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Q389 | Math Olympiad | Geometry | 2014 AMC 10B Problem 15 | Rectangle | Area
Q389 | Math Olympiad | Geometry | 2014 AMC 10B Problem 15 | Rectangle | Area
zhlédnutí: 44
Video
Q388 | Math Olympiad | Algebra | Function | Minimum | Axis of Symmetry | Absolute Value
zhlédnutí 15Před 9 hodinami
Q388 | Math Olympiad | Algebra | Function | Minimum | Axis of Symmetry | Absolute Value
Q387 | Math Olympiad | Algebra | Interval | Inequality | Range
zhlédnutí 16Před 9 hodinami
Q387 | Math Olympiad | Algebra | Interval | Inequality | Range
Q386 | Math Olympiad | Algebra | Piecewise Function | Case Work
zhlédnutí 104Před 14 hodinami
Q386 | Math Olympiad | Algebra | Piecewise Function | Case Work
Q385 | Math Olympiad | Algebra | Floor | Exponents | Range
zhlédnutí 54Před 14 hodinami
Q385 | Math Olympiad | Algebra | Floor | Exponents | Range
Q384 | Math Olympiad | Geometry | 2023 AMC 10A Problem 17 | Pythagorean Triples
zhlédnutí 83Před 14 hodinami
Q384 | Math Olympiad | Geometry | 2023 AMC 10A Problem 17 | Pythagorean Triples
Q382 | Math Olympiad | Algebra | Equal Sets | Set Properties
zhlédnutí 108Před dnem
Q382 | Math Olympiad | Algebra | Equal Sets | Set Properties
Q383 | Math Olympiad | Algebra | Logarithms | Argument | Base | Range
zhlédnutí 59Před dnem
Q383 | Math Olympiad | Algebra | Logarithms | Argument | Base | Range
Q381 | Math Olympiad | Algebra | Set Builder Notation | Complement | Inequality
zhlédnutí 140Před dnem
Q381 | Math Olympiad | Algebra | Set Builder Notation | Complement | Inequality
Q380 | Math Olympiad | Geometry | 2015 AMC 12B Problem 19 | Circle | Triangle Perimeter
zhlédnutí 360Před 14 dny
Q380 | Math Olympiad | Geometry | 2015 AMC 12B Problem 19 | Circle | Triangle Perimeter
Q379 | Math Olympiad | Algebra | 2015 AMC 12B Problem 8 | Logarithms
zhlédnutí 483Před 14 dny
Q379 | Math Olympiad | Algebra | 2015 AMC 12B Problem 8 | Logarithms
Q378 | Math Olympiad | Algebra | 2011 AMC 12A Problem 18 | Absolute Value | Extremization
zhlédnutí 85Před 14 dny
Q378 | Math Olympiad | Algebra | 2011 AMC 12A Problem 18 | Absolute Value | Extremization
Q377 | Math Olympiad | Algebra | Function | Substitution | Interval
zhlédnutí 31Před 14 dny
Q377 | Math Olympiad | Algebra | Function | Substitution | Interval
Q376 | Math Olympiad | Algebra | Logarithms | Base Change
zhlédnutí 214Před 14 dny
Q376 | Math Olympiad | Algebra | Logarithms | Base Change
Q375 | Math Olympiad | Geometry | 2019 AMC 12B Problem 12 | Pythagorean Theorem | Sine
zhlédnutí 70Před 14 dny
Q375 | Math Olympiad | Geometry | 2019 AMC 12B Problem 12 | Pythagorean Theorem | Sine
Q374 | Math Olympiad | Geometry | 2021 AMC 12A Problem 17 | Trapezoid | Similar Triangles
zhlédnutí 61Před 14 dny
Q374 | Math Olympiad | Geometry | 2021 AMC 12A Problem 17 | Trapezoid | Similar Triangles
Q373 | Math Olympiad | Algebra | 2021 AMC 12A Problem 7 | Minimum | Factor | Trivial Inequality
zhlédnutí 150Před 14 dny
Q373 | Math Olympiad | Algebra | 2021 AMC 12A Problem 7 | Minimum | Factor | Trivial Inequality
Q372 | Math Olympiad | Algebra | 2021 Fall AMC 10A Problem 14 | Graph | Quadratic | Absolute Value
zhlédnutí 24Před 14 dny
Q372 | Math Olympiad | Algebra | 2021 Fall AMC 10A Problem 14 | Graph | Quadratic | Absolute Value
Q371 | Math Olympiad | Geometry | 2021 Fall AMC 12A Problem 14 | Equilateral Hexagon | Area
zhlédnutí 60Před 14 dny
Q371 | Math Olympiad | Geometry | 2021 Fall AMC 12A Problem 14 | Equilateral Hexagon | Area
Q370 | Math Olympiad | Algebra | Odd | Even | Function | Adding Functions
zhlédnutí 152Před 14 dny
Q370 | Math Olympiad | Algebra | Odd | Even | Function | Adding Functions
Q369 | Math Olympiad | Algebra | Odd Function | Period | Interval
zhlédnutí 75Před 14 dny
Q369 | Math Olympiad | Algebra | Odd Function | Period | Interval
