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olympiad-amth-Lllama3-test-200

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<|begin_of_text|><|start_header_id|>system<|end_header_id|>You are a expert mathematician, solve the problems accordingly using relevant mathematical techniques<|eot_id|><|begin_of_text|><|start_header_id|>user<|end_header_id|> Given a parabola with equation $y = ax^2 + bx + c$ that has its vertex at $(p, p)$ and $y$-intercept at $(0, -p)$, where $p \neq 0$, determine the value of $b$.<|eot_id|><|start_header_id|>assistant<|end_header_id|> The key insight in this problem is to recognize that a parabola with vertex $(p, p)$ can be written in the form $y = a(x - p)^2 + p$. By comparing this form with the given equation $y = ax^2 + bx + c$, we can determine the value of $a$ using the $y$-intercept and then find the value of $b$. 1. Recognize that a parabola with vertex $(p, p)$ has the equation $y = a(x - p)^2 + p$. 2. Use the given $y$-intercept $(0, -p)$ to determine the value of $a$: - Substitute $x = 0$ and $y = -p$ into the equation $y = a(x - p)^2 + p$. - $-p = a(0 - p)^2 + p$ - $-p = ap^2 + p$ - $-2p = ap^2$ - $a = -\frac{2}{p}$ 3. Expand the equation $y = a(x - p)^2 + p$ using the value of $a$ found in step 2: - $y = -\frac{2}{p}(x^2 - 2px + p^2) + p$ - $y = -\frac{2}{p}x^2 + 4x - p$ 4. Compare the expanded equation with the given equation $y = ax^2 + bx + c$ to find the value of $b$. Therefore, the value of $b$ in the given parabola equation is $\boxed{4}$.<|eot_id|>

<|begin_of_text|><|start_header_id|>system<|end_header_id|>You are a expert mathematician, solve the problems accordingly using relevant mathematical techniques<|eot_id|><|begin_of_text|><|start_header_id|>user<|end_header_id|> Evaluate the logarithm of 64 with base 2, denoted as $\log_2 64$.<|eot_id|><|start_header_id|>assistant<|end_header_id|> The key insight is to recognize that the logarithm $\log_b x$ is the exponent to which the base $b$ must be raised to get the value $x$. In other words, if $b^y = x$, then $\log_b x = y$. 1. Recall the definition of logarithm: if $b^y = x$, then $\log_b x = y$. 2. In this case, we need to find the exponent $y$ such that $2^y = 64$. 3. Observe that $2^6 = 64$, as $2^6 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 64$. 4. Therefore, $\log_2 64 = 6$. $\log_2 64 = \boxed{6}$<|eot_id|>

