💡Alright, so I have this problem here where I need to figure out how many possible positive results there are when subtracting two 2-digit numbers. The numbers are set up like this: the top number is 5A and the bottom number is B5. I need to choose two different digits from 1 to 9 for A and B, and then subtract the bottom number from the top number. The question is asking how many of these possible results are positive.Okay, let me break this down. First, I need to understand what the numbers look like. The top number is 5A, which means it's a two-digit number where the tens digit is 5 and the units digit is A. Similarly, the bottom number is B5, so the tens digit is B and the units digit is 5. So, if I write this out, the top number is 50 + A, and the bottom number is 10*B + 5. When I subtract the bottom number from the top number, the expression becomes:(50 + A) - (10*B + 5)Let me simplify this:50 + A - 10B - 5 = A - 10B + 45So, the result of the subtraction is A - 10B + 45. We need this result to be positive. Therefore, the inequality we're looking at is:A - 10B + 45 > 0Which simplifies to:A > 10B - 45Now, since A and B are digits from 1 to 9, and they have to be different, I need to figure out for which values of B this inequality holds true and then count the corresponding valid A's.Let me consider the possible values of B one by one.Starting with B = 1:A > 10*1 - 45 => A > 10 - 45 => A > -35Since A is a digit from 1 to 9, and A ≠ B (which is 1), A can be 2, 3, 4, 5, 6, 7, 8, or 9. That's 8 possible values.Next, B = 2:A > 10*2 - 45 => A > 20 - 45 => A > -25Again, A can be any digit from 1 to 9 except 2. So, A can be 1, 3, 4, 5, 6, 7, 8, or 9. That's another 8 possible values.Moving on to B = 3:A > 10*3 - 45 => A > 30 - 45 => A > -15Same logic here. A can be any digit from 1 to 9 except 3. So, A can be 1, 2, 4, 5, 6, 7, 8, or 9. Again, 8 possible values.Now, B = 4:A > 10*4 - 45 => A > 40 - 45 => A > -5A can be any digit from 1 to 9 except 4. So, A can be 1, 2, 3, 5, 6, 7, 8, or 9. That's 8 possible values.Next, B = 5:A > 10*5 - 45 => A > 50 - 45 => A > 5Here, A has to be greater than 5. Since A is a digit from 1 to 9 and A ≠ 5, A can be 6, 7, 8, or 9. That's 4 possible values.For B = 6:A > 10*6 - 45 => A > 60 - 45 => A > 15But A is a single digit, so A can't be greater than 15. Therefore, there are no possible values for A when B = 6.Similarly, for B = 7:A > 10*7 - 45 => A > 70 - 45 => A > 25Again, A is a single digit, so no possible values here.Same goes for B = 8 and B = 9:For B = 8:A > 10*8 - 45 => A > 80 - 45 => A > 35No possible values.For B = 9:A > 10*9 - 45 => A > 90 - 45 => A > 45Still, no possible values.So, summarizing:- B = 1: 8 values- B = 2: 8 values- B = 3: 8 values- B = 4: 8 values- B = 5: 4 values- B = 6: 0 values- B = 7: 0 values- B = 8: 0 values- B = 9: 0 valuesAdding these up: 8 + 8 + 8 + 8 + 4 = 36Wait, but let me double-check. For B = 5, A has to be greater than 5, so A can be 6,7,8,9. That's 4 values. For B = 1,2,3,4, A can be any digit except B, which gives 8 values each. So, 4 + (8*4) = 4 + 32 = 36.But hold on, the options given are (A) 36, (B) 32, (C) 30, (D) 34, (E) 38.So, according to my calculation, it's 36, which is option A.But wait, let me think again. Is there any possibility that I'm overcounting? Because when B is 1,2,3,4, A can be any digit except B, but do I need to ensure that A is different from B? Yes, the problem states that two different digits are chosen, so A ≠ B. So, for each B, when I count A's, I have to exclude A = B.But in my earlier calculation, for B =1, A can be 2-9, which is 8 digits. Similarly for B=2, A can be 1,3-9, which is also 8 digits. Same for B=3 and B=4.For B=5, A has to be greater than 5, so 6-9, which is 4 digits. So, 4 + 8*4 = 36.Wait, but the answer is 36, which is option A. But I recall that sometimes in these problems, the total number of possible pairs is 9*8=72, and then the number of positive results is half of that, but that's only if the results are symmetrically distributed, which they aren't in this case because of the way the numbers are structured.Alternatively, maybe I'm missing something. Let me think differently.Suppose I consider all possible pairs of A and B where A ≠ B. There are 9 choices for A and 8 choices for B, so 72 total pairs.But not all of these will result in a positive difference. So, how many of these 72 pairs will satisfy A - 10B + 45 > 0.Alternatively, maybe I can think of it as for each B, how many A's satisfy A > 10B - 45.But I think my initial approach was correct. For B=1,2,3,4, A can be any digit except B, which gives 8 each, and for B=5, A has to be greater than 5, which gives 4. For B=6,7,8,9, no A satisfies the inequality.So, total positive results: 8+8+8+8+4=36.Therefore, the answer should be 36, which is option A.

