \dots$ contains infinitely many primes?
A frog sitting at the point $(1, 2)$ begins a sequence of jumps, where each jump is parallel to one of the coordinate axes and has length $1$, and the direction of each jump (up, down, right, or left) is chosen independently at random. The sequence ends when the frog reaches a side of the square with vertices $(0,0), (0,4), (4,4),$ and $(4,0)$. What is the probability that the sequence of jumps ends on a vertical side of the square$?$
$\textbf{(A) } \frac{1}{2} \qquad \textbf{(B) } \frac{5}{8} \qquad \textbf{(C) } \frac{2}{3} \qquad \textbf{(D) } \frac{3}{4} \qquad \textbf{(E) } \frac{7}{8}$
Let $(a_n)_{n\geq 1}$ be a sequence of positive real numbers with the property that
$$(a_{n+1})^2 + a_na_{n+2} \leq a_n + a_{n+2}$$
for all positive integers $n$. Show that $a_{2022}\leq 1$.
Given $3n$ cards, each of them will be written with a number from the following sequence:
$$2, 3, ..., n, n + 1, n + 3, n + 4, ..., 2n + 1, 2n + 2, 2n + 4, ..., 3n + 3$$
with each number used exactly once. Then every card is arranged from left to right in random order. Determine the probability such that for every $i$ with $1\le i \le 3n$, the number written on the $i$-th card, counted from the left, is greater than or equal to $i$.
The first two terms of a sequence are $2$ and $3$. Each next term thereafter is the sum of the nearestly previous two terms if their sum is not greather than $10, 0$ otherwise. The $2014$th term is:
(A): $0$, (B): $8$, (C): $6$, (D): $4$, (E) None of the above.
Is there an infinite sequence $ a_0, a_1, a_2, \cdots $ of nonzero real numbers such that for $ n = 1, 2, 3, \cdots $ the polynomial \[ p_n(x) = a_0 + a_1 x + a_2 x^2 + \cdots + a_n x^n \] has exactly $n$ distinct real roots?
Let $a_1,a_2,\dots$ and $b_1,b_2,\dots$ be sequences of positive real numbers such that $a_1=b_1=1$ and $b_n=b_{n-1}a_n-2$ for $n=2,3,\dots.$ Assume that the sequence $(b_j)$ is bounded. Prove that \[S=\sum_{n=1}^{\infty}\frac1{a_1\cdots a_n}\] converges, and evaluate $S.$
Let $n$ be a positive integer. For a permutation $a_1, a_2, \dots, a_n$ of the numbers $1, 2, \dots, n$ we define
$$b_k = \min_{1 \leq i \leq k} a_i + \max_{1 \leq j \leq k} a_j$$
We say that the permutation $a_1, a_2, \dots, a_n$ is [i]guadiana[/i] if the sequence $b_1, b_2, \dots, b_n$ does not contain two consecutive equal terms. How many guadiana permutations exist?
Consider two geometric sequences $16$, $a_1$, $a_2$, $ . . .$ and $56$, $b_1$, $b_2$, $. . . $ with the same common nonzero ratio. Given that $a_{2023} = b_{2020}$, compute $b_6 - a_6$.
For every $n \in N $, let $d(n)$ denote the sum of digits of $n$. It is easy to see that the sequence $d(n), d(d(n))$, $d(d(d(n))), ... $ will eventually become a constant integer between $1$ and $9$ (both inclusive). This number is called the digital root of $n$ . Denote it by $b(n)$. Then for how many natural numbers $k<1000 , \lim_{n \to \infty} b(k^n)$ exists.
Let $ b_1<b_2<b_3<\dots $ be the sequence of all natural numbers which are sum of squares of two natural numbers.
Prove that there exists infinite natural numbers like $m$ which $b_{m+1}-b_m=2015$ .
In a sequence of coin tosses, one can keep a record of instances in which a tail is immediately followed by a head, a head is immediately followed by a head, and etc. We denote these by TH, HH, and etc. For example, in the sequence HHTTHHHHTHHTTTT of 15 coin tosses we observe that there are two HH, three HT, four TH, and five TT subsequences. How many different sequences of 15 coin tosses will contain exactly two HH, three HT, four TH, and five TT subsequences?
In the Cartesian coordinate system, we define a [i]Bongo-line[/i] as a sequence of integer points $\alpha = (\ldots, A_{-1}, A_0, A_1, \ldots)$ such that:
- $A_iA_{i+1} = \sqrt{2}$ for every $i \in \mathbb{Z}$;
- the polyline $\ldots A_{-1}A_0A_1 \ldots$ has no self-intersections.
