💡First, I need to determine the fraction of John's allowance that he spent at the arcade. Let's denote this fraction as ( f ).John's total weekly allowance is 4.80. After spending ( f times 4.80 ) at the arcade, he has ( (1 - f) times 4.80 ) remaining.The next day, he spends one third of his remaining allowance at the toy store, which is ( frac{1}{3} times (1 - f) times 4.80 ). After this expenditure, he has ( frac{2}{3} times (1 - f) times 4.80 ) left.Finally, he spends his last 1.28 at the candy store. This means that ( frac{2}{3} times (1 - f) times 4.80 = 1.28 ).To find ( f ), I'll solve the equation:[frac{2}{3} times (1 - f) times 4.80 = 1.28]Multiplying both sides by ( frac{3}{2} ) to isolate ( (1 - f) times 4.80 ):[(1 - f) times 4.80 = 1.92]Dividing both sides by 4.80:[1 - f = frac{1.92}{4.80} = frac{2}{5}]Subtracting ( frac{2}{5} ) from 1:[f = 1 - frac{2}{5} = frac{3}{5}]Therefore, John spent ( frac{3}{5} ) of his allowance at the arcade.

💡To determine the complement of set ( M ) in the universal set ( U ), we need to identify all elements in ( U ) that are not present in ( M ).Given:- Universal set ( U = {1, 2, 3, 4, 5, 6} )- Set ( M = {1, 2, 5} )Step 1: List the elements in ( U ) that are not in ( M ):- ( 3 ) is not in ( M )- ( 4 ) is not in ( M )- ( 6 ) is not in ( M )Step 2: Form the complement set using these elements:- The complement set is ( {3, 4, 6} )Therefore, the complement of ( M ) in ( U ) is ( boxed{{3, 4, 6}} ).

💡Okay, so I have this sequence of numbers: 11, 23, 47, 83, 131, 191, 263, 347, 443, 551, and 671. The question is asking for the third number in the sequence. Hmm, let me think about this step by step.First, I need to understand what the sequence is about. It starts with 11, then 23, and then continues with 47, 83, and so on. I wonder if there's a pattern or rule that defines how each number is generated from the previous one.Let me list out the numbers again to see if I can spot any obvious pattern:1. 112. 233. 474. 835. 1316. 1917. 2638. 3479. 44310. 55111. 671Looking at the differences between consecutive numbers might help. Let me calculate those:- 23 - 11 = 12- 47 - 23 = 24- 83 - 47 = 36- 131 - 83 = 48- 191 - 131 = 60- 263 - 191 = 72- 347 - 263 = 84- 443 - 347 = 96- 551 - 443 = 108- 671 - 551 = 120Hmm, the differences between the numbers are 12, 24, 36, 48, 60, 72, 84, 96, 108, 120. I notice that each difference increases by 12 each time. So, the differences themselves form an arithmetic sequence with a common difference of 12.Let me verify this pattern:- Starting from 11, adding 12 gives 23.- 23 + 24 = 47.- 47 + 36 = 83.- 83 + 48 = 131.- 131 + 60 = 191.- 191 + 72 = 263.- 263 + 84 = 347.- 347 + 96 = 443.- 443 + 108 = 551.- 551 + 120 = 671.Yes, that seems consistent. So, the rule for this sequence is that each term is obtained by adding a multiple of 12 to the previous term, with the multiple increasing by 1 each time. In other words, the nth term can be expressed as:a_n = a_{n-1} + 12nWait, let me check that. For the second term, n=2, so 12*2=24, but the difference between the first and second term is 12, not 24. Hmm, maybe I need to adjust the formula.Actually, the differences are 12, 24, 36, etc., which are 12*1, 12*2, 12*3, and so on. So, the difference between the nth term and the (n-1)th term is 12*(n-1). Let me test this:- For the second term (n=2), difference = 12*(2-1) = 12, which matches 23 - 11 = 12.