💡Okay, so I have this problem where there are four identical equilateral triangles, each with a side length of 8 cm. They are stacked on top of each other, but each subsequent triangle is rotated by a certain angle around its center. The second one is rotated 45 degrees clockwise, the third by 90 degrees, and the top one by 135 degrees. I need to find the area of the resulting figure.First, I should probably visualize what this looks like. Four equilateral triangles stacked, each rotated by increasing angles. Since they are all equilateral, all sides are 8 cm, and all angles are 60 degrees. But when you rotate them, their orientations change, which might cause overlapping areas.I know the area of a single equilateral triangle can be calculated using the formula:[ text{Area} = frac{sqrt{3}}{4} times text{side}^2 ]So for one triangle with side length 8 cm:[ text{Area} = frac{sqrt{3}}{4} times 8^2 = frac{sqrt{3}}{4} times 64 = 16sqrt{3} text{ cm}^2 ]Since there are four such triangles, if there was no overlapping, the total area would be:[ 4 times 16sqrt{3} = 64sqrt{3} text{ cm}^2 ]But because each triangle is rotated, they must overlap each other. So the actual area of the resulting figure will be less than 64√3 cm². The question is, how much less?I need to figure out the overlapping areas. The second triangle is rotated by 45 degrees, the third by 90 degrees, and the top one by 135 degrees. Each rotation will cause different amounts of overlap.Maybe I should consider the positions of each triangle. The first triangle is not rotated, so it's in its standard position. The second is rotated 45 degrees, which is halfway between 0 and 90 degrees. The third is rotated 90 degrees, which is a quarter turn, and the fourth is rotated 135 degrees, which is three-quarters of a turn.I think the overlapping areas will be symmetrical in some way because of the rotations. Maybe each rotation creates similar overlapping regions, but I'm not entirely sure.Perhaps it's easier to calculate the total area without considering overlaps and then subtract the overlapping areas. But to do that, I need to know how much area is overlapped.Alternatively, maybe I can find the coordinates of the vertices of each triangle after rotation and then calculate the union of all these triangles. But that seems complicated because it involves coordinate geometry and might require more advanced techniques.Wait, maybe I can think about the figure as a combination of these triangles with certain overlaps. Since each triangle is rotated by 45 degrees more than the previous one, the overlaps might form smaller shapes, perhaps smaller triangles or other polygons.Let me try to sketch this mentally. The first triangle is in its original position. The second triangle is rotated 45 degrees, so it's kind of diamond-shaped relative to the first one. The third is rotated 90 degrees, which is a square-like orientation, and the fourth is rotated 135 degrees, which is another diamond but more rotated.I think the overlapping areas might form smaller equilateral triangles or maybe other regular polygons. If I can figure out the side lengths of these overlapping regions, I can calculate their areas and subtract them from the total.But I'm not sure about the exact positions and how much they overlap. Maybe I can consider that each rotation causes a certain proportion of the triangle to overlap with the one below it.Alternatively, perhaps the figure formed by these overlapping triangles is a larger regular polygon or some symmetric shape. Maybe a hexagon or something similar.Wait, another approach: since all triangles are identical and rotated by multiples of 45 degrees, the overall figure might have some rotational symmetry. Maybe I can calculate the area by considering the contributions of each triangle and then adjusting for overlaps.But this is getting a bit abstract. Maybe I should look for patterns or similar problems. I recall that when you stack rotated shapes, the overlapping areas can sometimes be calculated using trigonometric relationships or by considering the angles of rotation.Let me think about the angles. Each triangle is rotated by 45 degrees more than the previous one. So the angles between the triangles are 45, 90, and 135 degrees. These are all multiples of 45 degrees, which might help in calculating the overlaps.Perhaps I can calculate the area contributed by each triangle without overlapping and then sum them up. But I'm not sure how to calculate the non-overlapping parts.Wait, maybe I can use the principle of inclusion-exclusion. That is, calculate the total area as the sum of the areas of all triangles minus the areas where they overlap.But inclusion-exclusion can get complicated with four overlapping regions. I would need to calculate the areas of all pairwise overlaps, then add back in the areas where three triangles overlap, and so on. That seems quite involved.Alternatively, maybe the figure formed is a larger equilateral triangle or another shape whose area can be calculated directly.Wait, another thought: since each triangle is rotated by 45 degrees, the overall figure might have a star-like shape with points extending out. Maybe the area can be calculated by considering the outermost points of the triangles.But I'm not sure. Maybe I should try to calculate the coordinates of the vertices after each rotation and then find the convex hull or something like that.Let me try that. Let's place the first triangle with its base along the x-axis, with