💡Okay, so I need to figure out the fifth smallest three-digit number that can be formed using single-digit multiples of 3. Hmm, let's break this down step by step. First, what are the single-digit multiples of 3? Well, multiples of 3 are numbers that can be divided by 3 without leaving a remainder. So, starting from 3, the single-digit multiples would be 3, 6, and 9. Got it.Now, I need to form three-digit numbers using only these digits: 3, 6, and 9. Since we're dealing with three-digit numbers, the first digit (hundreds place) can't be zero, but in this case, we're only using 3, 6, and 9, so that's already taken care of. The goal is to find the fifth smallest such number. To do this, I think I should list out the numbers in ascending order and pick the fifth one. Let's start by listing the smallest possible numbers.The smallest three-digit number we can form with these digits is 333. That makes sense because 3 is the smallest digit we have. Next, to get the next smallest number, I should increment the last digit. So, changing the last digit from 3 to 6 gives us 336. Then, changing the last digit to 9 gives us 339.Now, to get the next set of numbers, I need to increment the middle digit. So, keeping the first digit as 3, changing the middle digit from 3 to 6, and resetting the last digit to 3 gives us 363. Following that, changing the last digit to 6 gives us 366. Wait, so far, I have:1. 3332. 3363. 3394. 3635. 366So, the fifth smallest number is 366. That seems straightforward, but let me double-check to make sure I haven't missed any numbers or made any mistakes in the order.Is there a number between 339 and 363 that I might have skipped? Let's see. After 339, the next logical step is to increment the middle digit from 3 to 6, which gives us 363. There's no number like 343 or 353 because we can only use 3, 6, and 9. So, 363 is indeed the next number after 339.Similarly, after 363, the next number would be 366 by changing the last digit from 3 to 6. So, it seems like my list is correct.Just to be thorough, let's list out all possible three-digit numbers using 3, 6, and 9, and then sort them to confirm the fifth one is 366.Possible numbers:- 333- 336- 339- 363- 366- 369- 393- 396- 399- 633- 636- 639- 663- 666- 669- 693- 696- 699- 933- 936- 939- 963- 966- 969- 993- 996- 999Now, sorting these numbers:1. 3332. 3363. 3394. 3635. 3666. 3697. 3938. 3969. 39910. 63311. 63612. 63913. 66314. 66615. 66916. 69317. 69618. 69919. 93320. 93621. 93922. 96323. 96624. 96925. 99326. 99627. 999Yes, the fifth number in this sorted list is indeed 366. So, my initial conclusion was correct. The fifth smallest three-digit number that can be formed using single-digit multiples of 3 is 366.

💡First, I need to determine the range of possible values for ( m ) that satisfy the Triangle Inequality Theorem.1. **Applying the Triangle Inequality:** - The sum of any two sides must be greater than the third side. - ( 7 + 11 > m ) implies ( m < 18 ). - ( 7 + m > 11 ) implies ( m > 4 ). - ( 11 + m > 7 ) implies ( m > -4 ) (which is always true since ( m > 4 )).2. **Determining the Range of ( m ):** - Combining the inequalities, ( m ) must be greater than 4 and less than 18. - Therefore, the possible whole number values for ( m ) are from 5 to 17, inclusive.3. **Calculating the Sum of Possible Values:** - The sum of integers from 5 to 17 can be calculated using the formula for the sum of an arithmetic series. - The sum is ( frac{(5 + 17) times 13}{2} = 132 ).Thus, the sum of all possible whole number values of ( m ) is 132.

