Practice (EndingDigits,TheDivideByNineMethod,MODBasic)
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There exist $5$ consecutive positive integers such that their sum is a square, and the sum of the middle three is a cube. What is the smallest one of these five numbers?
There are four wolves standing on the four corners of a square, and a rabbit standing at the center of that square. If a wolf can run at $1.4$ times of the rabbit's speed, but can only move along the sides of this square, can the rabbit escape to outside the square?
Let $a$ and $b$ be non-negative real numbers such that $a + b = 2$. Show that: $$\frac{1}{a^2+1}+\frac{1}{b^2 +1} \le \frac{2}{ab+1}$$
Let integers $a$, $b$ and $c$ satisfy $a + b + c = 0$, show that $\vert{a^3 + b^3 + c^3}\vert$ cannot be a prime number.
A set of three points is randomly chosen from the grid shown. Each three point set has the same probability of being chosen. What is the probability that the points lie on the same straight line?
A game is played with tokens according to the following rule. In each round, the player with the most tokens gives one token to each of the other players and also places one token in the discard pile. The game ends when some player runs out of tokens. Players $A$, $B$, and $C$ start with $15$, $14$, and $13$ tokens, respectively. How many rounds will there be in the game?
The average value of all the pennies, nickels, dimes, and quarters in Paula's purse is $20$ cents. If she had one more quarter, the average would be $21$ cents. How many dimes does she have in her purse?
Brenda and Sally run in opposite directions on a circular track, starting at diametrically opposite points. They first meet after Brenda has run 100 meters. They next meet after Sally has run 150 meters past their first meeting point. Each girl runs at a constant speed. What is the length of the track in meters?
A sequence of three real numbers forms an arithmetic progression with a first term of 9. If 2 is added to the second term and 20 is added to the third term, the three resulting numbers form a geometric progression. What is the smallest possible value for the third term of the geometric progression?
A white cylindrical silo has a diameter of 30 feet and a height of 80 feet. A red stripe with a horizontal width of 3 feet is painted on the silo, as shown, making two complete revolutions around it. What is the area of the stripe in square feet?
Points $E$ and $F$ are located on square $ABCD$ so that $\triangle BEF$ is equilateral. What is the ratio of the area of $\triangle DEF$ to that of $\triangle ABE$?
Two distinct lines pass through the center of three concentric circles of radii 3, 2, and 1. The area of the shaded region in the diagram is $\frac{8}{13}$ of the area of the unshaded region. What is the radian measure of the acute angle formed by the two lines? (Note: $\pi$ radians is $180$ degrees.)
Square $ABCD$ has side length $2$. A semicircle with diameter $\overline{AB}$ is constructed inside the square, and the tangent to the semicircle from $C$ intersects side $\overline{AD}$ at $E$. What is the length of $\overline{CE}$?
Circles $A$, $B$, and $C$ are externally tangent to each other and internally tangent to circle $D$. Circles $B$ and $C$ are congruent. Circle $A$ has radius $1$ and passes through the center of $D$. What is the radius of circle $B$?
Let $a_1,a_2,\cdots$, be a sequence with the following properties.
(i) $a_1=1$, and
(ii) $a_{2n}=n\cdot a_n$ for any positive integer $n$.
What is the value of $a_{2^{100}}$?
Three pairwise-tangent spheres of radius 1 rest on a horizontal plane. A sphere of radius 2 rests on them. What is the distance from the plane to the top of the larger sphere?
Solve the following equation in positive integers: $$x^2 +3x^2y^2 = 30y^2 + 517$$
Solve in positive integers the equation $x^3 - y^3 = xy + 61$.
Solve the equation in integers $(x^2-y^2)^2 = 1+16y$.
Show the equation $x^2 + y^2-8z^3 = 6$ has no integer solution.
Find all the integer solutions to the equation $xy - 10(x+ y)= 1$.
Solve in integers the equation $x^2 - xy +2x -3 y = 0$
Solve the equation in integers $x^2 +4xy + 5y^2 + 2x + 4y -7 =0$
Solve the equation in integers $x^2 - 2xy -3y^2 +3x-5y-6=0$
Show that the equation $x^4 + y^4 + z^4 = 2x^2y^2 + 2y^2 z^2 + 2z^2x^2 +24$ has no integer solution.