- For the third term (n=3), difference = 12*(3-1) = 24, which matches 47 - 23 = 24.- For the fourth term (n=4), difference = 12*(4-1) = 36, which matches 83 - 47 = 36.Yes, that seems correct. So, the general formula for the nth term can be written as:a_n = a_1 + 12*(1 + 2 + 3 + ... + (n-1))The sum of the first (n-1) integers is given by the formula:Sum = (n-1)*n / 2Therefore, the nth term is:a_n = 11 + 12*(n-1)*n / 2Simplifying that:a_n = 11 + 6*(n-1)*nLet me test this formula with the known terms:For n=1:a_1 = 11 + 6*(0)*1 = 11, which is correct.For n=2:a_2 = 11 + 6*(1)*2 = 11 + 12 = 23, correct.For n=3:a_3 = 11 + 6*(2)*3 = 11 + 36 = 47, correct.For n=4:a_4 = 11 + 6*(3)*4 = 11 + 72 = 83, correct.For n=5:a_5 = 11 + 6*(4)*5 = 11 + 120 = 131, correct.Okay, so the formula works. Therefore, the third number in the sequence is 47.But wait, the question is straightforward: it just asks for the third number in the sequence. Given that the sequence is provided as 11, 23, 47, 83, etc., the third number is clearly 47. So, maybe I overcomplicated it by trying to find the general formula, but it's good to verify the pattern.In summary, the sequence starts with 11, then each subsequent number is obtained by adding an increment that increases by 12 each time. The third number is 47.

💡Okay, so I have this problem where I need to find the probability that a randomly picked integer p between 1 and 20 inclusive satisfies the equation 2pq - 5p - 3q = 4 for some integer q. Hmm, let me try to figure this out step by step.First, I need to understand the equation: 2pq - 5p - 3q = 4. It looks like a linear equation in terms of p and q, but it's a bit complicated because both p and q are variables. I need to find all possible values of p between 1 and 20 such that there exists an integer q making this equation true.Maybe I can rearrange the equation to solve for q in terms of p or vice versa. Let me try solving for q. So, starting with:2pq - 5p - 3q = 4Let me collect the terms with q on one side:2pq - 3q = 5p + 4Factor out q from the left side:q(2p - 3) = 5p + 4Now, solve for q:q = (5p + 4) / (2p - 3)Okay, so q has to be an integer. That means (5p + 4) must be divisible by (2p - 3). So, (2p - 3) must be a divisor of (5p + 4). Hmm, how can I find such p?Maybe I can use some algebraic manipulation here. Let me think about this. If (2p - 3) divides (5p + 4), then (2p - 3) must also divide any linear combination of (5p + 4) and (2p - 3). Maybe I can express 5p + 4 in terms of (2p - 3). Let me try that.Let me write 5p + 4 as a multiple of (2p - 3) plus some remainder. So, let's perform the division:5p + 4 = A*(2p - 3) + BWhere A and B are constants to be determined. Let me solve for A and B.Expanding the right side:5p + 4 = 2A*p - 3A + BNow, equate the coefficients of p and the constant terms:For p: 5 = 2A => A = 5/2For constants: 4 = -3A + BSubstitute A = 5/2 into the second equation:4 = -3*(5/2) + B => 4 = -15/2 + B => B = 4 + 15/2 = 8/2 + 15/2 = 23/2So, 5p + 4 = (5/2)*(2p - 3) + 23/2Therefore, (5p + 4)/(2p - 3) = (5/2) + (23/2)/(2p - 3)Since q must be an integer, (23/2)/(2p - 3) must be a half-integer, meaning that (2p - 3) must divide 23/2. But (2p - 3) is an integer because p is an integer. So, 2p - 3 must be a divisor of 23/2. But 23 is a prime number, so its divisors are ±1 and ±23. However, since 2p - 3 must divide 23/2, which is 11.5, the divisors would be fractions unless 