Found problems: 105
2015 ASDAN Math Tournament, 10
An ant is walking on the edges of an icosahedron of side length $1$. Compute the length of the longest path that the ant can take if it never travels over the same edge twice, but is allowed to revisit vertices.
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2015 ASDAN Math Tournament, 2
Find the sum of the squares of the roots of $x^2-5x-7$.
2015 ASDAN Math Tournament, 35
Let $S$ be the set of positive integers less than $10^6$ that can be written as the sum of two perfect squares. Compute the number of elements in $S$. Your score will be given by $\max\{\lfloor75(\min\{(\tfrac{A}{C})^2,(\tfrac{C}{A})^2\}-\tfrac{2}{3})\rfloor,0\}$, where $A$ is your answer and $C$ is the actual answer.
2015 ASDAN Math Tournament, 19
Compute the number of $0\leq n\leq2015$ such that $6^n+8^n$ is divisible by $7$.
2015 Mexico National Olympiad, 2
Let $n$ be a positive integer and let $k$ be an integer between $1$ and $n$ inclusive. There is a white board of $n \times n$. We do the following process. We draw $k$ rectangles with integer sides lenghts and sides parallel to the ones of the $n \times n$ board, and such that each rectangle covers the top-right corner of the $n \times n$ board. Then, the $k$ rectangles are painted of black. This process leaves a white figure in the board.
How many different white figures are possible to do with $k$ rectangles that can't be done with less than $k$ rectangles?
Proposed by David Torres Flores
2015 ASDAN Math Tournament, 32
Let $ABC$ be a triangle with $AB=8$, $BC=7$, and $AC=11$. Let $\Gamma_1$ and $\Gamma_2$ be the two possible circles that are tangent to $AB$, $AC$, and $BC$ when $AC$ and $BC$ are extended, with $\Gamma_1$ having the smaller radius. $\Gamma_1$ and $\Gamma_2$ are tangent to $AB$ to $D$ and $E$, respectively, and $CE$ intersects the perpendicular bisector of $AB$ at a point $F$. What is $\tfrac{CF}{FD}$?
2015 ASDAN Math Tournament, 33
Compute the number of digits is $2015!$. Your score will be given by $\max\{\lfloor125(\min\{\tfrac{A}{C},\tfrac{C}{A}\}-\tfrac{1}{5})\rfloor,0\}$, where $A$ is your answer and $C$ is the actual answer.
2015 ASDAN Math Tournament, 12
Find the smallest positive integer solution to the equation $2^{2^k}\equiv k\pmod{29}$.
2015 ASDAN Math Tournament, 11
In the following diagram, each circle has radius $6$ and each circle passes through the center of the other two circles. Compute the area of the white center region and express your answer in terms of $\pi$.
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2015 ASDAN Math Tournament, 9
Regular tetrahedron $ABCD$ has center $O$ and side length $1$. Points $A'$, $B'$, $C'$, and $D'$ are defined by reflecting $A$, $B$, $C$, and $D$ about $O$. Compute the volume of the polyhedron with vertices $ABCDA'B'C'D'$.
2015 ASDAN Math Tournament, 25
Let $a_n$ be a sequence with $a_0=1$ and defined recursively by
$$a_{n+1}=\begin{cases}a_n+2&\text{if }n\text{ is even},\\2a_n&\text{if }n\text{ is odd.}\end{cases}$$
What are the last two digits of $a_{2015}$?
2015 Turkey Team Selection Test, 9
In a country with $2015$ cities there is exactly one two-way flight between each city. The three flights made between three cities belong to at most two different airline companies. No matter how the flights are shared between some number of companies, if there is always a city in which $k$ flights belong to the same airline, what is the maximum value of $k$?
2015 ASDAN Math Tournament, 13
The incircle of triangle $\triangle ABC$ is the unique inscribed circle that is internally tangent to the sides $\overline{AB}$, $\overline{BC}$, and $\overline{CA}$. How many non-congruent right triangles with integer side lengths have incircles of radius $2015$?
2015 China Team Selection Test, 6
There are some players in a Ping Pong tournament, where every $2$ players play with each other at most once. Given:
\\(1) Each player wins at least $a$ players, and loses to at least $b$ players. ($a,b\geq 1$)
\\(2) For any two players $A,B$, there exist some players $P_1,...,P_k$ ($k\geq 2$) (where $P_1=A$,$P_k=B$), such that $P_i$ wins $P_{i+1}$ ($i=1,2...,k-1$).
\\Prove that there exist $a+b+1$ distinct players $Q_1,...Q_{a+b+1}$, such that $Q_i$ wins $Q_{i+1}$ ($i=1,...,a+b$)
2015 Iran MO (2nd Round), 1
Consider a cake in the shape of a circle. It's been divided to some inequal parts by its radii. Arash and Bahram want to eat this cake. At the very first, Arash takes one of the parts. In the next steps, they consecutively pick up a piece adjacent to another piece formerly removed. Suppose that the cake has been divided to 5 parts. Prove that Arash can choose his pieces in such a way at least half of the cake is his.
2015 ASDAN Math Tournament, 1
Four unit circles are placed on a square of side length $2$, with each circle centered on one of the four corners of the square. Compute the area of the square which is not contained within any of the four circles.
2015 ASDAN Math Tournament, 9
You play a game with a biased coin, which has probability $\tfrac{3}{4}$ of landing heads. Each time you toss heads, you score $1$ point, while tossing tails earns no points. After any turn, you can stop playing the game and keep the points you currently have. However, if you are still playing when you toss tails for the second time, you lose all of your points. If you play to maximize your expected score, what is your expected score from playing this game?
2015 ASDAN Math Tournament, 8
You have $8$ friends, each of whom lives at a different vertex of a cube. You want to chart a path along the cube’s edges that will visit each of your friends exactly once. You can start at any vertex, but you must end at the vertex you started at, and you cannot travel on any edge more than once. How many different paths can you take?
2015 ASDAN Math Tournament, 9
Compute all pairs of nonzero real numbers $(x,y)$ such that
$$\frac{x}{x^2+y}+\frac{y}{x+y^2}=-1\qquad\text{and}\qquad\frac{1}{x}+\frac{1}{y}=1.$$
2015 ASDAN Math Tournament, 5
Compute the number of zeros at the end of $2015!$.
2015 ASDAN Math Tournament, 22
You flip a fair coin which results in heads ($\text{H}$) or tails ($\text{T}$) with equal probability. What is the probability that you see the consecutive sequence $\text{THH}$ before the sequence $\text{HHH}$?
2015 ASDAN Math Tournament, 8
In triangle $ABC$, point $D$ is on side $BC$ such that $AD$ is the angle bisector of $\angle BAC$. If $AB=12$, $AD=9$, and $AC=15$, compute $\cos\tfrac{\angle BAC}{2}$.
2015 ASDAN Math Tournament, 1
How many integers between $2$ and $100$ have only odd numbers in their prime factorizations?
2015 ASDAN Math Tournament, 3
Points $E$ and $F$ are chosen on sides $BC$ and $CD$ respectively of rhombus $ABCD$ such that $AB=AE=AF=EF$, and $FC,DF,BE,EC>0$. Compute the measure of $\angle ABC$.
2015 Turkey Team Selection Test, 5
We are going to colour the cells of a $2015 \times 2015$ board such that there are none of the following:
$1)$ Three cells with the same colour where two of them are in the same column, and the third is in the same row and to the right of the upper cell,
$2)$ Three cells with the same colour where two of them are in the same column, and the third is in the same row and to the left of the lower cell.
What is the minimum number of colours $k$ required to make such a colouring possible?