Let $\alpha = (\ldots, A_{-1}, A_0, A_1, \ldots)$ and $\beta = (\ldots, B_{-1}, B_0, B_1, \ldots)$ be two Bongo-lines such that there exists a bijection $f : \mathbb{Z} \to \mathbb{Z}$ such that $A_iA_{i+1}$ and $B_{f(i)}B_{f(i)+1}$ halve each other. Prove that all vertices of $\alpha$ and $\beta$ lie on two lines.
[i]Proposed by Pavle Martinović[/i]
[b]p1.[/b] Compute $17^2 + 17 \cdot 7 + 7^2$.
[b]p2.[/b] You have $\$1.17$ in the minimum number of quarters, dimes, nickels, and pennies required to make exact change for all amounts up to $\$1.17$. How many coins do you have?
[b]p3.[/b] Suppose that there is a $40\%$ chance it will rain today, and a $20\%$ chance it will rain today and tomorrow. If the chance it will rain tomorrow is independent of whether or not it rained today, what is the probability that it will rain tomorrow? (Express your answer as a percentage.)
[b]p4.[/b] A number is called boxy if the number of its factors is a perfect square. Find the largest boxy number less than $200$.
[b]p5.[/b] Alice, Bob, Carl, and Dave are either lying or telling the truth. If the four of them make the following statements, who has the coin?
[i]Alice: I have the coin.
Bob: Carl has the coin.
Carl: Exactly one of us is telling the truth.
Dave: The person who has the coin is male.[/i]
[b]p6.[/b] Vicky has a bag holding some blue and some red marbles. Originally $\frac23$ of the marbles are red. After Vicky adds $25$ blue marbles, $\frac34$ of the marbles are blue. How many marbles were originally in the bag?
[b]p7.[/b] Given pentagon $ABCDE$ with $BC = CD = DE = 4$, $\angle BCD = 90^o$ and $\angle CDE = 135^o$, what is the length of $BE$?
[b]p8.[/b] A Berkeley student decides to take a train to San Jose, stopping at Stanford along the way. The distance from Berkeley to Stanford is double the distance from Stanford to San Jose. From Berkeley to Stanford, the train's average speed is $15$ meters per second. From Stanford to San Jose, the train's average speed is $20$ meters per second. What is the train's average speed for the entire trip?
[b]p9.[/b] Find the area of the convex quadrilateral with vertices at the points $(-1, 5)$, $(3, 8)$, $(3,-1)$, and $(-1,-2)$.
[b]p10.[/b] In an arithmetic sequence $a_1$, $a_2$, $a_3$, $...$ , twice the sum of the first term and the third term is equal to the fourth term. Find $a_4/a_1$.
[b]p11.[/b] Alice, Bob, Clara, David, Eve, Fred, Greg, Harriet, and Isaac are on a committee. They need to split into three subcommittees of three people each. If no subcommittee can be all male or all female, how many ways are there to do this?
[b]p12.[/b] Usually, spaceships have $6$ wheels. However, there are more advanced spaceships that have $9$ wheels. Aliens invade Earth with normal spaceships, advanced spaceships, and, surprisingly, bicycles (which have $2$ wheels). There are $10$ vehicles and $49$ wheels in total. How many bicycles are there?
[b]p13.[/b] If you roll three regular six-sided dice, what is the probability that the three numbers showing will form an arithmetic sequence? (The order of the dice does matter, but we count both $(1,3, 2)$ and $(1, 2, 3)$ as arithmetic sequences.)
[b]p14.[/b] Given regular hexagon $ABCDEF$ with center $O$ and side length $6$, what is the area of pentagon $ABODE$?
[b]p15.[/b] Sophia, Emma, and Olivia are eating dinner together. The only dishes they know how to make are apple pie, hamburgers, hotdogs, cheese pizza, and ice cream. If Sophia doesn't eat dessert, Emma is vegetarian, and Olivia is allergic to apples, how many di erent options are there for dinner if each person must have at least one dish that they can eat?
[b]p16.[/b] Consider the graph of $f(x) = x^3 + x + 2014$. A line intersects this cubic at three points, two of which have $x$-coordinates $20$ and $14$. Find the $x$-coordinate of the third intersection point.
[b]p17.[/b] A frustum can be formed from a right circular cone by cutting of the tip of the cone with a cut perpendicular to the height. What is the surface area of such a frustum with lower radius $8$, upper radius $4$, and height $3$?
