Found problems: 85335
The NEMO (National Electronic Math Olympiad) is similar to the NIMO Summer Contest, in that there are fifteen problems, each worth a set number of points. However, the NEMO is weighted using Fibonacci numbers; that is, the $n^{\text{th}}$ problem is worth $F_n$ points, where $F_1 = F_2 = 1$ and $F_n = F_{n-1} + F_{n-2}$ for $n \ge 3$. The two problem writers are fair people, so they make sure that each of them is responsible for problems worth an equal number of total points. Compute the number of ways problem writing assignments can be distributed between the two writers.
[i]Proposed by Lewis Chen[/i]
For a three-digit prime number $p$, the equation $x^3+y^3=p^2$ has an integer solution. Calculate $p$.
Consider 2016 distinct points on a circle. It is allowed to move from one point to another on the circle by jumping 2 or 3 points forward in a clockwise direction. What is the minimum number of jumps required to visit all points and return to the starting point?
Find all pairs $ (a,b)$ of positive integers that satisfy the equation: $ a^{b^2} \equal{} b^a$.
If $x_1 + x_2 + \cdots + x_n = \sum_{i=1}^{n} x_i = \frac{1}{2}$ and $x_i > 0$ ; then prove that:
$ \frac{1-x_1}{1+x_1} \cdot \frac{1-x_2}{1+x_2} \cdots \frac{1-x_n}{1+x_n} = \prod_{i=1}^{n} \frac{1-x_i}{1+x_i} \geq \frac{1}{3}$
There is an ant on every vertex of a unit cube. At the time zero, ants start to move across the edges with the velocity of one unit per minute. If an ant reaches a vertex, it alternatively turns right and left (for the first time it will turn in a random direction). If two or more ants meet anywhere on the cube, they die! We know an ant survives after three minutes. Prove that there exists an ant that never dies!
Euclid places a morsel of food at the point $(0,0)$ and an ant at the point $(1,2)$. Every second, the ant walks one unit in one of the four coordinate directions. However, whenever the ant moves to $(x,\pm 3)$, Euclid's malicious brother Mobius picks it up and puts it at $(-x,\mp 2)$, and whenever it moves to $(\pm 2,y)$, his cousin Klein puts it at $(\mp 1,y)$. If $p$ and $q$ are relatively prime positive integers such that $\tfrac pq$ is the expected number of steps the ant takes before reaching the food, find $p+q$.
Let $G$ be a simple graph with $3n^2$ vertices ($n\geq 2$). It is known that the degree of each vertex of $G$ is not greater than $4n$, there exists at least a vertex of degree one, and between any two vertices, there is a path of length $\leq 3$. Prove that the minimum number of edges that $G$ might have is equal to $\frac{(7n^2- 3n)}{2}$.
Let $\, P(z) \,$ be a polynomial with complex coefficients which is of degree $\, 1992 \,$ and has distinct zeros. Prove that there exist complex numbers $\, a_1, a_2, \ldots, a_{1992} \,$ such that $\, P(z) \,$ divides the polynomial \[ \left( \cdots \left( (z-a_1)^2 - a_2 \right)^2 \cdots - a_{1991} \right)^2 - a_{1992}. \]
A $6$-inch-wide rectangle is rotated $90$ degrees about one of its corners, sweeping out an area of $45\pi$ square inches, excluding the area enclosed by the rectangle in its starting position. Find the rectangle’s length in inches.
A particle of unit mass moves on a straight line under the action of a force which is a function $f(v)$ of the velocity $v$ of the particle, but the form of the function is not known. A motion is observed, and the distance $x$ covered in time $t$ satisfies the formula $x= at^2 + bt+c$, where $a,b,c$ have numerical values determined by observation of the motion. Find the function $f(v)$ for the range of $v$ covered by the experiment.
A problem author for a math competition was looking through a tentative exam when he realized that
he could not use one of his proposed problems. Frustrated, he decided to take a nap instead, and slept
from $10:47\text{ AM}$ to $7:32\text{ PM}$. For how many minutes did he sleep?
In triangle $BEN$ shown below with its altitudes intersecting at $X$, $NA = 7$, $EA = 3$, $AX = 4$, and $NS = 8$. Find the area of $BEN$.
[img]https://cdn.artofproblemsolving.com/attachments/5/7/7e6dcbe6aa220821cb5020824b8aa6d4fc597d.png[/img]
Two medians of a triangle are perpendicular. Prove that the medians of the triangle are the sides of a right-angled triangle.
Prove that the sum of the squares of the areas of the projections of the faces of a rectangular parallelepiped on a plane is the same for all positions of the plane if and only if the parallelepiped is a cube.
Find all positive integers $N$ such that an $N\times N$ board can be tiled using tiles of size $5\times 5$ or $1\times 3$.
Note: The tiles must completely cover all the board, with no overlappings.
Suppose $ABC$ is a triangle, and define $B_1$ and $C_1$ such that $\triangle AB_1C$ and $\triangle AC_1B$ are isosceles right triangles on the exterior of $\triangle ABC$ with right angles at $B_1$ and $C_1$, respectively. Let $M$ be the midpoint of $\overline{B_1C_1}$; if $B_1C_1 = 12$, $BM = 7$ and $CM = 11$, what is the area of $\triangle ABC$?
