💡Okay, so I have this geometry problem here about a convex quadrilateral (ABCD). The problem states that (AB = AC = AD = 1 text{ cm}) and that (angle BAD = 100^circ). I need to find the measure of (angle BCD).Alright, let me try to visualize this quadrilateral first. Since (AB = AC = AD), point (A) is equidistant from points (B), (C), and (D). That suggests that (B), (C), and (D) lie on a circle centered at (A) with radius 1 cm. So, (A) is the center, and (B), (C), (D) are points on the circumference.Given that (angle BAD = 100^circ), this is the angle at point (A) between points (B) and (D). Since (AB = AD = 1 text{ cm}), triangle (ABD) is isosceles with (AB = AD). Therefore, the base angles at (B) and (D) should be equal.Wait, but actually, in triangle (ABD), sides (AB) and (AD) are equal, so the angles opposite them, which are (angle ABD) and (angle ADB), should be equal. Let me calculate those angles.The sum of angles in a triangle is (180^circ). So, in triangle (ABD):[angle BAD + angle ABD + angle ADB = 180^circ]Given (angle BAD = 100^circ), and (angle ABD = angle ADB), let's denote each as (x). Then:[100^circ + x + x = 180^circ][100^circ + 2x = 180^circ][2x = 80^circ][x = 40^circ]So, both (angle ABD) and (angle ADB) are (40^circ).Now, moving on to point (C). Since (AC = 1 text{ cm}), point (C) is also on the circle centered at (A) with radius 1 cm. So, points (B), (C), and (D) all lie on this circle.I need to find (angle BCD). This is the angle at point (C) between points (B) and (D). Hmm, how can I find this angle?Since (B), (C), and (D) are all on the circumference of the circle centered at (A), quadrilateral (ABCD) is cyclic? Wait, no, not necessarily. Because (ABCD) is convex, but unless all four points lie on a circle, it's not cyclic. However, in this case, points (B), (C), and (D) lie on a circle centered at (A), but (A) is not on that circle. So, (ABCD) isn't cyclic in the traditional sense.Maybe I can use the Law of Cosines or Law of Sines somewhere here. Let me think.First, let's consider triangle (ABC). Since (AB = AC = 1 text{ cm}), triangle (ABC) is also isosceles with (AB = AC). Therefore, the base angles at (B) and (C) are equal. Let me denote these angles as (y).Similarly, triangle (ACD) is isosceles with (AC = AD = 1 text{ cm}), so the base angles at (C) and (D) are equal. Let me denote these angles as (z).Wait, but I don't know the angles at (A) for triangles (ABC) and (ACD). Hmm, maybe I need more information.Alternatively, perhaps I can use coordinates to model this problem. Let me place point (A) at the origin ((0, 0)). Then, since (AB = AC = AD = 1 text{ cm}), points (B), (C), and (D) lie on the unit circle centered at (A).Let me assign coordinates to these points. Let's place point (B) at ((1, 0)). Then, since (angle BAD = 100^circ), point (D) will be at an angle of (100^circ) from point (B). So, the coordinates of (D) would be ((cos 100^circ, sin 100^circ)).Point (C) is another point on the unit circle, but I don't know its exact position yet. Since (AC = 1 text{ cm}), it's somewhere on the unit circle, but I need to find its coordinates such that the quadrilateral remains convex.Wait, maybe I can find the position of point (C) by considering the angles. Since (AB = AC = AD), the arcs (AB), (AC), and (AD) on the unit circle are equal? No, that's not necessarily true because the angles at (A) aren't necessarily equal.Wait, actually, in the circle centered at (A), the length of the chord is related to the central angle. Since (AB = AC = AD = 1 text{ cm}), the central angles corresponding to these chords should be equal. But wait, the chord length formula is (2r sin(theta/2)), where (theta) is the central angle. Since (r = 1), chord length is (2 sin(theta/2)). So, if all chords (AB), (AC), (AD) are equal, then their central angles should be equal.But in this case, (angle BAD = 100^circ), which is the central angle between points (B) and (D). So, if all chords (AB), (AC), (AD) are equal, then the central angles between (A) and each of (B), (C), (D) should be equal. But that would mean that points (B), (C), (D) are equally spaced around the circle, each separated by (120^circ). But wait, (angle BAD = 100^circ), which is not (120^circ). So, this seems contradictory.Hmm, maybe my assumption is wrong. Let me think again. The chord length is (2 sin(theta/2)), so if (AB = AC = AD = 1), then:[2 sin(theta/2) = 1][sin(theta/2) = 0.5][theta/2 = 30^circ][theta = 60^circ]So, the central angles corresponding to chords (AB), (AC), (AD) are each (60^circ). That means, from point (A), the angle between any two of these points (B), (C), (D) should be (60^circ). But wait, the problem states that (angle BAD = 100^circ), which is the angle between (B) and (D). So, this contradicts the earlier conclusion that the central angles should be (60^circ).This suggests that my initial assumption that all chords (AB), (AC), (AD) correspond to equal central angles is incorrect because the given angle (angle BAD = 100^circ) is larger than (60^circ). Therefore, I need to reconsider.Perhaps the central angles are not equal, but the chord lengths are equal. So, even though the central angles are different, the chord lengths (AB), (AC), (AD) are all 1 cm. That means, each of these chords subtends different central angles, but their lengths are the same.Wait, but chord length depends on the central angle. If the chord lengths are equal, then the central angles should be equal. So, if (AB = AC = AD), then the central angles must be equal. Therefore, there must be a mistake in my understanding.Wait, maybe the problem is not saying that (AB = AC = AD) as chords, but as line segments. So, in the quadrilateral (ABCD), sides (AB), (AC), and (AD) are all 1 cm. But in a quadrilateral, (AC) is a diagonal, not a side. So, perhaps the sides (AB), (BC), (CD), and (DA) are not all equal, but rather, (AB), (AC), and (AD) are equal.So, (AB = AC = AD = 1 text{ cm}). That means, from point (A), the lengths to points (B), (C), and (D) are all 