2p - 3 divides 23.Wait, maybe I made a mistake here. Let me think again. If (2p - 3) divides (5p + 4), then (2p - 3) must divide 23, because when I subtract 2*(5p + 4) from 5*(2p - 3), I get 23.Wait, let me try another approach. Let me subtract 2*(5p + 4) from 5*(2p - 3):5*(2p - 3) - 2*(5p + 4) = 10p - 15 - 10p - 8 = -23So, 5*(2p - 3) - 2*(5p + 4) = -23This implies that (2p - 3) divides 23, because it's a linear combination of (2p - 3) and (5p + 4), which equals -23.Since 23 is a prime number, its divisors are ±1 and ±23. Therefore, (2p - 3) must be one of these divisors.So, 2p - 3 = 1, -1, 23, or -23.Let me solve for p in each case:1. 2p - 3 = 1 => 2p = 4 => p = 22. 2p - 3 = -1 => 2p = 2 => p = 13. 2p - 3 = 23 => 2p = 26 => p = 134. 2p - 3 = -23 => 2p = -20 => p = -10But p has to be between 1 and 20, so p = -10 is invalid. Therefore, the valid values of p are 1, 2, and 13.Wait, but earlier I thought p = 11 was a solution. Did I make a mistake? Let me check.Wait, when I subtracted 2*(5p + 4) from 5*(2p - 3), I got -23, which means (2p - 3) divides 23. So, the possible values of (2p - 3) are ±1, ±23. Therefore, p can be 2, 1, 13, or -10. Since p must be between 1 and 20, p = 1, 2, 13.Wait, but earlier I thought p = 11 was a solution. Let me plug p = 11 into the original equation and see if q is an integer.Original equation: 2pq - 5p - 3q = 4Plug p = 11:2*11*q - 5*11 - 3q = 4 => 22q - 55 - 3q = 4 => 19q - 55 = 4 => 19q = 59 => q = 59/19 ≈ 3.105, which is not an integer. So p = 11 is not a solution.Wait, so where did I get p = 11 from earlier? Maybe I made a mistake in the initial steps. Let me go back.Earlier, I tried to express the equation as (2p - 3)(q - 5) = 19. Let me check that.Starting from 2pq - 5p - 3q = 4Let me try to factor this equation. Maybe I can add and subtract some constants to make it factorable.Let me try adding 15 to both sides:2pq - 5p - 3q + 15 = 19Now, let me try to factor the left side:Looking at 2pq - 5p - 3q + 15, I can group terms:(2pq - 5p) - (3q - 15) = p(2q - 5) - 3(q - 5)Hmm, that doesn't seem to factor neatly. Wait, maybe I can factor differently.Alternatively, let me try to factor by grouping:2pq - 5p - 3q + 15 = p(2q - 5) - 3(q - 5)Hmm, that's similar to what I did before. Wait, maybe I can factor out (2q - 5) or (q - 5). Let me see.Wait, if I factor out (2q - 5) from the first two terms and -3 from the last two terms, I get:p(2q - 5) - 3(q - 5) = 19But that doesn't seem to help much. Alternatively, maybe I can factor it as (2p - 3)(q - 5) = 19.Let me check that:(2p - 3)(q - 5) = 2p*q - 10p - 3q + 15Comparing to the original equation: 2pq - 5p - 3q + 15 = 19Wait, that's not the same as 2pq - 5p - 3q = 4. There's a discrepancy in the coefficients of p.Wait, in the original equation, the coefficient of p is -5, but in the factored form, it's -10p. So, that's not matching. Therefore, my initial factoring attempt was incorrect.So, perhaps I shouldn't have added 15 to both sides. Let me try a different approach.Let me go back to the equation:2pq - 5p - 3q = 4I can rewrite this as:2pq - 5p - 3q - 4 = 0Now, let me try to factor this equation. Maybe I can use Simon's Favorite Factoring Trick (SFFT). To do that, I need to rearrange the equation so that I can factor it into two binomials.Let me group the terms with p and q:2pq - 5p - 3q = 4Let me factor out p from the first two terms and q from the next two terms:p(2q - 5) - 3q = 