💡Alright, so I'm trying to figure out how many ladybugs are gathered in the meadow. The problem says that ladybugs with six spots always tell the truth, and those with four spots always lie. There are no other types of ladybugs. Let me break down the information given:1. The first ladybug says, "Each of us has the same number of spots on our back."2. The second ladybug says, "Altogether, there are 30 spots on our backs."3. The third ladybug contradicts the second one, saying, "No, altogether there are 26 spots on our backs."4. Each of the remaining ladybugs (if any) states, "Among these three, exactly one is telling the truth."Okay, so first, I need to figure out who is telling the truth and who is lying. Since only ladybugs with six spots tell the truth, and those with four spots lie, the statements can help me determine their types.Let's analyze the first statement: "Each of us has the same number of spots on our back." If this is true, then all ladybugs have the same number of spots. But if it's false, then not all ladybugs have the same number of spots.Now, the second and third ladybugs are giving different total numbers of spots: 30 and 26. Since they can't both be telling the truth, and only one of them can be truthful, this suggests that only one of these two is telling the truth, and the other is lying.Additionally, the remaining ladybugs (if any) say that among the first three, exactly one is telling the truth. So, this adds another layer to the problem.Let me consider the possibilities:1. If the first ladybug is telling the truth, then all ladybugs have the same number of spots. But if that's the case, then the second and third ladybugs would either both be telling the truth or both lying. However, they gave different total numbers, so they can't both be truthful or both be lying. Therefore, the first ladybug must be lying.2. Since the first ladybug is lying, not all ladybugs have the same number of spots. So, there must be a mix of six-spotted and four-spotted ladybugs.3. Now, among the second and third ladybugs, only one can be telling the truth. Let's consider both possibilities: a. Suppose the second ladybug is telling the truth: total spots are 30. If that's the case, then the third ladybug is lying, meaning the total is not 26. But we also know that among the first three, exactly one is telling the truth. Since the first ladybug is lying, and the second is truthful, the third must be lying. This fits. b. Suppose the third ladybug is telling the truth: total spots are 26. Then the second ladybug is lying, meaning the total is not 30. Again, since the first ladybug is lying, and the third is truthful, this also fits.But we need to determine which one is the case. Let's explore both scenarios.First, let's assume the second ladybug is truthful (total spots = 30). Let's denote the number of six-spotted ladybugs as S and four-spotted ladybugs as F. So, the total number of ladybugs is S + F.Each six-spotted ladybug contributes 6 spots, and each four-spotted ladybug contributes 4 spots. So, the total number of spots is 6S + 4F = 30.But we also know that among the first three ladybugs, exactly one is telling the truth. Since the first ladybug is lying, and the second is truthful, the third must be lying. Therefore, the third ladybug is lying, meaning the total spots are not 26.So, we have:6S + 4F = 30Also, the number of ladybugs is S + F. But we need more information to find S and F. Let's see if we can find another equation.Wait, we also know that the remaining ladybugs (if any) say that among the first three, exactly one is telling the truth. So, if there are more than three ladybugs, each of them is making a statement about the first three. But since we're assuming the second ladybug is truthful, and the first and third are lying, the remaining ladybugs are also lying because they are saying that exactly one of the first three is truthful, which is actually true. Wait, no, if the remaining ladybugs are lying, then their statement "Among these three, exactly one is telling the truth" is false. But in reality, exactly one is telling the truth, so their statement is true. But if they are lying, their statement should be false. This is a contradiction.Therefore, our assumption that the second ladybug is truthful leads to a contradiction because the remaining ladybugs would be lying about the number of truthful statements, but their statement would actually be true. Hence, the second ladybug cannot be truthful.Now, let's assume the third ladybug is truthful (total spots = 26). Then, the second ladybug is lying, meaning the total is not 30. Again, among the first three, exactly one is telling the truth (the third one), and the first and second are lying.So, we have:6S + 4F = 26Again, we need to find S and F. Let's see if we can find integer solutions for S and F.Let's rearrange the equation:6S + 4F = 26Divide both sides by 2:3S + 2F = 13Now, we need to find non-negative integers S and F such that 3S + 2F = 13.Let's try different values of S:- If S = 0: 2F = 13 → F = 6.5 (not an integer)- If S = 1: 3 + 2F = 13 → 2F = 10 → F = 5- If S = 2: 6 + 2F = 13 → 2F = 7 → F = 3.5 (not an integer)- If S = 3: 9 + 2F = 13 → 2F = 4 → F = 2- If S = 4: 12 + 2F = 13 → 2F = 1 → F = 0.5 (not an integer)- If S = 5: 15 + 2F = 13 → 2F = -2 (not possible)So, the possible solutions are:- S = 1, F = 5- S = 3, F = 2Now, let's check these solutions.First, S = 1, F = 5:Total ladybugs = 1 + 5 = 6But we have three ladybugs making statements, and the remaining 3 ladybugs are also making statements. Each of these remaining ladybugs says, "Among these three, exactly one is telling the truth." Since we've established that exactly one of the first three is telling the truth (the third ladybug), the remaining ladybugs are telling the truth by saying that. However, if the remaining ladybugs are truthful, they must have six spots. But in this case, S = 1, meaning only one ladybug has six spots, which contradicts the fact that the remaining three ladybugs are also truthful (and thus should have six spots). Therefore, this solution is invalid.Next, S = 3, F = 2:Total ladybugs = 3 + 2 = 5Here, the first three ladybugs are part of the total. So, among the five ladybugs, three have six spots, and two have four spots.Now, the first ladybug is lying (as established), so it has four spots. The second ladybug is lying, so it has four spots. The third ladybug is truthful, so it has six spots. The remaining two ladybugs (since total is five) are also making statements. Each of these remaining two says, "Among these three, exactly one is telling the truth." Since exactly one of the first three is truthful, their statement is true. Therefore, these two remaining ladybugs are truthful and must have six spots.But wait, we already have three six-spotted ladybugs (the third one and the two remaining), but according to our solution, S = 3. So, that fits: three six-spotted ladybugs and two four-spotted ladybugs.Let's verify the total number of spots:3 six-spotted ladybugs: 3 * 6 = 182 four-spotted ladybugs: 2 * 4 = 8Total spots: 18 + 8 = 26Which matches the third ladybug's statement. So, this solution is consistent.Therefore, the total number of ladybugs is 5.

