Determine the largest value of the quadratic expression \(-5x^2 + 40x - 12\) when \(x\) is a real number.

The largest value of the quadratic expression \(f(x) = 5x^2 + cx - 12 \) when \(x\) is a negative integer is 36. What is the value of \(c\)?

A parabola opening upwards doesn't have a maximum value for negative integer \(n\).

Answer: 68


Consider a real function \(g(y) = \frac{1}{1+\frac{1}{1+\frac{1}{y}}}.\) Find the sum of the three real values of \(y\) that make this function undefined.

Consider the real function
\[
g(y)=\frac{1}{1+\frac{1}{c+\frac{1}{y}}}.
\]
Suppose exactly two real values of \(y\) are excluded from its domain and their sum is \(\frac12\). Determine \(c\).

The expression can be undefined at \(y=0\), at \(y=-1/c\) when \(c\ne0\), and at \(y=-1/(c+1)\) when \(c\ne-1\). For generic \(c\), these are three distinct excluded values. Exactly two occur only when \(c=0\) or \(c=-1\): their sums are respectively \(-1\) and \(1\), never \(\frac12\). Hence no real \(c\) satisfies the stated conditions.

Answer: -\frac 32


Suppose \(m\) is a real number such that \(\frac{1}{2} - \frac{1}{3} = \frac{3}{m}\). Find \(m\).

Suppose \(m\), \(n\), and \(k\) are odd integers such that \(\frac{1}{2} - \frac{1}{k} = \frac{m}{n}\). Find a possible value for \(k\).

\[
\frac{1}{2} - \frac{m}{n} = \frac{n - 2m}{2n}, \quad \text{so } k = \frac{2n}{n - 2m}.
\]
Since \( m \) and \( n \) are odd, \( n - 2m \) must be odd, and since \( 2n \) is even, \( k \) must be even.
This contradicts the assumption that \( k \) is odd.
Therefore, such \( m \), \( n \), and \( k \) do not exist.

Answer: 18


Find the real number \(a\) such that \((2008+a)^2=a^2\).

If the equation \((q+a)^{2147483647}=a^{2147483647}\) in terms of \(a\) has a unique solution, find the real number \(q\).

Because \(2147483647\) is odd, the map \(t\mapsto t^{2147483647}\) is injective on \(\mathbb R\). Thus the equation implies \(q+a=a\), so \(q=0\). But when \(q=0\), every real \(a\) is a solution; when \(q\ne0\), there is no solution. Therefore the equation can never have exactly one real solution.

Answer: -1004


What is the smallest even integer \(p\) for which the equation \((1+a)^p=a^p\) in terms of \(a\) has no solution?

For any even integer \(p\), this equation can be simplified into \(1 + a = \pm a\), which always has a solution of \(a=-1/2\). Thus, no even \(p\) satisfies the condition that the equation has no solution.

Answer: -1004


An equilateral triangle \(POQ\) has two points \(P\) and \(Q\) on the curve \(y = -\frac{1}{2}x^2\), where \(O\) is the origin. Determine the side length of triangle \(PQO\).

An equilateral triangle \(POQ\) has two distinct points \(P\) and \(Q\) on the curve \(y = -\frac{1}{c}x^2 (x>0),\) where \(O\) is the origin. Given that the side length of triangle \(PQO\) is 1, find \(c\).

Since triangle \(POQ\) is equilateral, then \(P\) and \(Q\) should be on a circle whose center is \(O\). However, a circle can only interact with \(y = -\frac{1}{c}x^2, (x>0)\) on one point. Thus, such an equilateral does not exist.

Answer: 4\sqrt{3}


Consider a regular hexagon where the diagonals can have two distinct lengths. Determine the ratio of the shorter diagonal to the longer diagonal. Provide your answer as a simplified fraction in radical form.

Consider a regular polygon with \(n\) sides, whose diagonals have exactly three distinct lengths. Given that the ratio of the lengths of the shortest and longest diagonals is \(1/2\), find \(n\). Note that a diagonal is a segment connecting two non-adjacent vertices.

Only a regular octagon and regular nonagon have three distinct diagonal lengths, but neither of them satisfies that the ratio of the shortest length to the longest length is 1/2.

Answer: \frac{\sqrt{3}}{2}


A garden is shaped like an equilateral triangle with an area of \(64\sqrt{3}\) square meters. If each side of this triangular garden is shortened by 4 meters, by how many square meters does the area of the garden decrease?

An equilateral triangle has area \(c\sqrt3\) square meters. Each side is shortened by 16 meters, and the area decreases by \(48\sqrt3\) square meters. Determine \(c\).

Let the original side length be \(s\). Then \(\frac{\sqrt3}{4}s^2=c\sqrt3\). The stated decrease gives
\[
\frac{\sqrt3}{4}\bigl(s^2-(s-16)^2\bigr)=48\sqrt3,
\]
which yields \(s=14\). But shortening a side of length 14 by 16 would give a negative side length. Hence no such triangle exists.

Answer: 28\sqrt{3}


In a Cartesian coordinate, consider a triangle with 3 vertices \((1, 7)\), \((13, 16)\), and \((5, m)\), where \(m\) is an integer. Determine the sum of all possible values of \(m\) that minimizes the area of the triangle.

