Found problems: 105
2015 Purple Comet Problems, 10
In an abandoned chemistry lab Gerome found a two-pan balance scale and three 1-gram weights, three
5-gram weights, and three 50-gram weights. By placing one pile of chemicals and as many weights as
necessary on the pans of the scale, Gerome can measure out various amounts of the chemicals in the pile.
Find the number of different positive weights of chemicals that Gerome could measure.
2015 ASDAN Math Tournament, 15
Let $ABCD$ be a regular tetrahedron of side length $12$. Select points $E,F,G,H$ on $AC,BC,BD,AD$, respectively, such that $AE=BF=BG=AH=3$. Compute the area of quadrilateral $EFGH$.
2015 ASDAN Math Tournament, 5
Four men are each given a unique number from $1$ to $4$, and four women are each given a unique number from $1$ to $4$. How many ways are there to arrange the men and women in a circle such that no two men are next to each other, no two women are next to each other, and no two people with the same number are next to each other? Note that two configurations are considered to be the same if one can be rotated to obtain the other one.
2015 ASDAN Math Tournament, 34
Compute the number of natural numbers $1\leq n\leq10^6$ such that the least prime divisor of $n$ is $17$. Your score will be given by $\lfloor26\min\{(\tfrac{A}{C})^2,(\tfrac{C}{A})^2\}\rfloor$, where $A$ is your answer and $C$ is the actual answer.
2015 ASDAN Math Tournament, 10
The polynomial $f(x)=x^3-4\sqrt{3}x^2+13x-2\sqrt{3}$ has three real roots, $a$, $b$, and $c$. Find
$$\max\{a+b-c,a-b+c,-a+b+c\}.$$
2015 Belarus Team Selection Test, 1
Solve the equation in nonnegative integers $a,b,c$:
$3^a+2^b+2015=3c!$
I.Gorodnin
2015 ASDAN Math Tournament, 23
Two regular hexagons of side length $2$ are laid on top of each other such that they share the same center point and one hexagon is rotated $30^\circ$ about the center from the other. Compute the area of the union of the two hexagons.
2015 ASDAN Math Tournament, 24
Trains $A$ and $B$ are on the same track a distance $100$ miles apart heading towards one another, each at a speed of $50$ miles per hour. A fly starting out at the front of train $A$ flies towards train $B$ at a speed of $75$ miles per hour. Upon reaching train $B$, the fly turns around and flies towards train $A$, again at $75$ miles per hour. The fly continues flying back and forth between the two trains at $75$ miles per hour until the two trains hit each other. How many minutes does the fly spend closer to train $A$ than to train $B$ before getting squashed?
2015 ASDAN Math Tournament, 4
Let $\triangle ABC$ be a right triangle with hypotenuse $AC$. A square is inscribed in the triangle such that points $D,E$ are on $AC$, $F$ is on $BC$, and $G$ is on $AB$. Given that $AG=2$ and $CF=5$, what is the area of $\triangle BFG$?
2015 ASDAN Math Tournament, 3
Let $f(x)$ be a polynomial of finite degree satisfying
$$(x+9)f(x+1)=(x+3)f(x+3)$$
for all real $x$. If $f(0)=1$, find the value of $f(1)$.
2015 ASDAN Math Tournament, 13
A three-digit number $x$ in base $10$ has a units-digit of $6$. When $x$ is written is base $9$, the second digit of the number is $4$, and the first and third digit are equal in value. Compute $x$ in base $10$.
2015 Turkey Team Selection Test, 7
Find all the functions $f:R\to R$ such that \[f(x^2) + 4y^2f(y) = (f(x-y) + y^2)(f(x+y) + f(y))\] for every real $x,y$.
2015 ASDAN Math Tournament, 2
Jonah recently harvested a large number of lychees and wants to split them into groups. Unfortunately, for all $n$ where $3\leq n\leq8$, when the lychees are distributed evenly into $n$ groups, $n-1$ lychees remain. What is the smallest possible number of lychees that Jonah could have?
2015 ASDAN Math Tournament, 3
Simplify $\sqrt{7+\sqrt{33}}-\sqrt{7-\sqrt{33}}$.
2015 ASDAN Math Tournament, 7
What is the largest integer $n$ such that $n$ is divisible by every integer less than $\sqrt[3]{n}$?