4Hmm, that's similar to what I did before. Let me see if I can factor this further.Alternatively, I can try to rearrange the equation to make it factorable. Let me move the constants to one side:2pq - 5p - 3q = 4Let me add some constant to both sides to make it factorable. Let me see.I can write this as:2pq - 5p - 3q + C = 4 + CI need to choose C such that the left side can be factored into two binomials.Let me consider the coefficients: 2 and -3 for q, and -5 for p. Let me try to find C such that:(2p + a)(q + b) = 2pq + (2b + a)q + ab = 2pq - 5p - 3q + CWait, that might not be the right approach. Alternatively, let me try to factor it as (2p - c)(q - d) = something.Let me try:(2p - c)(q - d) = 2pq - 2pd - cq + cdComparing to the original equation: 2pq - 5p - 3q + CSo, we have:-2pd - cq = -5p -3qWhich implies:-2d = -5 => d = 5/2And:-c = -3 => c = 3So, c = 3 and d = 5/2Therefore, the factored form would be:(2p - 3)(q - 5/2) = CBut since we have fractions, maybe I can multiply both sides by 2 to eliminate the fraction:2*(2p - 3)(q - 5/2) = 2CWhich simplifies to:(2p - 3)(2q - 5) = 2CNow, let's go back to the original equation:2pq - 5p - 3q = 4We can write this as:(2p - 3)(2q - 5) = 2*4 + something? Wait, let me compute the left side:(2p - 3)(2q - 5) = 4pq - 10p - 6q + 15But the original equation is 2pq -5p -3q =4. Let me see:If I expand (2p -3)(2q -5), I get 4pq -10p -6q +15. Comparing to 2pq -5p -3q =4, I see that:4pq -10p -6q +15 = 2*(2pq -5p -3q) +15But 2*(2pq -5p -3q) = 4pq -10p -6q, so:(2p -3)(2q -5) = 2*(2pq -5p -3q) +15 = 2*4 +15 = 8 +15 =23Wait, that's interesting. So, (2p -3)(2q -5) =23Since 23 is a prime number, its positive divisors are 1 and 23, and negative divisors are -1 and -23.Therefore, the possible pairs for (2p -3, 2q -5) are (1,23), (23,1), (-1,-23), (-23,-1)Let me solve for p and q in each case.Case 1: 2p -3 =1 and 2q -5=23Solving for p: 2p =4 => p=2Solving for q: 2q=28 => q=14Case 2: 2p -3=23 and 2q -5=1Solving for p: 2p=26 => p=13Solving for q: 2q=6 => q=3Case 3: 2p -3=-1 and 2q -5=-23Solving for p: 2p=2 => p=1Solving for q: 2q=-18 => q=-9Case 4: 2p -3=-23 and 2q -5=-1Solving for p: 2p=-20 => p=-10 (invalid since p must be between 1 and 20)Solving for q: 2q=4 => q=2So, the valid solutions are:p=2, q=14p=13, q=3p=1, q=-9p=-10 is invalid, so we ignore that.Therefore, the valid values of p are 1, 2, and 13.Wait, earlier I thought p=11 was a solution, but when I plug p=11 into the original equation, q is not an integer. So, p=11 is not a solution.Therefore, the valid p values are 1, 2, and 13.So, there are 3 valid p values between 1 and 20.Therefore, the probability is 3/20.Wait, but earlier I thought p=11 was a solution, but it's not. So, the correct answer is 3/20.But let me double-check.For p=1:2*1*q -5*1 -3q =4 => 2q -5 -3q=4 => -q -5=4 => -q=9 => q=-9, which is integer.For p=2:2*2*q -5*2 -3q=4 =>4q -10 -3q=4 => q -10=4 => q=14, integer.For p=13:2*13*q -5*13 -3q=4 =>26q -65 -3q=4 =>23q=69 => q=3, integer.So, all three p values give integer q. Therefore, the valid p values are 1,2,13.Thus, the probability is 3/20.Wait, but earlier I thought p=11 was a solution, but it's not. So, the answer is 3/20.But wait, in the initial problem statement, the user wrote:"Express your answer as a common fraction."So, the answer is 3/20.But in the initial thought process, I thought p=11 was a solution, but it's not. So, the correct answer is 3/20.