one vertex at the origin. The coordinates of the first triangle can be calculated as follows:For an equilateral triangle with side length 8 cm, the height is:[ h = frac{sqrt{3}}{2} times 8 = 4sqrt{3} text{ cm} ]So the vertices of the first triangle (let's call it Triangle 1) are at:- (0, 0)- (8, 0)- (4, 4√3)Now, Triangle 2 is rotated 45 degrees clockwise about its center. The center of Triangle 1 is at (4, (4√3)/3) because the centroid of an equilateral triangle is at 1/3 the height.So the center is at (4, (4√3)/3). To rotate Triangle 2 by 45 degrees around this center, I need to apply a rotation matrix.The rotation matrix for 45 degrees clockwise is:[ R = begin{pmatrix} cos(45^circ) & sin(45^circ) -sin(45^circ) & cos(45^circ) end{pmatrix} ]Which is:[ R = begin{pmatrix} frac{sqrt{2}}{2} & frac{sqrt{2}}{2} -frac{sqrt{2}}{2} & frac{sqrt{2}}{2} end{pmatrix} ]So I need to translate the triangle so that its center is at the origin, apply the rotation, and then translate back.The vertices of Triangle 2 before rotation are the same as Triangle 1, but after rotation, they will be different.Wait, actually, Triangle 2 is identical to Triangle 1 but rotated. So its vertices after rotation will be the rotated versions of Triangle 1's vertices.But this is getting complicated. Maybe I can calculate the coordinates after rotation.Let me denote the center as C = (4, (4√3)/3).For each vertex V of Triangle 1, the rotated vertex V' is calculated as:[ V' = C + R times (V - C) ]So let's calculate this for each vertex.First vertex: (0, 0)V - C = (0 - 4, 0 - (4√3)/3) = (-4, -4√3/3)Apply rotation:x' = (-4)(√2/2) + (-4√3/3)(√2/2) = (-4√2/2) + (-4√6/6) = (-2√2) + (-2√6/3)y' = (-4)(-√2/2) + (-4√3/3)(√2/2) = (4√2/2) + (-4√6/6) = (2√2) + (-2√6/3)Then add back the center:x = (-2√2 - 2√6/3) + 4y = (2√2 - 2√6/3) + (4√3)/3This is getting messy. Maybe I should use approximate values to get an idea.But perhaps there's a better way. Maybe instead of calculating exact coordinates, I can consider the overall shape.Wait, another idea: since all triangles are rotated by 45-degree increments, the figure might have eight-fold symmetry or something similar. But I'm not sure.Alternatively, maybe the overlapping areas can be considered as smaller triangles whose areas can be calculated proportionally.Wait, if each triangle is rotated by 45 degrees, the angle between the sides of the triangles is 45 degrees. So the overlapping regions might form smaller triangles with angles of 45 degrees.But I'm not sure. Maybe I can think about the intersection points.Alternatively, perhaps the figure is a regular octagon or something similar, but I'm not certain.Wait, another approach: maybe the area can be calculated by considering the union of all four triangles. Since each triangle is rotated, their union might form a larger shape whose area can be calculated by considering the maximum extent in each direction.But without knowing the exact positions, it's hard to say.Wait, maybe I can consider that each rotation causes the triangle to cover new areas while overlapping with the previous ones. So the total area would be the sum of the areas of the four triangles minus the overlapping areas.But to calculate the overlapping areas, I need to know how much each triangle overlaps with the others.Alternatively, maybe the figure is a combination of the four triangles arranged in a symmetric pattern, and the area can be calculated by considering the overall dimensions.Wait, perhaps I can think about the height of the resulting figure. Since each triangle is rotated, the overall height might be the sum of the heights of the triangles minus the overlapping parts.But I'm not sure.Wait, another idea: maybe the figure is a regular tetrahedron-like shape, but in 2D. But that's not quite right.Alternatively, maybe the figure is a larger equilateral triangle formed by the union of the four smaller ones.But I'm not sure. Maybe I should try to calculate the area step by step.First, the area of one triangle is 16√3 cm². Four triangles would be 64√3 cm² without overlapping.Now, I need to subtract the overlapping areas. The question is, how much do they overlap?If I can figure out the area of overlap between two adjacent triangles, I can multiply that by the number of overlaps.But the overlaps might not all be the same because each triangle is rotated by a different angle.Wait, the second triangle is rotated 45 degrees, the third 90 degrees, and the fourth 135 degrees. So the overlaps between the first and second, second and third, and third and fourth might be different.But maybe the overlaps between the first and second, second and third, and third and fourth are similar in some way.Alternatively, maybe the figure is such that the overlapping areas form smaller triangles whose areas can be calculated.Wait, another thought: since each triangle is rotated by 45 degrees, the angle between the sides of the triangles is 45 degrees. So the overlapping regions might form smaller triangles with angles of 45 degrees.But I'm not sure. Maybe I can calculate the area of overlap between two triangles rotated by 45 degrees relative to each other.I recall that the area of overlap between two congruent equilateral triangles rotated by an angle θ can be calculated using some formula, but I don't remember it exactly.Alternatively, I can think about the intersection points of the triangles and calculate the area of the overlapping region.But this is getting too complicated. Maybe I should look for a simpler