Let $a$, $b$, $c$ be real numbers with $|a-b|=3$, $|b-c|=4$, and $|c-d|=5$. What is the sum of all possible values of $|a-d|$?
What is the greatest integer less than or equal to $$\dfrac{5^{200}+3^{200}}{5^{197}+3^{197}}$$
How many negative integers can be written in the form of $$a_4\cdot 5^4+a_3\cdot 5^3 +a_2\cdot 5^2+a_2\cdot 5^1+a_0\cdot 5^0$$ where $a_i \in \{-2,-1,0,1,2\}$ for $0\le i \le 4$.
What is the sum of all real numbers $x$ for which the median of the numbers $4$, $6$, $8$, $17$, and $x$ is equal to the mean of those five numbers?
What is the remainder when $2024^{4050}+2025$ is divided by $2024^{2025}+2024^{1013}+1$?
The number $2023$ is expressed in the form $$2023=\dfrac{a_1!a_2!…a_m!}{b_1!b_2!…b_n!}$$ where $a_1\ge a_2\ge … \ge a_m$ and $b_1\ge b_2\ge …\ge b_n$ are positive integers and $a_1+b_1$ is as small as possible. What is $|a_1-b_1|$?
Two increasing sequences of positive integers have different first terms. Each sequence has the property that each term beginning with the third term is the sum of the previous two terms, and the 7th term of each sequence is $N$. What is the smallest possible value of $N$?
In a certain card game, a player is dealt a hand of 13 cards from a deck of 52 distinct cards. The number of distinct (unordered) hands that can be dealt to the player can be written as $A35013559A00$. What is the digit of $A$?
In the following equation, each letter represents uniquely a different digit in base 10: $$XY\times YX=X20Y$$ Find the value of $XY$.
A digital clock is malfunctioning as it always skips digit 3. For example, if the clock is displaying $04:29$, the next minute its display would show $04:40$. The clock was reset as $00:00$ at midnight. Later that day, its display shows the current time as $12:56$. What should be the correct time?
3. If Sy can shovel snow from half of a driveway in 2 hours, and Ty can shovel snow from one quarter of the driveway in 2 hours, how many $minutes$ would it take to shovel the whole driveway working together at their respective constant rates?
4. Of the bottles that Viola collects, $80%$ are green. Of the green bottles, $30%$ held perfume and $45%$ held spices. If the remaining 25 green bottles held pills, how many bottles are in Viola’s collection?
6. Don and Juan had a total of $x$ cherries, but then Don ate 27 fewer than $x$ cherries and Juan ate 11 fewer than $x$ cherries. If they each ate at least 10 cheeries, and there was at least one cheery that was’nt eaten, then $x=$
10. Iko’s rectangular vegetable garden is $2x$ $m$ wide and $3x$ $m$ long. She wants to plant flowers to form a border of uniform width around the vegetable garden, and measures that the border will cover $14x^2$ $m^2$. How wider is the border of flowers going to be?
14. How many different ordered pairs of positive integers are there each of whose squares sum to $9797$? For example, for this ordered pair of positive integers, $(10,11)$, its squares sum to $10^2+11^2=121$. [Hint: The identity $(a^2+b^2)(c^2+d^2)=(ac+bd)^2+(ad-bc)^2$ can help].
16. If we define the $separation$ between two points in the $x-y$ plane as the length of the shortest path from one point to another along the axes and/or along lines parallel to axes, then there are exactly four points with integer coordinates whose separation from origin is 1. How many points with integral coordinates have a separation from the origin of 5?
17. From a point inside an equilateral triangle, if the distances to the three sides are $2\sqrt{3}$, $4\sqrt{3}$ and $5\sqrt{3}$, what is the area of the equilateral triangle?
19. The length of the sides of hexagon $H$ are 1, 2, 3, 4, 5, and 6. If no two consecutive sides of $H$ have consecutive-integer lengths, what is the maximum sum of the lenths of thrww consecutive sides of $H$?
22. Three congruent circles have their centers on the same diagonal of a square, with two of the circles each tangent to two sides of the square, and the third circle externally tangent to the other two circles, all as shown. If the length of a side of the square is $8$, what is the length of a radius of the one of the circles?
25. The area of the parallelogram shown is $44$. If the total area of the shaded region is $14$, what is the area of the region common to the two large unshaded triangles that share a common base?