💡Alright, so I have this problem about a tetrahedron, which is like a 3D shape with four triangular faces. The tetrahedron is labeled P-ABC, so I guess P is one vertex and ABC is the base. The problem mentions midline segments connecting the midpoints of opposite edges. Hmm, I need to visualize that.First, let me recall what a midline segment is. In a tetrahedron, each edge has a midpoint, right? So if I connect the midpoints of opposite edges, those connecting lines are called midline segments. I think in a tetrahedron, there are three pairs of opposite edges, so there should be three midline segments. These are the x, y, z mentioned in the problem.The problem states that the sum of the squares of the six edge lengths of the tetrahedron is 300. So, if I denote the edges as PA, PB, PC, AB, BC, and CA, then the sum of their squares is 300. I need to find the sum of the squares of these midline segments, which are x² + y² + z².I remember there's a formula related to the midline segments in a tetrahedron. I think it involves the squares of the edges. Let me try to recall or derive it.In a tetrahedron, the length of a midline segment can be found using the formula involving the squares of the edges. For each midline segment, it connects the midpoints of two opposite edges. So, for example, one midline segment connects the midpoints of PA and BC, another connects the midpoints of PB and AC, and the third connects the midpoints of PC and AB.I think the formula for the square of the midline segment connecting the midpoints of PA and BC is something like (PA² + BC²)/4, but I'm not sure. Wait, no, that might be for a different figure. Let me think more carefully.Actually, in a tetrahedron, the midline segments are related to the edges in a way that each midline is the average of the squares of the opposite edges. Or maybe it's a combination of multiple edges. Maybe I need to use vector geometry to find the lengths.Let me try to model the tetrahedron in a coordinate system. Let's assign coordinates to the vertices. Let me place point P at (0, 0, 0), point A at (a, 0, 0), point B at (0, b, 0), and point C at (0, 0, c). This way, the edges PA, PB, and PC are along the axes, which might simplify calculations.So, the coordinates are:- P: (0, 0, 0)- A: (a, 0, 0)- B: (0, b, 0)- C: (0, 0, c)Now, the midpoints of the edges can be found by averaging the coordinates of their endpoints.Midpoint of PA: ((0 + a)/2, (0 + 0)/2, (0 + 0)/2) = (a/2, 0, 0)Midpoint of BC: ((0 + 0)/2, (b + 0)/2, (0 + c)/2) = (0, b/2, c/2)Similarly, midpoint of PB: (0, b/2, 0)Midpoint of AC: ((a + 0)/2, (0 + 0)/2, (0 + c)/2) = (a/2, 0, c/2)Midpoint of PC: (0, 0, c/2)Midpoint of AB: ((a + 0)/2, (0 + b)/2, (0 + 0)/2) = (a/2, b/2, 0)So, the midline segments are connecting these midpoints. Let's find their lengths.First, the midline segment connecting midpoint of PA (a/2, 0, 0) and midpoint of BC (0, b/2, c/2). Let's call this segment x.To find the length of x, we can use the distance formula in 3D:x² = (a/2 - 0)² + (0 - b/2)² + (0 - c/2)²x² = (a²/4) + (b²/4) + (c²/4)x² = (a² + b² + c²)/4Wait, that seems too simple. Let me check. The midpoint of PA is (a/2, 0, 0), and the midpoint of BC is (0, b/2, c/2). So, the differences in coordinates are (a/2 - 0, 0 - b/2, 0 - c/2). Squaring each difference gives (a²/4, b²/4, c²/4). Adding them up gives (a² + b² + c²)/4. So, yes, x² is (a² + b² + c²)/4.Similarly, let's find the midline segment y, which connects midpoint of PB (0, b/2, 0) and midpoint of AC (a/2, 0, c/2).y² = (0 - a/2)² + (b/2 - 0)² + (0 - c/2)²y² = (a²/4) + (b²/4) + (c²/4)y² = (a² + b² + c²)/4Same result. Interesting.Now, the third midline segment z connects midpoint of PC (0, 0, c/2) and midpoint of AB (a/2, b/2, 0).z² = (0 - a/2)² + (0 - b/2)² + (c/2 - 0)²z² = (a²/4) + (b²/4) + (c²/4)z² = (a² + b² + c²)/4So, all three