approach.Wait, perhaps the figure formed by these four triangles is a larger equilateral triangle with side length 16 cm, but that seems too large.Alternatively, maybe the figure is a hexagon formed by the overlapping triangles.Wait, another idea: since each triangle is rotated by 45 degrees, the figure might have eight sides, making it an octagon. But I'm not sure.Alternatively, maybe the figure is a star with eight points.But I'm not sure. Maybe I should consider the answer choices. The options are 48√3, 52√3, 56√3, and 64√3.Since 64√3 is the total area without overlapping, and the figure must have less area, the answer must be one of the smaller options: 48√3, 52√3, or 56√3.Now, I need to figure out which one it is.Wait, maybe I can think about the figure as a combination of the four triangles with some overlapping. If each triangle overlaps with the one below it by a certain amount, the total overlapping area can be calculated.But I'm not sure how much each triangle overlaps with the others.Wait, another approach: maybe the figure is a regular tetrahedron in 2D, but that's not applicable here.Alternatively, maybe the figure is a larger equilateral triangle made up of the four smaller ones, but arranged in a certain way.Wait, I think I need to consider that each rotation causes the triangle to cover new areas while overlapping with the previous ones. So the total area would be the sum of the areas of the four triangles minus the overlapping areas.But without knowing the exact overlapping areas, it's hard to calculate.Wait, maybe I can consider that each rotation by 45 degrees causes the triangle to overlap with the previous one by a certain proportion.Alternatively, maybe the overlapping area between two triangles rotated by 45 degrees is a smaller equilateral triangle with side length half of the original.But I'm not sure.Wait, another idea: maybe the figure is a combination of the four triangles arranged in a symmetric pattern, and the area can be calculated by considering the overall dimensions.But I'm stuck. Maybe I should try to estimate.If the total area without overlapping is 64√3, and the overlapping areas are significant, the answer is likely to be 52√3 or 56√3.Given that the rotations are 45, 90, and 135 degrees, which are all multiples of 45, the figure might have a certain symmetry that allows the overlapping areas to be calculated more easily.Wait, another thought: maybe the figure is a regular octagon, and the area can be calculated using the formula for the area of a regular octagon.But I don't know the side length of the octagon.Alternatively, maybe the figure is a combination of the four triangles arranged in a way that their overlapping areas form a smaller equilateral triangle in the center.If that's the case, the total area would be the sum of the four triangles minus three times the area of the smaller triangle.But I'm not sure about the size of the smaller triangle.Wait, maybe the smaller triangle has a side length of 4 cm, which is half of the original 8 cm. Then its area would be:[ frac{sqrt{3}}{4} times 4^2 = 4sqrt{3} text{ cm}² ]If there are three such overlapping areas, the total overlapping area would be:[ 3 times 4sqrt{3} = 12sqrt{3} text{ cm}² ]Then the total area would be:[ 64sqrt{3} - 12sqrt{3} = 52sqrt{3} text{ cm}² ]Which is option B.But I'm not entirely sure if the overlapping areas are exactly three smaller triangles of 4√3 each. It might be an approximation, but given the answer choices, 52√3 seems plausible.Alternatively, maybe the overlapping areas are more than that, making the total area smaller, like 48√3. But I think 52√3 is more likely.So, after considering all these possibilities, I think the area of the resulting figure is 52√3 square cm.

💡Alright, so I've got this problem here about a new operation defined for real numbers x and y. The operation is given by x*y = ax + by - 1, where a and b are constants. They've given me two specific cases: 1*2 equals 4, and (-2)*3 equals 10. I need to find the value of a*b.Okay, let's break this down. First, I need to figure out what a and b are. Since the operation is defined in terms of a and b, and I have two equations from the given cases, I can set up a system of equations to solve for a and b.Starting with the first case: 1*2 = 4. Plugging into the operation formula, that would be a*1 + b*2 - 1 = 4. Simplifying, that's a + 2b - 1 = 4. Let me write that down as equation (1): a + 2b = 5.Next, the second case: (-2)*3 = 10. Using the same operation formula, that's a*(-2) + b*3 - 1 = 10. Simplifying, that's -2a + 3b - 1 = 10. Let me write that as equation (2): -2a + 3b = 11.Now I have two equations:1) a + 2b = 52) -2a + 3b = 11I need to solve this system of equations to find the values of a and b. I can use either substitution or elimination. Let's try elimination because the coefficients look manageable.If I multiply equation (1) by 2, I'll have:2a + 4b = 10Now, let's write equation (2) as it is:-2a + 3b = 11Now, if I add these two equations together, the a terms will cancel out:(2a - 2a) + (4b + 3b) = 10 + 11So, 0a + 7b = 21That simplifies to 7b = 21, so b = 3.Now that I have b, I can plug it back into equation (1) to find a.From equation (1): a + 2b = 5Plugging in b = 3: a + 2*3 = 5 => a + 6 = 5 => a = 5 - 6 => a = -1.So, a is -1 and b is 3.Now, the question asks for a*b. Using the operation defined, a*b would be a*a + b*b - 1, right? Wait, no. Wait, the operation is x*y = ax + by - 1. So, if I'm calculating a*b, that would be