💡Okay, so I'm trying to figure out how much personal income tax, or NDFL, I have to pay on the income I got from my securities in 2021. Let me see, I have a brokerage account, and I received some dividends and coupon incomes, and also made a profit from selling some shares. I need to calculate the total tax I owe.First, let's list out all the income I received:1. Dividends from PAO "Winning" - 50,000 rubles.2. Coupon income from government bonds OFZ - 40,000 rubles.3. Coupon income from corporate bonds of PAO "Reliable" - 30,000 rubles.4. Capital gain from selling 100 shares of PAO "Risky" at 200 rubles per share. I bought them at 150 rubles per share, and I held them for 4 months.Alright, so I need to calculate the tax on each of these incomes separately and then sum them up.Starting with the dividends: I remember that in Russia, dividends are taxed at 13%. So, for the 50,000 rubles from PAO "Winning", the tax would be 50,000 * 0.13.Next, the coupon income from government bonds OFZ. I think coupon income from government bonds is tax-exempt in Russia. So, I don't have to pay any tax on the 40,000 rubles from OFZ bonds.Then, the coupon income from corporate bonds of PAO "Reliable". I believe coupon income from corporate bonds is also taxed at 13%. So, for the 30,000 rubles, the tax would be 30,000 * 0.13.Lastly, the capital gain from selling the shares of PAO "Risky". I need to calculate the profit first. I sold 100 shares at 200 rubles each, so that's 100 * 200 = 20,000 rubles. I bought them at 150 rubles each, so that's 100 * 150 = 15,000 rubles. The capital gain is 20,000 - 15,000 = 5,000 rubles. Capital gains are also taxed at 13%, so the tax would be 5,000 * 0.13.Now, I need to add up all these taxes to get the total NDFL.Let me do the calculations step by step:1. Dividends tax: 50,000 * 0.13 = 6,500 rubles.2. Coupon income from OFZ: 40,000 rubles - tax-exempt, so 0 rubles.3. Coupon income from corporate bonds: 30,000 * 0.13 = 3,900 rubles.4. Capital gains tax: 5,000 * 0.13 = 650 rubles.Adding them up: 6,500 + 0 + 3,900 + 650 = 11,050 rubles.So, the total NDFL I owe for the income from my securities in 2021 is 11,050 rubles.

💡Okay, so I have this geometry problem here. Let me try to visualize it first. There's a triangle ABC with a circumscribed circle ω. That means the circle passes through all three vertices A, B, and C of the triangle. Now, tangents are drawn at points A and B on this circle, and these tangents intersect at point K. So, K is the point where the two tangents meet. Next, point M is the midpoint of side AC. That should be straightforward—just the halfway point between A and C. Then, there's a line passing through point K and parallel to AC, and this line intersects side BC at point L. The question is asking for the measure of angle AML. Hmm, okay.Let me try to draw this step by step. I imagine triangle ABC with the circumcircle ω. Points A and B have tangents, and those tangents meet at K. Since tangents from a single point to a circle are equal in length, KA equals KB. That might be useful later.Point M is the midpoint of AC, so AM equals MC. Then, from K, we draw a line parallel to AC, and this line meets BC at L. So, KL is parallel to AC. That should mean that triangle KBL is similar to triangle ABC because of the parallel lines, but I need to think about that.Wait, actually, since KL is parallel to AC, by the basic proportionality theorem (Thales' theorem), the line KL divides sides AB and BC proportionally. But in this case, KL is drawn from point K, which is outside the triangle. Hmm, maybe I need to use some properties of similar triangles or cyclic quadrilaterals here.Let me recall that the tangent from a point to a circle is perpendicular to the radius at the point of contact. So, OA is perpendicular to KA, and OB is perpendicular to KB, where O is the center of the circle ω. But I'm not sure if the center is needed here.Since KA and KB are tangents, triangle KAB is isosceles with KA = KB. So, angles at A and B in triangle KAB are equal. Let me denote angle KAB as α. Then angle KBA is also α. Therefore, angle AKB is 180° - 2α.Now, since KL is parallel to AC, the corresponding angles should be equal. So, angle BKL should be equal to angle BAC. Wait, is that correct? Let