<|begin_of_text|><|start_header_id|>system<|end_header_id|>You are a expert mathematician, solve the problems accordingly using relevant mathematical techniques<|eot_id|><|begin_of_text|><|start_header_id|>user<|end_header_id|> Two circles are given in the coordinate plane. One circle is centered at $(-3,2)$ and the other circle is centered at $(0,-1)$. The circles are internally tangent, meaning they touch each other at exactly one point. The diagram is shown below:[asy]import graph; size(7cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-9.34,xmax=9.27,ymin=-9.36,ymax=7.89;Label laxis; laxis.p=fontsize(10);xaxis(xmin,xmax,Ticks(laxis,Step=2.0,Size=2,OmitTick(0)),Arrows(6),above=true); yaxis(ymin,ymax,Ticks(laxis,Step=2.0,Size=2,OmitTick(0)),Arrows(6),above=true); draw(circle((0,-1),7.07)); draw(circle((-3,2),2.83));dot((0,-1),ds); label("$(0, -1)$",(0.23,-1.87),SE*lsf); dot((-3,2),ds); label("$(-3, 2)$",(-2.82,2.29),N*lsf); dot((1,6),ds); label("$(1, 6)$",(1.2,6.3),NE*lsf); dot((-5,4),ds);clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);[/asy]The equation of the smaller circle can be written in the general form:$x^2 + y^2 + Dx + Ey + F = 0$The problem is to find the value of $D + E + F$.<|eot_id|><|start_header_id|>assistant<|end_header_id|> The key insight to solve this problem is to use the distance formula to find the radii of the circles. Since the circles are internally tangent, the distance between their centers is equal to the difference of their radii. Once we have the radius of the smaller circle, we can find its equation by expanding $(x+3)^2+(y-2)^2 = r^2$, where $r$ is the radius of the smaller circle. Comparing this equation with the general form $x^2 + y^2 + Dx + Ey + F = 0$, we can identify the values of $D$, $E$, and $F$. 1) Find the radius of the larger circle using the distance formula between its center $(0,-1)$ and the point $(1,6)$: $\sqrt{(6-(-1))^2 + (1-0)^2} = \sqrt{49 + 1} = 5\sqrt{2}$ 2) Find the distance between the centers of the two circles using the distance formula: $\sqrt{(-3-0)^2 + (2-(-1))^2} = \sqrt{3^2 + 3^2} = 3\sqrt{2}$ 3) The radius of the smaller circle is equal to the difference between the radius of the larger circle and the distance between their centers: $5\sqrt{2} - 3\sqrt{2} = 2\sqrt{2}$ 4) The square of the radius of the smaller circle is: $(2\sqrt{2})^2 = 8$ 5) The equation of the smaller circle is given by: $(x+3)^2+(y-2)^2 = 8$ 6) Expand the equation: $x^2 + 6x + 9 + y^2 - 4y + 4 = 8$ $x^2 + y^2 + 6x - 4y + 5 = 0$ 7) Comparing with the general form $x^2 + y^2 + Dx + Ey + F = 0$, we have: $D = 6$, $E = -4$, $F = 5$ 8) Therefore, $D + E + F = 6 - 4 + 5 = 7$ $D + E + F = \boxed{7}$<|eot_id|>

<|begin_of_text|><|start_header_id|>system<|end_header_id|>You are a expert mathematician, solve the problems accordingly using relevant mathematical techniques<|eot_id|><|begin_of_text|><|start_header_id|>user<|end_header_id|> Today a father's age is five times his son's age. Exactly three years ago, the sum of their ages was 30. How old is the son today?<|eot_id|><|start_header_id|>assistant<|end_header_id|> The key insight in this problem is to use the relationship between the father's and son's ages today ($y = 5x$) to simplify the equation that represents the sum of their ages three years ago. By substituting $y$ with $5x$ in the second equation, we can solve for $x$ and find the son's age today. 1. Substitute the first equation ($y = 5x$) into the second equation ($(x - 3) + (y - 3) = 30$): $(x - 3) + (5x - 3) = 30$ 2. Simplify the equation: $x - 3 + 5x - 3 = 30$ $6x - 6 = 30$ 3. Add 6 to both sides of the equation: $6x = 36$ 4. Divide both sides by 6 to solve for $x$: $x = 6$ The son is $\boxed{6}$ years old today.<|eot_id|>

<|begin_of_text|><|start_header_id|>system<|end_header_id|>You are a expert mathematician, solve the problems accordingly using relevant mathematical techniques<|eot_id|><|begin_of_text|><|start_header_id|>user<|end_header_id|> Given a line with the equation $\frac{y}{3}+\frac{2x}{5}=2$, find the slope of line $j$, which is perpendicular to the given line.<|eot_id|><|start_header_id|>assistant<|end_header_id|> The key insight to solve this problem is to recognize that the slopes of two perpendicular lines are negative reciprocals of each other. This means that if we can find the slope of the given line, we can easily determine the slope of line $j$ by taking the negative reciprocal of the given line's slope. 1. Convert the given equation $\frac{y}{3}+\frac{2x}{5}=2$ to slope-intercept form ($y=mx+b$) to find the slope of the given line. - Multiply both sides of the equation by 3: $y+\frac{6x}{5}=6$ - Subtract $\frac{6x}{5}$ from both sides: $y=-\frac{6x}{5}+6$ - The equation is now in slope-intercept form, where the coefficient of $x$ is the slope. The slope of the given line is $-\frac{6}{5}$. 2. Find the slope of line $j$ by taking the negative reciprocal of the given line's slope. - The negative reciprocal of $-\frac{6}{5}$ is $\frac{5}{6}$. The slope of line $j$ is $\frac{5}{6}$.<|eot_id|>