Q368 | Math Olympiad | Geometry | 2021 Fall AMC 12A Problem 21 | Isosceles Trapezoid
zhlédnutí 85Před 14 dny
Q368 | Math Olympiad | Geometry | 2021 Fall AMC 12A Problem 21 | Isosceles Trapezoid
Q367 | Math Olympiad | Algebra | Function | Odd | Parity | Domain
zhlédnutí 158Před 14 dny
Q367 | Math Olympiad | Algebra | Function | Odd | Parity | Domain
Q366 | Math Olympiad | Algebra | Function | Period
zhlédnutí 105Před 14 dny
Q366 | Math Olympiad | Algebra | Function | Period
Q365 | Math Olympiad | Geometry | 2016 AMC 12B Problem 17 | Heron Formula | Angle Bisector Theorem
zhlédnutí 105Před 14 dny
Q365 | Math Olympiad | Geometry | 2016 AMC 12B Problem 17 | Heron Formula | Angle Bisector Theorem
Q364 | Math Olympiad | Algebra | Odd Function | Casework
zhlédnutí 156Před 14 dny
Q364 | Math Olympiad | Algebra | Odd Function | Casework
Q363 | Math Olympiad | Algebra | Logarithms | Base Change Property
zhlédnutí 85Před 14 dny
Q363 | Math Olympiad | Algebra | Logarithms | Base Change Property
Q362 | Math Olympiad | Algebra | Sum of Functions | Period
zhlédnutí 253Před 21 dnem
Q362 | Math Olympiad | Algebra | Sum of Functions | Period
Q361 | Math Olympiad | Algebra | Function | Period
zhlédnutí 42Před 21 dnem
Q361 | Math Olympiad | Algebra | Function | Period
Q360 | Math Olympiad | Geometry | 2016 AMC 12A Problem 12 | Angle Bisector Theorem
zhlédnutí 66Před 21 dnem
Q360 | Math Olympiad | Geometry | 2016 AMC 12A Problem 12 | Angle Bisector Theorem
van Aubel's theorem. AF/FD= 8/7 + 6/7=2
Why is the point M the circle of center?
why does the 5 cancel the log in base 5 ?
we know e^lnx = x. this is equivalent to e^log_e(x) = x. in general, for any base b, b^log_b(x) = x similarly, 5^log_5(x) = x
@@MilesIsReal ok thanks
Knew answer in 5 seconds
Nice! Took me nearly 8 seconds.
same for me
cool solution
didn't understood a shit
If you didn’t understand the solution, feel free to ask me any questions about it.
It may not be obvious for some how you got from the x^2(y^2+1)+(y^2+1) step to (x^2+1)(y^2+1). You should also mention that the (y^2+1) multiplier was also factored out from both terms leaving (x^2+1). Just for clarity’s sake.
I will take note of that next time.
why does f(-x) the same as -f(x)? i want to learn but i cant grasp this part yet
This is a direct result of the definition of an odd function. Odd functions are defined as functions such that f(-x) = -f(x).
@@dylansalus9159 oh thanks bro🙏
The thumbnail does not match the video presentation. For those interested in the thumbnail's solution: g(x)=1/(1-x^2) and f(x)=x/(x^2-1)
Thanks for pointing that out. I fixed the thumbnail to match the video.
This function is periodic over the integers with a period of 2. You can start evaluating f(6) to move forward to see a period of 2, and you can also move backward with a little more effort. The assumption is that this function is defined over the integers.
Cos(EBC)=Sin(ABE). S=5^2-0.5*3*4-0.5*3*5*(3/5).
how do you know the period is 6?
The value??
BX=40/sin(60°-a)sina,s=20√3/sin (60°-a) FZ=41(√3-1)-40/sin(60°-a) sina a=-arccos((163-41√3)/2/√(83 24-41*82√3)+arccos(-40/√(8324- 41*82√3)) 11896 285/1487× s=20√3/sin120°=40
{y=X²-4X,0=(x-2)⁴-12(x-2)²-(x-2)+32; (z-2)³-17 3/4(z-2)-27 33/64=0 z=2+(880 1/2+√(1761²-71³*256/2 7)/2)¹/³/4+(..-..)..;2-(880 1/2+√ (1761²-71³*256/27)/2)¹/³/8-(..-..)..±√(3/4((880 1/2+√(1761²-71³*25 6/27)/2)¹/³/4+(..-..)..)²-17,75)i
x³(-√5x+x²+1)=0
you can find the inverse of the first one, f(x) = ax + b => f^-1(x) = (x - b)/a. Then equal to the other inverse, (x - b) / a = bx + a <=> x - b = abx + a^2, you find that: x = abx <=> ab = 1, and a^2 = -b. Furthermore, b = -a^2 and replacing in the other one: a(-a^2) = 1 <=> a = -1. Finally, (-1)^2 = - b, b = - 1, so a + b = -1 + (-1) = -2.