a*a + b*b - 1? Wait, no, hold on.Wait, no, x*y is ax + by - 1. So, if x is a and y is b, then a*b = a*a + b*b - 1. Wait, but that seems a bit off. Let me double-check.Wait, no, actually, x*y is defined as a*x + b*y - 1. So, if I'm calculating a*b, where x is a and y is b, then it's a*a + b*b - 1.Wait, but that seems a bit confusing because a and b are constants, so a*b would just be a constant operation. But let's see.Wait, actually, maybe I'm overcomplicating it. Since a and b are constants, and the operation is x*y = a*x + b*y - 1, then a*b would be a*a + b*b - 1.But let me think again. If x is a and y is b, then x*y = a*x + b*y - 1, so a*b = a*a + b*b - 1.But let me confirm this because sometimes these operations can be confusing.Alternatively, maybe a*b is just the operation applied to a and b, so it's a*a + b*b - 1.But let's compute that with the values we have: a = -1 and b = 3.So, a*b = (-1)*(-1) + 3*3 - 1 = 1 + 9 - 1 = 9.Wait, that seems straightforward. So, a*b equals 9.But let me make sure I didn't make a mistake in interpreting the operation.The operation is x*y = a*x + b*y - 1. So, when they ask for a*b, they're asking for the operation applied to a and b, which would be a*a + b*b - 1.Yes, that makes sense. So, plugging in a = -1 and b = 3, we get:a*b = (-1)*(-1) + 3*3 - 1 = 1 + 9 - 1 = 9.So, the answer should be 9.But just to be thorough, let me go through the steps again to ensure I didn't make any errors.First, from 1*2 = 4:a*1 + b*2 - 1 = 4 => a + 2b = 5.From (-2)*3 = 10:a*(-2) + b*3 - 1 = 10 => -2a + 3b = 11.Solving the system:Equation (1): a + 2b = 5Equation (2): -2a + 3b = 11Multiply equation (1) by 2: 2a + 4b = 10Add to equation (2): (2a - 2a) + (4b + 3b) = 10 + 11 => 7b = 21 => b = 3.Substitute back into equation (1): a + 2*3 = 5 => a + 6 = 5 => a = -1.So, a = -1, b = 3.Then, a*b = a*a + b*b - 1 = (-1)*(-1) + 3*3 - 1 = 1 + 9 - 1 = 9.Yes, that seems correct. I don't see any mistakes in the calculations.Therefore, the value of a*b is 9.

💡First, I need to determine the work rates of A, B, and C individually. I'll start by finding A's work rate.A can complete the work in 4 hours, so A's work rate is ( frac{1}{4} ) of the work per hour.Next, I know that A and C together can finish the work in 2 hours. Using A's work rate, I can find C's work rate.Once I have C's work rate, I can use the information that B and C together can complete the work in 3 hours to find B's work rate.Finally, with B's work rate known, I can determine how long B alone will take to complete the work.

💡Okay, so I have this problem where there's a function ( g(x) ) that satisfies the equation ( g(x + y) = g(x) + g(y) ) for all real numbers ( x ) and ( y ). Also, they tell me that ( g(6) = 8 ). I need to find ( g(7) ).Hmm, the equation ( g(x + y) = g(x) + g(y) ) looks familiar. I think that's called Cauchy's functional equation. From what I remember, functions that satisfy this are linear functions of the form ( g(x) = kx ), where ( k ) is a constant. Is that right? Let me check.If ( g(x) = kx ), then ( g(x + y) = k(x + y) = kx + ky = g(x) + g(y) ). Yeah, that works. So, it seems like ( g(x) ) is a linear function. That makes things easier because if I can find the constant ( k ), I can write ( g(x) ) explicitly and then find ( g(7) ).They gave me ( g(6) = 8 ). If ( g(x) = kx ), then ( g(6) = 6k = 8 ). So, I can solve for ( k ):( 6k = 8 )Divide both sides by 6:( k = frac{8}{6} = frac{4}{3} )Okay, so ( k = frac{4}{3} ). Therefore, the function ( g(x) ) is ( g(x) = frac{4}{3}x ).Now, I need to find ( g(7) ). Plugging ( x = 7 ) into the function:( g(7) = frac{4}{3} times 7 = frac{28}{3} )Wait, that seems straightforward. But let me make sure I didn't miss anything. The problem didn't specify that ( g ) is continuous or anything, but since it's defined for all real numbers and satisfies Cauchy's equation, under certain conditions, it's linear. I think in this case, since we're dealing with real numbers and given a specific value, it's safe to assume it's linear.Alternatively, I can approach it without assuming linearity. Let's try that.Given ( g(x + y) = g(x) + g(y) ) for all real ( x, y ). This is Cauchy's functional equation. One of the properties of such functions is that if we can find ( g(1) ), we can find ( g(n) ) for any integer ( n ) by induction.So, let's find ( g(1) ). Since ( g(6) = 8 ), and ( 6 = 1 + 1 + 1 + 1 + 1 + 1 ), we can write:( g(6) = g(1 + 1 + 1 + 1 + 1 + 1) = g(1) + g(1) + g(1) + g(1) + g(1) + g(1) = 6g(1) )Given that ( g(6) = 8 ), we have:( 6g(1) = 8 )So, ( g(1) = frac{8}{6} = frac{4}{3} ), which matches what I found earlier.Now, to find ( g(7) ), since ( 7 = 6 + 1 ), we can use the functional equation again:( g(7) = g(6 + 1) = g(6) + g(1) = 8 + frac{4}{3} )Adding these together:( 8 + frac{4}{3} = frac{24}{3} + frac{4}{3} = frac{28}{3} )So, ( g(7) = frac{28}{3} ).Just to double-check, let's see if this makes sense. If ( g(x) = frac{4}{3}x ), then:- ( g(1) = frac{4}{3} )- ( g(2) = frac{8}{3} )- ( g(3) = 4 )- ( g(4) = frac{16}{3} )- ( g(5) = frac{20}{3} )- ( g(6) = 8 )- ( g(7) = frac{28}{3} )All these values satisfy the functional equation ( g(x + y) = g(x) + g(y) ). For example, ( g(2) = g(1 + 1) = g(1) + g(1) = frac{4}{3} + frac{4}{3} = frac{8}{3} ), which matches. Similarly, ( g(3) = g(2 + 1) = g(2) + g(1) = frac{8}{3} + frac{4}{3} = frac{12}{3} = 4 ), which also matches. So, it seems consistent.Therefore, I'm confident that ( g(7) = frac{28}{3} ).**Final Answer**The value of ( g(7) ) is boxed{dfrac{28}{3}}.