<|begin_of_text|><|start_header_id|>system<|end_header_id|>You are a expert mathematician, solve the problems accordingly using relevant mathematical techniques<|eot_id|><|begin_of_text|><|start_header_id|>user<|end_header_id|> Let $a$ be the smallest integer satisfying the inequality $x^2 - 15 < 2x$, and let $b$ be the largest integer satisfying the same inequality. Find the value of $b-a$.<|eot_id|><|start_header_id|>assistant<|end_header_id|> The key insight in this solution is to manipulate the inequality to a form where we can factor it. By subtracting $2x$ from both sides, we get a quadratic inequality that can be factored. The factors help us determine the range of values for $x$ that satisfy the inequality. Once we have this range, we can easily find the smallest and largest integers within this range, which are the values of $a$ and $b$ respectively. 1. Subtract $2x$ from both sides of the inequality: $x^2 - 15 < 2x$ $x^2 - 2x - 15 < 0$ 2. Factor the quadratic expression: $x^2 - 2x - 15 = (x-5)(x+3) < 0$ 3. For the product of two factors to be negative, one factor must be positive and the other must be negative. This gives us the range of values for $x$: $-3 < x < 5$ 4. Find the smallest integer $a$ and the largest integer $b$ within this range: $a = -2$ $b = 4$ 5. Calculate $b-a$: $b-a = 4 - (-2) = 6$ $b-a = \boxed{6}$<|eot_id|>

<|begin_of_text|><|start_header_id|>system<|end_header_id|>You are a expert mathematician, solve the problems accordingly using relevant mathematical techniques<|eot_id|><|begin_of_text|><|start_header_id|>user<|end_header_id|> An airplane climbs 100 feet during the first second after takeoff. In each succeeding second, it climbs 100 feet more than it climbed during the previous second. The problem is to find the number of seconds it takes for the plane to reach an altitude of 12,000 feet above its takeoff height.<|eot_id|><|start_header_id|>assistant<|end_header_id|> The key insight in this problem is to recognize that the altitude gained by the airplane in each second follows an arithmetic sequence. The first term of the sequence is 100 feet (the altitude gained in the first second), and the common difference is also 100 feet (the additional altitude gained in each succeeding second compared to the previous second). By finding the sum of this arithmetic sequence, we can determine the total altitude gained after a given number of seconds. 1. Let $t$ be the number of seconds after takeoff. 2. The altitude gained in the first second is 100 feet. 3. The altitude gained in each succeeding second increases by 100 feet compared to the previous second. 4. The altitude gained in $t$ seconds can be expressed as an arithmetic sequence: $100, 200, 300, \dots, 100t$. 5. The sum of this arithmetic sequence is given by: $100(1 + 2 + \dots + t) = 100 \cdot \frac{t(t + 1)}{2} = 50t(t + 1)$. 6. To find the number of seconds required to reach an altitude of 12,000 feet, we solve the inequality: $50t(t + 1) \ge 12000$. 7. Dividing both sides by 50, we get: $t(t + 1) \ge 240$. 8. The smallest value of $t$ that satisfies this inequality is 15, since $15 \cdot 16 = 240$. It takes $\boxed{15}$ seconds for the airplane to reach an altitude of 12,000 feet above its takeoff height.<|eot_id|>