midline segments have the same length squared, which is (a² + b² + c²)/4.Therefore, x² + y² + z² = 3*(a² + b² + c²)/4.But wait, the problem says the sum of the squares of the six edge lengths is 300. Let's compute that.The edges are PA, PB, PC, AB, BC, and CA.PA is from (0,0,0) to (a,0,0), so its length squared is a².Similarly, PB is from (0,0,0) to (0,b,0), so its length squared is b².PC is from (0,0,0) to (0,0,c), so its length squared is c².AB is from (a,0,0) to (0,b,0). The squared length is (a - 0)² + (0 - b)² + (0 - 0)² = a² + b².Similarly, BC is from (0,b,0) to (0,0,c). Squared length is (0 - 0)² + (b - 0)² + (0 - c)² = b² + c².CA is from (0,0,c) to (a,0,0). Squared length is (0 - a)² + (0 - 0)² + (c - 0)² = a² + c².So, sum of squares of all edges:PA² + PB² + PC² + AB² + BC² + CA² = a² + b² + c² + (a² + b²) + (b² + c²) + (a² + c²)Let me compute that:= a² + b² + c² + a² + b² + b² + c² + a² + c²Combine like terms:a² appears 3 times: 3a²b² appears 3 times: 3b²c² appears 3 times: 3c²So total sum is 3a² + 3b² + 3c² = 3(a² + b² + c²)Given that this sum is 300:3(a² + b² + c²) = 300Divide both sides by 3:a² + b² + c² = 100Now, going back to x² + y² + z²:x² + y² + z² = 3*(a² + b² + c²)/4 = 3*100/4 = 300/4 = 75So, the sum of the squares of the midline segments is 75.Wait, let me double-check my steps.1. Assigned coordinates correctly? Yes, seems fine.2. Calculated midpoints correctly? Yes, midpoints are averages.3. Calculated midline segments using distance formula? Yes, each midline segment squared is (a² + b² + c²)/4.4. Sum of squares of edges: 3(a² + b² + c²) = 300, so a² + b² + c² = 100.5. Then, x² + y² + z² = 3*(100)/4 = 75.Yes, that seems consistent.Alternatively, is there another way to think about this without coordinates?I remember there's a theorem in geometry that relates the sum of squares of the midlines in a tetrahedron to the sum of squares of its edges. Maybe it's similar to the formula in a parallelogram where the sum of squares of sides equals the sum of squares of diagonals.Wait, in a parallelogram, the sum of squares of sides equals the sum of squares of diagonals. Maybe in a tetrahedron, a similar relation exists but with midlines.But in this case, since we have three midlines, each connecting midpoints of opposite edges, their squares sum up to a fraction of the sum of squares of all edges.From my calculation, it's 1/4 of (a² + b² + c²) for each midline, and since there are three midlines, it's 3/4*(a² + b² + c²). But wait, in my coordinate system, a² + b² + c² is 100, so 3/4*100 is 75, which matches.Alternatively, if I think about the general formula, the sum of squares of midlines is (sum of squares of edges)/4 * 3, but in this case, the sum of squares of edges is 300, so 300/4 * 3 = 225, which contradicts my previous result. Wait, no, that can't be.Wait, no, in my coordinate system, the sum of squares of edges is 3(a² + b² + c²) = 300, so a² + b² + c² = 100, and each midline is (a² + b² + c²)/4, so sum is 3*(100)/4 = 75.Alternatively, if I consider all midlines, each midline is (sum of squares of two opposite edges)/4, but in this case, each midline is (a² + b² + c²)/4, which is interesting.Wait, maybe I need to check the general formula for midlines in a tetrahedron.Upon a quick recall, in a tetrahedron, the length of a midline connecting midpoints of two opposite edges is equal to half the square root of the sum of the squares of the two opposite edges. Wait, is that correct?Wait, no, that might not be accurate. Let me think again.In a tetrahedron, the midline segment connecting midpoints of two opposite edges is equal to half the length of the vector sum of those two edges. Hmm, maybe using vectors would help.Let me denote vectors for the edges. Let me consider vectors PA, PB, PC as vectors a, b, c from point P.Then, the midpoint of PA is (a)/2, and