me think. If KL is parallel to AC, then angle BKL is equal to angle BAC because they are corresponding angles. Similarly, angle KBL is equal to angle ABC.Hmm, maybe I can relate these angles to find some similar triangles or cyclic quadrilaterals.Wait, another thought: since KL is parallel to AC, and M is the midpoint of AC, maybe L is the midpoint of BC? But that might not necessarily be true because KL is drawn from point K, which is outside the triangle. So, it's not a midline of the triangle unless K is at a specific position.Alternatively, maybe I can use coordinate geometry to solve this. Let me assign coordinates to the points and see if I can compute the necessary angles.Let's place point A at (0, 0) and point C at (2c, 0) so that M, the midpoint, is at (c, 0). Let me assign point B at (d, e). Then, the circumcircle ω can be determined, but that might get complicated. Alternatively, maybe I can use vectors or some other method.Wait, perhaps using vectors is a good idea. Let me denote vectors for points A, B, C, and so on. But I'm not sure if that's the most straightforward approach here.Let me go back to the properties of the circle and tangents. Since K is the intersection of the tangents at A and B, it's the ex-pole of line AB with respect to circle ω. That might be too advanced, though.Alternatively, maybe I can use power of a point. The power of point K with respect to ω is equal to KA² = KB². That's because K lies on the tangents from A and B. So, KA = KB, which we already established.Now, since KL is parallel to AC, the triangles KBL and ABC are similar. Because of the parallel lines, the corresponding angles are equal. So, triangle KBL ~ triangle ABC by AA similarity.Therefore, the ratio of sides is preserved. So, KB/AB = KL/AC = BL/BC. But KB = KA, and we might need to find some relationship between these lengths.Wait, but I'm not sure how this helps me find angle AML. Maybe I need to look at triangle AML specifically.Point M is the midpoint of AC, and we need to find angle AML. So, points A, M, and L. Maybe I can find some properties about triangle AML or quadrilateral AMCL.Wait, since KL is parallel to AC, and M is the midpoint of AC, is there a way to relate point L to M? Maybe through midsegments or something.Alternatively, maybe I can use the fact that AM is a median and KL is parallel to AC to find some proportional segments.Wait, another idea: since KL is parallel to AC, then the distance from K to AC is the same as the distance from L to AC. But I'm not sure if that helps.Wait, let me think about the cyclic quadrilateral. Since K is the intersection of tangents at A and B, points A, B, K, and the center O lie on a circle? No, wait, that's not necessarily true. But perhaps quadrilateral AKBC is cyclic? No, because K is outside the circle.Wait, but earlier I thought that triangle KBL is similar to triangle ABC. So, maybe I can find some ratio or something.Alternatively, maybe I can use harmonic division or projective geometry concepts, but that might be overcomplicating.Wait, let me try to use coordinates. Let me place point A at (0, 0), point C at (2, 0), so M is at (1, 0). Let me assign point B at (0, b) for some b. Then, the circumcircle ω can be found.First, find the circumcircle of triangle ABC with points A(0,0), B(0,b), and C(2,0). The circumcircle can be found by finding the perpendicular bisectors of AB and AC.The perpendicular bisector of AB: AB is vertical from (0,0) to (0,b). The midpoint is (0, b/2), and the perpendicular bisector is the horizontal line y = b/2.The perpendicular bisector of AC: AC is from (0,0) to (2,0). The midpoint is (1,0), and the perpendicular bisector is the vertical line x = 1.So, the intersection of x=1 and y=b/2 is the center of the circle, which is (1, b/2). The radius is the distance from center to A: sqrt((1-0)^2 + (b/2 - 0)^2) = sqrt(1 + b²/4).Now, the tangent at A(0,0) to circle ω. The tangent at a point on a circle is perpendicular to the radius at that point. The radius OA is from (1, b/2) to (0,0), which has slope (b/2 - 0)/(1 - 0) = b/2. Therefore, the tangent at A is perpendicular to OA, so its slope is -2/b.Similarly, the tangent at B(0,b) has radius OB from (1, b/2) to (0,b). The slope of OB is (b - b/2)/(0 - 