1 cm. So, points (B), (C), and (D) lie on a circle of radius 1 cm centered at (A).Given that, the angle (angle BAD = 100^circ) is the angle between points (B), (A), and (D). So, the arc from (B) to (D) on the circle centered at (A) is (100^circ).Now, since (AB = AC = AD), points (B), (C), and (D) are all on the circle, but their positions relative to each other are not specified. So, point (C) can be anywhere on the circle, but the quadrilateral must be convex.I need to find (angle BCD). That is, the angle at point (C) between points (B), (C), and (D).Hmm, maybe I can use the fact that in circle geometry, the angle subtended by an arc at the center is twice the angle subtended at the circumference. But since (A) is the center, and (B), (C), (D) are on the circumference, perhaps I can relate the central angles to the angles at (C).Wait, let me think. The central angle (angle BAD = 100^circ) subtends arc (BD). The angle at point (C), which is (angle BCD), subtends the same arc (BD). But since (C) is on the circumference, the angle (angle BCD) should be half of the central angle subtended by arc (BD).But wait, the central angle is (100^circ), so the inscribed angle should be (50^circ). But that would mean (angle BCD = 50^circ). However, I'm not sure if this is correct because point (C) is not necessarily on the same circle as the center (A). Wait, actually, point (C) is on the circle centered at (A), so the same circle.Wait, no, in this case, point (C) is on the circle centered at (A), so the angle (angle BCD) is an inscribed angle subtended by arc (BD). Therefore, (angle BCD = frac{1}{2} times angle BAD = frac{1}{2} times 100^circ = 50^circ).But wait, that seems too straightforward. Let me verify.Alternatively, perhaps I need to consider the triangle (BCD). To find (angle BCD), I might need to know the lengths of sides (BC), (CD), and (BD).I know (AB = AC = AD = 1 text{ cm}). I can find (BD) using the Law of Cosines in triangle (ABD).In triangle (ABD), sides (AB = AD = 1 text{ cm}), and angle (angle BAD = 100^circ). So, using the Law of Cosines:[BD^2 = AB^2 + AD^2 - 2 times AB times AD times cos(angle BAD)][BD^2 = 1^2 + 1^2 - 2 times 1 times 1 times cos(100^circ)][BD^2 = 2 - 2 cos(100^circ)][BD = sqrt{2 - 2 cos(100^circ)}]I can calculate (cos(100^circ)) numerically, but maybe I can leave it as is for now.Now, to find (BC) and (CD), I need more information. Since (AC = 1 text{ cm}), and points (B) and (D) are also 1 cm from (A), perhaps triangles (ABC) and (ACD) are also isosceles.In triangle (ABC), sides (AB = AC = 1 text{ cm}), so it's isosceles with base (BC). Similarly, in triangle (ACD), sides (AC = AD = 1 text{ cm}), so it's isosceles with base (CD).But I don't know the angles at (A) for triangles (ABC) and (ACD). However, since all points (B), (C), (D) are on the unit circle centered at (A), the central angles between (A) and each pair of points can be determined.Wait, the total angle around point (A) is (360^circ). We know that (angle BAD = 100^circ), so the remaining angle around (A) is (360^circ - 100^circ = 260^circ). This remaining angle is split between (angle BAC) and (angle CAD).But since (AB = AC = AD), the arcs (AB), (AC), and (AD) should be equal? Wait, no, because the central angles correspond to the chords, and since all chords are equal, the central angles should be equal. But earlier, we saw that (angle BAD = 100^circ), which is larger than (60^circ), which would be the case if all central angles were equal.This is confusing. Maybe I need to approach this differently.Let me consider the coordinates again. Let me place point (A) at ((0, 0)). Let me place point (B) at ((1, 0)). Then, since (angle BAD = 100^circ), point (D) will be at an angle of (100^circ) from point (B). So, the coordinates of (D) would be ((cos 100^circ, sin 100^circ)).Now, point (C) is another point on the unit circle, but I need to determine its coordinates such that the quadrilateral remains convex. Since (AB = AC = AD = 1 text{ cm}), point (C) must also lie on the unit circle.Let me denote the angle between (AB) and (AC) as (theta). Then, the coordinates of (C) would be ((cos theta, sin theta)).Now, since the quadrilateral is convex, point (C) must lie in such a way that the entire quadrilateral doesn't intersect itself. Given that (B) is at ((1, 0)) and (D) is at ((cos 100^circ, sin 100^circ)), point (C) should lie somewhere between (B) and (D) on the unit circle to maintain convexity.But I'm not sure about that. Maybe it's better to consider the angles.Since (AB = AC = AD = 1 text{ cm}), the central angles from (A) to each of (B), (C), and (D) are equal. Wait, but earlier, we saw that (angle BAD = 100^circ), which is the angle between (B) and (D). If all central angles are equal, then the angle between (B) and (C) and between (C) and (D) should be equal as well.But if the total central angle around (A) is (360^circ), and we have three equal central angles, each would be (120^circ). But (angle BAD = 100^circ), which is less than (120^circ). So, this suggests that my initial assumption is wrong.Wait, perhaps the central angles are not equal. Since (AB = AC = AD), the chords are equal, but the central angles can be different if the arcs are different. Wait, no, equal chords correspond to equal central angles in a circle. So, if (AB = AC = AD), then the central angles must be equal.But then, if all central angles are equal, each would be (120^circ), but (angle BAD = 100^circ), which contradicts that. Therefore, there must be a mistake in my understanding.Wait, maybe the problem is not saying that (AB = AC = AD) as chords, but as sides of the quadrilateral. But in a quadrilateral, (AC) is a diagonal, not a side. So, the sides are (AB), (BC), (CD), and (DA), while (AC) is a diagonal.So, the problem states that (AB = AC = AD = 1 text{ cm}). So, sides (AB), (AC), and (AD) are all 1 cm. But in a quadrilateral, (AC) is a diagonal, not a side. So, this is a bit confusing.Wait, maybe the problem is mistyped, and it should say (AB = BC = CD = DA = 1 text{ cm}), making it a rhombus. But no, the problem clearly states (AB = AC = AD = 1 text{ cm}).Alternatively, perhaps the quadrilateral is such that from point (A), the lengths to (B), (C), and (D) are all 1 cm, but (B), (C), and (D) are not necessarily on a circle centered at (A). But that contradicts the earlier thought.Wait, no, if (AB = AC = AD = 1 text{ cm}), then points (B), (C), and (D) must lie on a sphere (in 3D) or a circle (in 2D) centered at (A) with radius 1 cm.But since we're dealing with a convex quadrilateral, it's in 2D, so they lie on a circle.So, given that, points (B), (C), and (D) lie on a circle centered at (A) with radius 1 cm. The angle (angle BAD = 100^circ) is the central angle between points (B) and (D). Therefore, the arc (BD) measures (100^circ).Now, point (C) is another point on this circle. To find (angle BCD), which is the angle at point (C) between points (B), (C), and (D).In circle geometry, the measure of an inscribed angle is half the measure of its intercepted arc. So, (angle BCD) intercepts arc (BD), which is (100^circ). Therefore, (angle BCD = frac{1}{2} times 100^circ = 50^circ).Wait, but that seems too simple. Let me verify.Alternatively, perhaps I need to consider the triangle (BCD). To find (angle BCD), I might need to use the Law of Cosines or Law of Sines.I know (AB = AC = AD = 1 text{ cm}), so (AC = 1 text{ cm}). I can find (BC) and (CD) using the Law of Cosines in triangles (ABC) and (ACD).In triangle (ABC), sides (AB = AC = 1 text{ cm}), so it's isosceles. Let me denote the angle at (A) as (angle BAC = alpha). Then, the base angles at (B) and (C) are equal, each being (frac{180^circ - alpha}{2}).Similarly, in triangle (ACD), sides (AC = AD = 1 text{ cm}), so it's isosceles. Let me denote the angle at (A) as (angle CAD = beta). Then, the base angles at (C) and (D) are equal, each being (frac{180^circ - beta}{2}).Now, since the total angle around point (A) is (360^circ), we have:[angle BAC + angle CAD + angle BAD = 360^circ][alpha + beta + 100^circ = 360^circ][alpha + beta = 260^circ]But I don't know the values of (alpha) and (beta) individually.However, in triangle (ABC), the sides (AB = AC = 1 text{ cm}), so it's isosceles, and the base (BC) can be found using the Law of Cosines:[BC^2 = AB^2 + AC^2 - 2 times AB times AC times cos(alpha)][BC^2 = 1 + 1 - 2 times 1 times 1 times cos(alpha)][BC^2 = 2 - 2 cos(alpha)][BC = sqrt{2 - 2 cos(alpha)}]Similarly, in triangle (ACD):[CD^2 = AC^2 + AD^2 - 2 times AC times AD times cos(beta)][CD^2 = 1 + 1 - 2 times 1 times 1 times cos(beta)][CD^2 = 2 - 2 cos(beta)][CD = sqrt{2 - 2 cos(beta)}]Now, in triangle (BCD), I can use the Law of Cosines to find (angle BCD). The Law of Cosines states:[BD^2 = BC^2 + CD^2 - 2 times BC times CD times cos(angle BCD)]I already have expressions for (BC), (CD), and (BD). Let me substitute them in.First, recall that (BD^2 = 2 - 2 cos(100^circ)).So,[2 - 2 cos(100^circ) = (2 - 2 cos(alpha)) + (2 - 2 cos(beta)) - 2 times sqrt{2 - 2 cos(alpha)} times sqrt{2 - 2 cos(beta)} times cos(angle BCD)]Simplify the left side:[2 - 2 cos(100^circ) = 4 - 2 (cos(alpha) + cos(beta)) - 2 sqrt{(2 - 2 cos(alpha))(2 - 2 cos(beta))} cos(angle BCD)]This seems complicated. Maybe I can find a relationship between (alpha) and (beta).Earlier, we had:[alpha + beta = 260^circ]So, (beta = 260^circ - alpha).Let me substitute (beta) in terms of (alpha):[2 - 2 cos(100^circ) = 4 - 2 (cos(alpha) + cos(260^circ - alpha)) - 2 sqrt{(2 - 2 cos(alpha))(2 - 2 cos(260^circ - alpha))} cos(angle BCD)]This is getting very complex. Maybe there's a trigonometric identity I can use for (cos(260^circ - alpha)).Recall that (cos(260^circ - alpha) = cos(260^circ) cos(alpha) + sin(260^circ) sin(alpha)).Calculate (cos(260^circ)) and (sin(260^circ)):[cos(260^circ) = cos(180^circ + 80^circ) = -cos(80^circ)][sin(260^circ) = sin(180^circ + 80^circ) = -sin(80^circ)]So,[cos(260^circ - alpha) = -cos(80^circ) cos(alpha) - sin(80^circ) sin(alpha)]This substitution might not simplify things much. Maybe I need a different approach.Let me go back to the circle idea. Since points (B), (C), and (D) lie on a circle centered at (A), and (angle BAD = 100^circ), the arc (BD) is (100^circ). The angle (angle BCD) is an inscribed angle subtended by arc (BD), so it should be half of the central angle.Therefore,[angle BCD = frac{1}{2} times angle BAD = frac{1}{2} times 100^circ = 50^circ]But earlier, I thought this might be too straightforward. Let me verify with coordinates.Let me place point (A) at ((0, 0)), point (B) at ((1, 0)), and point (D) at ((cos 100^circ, sin 100^circ)). Let me choose point (C) somewhere on the unit circle such that the quadrilateral is convex.Let me assume point (C) is at an angle of (50^circ) from point (B). So, its coordinates would be ((cos 50^circ, sin 50^circ)).Now, let me calculate vectors (CB) and (CD):Vector (CB = B - C = (1 - cos 50^circ, 0 - sin 50^circ))Vector (CD = D - C = (cos 100^circ - cos 50^circ, sin 100^circ - sin 50^circ))Now, the angle (angle BCD) is the angle between vectors (CB) and (CD). I can use the dot product formula to find this angle.The dot product of vectors (CB) and (CD) is:[(1 - cos 50^circ)(cos 100^circ - cos 50^circ) + (0 - sin 50^circ)(sin 100^circ - sin 50^circ)]This seems complicated, but let me compute it numerically.First, calculate the components:[1 - cos 50^circ approx 1 - 0.6428 = 0.3572][cos 100^circ - cos 50^circ approx -0.1736 - 0.6428 = -0.8164][0 - sin 50^circ approx -0.7660][sin 100^circ - sin 50^circ approx 0.9848 - 0.7660 = 0.2188]Now, compute the dot product:[0.3572 times (-0.8164) + (-0.7660) times 0.2188 approx -0.2914 - 0.1683 = -0.4597]Now, find the magnitudes of vectors (CB) and (CD):Magnitude of (CB):[sqrt{(1 - cos 50^circ)^2 + (-sin 50^circ)^2} approx sqrt{0.3572^2 + 0.7660^2} approx sqrt{0.1276 + 0.5868} approx sqrt{0.7144} approx 0.8453]Magnitude of (CD):[sqrt{(cos 100^circ - cos 50^circ)^2 + (sin 100^circ - sin 50^circ)^2} approx sqrt{(-0.8164)^2 + (0.2188)^2} approx sqrt{0.6665 + 0.0479} approx sqrt{0.7144} approx 0.8453]So, both vectors (CB) and (CD) have magnitudes of approximately (0.8453).Now, the dot product formula is:[cos(theta) = frac{text{dot product}}{|text{CB}| |text{CD}|}][cos(theta) = frac{-0.4597}{0.8453 times 0.8453} approx frac{-0.4597}{0.7144} approx -0.643]Therefore,[theta approx cos^{-1}(-0.643) approx 130^circ]Wait, that's different from the earlier result of (50^circ). So, which one is correct?Hmm, I think the confusion arises because point (C) is not necessarily positioned such that arc (BD) is (100^circ) as seen from (C). In the coordinate system, when I placed point (C) at (50^circ), it created a different configuration.But in reality, since points (B), (C), and (D) are all on the unit circle centered at (A), the angle (angle BCD) should indeed be half the central angle subtended by arc (BD), which is (100^circ). Therefore, (angle BCD = 50^circ).But the coordinate calculation gave me (130^circ). This discrepancy suggests that my assumption about the position of point (C) was incorrect.Wait, perhaps point (C) is not between (B) and (D) on the circle, but on the opposite side. Let me try placing point (C) at an angle of (100^circ + theta) instead of (50^circ).Alternatively, maybe I need to consider that the angle (angle BCD) is actually the external angle corresponding to the arc (BD). In circle geometry, the external angle is equal to half the difference of the intercepted arcs.Wait, let me recall. The measure of an angle formed by two chords intersecting at a point on the circumference is half the sum of the measures of the intercepted arcs. If the angle is outside the circle, it's half the difference.In this case, point (C) is on the circumference, so (angle BCD) is formed by two chords (CB) and (CD). The intercepted arcs are arc (BD) and the opposite arc.Wait, the total circumference is (360^circ), so the opposite arc would be (360^circ - 100^circ = 260^circ).Therefore, the measure of (angle BCD) is half the difference of the intercepted arcs:[angle BCD = frac{1}{2} ( text{arc } BD - text{arc opposite } BD )][angle BCD = frac{1}{2} (100^circ - 260^circ) = frac{1}{2} (-160^circ) = -80^circ]But angles can't be negative, so maybe I took the difference in the wrong order.Alternatively,[angle BCD = frac{1}{2} ( text{arc opposite } BD - text{arc } BD )][angle BCD = frac{1}{2} (260^circ - 100^circ) = frac{1}{2} (160^circ) = 80^circ]But this still doesn't match the earlier results.Wait, perhaps I'm confusing the internal and external angles. Let me clarify.When the angle is inside the circle, the measure is half the sum of the intercepted arcs. When it's outside, it's half the difference.In this case, point (C) is on the circumference, so the angle is inside the circle. Therefore, the measure of (angle BCD) is half the sum of the intercepted arcs.But what are the intercepted arcs? Arc (BD) and arc (BC)?Wait, no. When two chords intersect at a point on the circumference, the angle is half the sum of the measures of the intercepted arcs.In this case, chords (CB) and (CD) intersect at point (C), intercepting arcs (BD) and the opposite arc.Wait, the opposite arc would be the arc that is not between (B) and (D), which is (260^circ).Therefore,[angle BCD = frac{1}{2} ( text{arc } BD + text{arc opposite } BD )][angle BCD = frac{1}{2} (100^circ + 260^circ) = frac{1}{2} (360^circ) = 180^circ]But that can't be right because the quadrilateral is convex, so (angle BCD) must be less than (180^circ).This is getting more confusing. Maybe I need to revisit the problem.Given that (AB = AC = AD = 1 text{ cm}) and (angle BAD = 100^circ), we need to find (angle BCD).Since (AB = AC = AD), point (A) is the circumcenter of triangle (BCD). Wait, no, because (A) is equidistant from (B), (C), and (D), but it's not necessarily the circumcenter unless the triangle is equilateral or something.Wait, actually, in this case, since (AB = AC = AD), point (A) is the circumcenter of triangle (BCD). Therefore, triangle (BCD) is inscribed in a circle centered at (A) with radius 1 cm.In that case, the central angle (angle BAD = 100^circ) subtends arc (BD). The inscribed angle (angle BCD) subtends the same arc (BD), so it should be half of the central angle.Therefore,[angle BCD = frac{1}{2} times 100^circ = 50^circ]But earlier, when I used coordinates, I got (130^circ). This suggests that my coordinate approach might have been flawed because I didn't account for the correct position of point (C).Alternatively, perhaps the angle (angle BCD) is actually (180^circ - 50^circ = 130^circ) because of the orientation of the quadrilateral.Wait, if the central angle is (100^circ), the inscribed angle could be (50^circ) or (180^circ - 50^circ = 130^circ) depending on the position of point (C).Since the quadrilateral is convex, point (C) must lie such that (angle BCD) is less than (180^circ). But whether it's (50^circ) or (130^circ) depends on the specific configuration.Given that in my coordinate example, I got (130^circ), and considering that the central angle is (100^circ), the inscribed angle could be (130^circ) if point (C) is on the opposite side of the circle.Wait, let me think again. The measure of an inscribed angle is half the measure of its intercepted arc. If the arc (BD) is (100^circ), then the inscribed angle (angle BCD) is (50^circ). However, if point (C) is on the opposite side of the circle, the intercepted arc would be the major arc (BD), which is (360^circ - 100^circ = 260^circ). Then, the inscribed angle would be half of that, which is (130^circ).Since the quadrilateral is convex, point (C) must lie on the same side as the major arc (BD), making (angle BCD = 130^circ).Therefore, the correct measure of (angle BCD) is (130^circ).I think this makes sense now. The key was realizing that depending on the position of point (C), the intercepted arc could be the minor arc (BD) or the major arc (BD). Since the quadrilateral is convex, point (C) must lie on the major arc side, resulting in (angle BCD = 130^circ).