💡Alright, so I'm trying to solve this problem about a 100x100 table where each cell is painted one of four colors. The condition is that each row and each column has exactly 25 cells of each color. I need to prove that there exist two rows and two columns such that all four cells at their intersections are painted in different colors.Hmm, okay. Let me start by understanding the problem better. We have a large grid, 100x100, and each row and column is balanced with exactly 25 cells of each of the four colors. So, in any given row, there are 25 cells of color 1, 25 of color 2, and so on for colors 3 and 4. The same goes for each column.The goal is to show that somewhere in this grid, there's a 2x2 subgrid where all four cells are different colors. That is, if I pick two rows and two columns, the four cells where they intersect are all four different colors.I think the way to approach this is by contradiction. Let me assume the opposite: that in every possible 2x2 subgrid, there are at least two cells of the same color. If I can show that this assumption leads to some kind of contradiction, then the original statement must be true.So, suppose that in every 2x2 subgrid, there are at least two cells of the same color. That means that for any two rows and any two columns, the four cells formed by their intersections cannot all be different colors.Let me think about how this affects the structure of the grid. If every 2x2 subgrid has at least two cells of the same color, then the coloring must have some kind of regularity or repetition. Maybe this implies that certain color arrangements are forced, which might conflict with the initial condition of having exactly 25 cells of each color in every row and column.Another idea is to consider the number of color pairs in rows and columns. Since each row has 25 cells of each color, the number of pairs of different colors in a row can be calculated. Similarly, for columns.Wait, let's formalize this a bit. In any row, there are 25 cells of each color, so the number of pairs of cells with different colors in a row is 25*25 for each pair of colors. Since there are four colors, the number of different color pairs in a row would be C(4,2) = 6. So, for each row, the number of different color pairs is 6*(25*25) = 6*625 = 3750.But wait, that might not be the right way to count. Actually, in a single row, the number of pairs of cells is C(100,2) = 4950. But since each color appears exactly 25 times, the number of monochromatic pairs (pairs of the same color) is 4*C(25,2) = 4*(300) = 1200. Therefore, the number of dichromatic pairs (pairs of different colors) is 4950 - 1200 = 3750, which matches my earlier calculation.So, in each row, there are 3750 pairs of cells with different colors. Similarly, in each column, there are also 3750 such pairs.Now, if I consider two arbitrary rows, how do the color pairs in these rows interact? If my assumption is that every 2x2 subgrid has at least two cells of the same color, then for any two rows, the pairs of cells in the same column must not form a 2x2 subgrid with all four different colors.Wait, maybe I need to think about the overlap between the color pairs in different rows and columns.Alternatively, perhaps I can use the pigeonhole principle. Since each row has a fixed number of each color, and each column as well, there must be some overlap that forces a 2x2 subgrid with all different colors.Let me try to think about it differently. Suppose I fix two rows. In each of these rows, there are 25 cells of each color. Now, if I look at the columns where these two rows have certain colors, maybe I can find a column where the two rows have different colors, and then another column where they have different colors as well, such that all four cells are different.But I'm not sure how to formalize this.Wait, another approach: consider the number of possible colorings for two rows. Since each row has 25 cells of each color, the number of ways to arrange the colors in two rows is quite large. But if we assume that no two rows and two columns form a 2x2 subgrid with all four different colors, then there must be some restriction on how the colors can be arranged.Perhaps I can count the number of possible color pairs between two rows and see if it's possible to avoid having all four colors in some 2x2 subgrid.Let me consider two rows. Each row has 25 cells of each color. So, for each color, there are 25 cells in the first row and 25 in the second row. The number of columns where both rows have the same color is limited by the fact that each column can only have one cell from each row.Wait, actually, each column has one cell from each row, so for each column, the two cells in the two rows can be of the same color or different colors.If I assume that in every pair of rows and columns, the 2x2 subgrid doesn't have all four different colors, then for any two rows, the number of columns where the two rows have the same color must be significant.But I'm not sure how to quantify this.Alternatively, maybe I can use double counting. Let's count the number of ordered pairs (row1, row2, column1, column2) such that the four cells at the intersections are all different colors.If I can show that this number is positive, then such a configuration exists.But I need to relate this to the given conditions.Wait, another idea: consider the number of colorings in the grid. Since each row and column is balanced, the grid is a kind of Latin square, but with four colors instead of four symbols, and each color appearing 25 times per row and column.In such a structure, certain properties must hold. Maybe I can use combinatorial arguments to show that a 2x2 subgrid with all four colors must exist.Alternatively, think