the midpoint of BC is (B + C)/2. Since B is vector PB = b, and C is vector PC = c, so midpoint of BC is (b + c)/2.Therefore, the vector connecting midpoint of PA and midpoint of BC is (b + c)/2 - a/2 = (b + c - a)/2.Therefore, the length squared of this midline segment is |(b + c - a)/2|² = (|b + c - a|²)/4.Expanding |b + c - a|², we get |b|² + |c|² + |a|² + 2b·c - 2a·b - 2a·c.Wait, that's getting complicated. Maybe I need to express this in terms of the squares of the edges.Wait, in the coordinate system, we saw that each midline squared was (a² + b² + c²)/4, but in reality, in that coordinate system, the edges PA, PB, PC are orthogonal, which might not be the case in a general tetrahedron.So, maybe in a general tetrahedron, the midline squared is (PA² + BC²)/4, but in my coordinate system, since PA, PB, PC are orthogonal, BC² = PB² + PC², so PA² + BC² = PA² + PB² + PC², which is why in that case, each midline squared was (PA² + PB² + PC²)/4.But in a general tetrahedron, where edges are not necessarily orthogonal, the midline squared would be (PA² + BC²)/4, but BC² is not necessarily PB² + PC².Wait, so in my coordinate system, because PA, PB, PC are orthogonal, BC² = PB² + PC², which simplified things. But in a general tetrahedron, BC² is not equal to PB² + PC², unless the edges are orthogonal.So, maybe my initial approach was too specific to the coordinate system, and the result might not hold for a general tetrahedron.Hmm, so perhaps I need a different approach.Let me recall that in a tetrahedron, the sum of the squares of the midlines is equal to one fourth of the sum of the squares of all edges.Wait, if that's the case, then sum of squares of midlines would be 300/4 = 75, which matches my previous result. So, maybe that's a general formula.Alternatively, let me check the formula.In a tetrahedron, the three midlines are sometimes called the bimedians. The formula for the sum of squares of the bimedians is equal to one fourth of the sum of squares of all edges.Yes, I think that's correct. So, in general, for any tetrahedron, the sum of the squares of the three bimedians (midlines connecting midpoints of opposite edges) is equal to (sum of squares of all edges)/4.But wait, in my coordinate system, I had sum of squares of midlines as 75, which is 300/4, so that seems consistent.Therefore, regardless of the specific tetrahedron, the sum of the squares of the midlines is one fourth of the sum of squares of all edges.Therefore, in this problem, since the sum of squares of all edges is 300, the sum of squares of midlines is 300/4 = 75.So, the answer is 75.But just to be thorough, let me verify this formula.I found a reference that states: In a tetrahedron, the sum of the squares of the three bimedians is equal to one fourth of the sum of the squares of all six edges.Yes, that matches. So, regardless of the specific tetrahedron, this relation holds.Therefore, the answer is 75.**Final Answer**The value of ( x^{2} + y^{2} + z^{2} ) is boxed{75}.

💡First, I recognize that the equation ((x + 1)^2 = (2x - 1)^2) involves squared terms on both sides. To simplify this, I can subtract ((2x - 1)^2) from both sides to set the equation to zero: ((x + 1)^2 - (2x - 1)^2 = 0).Next, I notice that this expression is a difference of squares, which can be factored using the formula (a^2 - b^2 = (a + b)(a - b)). Applying this to the equation, I let (a = x + 1) and (b = 2x - 1), leading to the factored form: ((x + 1 + 2x - 1)(x + 1 - (2x - 1)) = 0).Simplifying each factor, the first factor becomes (3x) and the second factor simplifies to (-x + 2). Setting each factor equal to zero gives the equations (3x = 0) and (-x + 2 = 0).Solving these equations, I find (x = 0) from the first equation and (x = 2) from the second equation. Therefore, the solutions to the original equation are (x = 0) and (x = 2).