💡Okay, so I have this problem here about four positive integers. When each of them is divided by 53, they leave remainders 33, 26, 18, and 6 respectively. Then, each of these integers is multiplied by 2, and I need to find the remainder when the sum of these products is divided by 53. Hmm, let me try to break this down step by step.First, let's understand what it means for a number to have a certain remainder when divided by 53. If a number, say, a, when divided by 53 leaves a remainder of 33, that means a can be expressed as 53 times some integer plus 33. Similarly for the other numbers. So, mathematically, I can write:a = 53k + 33 b = 53m + 26 c = 53n + 18 d = 53p + 6 Where k, m, n, p are some integers. Now, the problem says each of these integers is multiplied by 2. So, let's compute 2a, 2b, 2c, and 2d.2a = 2*(53k + 33) = 106k + 66 2b = 2*(53m + 26) = 106m + 52 2c = 2*(53n + 18) = 106n + 36 2d = 2*(53p + 6) = 106p + 12 So, each of these products is 106 times some integer plus the doubled remainder. Now, if I add all these together, I get:2a + 2b + 2c + 2d = (106k + 66) + (106m + 52) + (106n + 36) + (106p + 12) = 106(k + m + n + p) + (66 + 52 + 36 + 12) Let me compute the sum of the constants: 66 + 52 is 118, plus 36 is 154, plus 12 is 166. So, the total sum is 106(k + m + n + p) + 166.Now, the question is asking for the remainder when this sum is divided by 53. So, I need to compute (106(k + m + n + p) + 166) mod 53.But 106 is exactly 2*53, so 106 is congruent to 0 mod 53. That means 106(k + m + n + p) is also congruent to 0 mod 53 because it's a multiple of 53. So, the entire expression simplifies to 0 + 166 mod 53, which is just 166 mod 53.Now, I need to compute 166 divided by 53 and find the remainder. Let's see, 53*3 is 159, and 166 - 159 is 7. So, 166 mod 53 is 7.Wait, let me double-check that. 53*3 is indeed 159, and 166 minus 159 is 7. So yes, the remainder is 7.Alternatively, I could have approached this problem by considering each multiplication modulo 53 first before adding them up. Let me try that method to verify.Starting with the remainders: 33, 26, 18, 6.Multiplying each by 2:2*33 = 66 2*26 = 52 2*18 = 36 2*6 = 12 Now, compute each of these modulo 53:66 mod 53 is 66 - 53 = 13 52 mod 53 is 52 (since 52 is less than 53) 36 mod 53 is 36 12 mod 53 is 12 Now, add these results: 13 + 52 + 36 + 12.13 + 52 is 65, plus 36 is 101, plus 12 is 113.Now, compute 113 mod 53. 53*2 is 106, so 113 - 106 is 7. So again, the remainder is 7.Hmm, both methods give me the same result, which is reassuring. So, whether I handle the modulo operation after adding all the terms or before, I end up with the same remainder of 7.Just to make sure I haven't missed anything, let me recap:1. Each original number can be expressed as 53 times some integer plus its respective remainder.2. Multiplying each by 2 gives 106 times that integer plus twice the remainder.3. Since 106 is a multiple of 53, those terms vanish when taking modulo 53.4. Therefore, only the doubled remainders contribute to the final sum modulo 53.5. Adding the doubled remainders and taking modulo 53 gives the final result.Alternatively, since modulo operations are linear, I can first compute each multiplication modulo 53 and then add them up, which is a more straightforward approach and avoids dealing with larger numbers.Either way, the result is consistent. So, I feel confident that the remainder is indeed 7.I wonder if there's a quicker way to do this without having to compute each step. Maybe by recognizing that multiplying by 2 is equivalent to scaling the remainders, and since 2 and 53 are coprime (their greatest common divisor is 1), the scaling doesn't affect the modulus in any tricky way. So, I can safely multiply each remainder by 2 and then add them up modulo 53.Another thought: since 53 is a prime number, that might have some implications, but in this case, it doesn't directly affect the calculation since we're dealing with remainders and linear operations.Also, just to check, if I had added the original remainders first and then multiplied by 2, would that give the same result? Let's see:Original remainders: 33, 26, 18, 6.Sum: 33 + 26 = 59, plus 18 = 77, plus 6 = 83.Multiply by 2: 83*2 = 166.166 mod 53 is 7, same as before. So, that method also works. Interesting, so whether I add first or multiply first, the result is the same because multiplication distributes over addition. So, that's another way to approach it, which might be quicker.But in this case, since the numbers are small, all methods are manageable. It's good to know multiple approaches to solve the same problem, as it can sometimes lead to insights or shortcuts.In summary, the key steps are:1. Recognize that each number can be expressed in terms of 53 plus a remainder.2. Multiply each by 2, which affects both the multiple of 53 and the remainder.3. Since the multiple of 53 will be 0 modulo 53, only the doubled remainders matter.4. Add the doubled remainders and take modulo 53 to find the final remainder.Alternatively, add the original remainders, multiply by 2, and then take modulo 53.Either way, the result is 7. So, I think that's solid.Just to ensure I haven't made any arithmetic errors, let me recompute the sum of the doubled remainders:2*33 = 66 2*26 = 52 2*18 = 36 2*6 = 12 Adding these: 66 + 52 = 118, 118 + 36 = 154, 154 + 12 = 166.166 divided by 53: 53*3 = 159, 166 - 159 = 7. Yep, that's correct.Alternatively, adding the original remainders: 33 + 26 = 59, 59 + 18 = 77, 77 + 6 = 83. 83*2 = 166, same as before. 166 mod 53 is 7.So, all methods converge to the same answer, which is 7. Therefore, I can confidently say that the remainder is 7.**Final Answer**The remainder is boxed{7}.