[b]p18.[/b] A quadrilateral $ABCD$ is de ned by the points $A = (2,-1)$, $B = (3, 6)$, $C = (6, 10)$ and $D = (5,-2)$. Let $\ell$ be the line that intersects and is perpendicular to the shorter diagonal at its midpoint. What is the slope of $\ell$?
[b]p19.[/b] Consider the sequence $1$, $1$, $2$, $2$, $3$, $3$, $3$, $5$, $5$, $5$, $5$, $5$, $...$ where the elements are Fibonacci numbers and the Fibonacci number $F_n$ appears $F_n$ times. Find the $2014$th element of this sequence. (The Fibonacci numbers are defined as $F_1 = F_2 = 1$ and for $n > 2$, $F_n = F_{n-1}+F_{n-2}$.)
[b]p20.[/b] Call a positive integer top-heavy if at least half of its digits are in the set $\{7, 8, 9\}$. How many three digit top-heavy numbers exist? (No number can have a leading zero.)
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Let \( P(x) \) be a polynomial with natural coefficients. We denote by \( d(n) \) the number of positive divisors of the natural number \( n \), and by \( \sigma(n) \), the sum of these divisors. The sequence \( a_n \) is defined as follows:
\[
a_{n+1} \in \left\{
\begin{array}{ll}
\sigma(P(d(a_n))) \\
d(P(\sigma(a_n)))
\end{array}
\right.
\]
That is, \( a_{n+1} \) is one of the two terms above. Show that there exists a constant \( C \), depending on \( a_1 \) and \( P(x) \), such that for all \( i \), \( a_i < C \); in other words, show that the sequence \( a_n \) is bounded.
Define the sequence $a_1, a_2,...$ as follows: $a_1 = 1$, and for every $n \ge 2$, $a_n = n - 2$ if $a_{n-1} = 0$ and $a_n = a_{n-1} - 1$, otherwise. Find the number of $1 \le k \le 2016$ such that there are non-negative integers $r, s$ and a positive integer $n$ satisfying $k = r + s$ and $a_{n+r} = a_n + s$.
Consider $(f_n)_{n\ge 0}$ the Fibonacci sequence, defined by $f_0 = 0$, $f_1 = 1$, $f_{n+1} = f_n + f_{n-1}$ for all positive integers $n$. Solve the following equation in positive integers $$nf_nf_{n+1} = (f_{n+2} - 1)^2.$$
.
Denote by $a_n$ the greatest number that is not divisible by $3$ and that divides $n$. Consider the sequence $s_0 = 0, s_n = a_1 +a_2+\cdots+a_n, n \in \mathbb N$. Denote by $A(n)$ the number of all sums $s_k \ (0 \leq k \leq 3^n, k \in \mathbb N_0)$ that are divisible by $3$. Prove the formula
\[A(n) = 3^{n-1} + 2 \cdot 3^{(n/2)-1} \cos \left(\frac{n\pi}{6}\right), \qquad n\in \mathbb N_0.\]
Let be a sequence of continuous functions $ \left( f_n \right)_{n\ge 1} :[0,1]\longrightarrow\mathbb{R} $ satisfying the following properties:
$ \text{a) } $ for any natural $ n $ and $ x\in [1/n,1] ,$ it follows $ \left| f_n(x) \right|\leqslant 1/n. $
$ \text{b) } $ for any natural $ n, $ it follows $ \int_0^1 f_n^2(t)dt\leqslant 1. $
Then, $\lim_{n\to 0} \int_0^1\left| f_n(t) \right| dt=0 $
[i]Cristinel Mortici[/i]
In the five-sided star shown, the letters $A,B,C,D,$ and $E$ are replaced by the numbers $3,5,6,7,$ and $9$, although not necessarily in this order. The sums of the numbers at the ends of the line segments $\overline{AB}$,$\overline{BC}$,$\overline{CD}$,$\overline{DE}$, and $\overline{EA}$ form an arithmetic sequence, although not necessarily in this order. What is the middle term of the arithmetic sequence?
[asy]
size(150);
defaultpen(linewidth(0.8));
string[] strng = {'A','D','B','E','C'};
pair A=dir(90),B=dir(306),C=dir(162),D=dir(18),E=dir(234);
draw(A--B--C--D--E--cycle);
for(int i=0;i<=4;i=i+1)
{
path circ=circle(dir(90-72*i),0.125);
unfill(circ);
draw(circ);
label("$"+strng[i]+"$",dir(90-72*i));
}
[/asy]
$ \textbf{(A)}\ 9\qquad
\textbf{(B)}\ 10\qquad
\textbf{(C)}\ 11\qquad
\textbf{(D)}\ 12\qquad
\textbf{(E)}\ 13$