about the total number of possible color pairs in the grid. Since each row has 25 cells of each color, the number of color pairs (i.e., two cells in the same row with different colors) is significant.But I'm not sure how to connect this to the existence of a 2x2 subgrid with all four colors.Wait, maybe I can use the concept of Ramsey numbers. Ramsey theory deals with conditions under which order must appear. Specifically, it might be relevant here because we're dealing with a structured grid and looking for a particular substructure.Ramsey numbers tell us the minimum number of vertices a graph must have to ensure that a particular subgraph exists. In this case, our "graph" is the grid, and the subgraph we're looking for is a 2x2 subgrid with all four colors.But I'm not sure if Ramsey numbers directly apply here, as the coloring isn't arbitrary—it's constrained by the row and column conditions.Alternatively, maybe I can model this as a hypergraph where each edge corresponds to a color, and then look for a particular configuration.But perhaps that's overcomplicating it.Let me try a different approach. Suppose I fix two rows. In each row, there are 25 cells of each color. Now, consider the columns where the two rows have the same color. Let's say in column c, both rows have color 1. Then, in column c, the two cells are both color 1, so any 2x2 subgrid involving these two rows and another column cannot have all four colors if the other column also has color 1 in both rows.But if I can find two columns where the two rows have different colors, and those colors are all distinct, then I have my 2x2 subgrid with all four colors.Wait, so maybe I can count the number of columns where the two rows have the same color and show that it's not possible for all columns to have the same color in both rows.But each row has 25 cells of each color, so the number of columns where both rows have color 1 is at most 25, because each row can only have 25 cells of color 1. Similarly, for color 2, 3, and 4.Therefore, the total number of columns where the two rows have the same color is at most 4*25 = 100. But since there are 100 columns, this means that every column must have the same color in both rows. But that's impossible because each row has 25 cells of each color, so they can't all overlap completely.Wait, that seems contradictory. If each row has 25 cells of each color, and there are 100 columns, then the number of columns where both rows have color 1 is at most 25, because each row can only have 25 cells of color 1. Similarly for the other colors.Therefore, the total number of columns where the two rows have the same color is at most 4*25 = 100, which is exactly the number of columns. So, it's possible that every column has the same color in both rows, but that would mean that both rows are identical, which contradicts the fact that each row has 25 cells of each color.Wait, no, actually, if both rows are identical, then each column would have the same color in both rows, but each column must have exactly 25 cells of each color. So, if two rows are identical, then each column would have two cells of the same color, which would mean that each column has 50 cells of that color, which contradicts the condition that each column has exactly 25 cells of each color.Therefore, it's impossible for two rows to be identical, and hence, there must be at least some columns where the two rows have different colors.But how does this help me find a 2x2 subgrid with all four colors?Maybe I can consider that for any two rows, there are columns where they have different colors, and among these, there must be some pair of columns where the colors in the two rows are all distinct.Let me formalize this. Suppose I have two rows, R1 and R2. For each column c, let’s denote the color in R1 as C1(c) and in R2 as C2(c). We know that for each color, there are 25 columns where C1(c) = color and 25 columns where C2(c) = color.Now, if I consider the pairs (C1(c), C2(c)) for all columns c, there are 100 such pairs. Each pair is an ordered pair of colors.If I assume that no two columns c1 and c2 exist such that {C1(c1), C1(c2)} and {C2(c1), C2(c2)} are all distinct, then this imposes some restrictions on the pairs (C1(c), C2(c)).Specifically, for any two columns c1 and c2, the four cells R1(c1), R1(c2), R2(c1), R2(c2) must have at least two cells of the same color. This means that the pairs (C1(c1), C1(c2)) and (C2(c1), C2(c2)) cannot both be such that all four colors are present.Wait, maybe I can model this as a graph where each column is a vertex, and the edges represent the color pairs between two rows. Then, the condition is that no two edges form a complete graph with all four colors.But I'm not sure.Alternatively, perhaps I can use the concept of Latin squares. In a Latin square, each symbol appears exactly once in each row and column, but here we have four symbols each appearing 25 times.But maybe some properties from Latin squares can be applied here.Wait, another idea: consider the total number of colorings. Since each row and column is balanced, the grid is a kind of orthogonal array. Maybe I can use properties of orthogonal arrays to show that a 2x2 subgrid with all four colors must exist.But I'm not too familiar with orthogonal arrays, so maybe I should stick to more basic combinatorial arguments.Let me try counting the number of possible colorings for two rows and two columns. If I fix two rows and two columns, there are four cells, each of which can be one of four colors. However, due to the constraints on the rows and columns, the number of valid colorings is limited.Specifically, each row has 25 cells of each color, so in the two rows, each color appears 50 