💡Okay, so I have this problem about sets, and I need to find some intersections, complements, unions, and stuff. Let me see if I can figure this out step by step. First, let's understand the sets we're dealing with. Set A is defined as all positive integers less than 11. So, that should be the numbers from 1 to 10, right? So, A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. Got that down.Set B is given as {1, 2, 3, 4}. That's straightforward. Set C is {3, 4, 5, 6, 7}. Okay, so C starts at 3 and goes up to 7. Now, the problem has two parts, each asking for two different set operations. Let's tackle them one by one.**Part 1:**1. **A ∩ C**: This is the intersection of sets A and C. The intersection means all elements that are common to both sets. So, since A is {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and C is {3, 4, 5, 6, 7}, the common elements are 3, 4, 5, 6, and 7. So, A ∩ C should be {3, 4, 5, 6, 7}. That seems right.2. **Complement of B in A**: This is denoted as ∁ₐB. The complement of a set B with respect to A means all elements that are in A but not in B. So, since B is {1, 2, 3, 4}, and A is {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, the complement should be all the elements in A that aren't in B. That would be 5, 6, 7, 8, 9, 10. So, ∁ₐB = {5, 6, 7, 8, 9, 10}. That makes sense.**Part 2:**1. **Complement of (B ∪ C) in A**: First, I need to find the union of B and C, which is all elements that are in either B or C or both. B is {1, 2, 3, 4} and C is {3, 4, 5, 6, 7}, so combining these, we get {1, 2, 3, 4, 5, 6, 7}. Now, the complement of this union with respect to A would be all elements in A that are not in this union. A is up to 10, so the elements not in {1, 2, 3, 4, 5, 6, 7} are 8, 9, 10. Therefore, ∁ₐ(B ∪ C) = {8, 9, 10}.2. **A ∪ (B ∩ C)**: First, find the intersection of B and C. B is {1, 2, 3, 4} and C is {3, 4, 5, 6, 7}, so the common elements are 3 and 4. So, B ∩ C = {3, 4}. Now, take the union of A and this intersection. But since A already contains all elements from 1 to 10, and {3, 4} are already in A, the union will just be A itself. So, A ∪ (B ∩ C) = A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}.Wait, let me double-check to make sure I didn't make any mistakes.For A ∩ C: Yes, both sets have 3,4,5,6,7. So that's correct.For ∁ₐB: A has up to 10, and B goes up to 4, so the complement should start from 5. Yep, that's right.For ∁ₐ(B ∪ C): The union of B and C is up to 7, so the complement in A would be 8,9,10. Correct.For A ∪ (B ∩ C): Since A already includes everything, the union doesn't change. So, that's correct too.I think I got all the answers right. Let me just write them down neatly.**Final Answer**1. ( A cap C = boxed{{3, 4, 5, 6, 7}} ) and ( complement_{A} B = boxed{{5, 6, 7, 8, 9, 10}} ).2. ( complement_{A}(B cup C) = boxed{{8, 9, 10}} ) and ( A cup (B cap C) = boxed{{1, 2, 3, 4, 5, 6, 7, 8, 9, 10}} ).

💡Alright, so I've got this problem about a sequence {a_n} where the sum of the first n terms is S_n. They give me that a_1 is 2, and there's this relation S_{n+1} = 4a_n - 2 for n = 1, 2, 3, and so on. The problem has three parts: I need to find a_2 and a_3, prove that the sequence {a_n - 2a_{n-1}} is constant, and then prove an inequality involving a sum of fractions. Let me try to tackle each part step by step.Starting with part (I): Find a_2 and a_3.Okay, so S_n is the sum of the first n terms. That means S_1 = a_1 = 2. Then S_2 = a_1 + a_2, S_3 = a_1 + a_2 + a_3, etc. The given relation is S_{n+1} = 4a_n - 2. So for n=1, S_2 = 4a_1 - 2. Since a_1 is 2, plugging that in: S_2 = 4*2 - 2 = 8 - 2 = 6. But S_2 is also a_1 + a_2, which is 2 + a_2. So 2 + a_2 = 6, which means a_2 = 4.Now, to find a_3, I can use the same relation for n=2: S_3 = 4a_2 - 2. We already know a_2 is 4, so S_3 = 4*4 - 2 = 16 - 2 = 14. But S_3 is also a_1 + a_2 + a_3, which is 2 + 4 + a_3 = 6 + a_3. So 6 + a_3 = 14, which means a_3 = 8.Wait, that seems straightforward. So a_2 is 4 and a_3 is 8. Let me just double-check that. If a_1 is 2, a_2 is 4, then S_2 is 6, which matches 4a_1 - 2. Then S_3 is 14, which is 4a_2 - 2, which is 16 - 2 = 14. So that's correct. Okay, part (I) seems done.Moving on to part (II): Prove that the sequence {a_n - 2a_{n-1}} is a constant sequence.Hmm, so I need to show that a_n - 2a_{n-1} is the same for all n. Let's see. Maybe I can find a recurrence relation for a_n.Given that S_{n+1} = 4a_n - 2, and S_{n+1} is also equal to S_n + a_{n+1}. So, S_{n+1} = S_n + a_{n+1} = 4a_n - 2.But S_n is also given by the same formula for n-1: S_n = 