10/√(y²-100)(18-y) x=10√(y²-100)/y 5√(5+400/y²) 3/√(5+400/y²)=1,8√(y ²-100)/(18-y) 0=y⁴+45y³-450y²-18000 у=10,88 DF=16 324/536 x=19 12/17 K=653 (78 10/17)/134 ..=922(~✓)
10/√(y²-100)(18-y) x=10√(y²-100)/y 5√(5+400/y²) 3/√(5+400/y²)=1,8√(y ²-100)/(18-y) 0=y⁴+45y³-450y²-18000
x<log3(1/2) 0=3(3^x)²+2*3^x-1 3^x=(-2±√16)/6=1/3;-1>0 x=-1✓
a=10log5(2) b=10lg(2) 1/10log2(5)-1/10log2(10)=-1/10
S∆=√(12*3*4*5)=12√5 h=3√5 PQ=17/15√5
2log³10+6(log10-1)log²2=2 Ну в этом случае да.
just for fun. let O be center of circle. tan(OCB) = 1/2 thus tan(ECB) = 4/3 thus cos(DCE) = cos(90 - ECB) = sin(ECB) = 4/5 EC = 2 / cos(DCE) = 2 / (4/5) = 5/2
its crazy how the first step is thought of
I found another way to find the solution. Step 1) Find the inverse f^-1(x) of f(x) while assuming that the given f^-1(x) is different to the actual f^-1(x). Let's call the given f^-1(x) from now on g(x). Well, you'll get f^-1(x) = 1/a * x - b/a as the inverse of f(x) . Step 2) Equate the calculated f^-1(x) and g(x), which means that 1/a * x - b/a has to be equal to bx + a. Step 3) That leads to a system of equations with I) 1/a = b and II) -b/a = a, which is a solvable system of equation. You'll get a = -1 and b = -1. Step 4) Calculate a + b = -1 + (-1) = -2 (sry for my bad english)
b(ax+b)+a=x, abx+b²+a=x. ab=1, b²+a=0. a=1/b, b²+1/b=0. b isn't 0 or ∞, so b³+1=0. Meaning b is one of the 3 cube roots of -1. Meanwhile, a, being its reciprocal, is b's conjugate. Either a=b=-1, or a=(1±√(3)i)/2 and b=(1-±√(3)i)/2. Meaning either a+b=-2, or a+b=1. The two solutions are -2 and 1.
Easy ass shiiiiit bruh was this Olympiad considered hard or moderate??
it goes amc12 to aime to usamo to imo this is the easiest level of olympiad
I didn't got the last part, last row.
If a and b are not real, you will get three solutions. Sums would be -2, 1, 1.
Exactly how I did it.
Well done. I got 1/3 as well.
what???????
(2)^2=4 360°ABCD/4 = 90°ABCD 3^30 3^5^6 3^5^13^2 1^1^13^2 3^2 (ABCD ➖ 3ABCD+2)
I did it a completely different way, but I got the same answer. Nice video!
absolutely i loved this question🙏🏼
Use tan I'm guessing after finding out that KLB = LCM = MDN = NAK and notice that its also a square
I would Never find this solution
Wow. Good this was fun
V nice sol.!! Here's another fun but longer solution w. indices. log x(w) = 24 | x^24 = w log y(w) = 40 | y^40 = w log xyz(w) = 12 | (xyz)^12 = w log z(w) = ? | z^? = w therefore: x^24 = w | x^12 = w^(1/2) [1] y^40 = w | (y^12)^(10/3) = w | y^12 = w^(3/10 ) [2] x^12 * y^12 * z^12 = w [3] substituting [1] and [2] into [3]: w^(1/2) * w^(3/10) * z^12 = w w^(4/5) * z^12 = w^1 z^12 = w^(1/5) z^60 = w therefore: log z(w) = 60
ez
Nice explanation 👌
math is mathing
Is this seriously Olympiad level?!
1:06 Shouldn't BG = 2?
On the graph it says BG=2.
this is a romanian baccalaureate mock exam question...
This was hard
Very cool!
Nice synthetic proof 🎉
32 = r * SQRT(2) thus r = 16 * SQRT(2) the area of the triangle is 16*32/2 is 256. Since the triangle is anchored to the center of the Square we would like to know how much space to rotate it . The chord of 90° is SQRT(2) on the unit circle the bisector and halfchords are r SQRT(2)/2. Thus thus is bisector SQRT(2)/2 * 16 *SQRT(2) = 16. While we are here the area between a chord and the origin is bisector * halfchord = 256. That’s rather obvious but it proves our math so far is sound. If we inscribed the square into a circle the the radius is 21/SQRT(2) or 10.5 * SQRT(2) So this is smaller than 16. This means we can rotate the triangle so that its sides are orthogonal to the sides of the square, done. The square is 10.5 x 10.5 = 21^2/4 The shaded area is 256 - 21^2/4 = 21x21/4 = 441/4 = 110.25 256-110.25 = 145.75 This is the complete answer.