<|begin_of_text|><|start_header_id|>system<|end_header_id|>You are a expert mathematician, solve the problems accordingly using relevant mathematical techniques<|eot_id|><|begin_of_text|><|start_header_id|>user<|end_header_id|> Given the expression $24-(2x-y)$, evaluate it when $x=4$ and $y=3$.<|eot_id|><|start_header_id|>assistant<|end_header_id|> The key insight is to substitute the given values for $x$ and $y$ into the expression and then simplify it step by step using the order of operations (PEMDAS - Parentheses, Exponents, Multiplication and Division from left to right, Addition and Subtraction from left to right). 1. Substitute the given values for $x$ and $y$ into the expression: $24-(2x-y)$ becomes $24-(2\cdot 4-3)$ 2. Simplify the expression inside the parentheses: $2\cdot 4-3 = 8-3 = 5$ So, the expression becomes $24-(5)$ 3. Perform the subtraction: $24-5 = 19$ The value of the expression $24-(2x-y)$ when $x=4$ and $y=3$ is $\boxed{19}$.<|eot_id|>

<|begin_of_text|><|start_header_id|>system<|end_header_id|>You are a expert mathematician, solve the problems accordingly using relevant mathematical techniques<|eot_id|><|begin_of_text|><|start_header_id|>user<|end_header_id|> Find the value of $x$ that satisfies the equation $4^x = 8$. Express your answer as a common fraction.<|eot_id|><|start_header_id|>assistant<|end_header_id|> The key insight in this problem is to rewrite both 4 and 8 as powers of 2. This allows us to simplify the equation and solve for $x$ by setting the exponents equal to each other. 1. Rewrite 4 as $2^2$ and 8 as $2^3$: $$(2^2)^x = 2^3$$ 2. Simplify the left-hand side using the power rule $(a^b)^c = a^{bc}$: $$2^{2x} = 2^3$$ 3. Set the exponents equal to each other: $$2x = 3$$ 4. Solve for $x$ by dividing both sides by 2: $$x = \frac{3}{2}$$ $x = \frac{3}{2}$<|eot_id|>

<|begin_of_text|><|start_header_id|>system<|end_header_id|>You are a expert mathematician, solve the problems accordingly using relevant mathematical techniques<|eot_id|><|begin_of_text|><|start_header_id|>user<|end_header_id|> Given the equation of a parabola $y=2x^2-4x+4$, the vertex of the parabola is at the point $(m,n)$. The task is to find the value of $m$.<|eot_id|><|start_header_id|>assistant<|end_header_id|> The key insight to solve this problem is to recognize that the equation of the parabola is not in vertex form. By completing the square, we can rewrite the equation in vertex form $y=a(x-h)^2+k$, where $(h,k)$ is the vertex of the parabola. Once the equation is in vertex form, we can easily identify the $x$-coordinate of the vertex, which is the value of $m$. 1. Factor out the coefficient of $x^2$ from the first two terms of the equation: $y = 2(x^2-2x) + 4$ 2. Complete the square inside the parentheses by adding and subtracting the square of half the coefficient of $x$: $y = 2(x^2-2x+1-1) + 4$ Note: We add and subtract 1 to maintain the equality. 3. Factor the perfect square trinomial inside the parentheses: $y = 2(x-1)^2 - 2 + 4$ 4. Simplify the equation: $y = 2(x-1)^2 + 2$ 5. Identify the vertex of the parabola from the vertex form of the equation: The equation is now in the form $y=a(x-h)^2+k$, where $(h,k)$ is the vertex. Therefore, the vertex is $(1,2)$, and $m=1$. The $x$-coordinate of the vertex, $m$, is equal to $\boxed{1}$.<|eot_id|>

<|begin_of_text|><|start_header_id|>system<|end_header_id|>You are a expert mathematician, solve the problems accordingly using relevant mathematical techniques<|eot_id|><|begin_of_text|><|start_header_id|>user<|end_header_id|> Given two squares, Square A and Square B, both with dimensions $2009$ by $2009$, we want to find the minimum value of $x$ such that:1. Increasing both the length and width of Square A by $x$,2. Decreasing both the length and width of Square B by $x$,the difference in area between the two new squares is at least as great as the area of the original $2009$ by $2009$ square.<|eot_id|><|start_header_id|>assistant<|end_header_id|> The key insight in this problem is to express the areas of the modified squares in terms of the original dimensions and the change $x$. By setting up an inequality comparing the difference in areas to the original area, we can solve for the minimum value of $x$ that satisfies the condition. 1. Express the new area of Square A as $(2009+x)^2$. 2. Express the new area of Square B as $(2009-x)^2$. 3. Calculate the difference in area between the new squares: \begin{align*} &(2009+x)^2-(2009-x)^2\\ &\qquad=(2009+x+2009-x)(2009+x-2009+x) \\ &\qquad=(2\cdot 2009)(2x) \end{align*} 4. Set up an inequality to ensure the difference in area is at least as great as the original area: $$2(2009)2(x)\geq 2009^2$$ 5. Solve the inequality for $x$: $$x\geq \frac{2009}{4}$$ The minimum value of $x$ such that the difference in area between the two new squares is at least as great as the area of a $2009$ by $2009$ square is $\boxed{\frac{2009}{4}}$.<|eot_id|>