4a_{n-1} - 2. So substituting that into the equation above: 4a_{n-1} - 2 + a_{n+1} = 4a_n - 2.Simplifying this: 4a_{n-1} - 2 + a_{n+1} = 4a_n - 2. Subtract 4a_{n-1} - 2 from both sides: a_{n+1} = 4a_n - 4a_{n-1}.So, the recurrence relation is a_{n+1} = 4a_n - 4a_{n-1}. Let me write that as a_{n+1} - 4a_n + 4a_{n-1} = 0.Hmm, this is a linear recurrence relation. Maybe I can find its characteristic equation. The characteristic equation would be r^2 - 4r + 4 = 0. Solving this: r = [4 ± sqrt(16 - 16)] / 2 = [4 ± 0]/2 = 2. So, a repeated root at r=2.Therefore, the general solution is a_n = (C1 + C2*n)*2^n. Hmm, interesting. So the sequence is a linear function times 2^n.But wait, maybe I don't need the general solution for part (II). The question is about a_n - 2a_{n-1}. Let me compute that.From the recurrence relation: a_{n+1} = 4a_n - 4a_{n-1}. Let me rearrange this: a_{n+1} - 2a_n = 2a_n - 4a_{n-1} = 2(a_n - 2a_{n-1}).So, a_{n+1} - 2a_n = 2(a_n - 2a_{n-1}).This suggests that the sequence {a_n - 2a_{n-1}} is a geometric sequence with common ratio 2. Wait, but the problem says it's a constant sequence. Hmm, maybe I made a mistake.Wait, let's compute a_n - 2a_{n-1} for n=2 and n=3 to see.From part (I), a_1=2, a_2=4, a_3=8.Compute a_2 - 2a_1 = 4 - 2*2 = 4 - 4 = 0.Compute a_3 - 2a_2 = 8 - 2*4 = 8 - 8 = 0.So, both are 0. Maybe the sequence is always 0? Let me check for n=4.Using the recurrence relation: a_4 = 4a_3 - 4a_2 = 4*8 - 4*4 = 32 - 16 = 16.Then a_4 - 2a_3 = 16 - 2*8 = 16 - 16 = 0. So, it's 0 again.So, it seems that a_n - 2a_{n-1} is always 0. Therefore, the sequence is constant, with each term equal to 0.But wait, in my earlier step, I had a_{n+1} - 2a_n = 2(a_n - 2a_{n-1}). If a_n - 2a_{n-1} is 0, then a_{n+1} - 2a_n is also 0. So, if it's 0 for n=2, it's 0 for all n.Therefore, the sequence {a_n - 2a_{n-1}} is indeed a constant sequence, with each term equal to 0.Okay, that makes sense. So part (II) is proven.Now, part (III): Prove that (a_1 -1)/(a_2 -1) + (a_2 -1)/(a_3 -1) + ... + (a_n -1)/(a_{n+1} -1) < n/2.Hmm, okay. Let's first compute the terms for small n to see the pattern.From part (I), a_1=2, a_2=4, a_3=8. So:(a_1 -1)/(a_2 -1) = (2-1)/(4-1) = 1/3 ≈ 0.333(a_2 -1)/(a_3 -1) = (4-1)/(8-1) = 3/7 ≈ 0.428So, for n=2, the sum is approximately 0.333 + 0.428 ≈ 0.761, which is less than 2/2=1.Similarly, a_4=16, so (a_3 -1)/(a_4 -1) = (8-1)/(16-1)=7/15≈0.466. Adding that to the previous sum: ≈0.761 + 0.466≈1.227, which is less than 3/2=1.5.So, it seems each term is less than 1/2, and their sum is less than n/2.Wait, let me check: 1/3 ≈0.333 <0.5, 3/7≈0.428 <0.5, 7/15≈0.466 <0.5, and so on. So each term is less than 1/2, so the sum of n terms is less than n*(1/2)=n/2.But wait, let me verify this more formally.From part (II), we know that a_n - 2a_{n-1}=0, so a_n=2a_{n-1}. Therefore, the sequence {a_n} is a geometric sequence with ratio 2. Since a_1=2, then a_n=2^n.Wait, let's check: a_1=2=2^1, a_2=4=2^2, a_3=8=2^3, a_4=16=2^4, etc. Yes, that's correct.So, a_n=2^n. Therefore, a_n -1=2^n -1.So, each term in the sum is (a_k -1)/(a_{k+1} -1) = (2^k -1)/(2^{k+1} -1).We need to show that the sum from k=1 to n of (2^k -1)/(2^{k+1} -1) < n/2.Hmm, let's compute (2^k -1)/(2^{k+1} -1). Let's simplify this fraction.Note that 2^{k+1} -1 = 2*2^k -1 = 2*(2^k) -1.So, (2^k -1)/(2^{k+1} -1) = (2^k -1)/(2*2^k -1).Let me write this as (2^k -1)/(2*2^k -1) = [2^k -1]/[2*(2^k) -1].Let me factor out 2^k from the denominator: 2*(2^k) -1 = 2^{k+1} -1.Wait, maybe another approach. Let me write the fraction as:(2^k -1)/(2^{k+1} -1) = [2^k -1]/[2*2^k -1] = [2^k -1]/[2^{k+1} -1].Let me see if I can write this as 1/2 * [something].Let me try to write it as:(2^k -1)/(2^{k+1} -1) = (2^k -1)/(2*2^k -1) = [ (2^k -1) ] / [2*(2^k) -1].Let me factor numerator and denominator:Numerator: 2^k -1.Denominator: 2*(2^k) -1 = 2^{k+1} -1.Wait, maybe I can write the fraction as:(2^k -1)/(2^{k+1} -1) = [ (2^k -1) ] / [2*(2^k) -1] = [ (2^k -1) ] / [2*(2^k) -1].Let me factor 2^k from the denominator:= [ (2^k -1) ] / [2^k*(2) -1] = [ (2^k -1) ] / [2*2^k -1].Hmm, not sure if that helps. Maybe another approach: Let's consider the fraction:(2^k -1)/(2^{k+1} -1) = [2^k -1]/[2*2^k -1] = [2^k -1]/[2^{k+1} -1].Let me divide numerator and denominator by 2^k:= [1 - 1/2^k]/[2 - 1/2^k].So, (2^k -1)/(2^{k+1} -1) = [1 - 1/2^k]/[2 - 1/2^k].Let me denote x = 1/2^k, then the fraction becomes (1 - x)/(2 - x).Let me compute this:(1 - x)/(2 - x) = [1 - x]/[2 - x] = [ (2 - x) -1 ]/[2 - x] = 1 - [1/(2 - x)].Wait, that might not be helpful. Alternatively, let's compute:(1 - x)/(2 - x) = [1 - x]/[2 - x] = [ (2 - x) -1 ]/[2 - x] = 1 - [1/(2 - x)].Hmm, maybe not helpful. Alternatively, let's compute:(1 - x)/(2 - x) = [1 - x]/[2 - x] = [ (2 - x) -1 ]/[2 - x] = 1 - [1/(2 - x)].Wait, maybe another way: Let's write it as:(1 - x)/(2 - x) = [1 - x]/[2 - x] = [ (2 - x) -1 ]/[2 - x] = 1 - [1/(2 - x)].Hmm, perhaps not helpful. Maybe I can write it as:(1 - x)/(2 - x) = [1 - x]/[2 - x] = [ (2 - x) -1 ]/[2 - x] = 1 - [1/(2 - x)].Wait, maybe I can compare this to 1/2.Let me see: Is (1 - x)/(2 - x) < 1/2?Let's check:(1 - x)/(2 - x) < 1/2Multiply both sides by (2 - x), which is positive since x=1/2^k >0, so 2 - x >0.So, 1 - x < (1/2)(2 - x)Simplify RHS: (1/2)(2 - x) = 1 - x/2.So, inequality becomes:1 - x < 1 - x/2Subtract 1 from both sides:-x < -x/2Multiply both sides by -1 (inequality sign reverses):x > x/2Which is true for x >0, which it is since x=1/2^k >0.Therefore, (1 - x)/(2 - x) < 1/2.Thus, (2^k -1)/(2^{k+1} -1) < 1/2.Therefore, each term in the sum is less than 1/2, so the sum of n terms is less than n*(1/2)=n/2.Hence, the inequality holds.Wait, that seems to work. Let me recap:We showed that each term (a_k -1)/(a_{k+1} -1) = (2^k -1)/(2^{k+1} -1) < 1/2.Therefore, summing from k=1 to n, we get the sum < n*(1/2)=n/2.Hence, the inequality is proven.But let me double-check for n=1: (a_1 -1)/(a_2 -1)=1/3≈0.333 <1/2=0.5. Correct.For n=2: 1/3 + 3/7≈0.333 +0.428≈0.761 <1. Correct.For n=3: 1/3 +3/7 +7/15≈0.333 +0.428 +0.466≈1.227 <1.5. Correct.So, it seems to hold.Therefore, part (III) is proven.

💡Alright, so I'm trying to solve this problem from the ancient Chinese mathematical text "Nine Chapters on the Mathematical Art." The problem is about grain and rice production rates, and it involves setting up a system of equations to find out how much rice was originally in a bucket. Let me try to break this down step by step.First, the problem states that the rate of grain to rice is 50 to 30. That means for every 50 dou of grain, you get 30 dou of rice. So, the rice production rate is 30/50, which simplifies to 3/5. This is important because it tells me how much rice I can expect from a given amount of grain.Next, the problem mentions a 10-dou bucket that already contains some rice, but the quantity is unknown. Let's call the original amount of rice in the bucket "y" dou. Since the bucket's total capacity is 10 dou, the remaining space must be filled with grain. Let's denote the amount of grain added as "x" dou. So, the total content of the bucket before threshing is x (grain) + y (rice) = 10 dou.After filling the bucket with grain and threshing it, the total rice obtained is 7 dou. This means that the original rice "y" plus the rice produced from the grain "x" equals 7 dou. Since we know the production rate of grain to rice is 3/5, the amount of rice produced from "x" dou of grain would be (3/5)x.So, putting it all together, I have two equations:1. The total volume of the bucket: x + y = 102. The total rice after threshing: (3/5)x + y = 7Now, looking at the options provided, I need to see which one matches these equations.Option A:[left{begin{array}{l}x + y = 10 x + frac{3}{5}y = 7end{array}right.]Hmm, this looks similar but not exactly the same as what I derived. In my second equation, it's (3/5)x + y = 7, but in option A, it's x + (3/5)y = 7. That's different because the coefficient is applied to y instead of x.Option B:[left{begin{array}{l}x + y = 10 frac{3}{5}x + y = 7end{array}right.]This matches exactly what I derived. The first equation is the total volume, and the second equation accounts for the rice produced from grain and the original rice.Option C:[left{begin{array}{l}x + 7 = 7 x + frac{5}{3}y = 10end{array}right.]This doesn't make sense. The first equation simplifies to x = 0, which contradicts the problem statement because we know grain was added to the bucket.Option D:[left{begin{array}{l}x + y = 7 frac{5}{3}x + y = 10end{array}right.]This also doesn't align with my equations. The first equation here suggests that the total rice after threshing is 7 dou, but it incorrectly sets x + y = 7 instead of accounting for the production rate.So, after reviewing all the options, Option B is the correct system of equations that accurately represents the problem.

💡First, I need to determine the Simple Interest (SI) by subtracting the Principal (P) from the Amount (A).Next, I'll use the simple interest formula to solve for the Rate (R).I'll plug in the known values into the formula and perform the necessary calculations to find the rate.Finally, I'll present the calculated rate as the solution.