[b]p1.[/b] Let $a$ be an integer. How many fractions $\frac{a}{100}$ are greater than $\frac17$ and less than $\frac13$ ?.
[b]p2.[/b] Justin Bieber invited Justin Timberlake and Justin Shan to eat sushi. There were $5$ different kinds of fish, $3$ different rice colors, and $11$ different sauces. Justin Shan insisted on a spicy sauce. If the probability of a sushi combination that pleased Justin Shan is $6/11$, then how many non-spicy sauces were there?
[b]p3.[/b] A palindrome is any number that reads the same forward and backward (for example, $99$ and $50505$ are palindromes but $2020$ is not). Find the sum of all three-digit palindromes whose tens digit is $5$.
[b]p4.[/b] Isaac is given an online quiz for his chemistry class in which he gets multiple tries. The quiz has $64$ multiple choice questions with $4$ choices each. For each of his previous attempts, the computer displays Isaac's answer to that question and whether it was correct or not. Given that Isaac is too lazy to actually read the questions, the maximum number of times he needs to attempt the quiz to guarantee a $100\%$ can be expressed as $2^{2^k}$. Find $k$.
[b]p5.[/b] Consider a three-way Venn Diagram composed of three circles of radius $1$. The area of the entire Venn Diagram is of the form $\frac{a}{b}\pi +\sqrt{c}$ for positive integers $a$, $b$, $c$ where $a$, $b$ are relatively prime. Find $a+b+c$. (Each of the circles passes through the center of the other two circles)
[b]p6.[/b] The sum of two four-digit numbers is $11044$. None of the digits are repeated and none of the digits are $0$s. Eight of the digits from $1-9$ are represented in these two numbers. Which one is not?
[b]p7.[/b] Al wants to buy cookies. He can buy cookies in packs of $13$, $15$, or $17$. What is the maximum number of cookies he can not buy if he must buy a whole number of packs of each size?
[b]p8.[/b] Let $\vartriangle ABC$ be a right triangle with base $AB = 2$ and hypotenuse $AC = 4$ and let $AD$ be a median of $\vartriangle ABC$. Now, let $BE$ be an altitude in $\vartriangle ABD$ and let $DF$ be an altitude in $\vartriangle ADC$. The quantity $(BE)^2 - (DF)^2$ can be expressed as a common fraction $\frac{a}{b}$ in lowest terms. Find $a + b$.
[b]p9.[/b] Let $P(x)$ be a monic cubic polynomial with roots $r$, $s$, $t$, where $t$ is real. Suppose that $r + s + 2t = 8$, $2rs + rt + st = 12$ and $rst = 9$. Find $|P(2)|$.
[b]p10.[/b] Let S be the set $\{1, 2,..., 21\}$. How many $11$-element subsets $T$ of $S$ are there such that there does not exist two distinct elements of $T$ such that one divides the other?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Let $a_1,a_2,\cdots ,a_n$ be $n$ positive numbers such that $\sum \limits_{i=1}^n\sqrt{a_i}=\sqrt{n}$. Then$$\prod \limits_{i=1}^{n-1}\left(1+\frac{1}{a_i}\right)^{a_{i+1}}\left(1+\frac{1}{a_n}\right)^{a_1}\geq 1+\frac{n}{\sum \limits_{i=1}^na_i}$$
A player is playing the following game. In each turn he flips a coin and guesses the outcome. If his guess is correct, he gains $ 1$ point; otherwise he loses all his points. Initially the player has no points, and plays the game
until he has $ 2$ points.
(a) Find the probability $ p_{n}$ that the game ends after exactly $ n$ flips.
(b) What is the expected number of flips needed to finish the game?
Find the second smallest prime factor of $18!+1.$
[i]Proposed by Kaylee Ji[/i]
Let $(a_n)_{n \in \mathbb{N}}$ be a sequence of real numbers such that $$2(a_1+a_2+…+a_n)=na_{n+1}~\forall~n \ge 1.$$
$\textbf{a)}$ Prove that the given sequence is an arithmetic progression.
$\textbf{b)}$ If $\lfloor a_1 \rfloor + \lfloor a_2 \rfloor +…+ \lfloor a_n \rfloor = \lfloor a_1+a_2+…+a_n \rfloor~\forall~ n \in \mathbb{N},$ prove that every term of the sequence is an integer.
Show that there are no triples of positive integers $(x,y,z)$ satisfying the equation \[x^2= 5^y+3^z\]
Line segment $AB$ has length $4$ and midpoint $M$. Let circle $C_1$ have diameter $AB$, and let circle $C_2$ have diameter $AM$. Suppose a tangent of circle $C_2$ goes through point $ B$ to intersect circle $C_1$ at $N$. Determine the area of triangle $AMN$.
Let $ABC$ be a triangle with $AB=2$, $BC=3$, and $AC=4$. Consider all lines $XY$ such that $X$ lies on $AC$, $Y$ lies on $BC$, and $\triangle XYC$ has area equal to half that of $\triangle ABC$. What is the minimum possible length of $XY$?