<|begin_of_text|><|start_header_id|>system<|end_header_id|>You are a expert mathematician, solve the problems accordingly using relevant mathematical techniques<|eot_id|><|begin_of_text|><|start_header_id|>user<|end_header_id|> Given a line segment with one endpoint at $(-9, 1)$ and midpoint at $(3, -5)$, find the coordinates of the other endpoint, denoted as $(x, y)$.<|eot_id|><|start_header_id|>assistant<|end_header_id|> The midpoint formula is used to solve this problem. The midpoint formula states that for a line segment with endpoints $(x_1, y_1)$ and $(x_2, y_2)$, the midpoint is located at $\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$. By applying this formula and equating the resulting expression to the given midpoint coordinates, we can solve for the unknown endpoint $(x, y)$. 1. Let the unknown endpoint be $(x, y)$ and the known endpoint be $(-9, 1)$. 2. Apply the midpoint formula: $\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) = (3, -5)$ - Substitute the known values: $\left(\frac{-9 + x}{2}, \frac{1 + y}{2}\right) = (3, -5)$ 3. Equate the x-coordinates and solve for $x$: - $\frac{-9 + x}{2} = 3$ - $-9 + x = 6$ - $x = 15$ 4. Equate the y-coordinates and solve for $y$: - $\frac{1 + y}{2} = -5$ - $1 + y = -10$ - $y = -11$ The coordinates of the unknown endpoint $(x, y)$ are $\boxed{(15, -11)}$.<|eot_id|>

<|begin_of_text|><|start_header_id|>system<|end_header_id|>You are a expert mathematician, solve the problems accordingly using relevant mathematical techniques<|eot_id|><|begin_of_text|><|start_header_id|>user<|end_header_id|> Let $a$ and $b$ be the solutions of the equation $2x^2-10x+5=0$. The problem is to find the value of the expression $(2a-3)(4b-6)$.<|eot_id|><|start_header_id|>assistant<|end_header_id|> The key insight here is to recognize that the expression $(2a-3)(4b-6)$ can be expanded and simplified using the sum and product of the roots of the given quadratic equation. By Vieta's formulas, we know that for a quadratic equation $ax^2 + bx + c = 0$ with roots $p$ and $q$, the sum of the roots is given by $-\frac{b}{a}$ and the product of the roots is given by $\frac{c}{a}$. 1. Expand the expression $(2a-3)(4b-6)$: $(2a-3)(4b-6) = 8ab - 12a - 12b + 18$ 2. Recognize that $a$ and $b$ are the roots of the equation $2x^2-10x+5=0$. 3. Use Vieta's formulas to find the sum and product of the roots: - Sum of roots: $-\frac{b}{a} = -\frac{-10}{2} = 5$ - Product of roots: $\frac{c}{a} = \frac{5}{2}$ 4. Substitute the sum and product of the roots into the expanded expression: $8ab - 12a - 12b + 18 = 8 \cdot \frac{5}{2} - 12 \cdot 5 + 18$ 5. Simplify the expression: $8 \cdot \frac{5}{2} - 12 \cdot 5 + 18 = 20 - 60 + 18 = -22$ Therefore, the value of $(2a-3)(4b-6)$ is $\boxed{-22}$.<|eot_id|>