1) = (b/2)/(-1) = -b/2. Therefore, the tangent at B is perpendicular to OB, so its slope is 2/b.Now, the tangent at A has equation y = (-2/b)x + c. Since it passes through A(0,0), c=0. So, equation is y = (-2/b)x.Similarly, tangent at B has equation y = (2/b)x + c. It passes through B(0,b), so plugging in: b = (2/b)(0) + c => c = b. So, equation is y = (2/b)x + b.Now, find point K where these two tangents intersect. Set (-2/b)x = (2/b)x + b.So, (-2/b)x - (2/b)x = b => (-4/b)x = b => x = -b²/4.Then, y = (-2/b)(-b²/4) = (2/b)(b²/4) = b/2.So, point K is at (-b²/4, b/2).Now, we need to find point L, which is the intersection of line KL (parallel to AC) with BC.First, AC is from (0,0) to (2,0), so it's horizontal. Therefore, KL is also horizontal because it's parallel to AC. So, KL has the same y-coordinate as K, which is b/2.So, the line KL is y = b/2.Now, find point L where y = b/2 intersects BC.Point B is at (0,b) and point C is at (2,0). The equation of BC can be found.Slope of BC is (0 - b)/(2 - 0) = -b/2.Equation of BC: y - b = (-b/2)(x - 0) => y = (-b/2)x + b.Set y = b/2:b/2 = (-b/2)x + b => (-b/2)x = -b/2 => x = 1.So, point L is at (1, b/2).Wait, that's interesting. So, point L is at (1, b/2). Now, point M is the midpoint of AC, which is at (1,0). So, points M(1,0) and L(1, b/2) lie on the vertical line x=1.Therefore, segment ML is vertical from (1,0) to (1, b/2). So, ML is vertical.Now, point A is at (0,0), point M is at (1,0), and point L is at (1, b/2). So, triangle AML has points A(0,0), M(1,0), and L(1, b/2).So, angle AML is the angle at point M between points A, M, and L.Since MA is from M(1,0) to A(0,0), which is along the x-axis to the left. ML is from M(1,0) to L(1, b/2), which is straight up along the y-axis.Therefore, angle AML is the angle between the negative x-axis and the positive y-axis at point M. That is, it's a right angle, 90 degrees.Wait, that seems too straightforward. Let me double-check.Point A is at (0,0), M is at (1,0), L is at (1, b/2). So, vector MA is (-1, 0) and vector ML is (0, b/2). The angle between (-1,0) and (0, b/2) is indeed 90 degrees because their dot product is (-1)(0) + (0)(b/2) = 0. So, the vectors are perpendicular.Therefore, angle AML is 90 degrees.Hmm, so in this coordinate setup, it's 90 degrees. But does this hold for any triangle ABC? Because I chose specific coordinates where A is at (0,0), C at (2,0), and B at (0,b). But the problem didn't specify any particular triangle, so it should hold generally.Wait, but I think the result is general because regardless of the coordinates, the construction leads to ML being perpendicular to MA. So, angle AML is 90 degrees.Therefore, the answer is 90 degrees.

💡Okay, so I have this problem here: If t equals some fraction multiplied by (k minus 32), and if t is 75, then k is 167. I need to find out what that fraction is. Hmm, let's break this down step by step.First, let's write down what we know. The equation is t equals a fraction times (k minus 32). So, in mathematical terms, that would be:t = f * (k - 32)Where f is the fraction we're trying to find. We're given that t is 75 and k is 167. So, plugging those values into the equation:75 = f * (167 - 32)Okay, let's do the subtraction inside the parentheses first. 167 minus 32 is... let's see, 167 minus 30 would be 137, and then minus 2 more is 135. So now the equation simplifies to:75 = f * 135Now, I need to solve for f. To isolate f, I should divide both sides of the equation by 135. So:f = 75 / 135Alright, now I have the fraction 75 over 135. I think I can simplify this fraction. Let's see if both numbers can be divided by the same number. 75 and 135, do they have a common divisor? Well, 75 divided by 5 is 15, and 135 divided by 5 is 27. So, simplifying:f = 15 / 27Wait, can I simplify this further? 15 and 27 are both divisible by 3. So, dividing numerator and denominator by 3:f = 5 / 9Okay, so the fraction is 5 over 9. Let me double-check to make sure I didn't make any mistakes. Starting from the beginning:t = f * (k - 32)Given t = 75 and k = 167, so:75 = f * (167 - 32)167 minus 32 is 135, so:75 = f * 135Dividing both sides by 135:f = 75 / 135Simplify by dividing numerator and denominator by 15:75 ÷ 15 = 5135 ÷ 15 = 9So, f = 5/9Yep, that seems correct. I think I've got it.