28. The only possible scores on an exam are the 16 integers from 0 to 15. The most frequent score earned by the 100 students who took the exam was 0 (a score achieved by $k$ students). If no other score was earned as frequently, what the least poissble value of $k$?
29. Two isosceles triangles with supplementary vertex angles share a common base. The lengths of the legs of one triangle are 12 and the other triangle are 5. What is the sum of the lengths of the altitudes that can be drawn to the common base of the triangles?
30. Two numbers are called reversal numbers if one is obtained from the other by reversing the order of digits. For example, 123 and 321. Are there two reversal numbers whose product is $92,565$?
Solution: Rotating $\triangle{ABE}$ $90^\circ$ degree clockwise around $A$, we have $\triangle{AFE}\cong\triangle{AEG}$. $\angle{AFE}=65^\circ$ and the perimeter of $\triangle{CEF}$ is $2$.
Posted inGeometry|Comments Off on Geometry Challenge – 14
How many pairs of positive integers $(x,y)$ are there such that $x<y$ and $\dfrac{x^2+y^2}{x+y}$ is an integer which is a divisor of $2835$? BIMC 2018
Solution: Based on the result of Number Theory Challenge – 1, if $x^2+y^2=pq$, where $p=4k+3$ is a prime number, we have $x\equiv y\equiv 0\pmod{p}$. Therefore, if $(x,y)$ is a solution for $\dfrac{x^2+y^2}{x+y}$ as an integer, then $(\dfrac{x}{p},\dfrac{y}{p})$ is also a solution.
Because $2835=3^{4}\cdot 5\cdot 7$, and $3$ and $7$ are prime numbers in the format of $4k+3$, therefore, if we find all solutions for $$\dfrac{x^2+y^2}{x+y}=q$$ where $q$ is not divisible by $3$ nor $7$, we can find all solutions. Therefore, we only need to consider two cases, where $q=1$ or $q=5$.
Case 1: if $q=1$, we have $\dfrac{x^2+y^2}{x+y}=1$, i.e. $$(2x-1)^2+(2y-1)^2=2$$ The above equation have no integer solution with $0<x<y$.
Case 2: if $q=5$, we have $\dfrac{x^2+y^2}{x+y}=5$, i.e. $$(2x-5)^2+(2y-5)^2=50$$ The two perfect square numbers on the left side of the above equation must be $1$ and $49$.
Because $0<x<y$, we have $$2x-5=\pm 1$$ $$2y-5=\ \ \ 7$$
which leads two solutions of $(x,y)$ as $(2, 6)$ or $(3,6)$.
Therefore, the total number of solutions is $\boxed{2\cdot(4+1)\cdot(1+1)=20}$.
Posted inNumber Theory|Comments Off on Number Theory Challenge – 2
Let $n=4k+3$ is a prime number and $a^2+b^2\equiv 0\pmod{n}$, prove that $a\equiv b \equiv 0\pmod{n}$.
Proof: If $a\equiv 0\pmod{n}$, because $a^2+b^2\equiv 0\pmod{n}$, $b\equiv 0\pmod{n}$.
If $a\not\equiv 0\pmod{n}$, because $n$ is a prime number, according to Fermat’s Little Theorem, we have $$a^{n-1}\equiv 1\pmod{n}\tag{1}$$
Because $a^2+b^2\equiv 0\pmod{n}$, we have $b\not\equiv 0\pmod{n}$. Similarily, we $$b^{n-1}\equiv 1\pmod{n}\tag{2}$$ Therefore, we have the following: $$a^{n-1}+b^{n-1}\equiv 2\pmod{n}\tag{3}$$
Because $n=4k+3$, and $a^2+b^2\equiv 0\pmod{n}$, we have $$a^{n-1}+b^{n-1}\equiv a^{4k+2}+b^{4k+2}\equiv (a^2)^{2k+1}+(b^2)^{2k+1}$$ $$\equiv (a^2+b^2)\sum_{i=0}^{2k}(-1)^{i}(a^2)^{2k-i}(b^2)^i\equiv 0\pmod{n}\tag{4}$$
Therefore the assumption of $a\not\equiv 0\pmod{n}$ leads conflicting equation $(3)$ and $(4)$. So, we can only have $\boxed{a\equiv b\equiv 0\pmod{n}}$.
Posted inNumber Theory|Comments Off on Number Theory Challenge – 1