times in total across the 200 cells. Similarly, each column has 25 cells of each color, so in the two columns, each color appears 50 times across the 200 cells.But I'm not sure how this helps.Wait, maybe I can use an averaging argument. Since each row and column has a balanced number of colors, the overall distribution must allow for some variation that leads to a 2x2 subgrid with all four colors.Alternatively, think about the number of possible color pairs in the grid. For any two rows, the number of columns where they have the same color is limited, as we saw earlier. Therefore, there must be a significant number of columns where the two rows have different colors.If I can show that among these differing columns, there must be two columns where the color pairs are all distinct, then I have my 2x2 subgrid.Let me try to formalize this. Suppose I have two rows, R1 and R2. Let S be the set of columns where R1 and R2 have different colors. Since each row has 25 cells of each color, the number of columns where R1 has color 1 is 25, and similarly for R2. The number of columns where both R1 and R2 have color 1 is at most 25, but since each column can only have one cell from each row, the overlap is limited.Wait, actually, the number of columns where R1 has color 1 is 25, and the number of columns where R2 has color 1 is also 25. The maximum number of columns where both R1 and R2 have color 1 is 25, but since there are 100 columns, the number of columns where R1 has color 1 and R2 has a different color is 25 - overlap.But I'm not sure.Alternatively, the total number of columns where R1 and R2 have the same color is the sum over all colors of the number of columns where both have that color. Since each color appears 25 times in each row, the maximum number of overlapping columns for each color is 25, but since there are four colors, the total number of overlapping columns is at most 4*25 = 100, which is the total number of columns. Therefore, it's possible that every column has the same color in both rows, but as we saw earlier, this leads to a contradiction because each column would then have two cells of the same color, violating the column constraint.Therefore, the number of columns where R1 and R2 have the same color must be less than 100. Hence, there must be at least some columns where R1 and R2 have different colors.But how many such columns are there? If the number of overlapping columns is less than 100, then the number of differing columns is more than 0. But we need more than that; we need to find two columns where the color pairs are all distinct.Wait, maybe I can use the pigeonhole principle here. For two rows, R1 and R2, consider the color pairs (C1(c), C2(c)) for each column c. There are 100 such pairs, and each pair is an ordered pair of colors from the set {1,2,3,4}.There are 4*4 = 16 possible ordered pairs. If we assume that no two columns c1 and c2 have color pairs that form a 2x2 subgrid with all four colors, then certain pairs must be avoided.Specifically, if we have two columns c1 and c2, and the color pairs are (a,b) and (c,d), then to avoid having all four colors, at least one color must repeat. That is, either a = c, a = d, b = c, or b = d.Therefore, the set of color pairs must be such that any two pairs share at least one common color.This is similar to a graph where each color pair is a vertex, and edges connect pairs that share a common color. Then, our assumption is that the color pairs form a clique in this graph.But I'm not sure if this helps.Alternatively, perhaps I can consider that if the color pairs are such that any two pairs share a common color, then the number of distinct color pairs is limited.Wait, actually, if any two color pairs share a common color, then the set of color pairs must all share at least one common color. Because if you have two pairs that don't share a color, then you could form a 2x2 subgrid with all four colors.Therefore, under our assumption, all color pairs must share at least one common color. Let's say they all share color 1. Then, every column c has either C1(c) = 1 or C2(c) = 1.But wait, in row R1, there are only 25 columns where C1(c) = 1, and similarly, in row R2, there are only 25 columns where C2(c) = 1. Therefore, the total number of columns where either C1(c) = 1 or C2(c) = 1 is at most 25 + 25 = 50.But there are 100 columns, so this is impossible. Therefore, our assumption that all color pairs share a common color is false.Hence, there must be at least two columns where the color pairs do not share a common color, meaning that the four cells form a 2x2 subgrid with all four different colors.Wait, that seems promising. Let me recap:1. Assume that in every 2x2 subgrid, there are at least two cells of the same color.2. Consider two arbitrary rows, R1 and R2.3. For each column c, define the color pair (C1(c), C2(c)).4. If our assumption holds, then any two color pairs must share at least one common color.5. This implies that all color pairs must share a common color, say color 1.6. However, the number of columns where C1(c) = 1 or C2(c) = 1 is limited to 25 + 25 = 50, which is less than 100.7. Therefore, our assumption leads to a contradiction, implying that there must exist two columns where the color pairs do not share a common color, forming a 2x2 subgrid with all four different colors.Yes, that makes sense. Therefore, such a 2x2 subgrid must exist.I think this is a valid approach. By assuming the contrary and using the pigeonhole principle and counting arguments, we arrive at a contradiction, which proves the original statement.So, in conclusion, there must exist two rows and two columns such that all four cells at their intersections are painted in different colors.