2015 ASDAN Math Tournament, 7
Nine identical spheres of radius $r$ are packed into a unit cube. One sphere is centered at the center of the cube and is tangent to the other eight spheres, each of which is located in a corner of the cube and is tangent to three faces of the cube. Compute the radius of the spheres $r$.
2015 ASDAN Math Tournament, 1
Given that $xy+x+y=5$ and $x+1=2$, compute $y+1$.
2015 ASDAN Math Tournament, 9
Compute the sum of the digits of $101^6$.
2015 ASDAN Math Tournament, 29
Suppose that the following equations hold for positive integers $x$, $y$, and $n$, where $n>18$:
\begin{align*}
x+3y&\equiv7\pmod{n}\\
2x+2y&\equiv18\pmod{n}\\
3x+y&\equiv7\pmod{n}
\end{align*}
Compute the smallest nonnegative integer $a$ such that $2x\equiv a\pmod{n}$.
2015 ASDAN Math Tournament, 2
There is a rectangular field that measures $20\text{m}$ by $15\text{m}$. Xiaoyu the butterfly is sitting at the perimeter of the field on one of the $20\text{m}$ sides such that he is $6\text{m}$ from a corner. He flies in a straight line to another point on the perimeter. His flying path splits the field into two parts with equal area. How far in meters did Xiaoyu fly?
2015 ASDAN Math Tournament, 14
A standard deck of $52$ cards is shuffled and randomly arranged in a queue, with each card having a suit $(\diamondsuit,\clubsuit,\heartsuit,\spadesuit)$ and a rank $(\text{Ace},2,3,4,5,6,7,8,9,10,\text{Jack},\text{Queen},\text{ King})$. For example, a card with the $\diamondsuit$ suit and the $7$ rank would be denoted as $\diamondsuit7$, and a card with the $\spadesuit$ and the $\text{Ace}$ rank would be denoted as $\spadesuit\text{Ace}$. In the queue, there exists a card with a rank of $\text{Ace}$ that appears for the first time in the queue. Let the card immediately following the above card be denoted as card $C$. Is the probability that $C$ is a $\spadesuit\text{A}$ higher than, equal to, or lower than the probability that $C$ is a $\clubsuit2$?
2015 ASDAN Math Tournament, 36
A blue square of side length $10$ is laid on top of a coordinate grid with corners at $(0,0)$, $(0,10)$, $(10,0)$, and $(10,10)$. Red squares of side length $2$ are randomly placed on top of the grid, changing the color of a $2\times2$ square section red. Each red square when placed lies completely within the blue square, and each square's four corners take on integral coordinates. In addition, randomly placed red squares may overlap, keeping overlapped regions red. Compute the expected value of the number of red squares necessary to turn the entire blue square red, rounded to the nearest integer. Your score will be given by $\lfloor25\min\{(\tfrac{A}{C})^2,(\tfrac{C}{A})^2\}\rfloor$, where $A$ is your answer and $C$ is the actual answer.
2015 ASDAN Math Tournament, 5
Laurie loves multiplying numbers in her head. One day she decides to multiply two $2$-digit numbers $x$ and $y$ such that $x\leq y$ and the two numbers collectively have at least three distinct digits. Unfortunately, she accidentally remembers the digits of each number in the opposite order (for example, instead of remembering $51$ she remembers $15$). Surprisingly, the product of the two numbers after flipping the digits is the same as the product of the two original numbers. How many possible pairs of numbers could Laurie have tried to multiply?
2015 Iran MO (2nd Round), 3
Consider a triangle $ABC$ . The points $D,E$ are on sides $AB,AC$ such that $BDEC$ is a cyclic quadrilateral. Let $P$ be the intersection of $BE$ and $CD$. $H$ is a point on $AC$ such that $\angle PHA = 90^{\circ}$. Let $M,N$ be the midpoints of $AP,BC$. Prove that: $ ACD \sim MNH $.
2015 ASDAN Math Tournament, 27
In triangle $ABC$, $D$ is a point on $AB$ between $A$ and $B$, $E$ is a point on $AC$ between $A$ and $C$, and $F$ is a point on $BC$ between $B$ and $C$ such that $AF$, $BE$, and $CD$ all meet inside $\triangle ABC$ at a point $G$. Given that the area of $\triangle ABC$ is $15$, the area of $\triangle ABE$ is $5$, and the area of $\triangle ACD$ is $10$, compute the area of $\triangle ABF$.