Find this: $ (1+\frac{1}{5})(1+\frac{1}{6})...(1+\frac{1}{2023})(1+\frac{1}{2024}) $

There is an isosceles triangle which follows the following: $ \bar{AB}=\bar{AC}=5, \bar{BC}=6 $ D,E are points on $ \bar{AC} $ which follows $ \bar{AD}=1, \bar{EC}=2 $ If the extent of $ \triangle $ BDE = S, Find 15S.

Find the number of positive integers (m,n) which follows the following: 1) m<n 2) The sum of even numbers between 2m and 2n is 100 greater than the sum of odd numbers between 2m and 2n.

There is a shape like this (Attachment down below) Find the number of triangles made by choosing 3 vertex from the 8 vertex in the attachment.

Find the addition of all positive integers n that follows the following: $ \frac{\sqrt{n}}{2} + \frac{30}{\sqrt{n}} $ is an integer.

Find the number of $ x $ which follows the following : $ x-\frac{1}{x}=[x]-[\frac{1}{x}] $ $ ( \frac{1}{100} \le x \le {100} ) $

There are four collinear spots: $ A,B,C,D $ $ \bar{AB}=\bar{BC}=\frac{\bar{CD}}{4}=\sqrt{5} $ There are two circles; One which has $ \bar{AC} $ as a diameter, and the other having $ \bar{BD} $ as a diameter. Let's put $ \odot (AC) \cap \odot (BD) = E,F $ Let's put the area of $ EAFD $ $ S $ Find $ S^2 $.

Find the number of 4 digit positive integers '$n$' that follow these. 1) the number of digit $ \le $ 6 2) $ 3 \mid n$, but $ 6 \nmid n $

Find the number of positive integers that are equal to or equal to 1000 that have exactly 6 divisors that are perfect squares

Find the number of cases in which one of the numbers 1, 2, 3, 4, and 5 is written at each vertex of an equilateral triangle so that the following conditions are satisfied. (However, the same number is counted as one when rotated, and the same number can be written multiple times.) $ \bigstar $ The product of the two numbers written at each end of the sides of an equilateral triangle is an even number.

There is a square $ ABCD. $ $ P $ is on $\bar{AB}$ , and $Q$ is on $ \bar{AD} $ They follow $ \bar{AP}=\bar{AQ}=\frac{\bar{AB}}{5} $ Let $ H $ be the foot of the perpendicular point from $ A $ to $ \bar{PD} $ If $ |\triangle APH|=20 $, Find the area of $ \triangle HCQ $.

For reals $x,y$, find the maximum of A. $ A=\frac{-x^2-y^2-2xy+30x+30y+75}{3x^2-12xy+12y^2+12} $

Find the number of positive integer n, which follows the following $ \bigstar $ $ n=[\frac{m^3}{2024}] $ $n$ has a positive integer $m$ that follows this equation ($ m \le 1000$)

Find the number of positive integer $x$ that has $ {a}_{1},{a}_{2},\cdot \cdot \cdot {a}_{20} $ which follows the following ($x \ge 1000$) 1) $ {a}_{1}=2, {a}_{2}=1, {a}_{3}=x $ 2) for positive integer $n$, ($ 4 \le n \le 20 $), $ {a}_{n}={a}_{n-3}+\frac{(-2)^n}{{a}_{n-1}{a}_{n-2}} $

In the following illustration, starting from point X, we move one square along the segment until we arrive at point Y. Calculate the number of times a point has passed once and does not pass again, from X to Y. (However, starting point X is considered to have passed.)

There is an Equilateral trapezoid $ ABCD. $ $ \bar{AB} =60, \bar{BC}=\bar{DA}= 36, \bar{CD}=108. $ $ M $ is the middle point of $ \bar {AB} $, and point $P$ on $ \bar{AM} $ follows that $ \bar {AP} $ =10. The foot of perpendicular dropped from $P$ to $ \bar {BD} $ is $E$. $ \bar{AC} \cap \bar{BD} $ is $ F $. Point $X$ is on $ \bar {AF} $ which follows $ \bar{MX}=\bar{ME} $ Find $ \bar{AX} \times \bar{AF} $

Find the number of $n$ that follow the following: $ \bigstar $ The number of integers $ (x,y,z) $ following this equation is not a multiple of 4. $ 2n=x^2+2y^2+2x^2+2xy+2yz $

As shown in the following figure, there is a line segment consisting of five line segments $AB, BC, CD, DE, and EA$ and $10$ intersection points of these five line segments. Find the number of ways to write $1$ or $2$ at each of the $10$ vertices so that the following conditions are satisfied. $\bigstar$ The sum of the four numbers written on each line segment $AB, BC, CD, DE, and EA$ is the same.

For all integers $ {a}_{0},{a}_{1}, \cdot\cdot\cdot {a}_{100}$, find the maximum of ${a}_{5}-2{a}_{40}+3{a}_{60}-4{a}_{95} $ $\bigstar$ 1) ${a}_{0}={a}_{100}=0$ 2) for all $i=0,1,\cdot \cdot \cdot 99, $ $|{a}_{i+1}-{a}_{i}|\le1$ 3) $ {a}_{10}={a}_{90} $

There is a $\triangle ABC$ which $\angle C=90$, and $\bar{AB}=36$ On the circumcircle of $\triangle ABC$, there is $\overarc{BC}$ which does not include point $A$. D is on $\overarc{BC}$. It satisfies $2\times\angle CAD = \angle BAD $ $E: \bar{AD}\cap\bar{BC} $ $ \bar{AE}=20 $ Find $ \bar{BD}^2 $