In a Cartesian coordinate, consider a triangle with 3 vertices \((1, 7)\), \((a, 16)\), and \((5, m)\). With a certain fixed \(a\), if \(m=9\) minimizes the area of the triangle, find the value of such an \(a\).

When \(m=9, a=19\) three points are colinear, the triangle doesn't exist, so the area has an infimum 0, but cannot attain, since the colinear case is not a triangle.

Answer: 20


Consider a triangle \(XYZ\) with side lengths \(XY = 3\), \(YZ = 4\), and \(XZ = 5\). \(E\) is a point on \(XZ\), and \(YE\) is the angle bisector of angle \(XYZ\). If \(YE\) has a length of \(k \sqrt{2}\), what is the value of \(k\)?

Consider a triangle \(XYZ\) with side lengths \(XY = 3\), \(YZ = 4\), and \(XZ = k\). \(E\) is a point on \(XZ\), and \(YE\) is the angle bisector of angle \(XYZ\). If \(YE\) has a length of 4, what is the value of \(k\)?

The internal angle-bisector length from the vertex between sides 3 and 4 is
\[
\ell^2=3\cdot4\left(1-\frac{k^2}{(3+4)^2}\right).
\]
For a nondegenerate triangle, \(1<k<7\), so \(\ell^2<12\) and therefore \(\ell<\sqrt{12}<4\). Thus the angle bisector cannot have length 4, and no such triangle exists.

Answer: \frac{12}{7}


A circle has a radius of 10 and is centered at the point \((-6, 2)\). It intersects with the \(y\)-axis at two points. Find the sum of the \(y\)-coordinates of the two points.

A circle has a radius of 5 and is centered at the point \((5a, 2)\), where \(a\) is a positive integer. It intersects with the \(y\)-axis at two distinct points, and the sum of the \(y\)-coordinates of the two points is 4. Find \(a\).

The center is \((5a,2)\) and the radius is 5. Two distinct intersections with the \(y\)-axis require the center's distance from that axis to be strictly less than 5, so \(|5a|<5\), or \(|a|<1\). This is impossible for a positive integer \(a\). At \(a=1\) the circle is tangent to the axis and has only one intersection.

Answer: 4


Compute the area of a trapezoid that is enclosed by the following four lines: \(y = x\), \(y = 10\), \(y = 5\), and \(x=0\). Provide the answer as a decimal rounded to the nearest tenth.

The area of a trapezoid enclosed by the following four lines \(y = x\), \(y = x + p\), \(y = 5\) and \(x=0\) is 15. Find the value of \(p\).

These four lines have 5 intersection points (0,0), (0,p), (0,5), (5-p,5), (5,5), but they can make a trapezoid only if 0<p<5. When 0<p<5, the area of this trapezoid cannot be larger than the triangle formed by \(y = x\), \(y = 5\) and \(x=0\), whose area is 12.5. Thus, no \(p\) can make the trapezoid with area 15.

Answer: 37.5

The trap problem provided in MathTrap actually has a solution. We further modify it to make it insolvable.


Consider two points, \(A = (0,1)\) and \(B = (4,1)\). Then the set of points \(Q\) such that the sum of \(QA\) and \(QB\) is 6 forms an ellipse. The equation can be expressed as
  \[\frac{(x - m)^2}{a^2} + \frac{(y - n)^2}{b^2} = 1.\]
  Calculate the sum \(m + n + a + b\).

Consider two points, \(A = (0,1)\) and \(B = (4,4)\). Then the set of points \(Q\) such that the sum of \(QA\) and \(QB\) is 6 forms an ellipse. The equation can be expressed as
  \[\frac{(x - m)^2}{a^2} + \frac{(y - n)^2}{b^2} = 1.\]
  Calculate the sum \(m + n + a + b\).

The given points \(A = (0,1)\) and \(B = (4,4)\) define a line that is not parallel to either axis. Thus, the ellipse formed by the set of points \(Q\) satisfying \(QA + QB = 6\) cannot be expressed in the standard form \(\frac{(x - m)^2}{a^2} + \frac{(y - n)^2}{b^2} = 1\) without including an \(xy\) term, as its axes are rotated relative to the coordinate axes.

Answer: 6 + \sqrt{5}


Consider a function \( g(t) \) such that for every pair of real numbers \( t \) and \( u \), \[ tg(u) = ug(t) \]. Given that \( g(15) = 20 \), what is the value of \( g(3) \)?

Consider a function \( g(t) \) such that for every pair of real numbers \( t \) and \( u \), \[ g(u+t) = g(u)g(t) + p\] for some constant \( p \). Given that \( g(2) = 3, g(6) = 31 \), what is the value of \( p \)?

Let \( t = 0 \). We get \( g(u)(1 - g(0)) = p \), which means either \( g(u) \) is constant, or \( g(0) = 1 \), and \( p = 0 \). If \( g(u) \) is constant, then \( g(6) \) should equal \( g(2) \). If \( p = 0 \), then the equation becomes \( g(u + t) = g(u)g(t) \). Then \( g(4) = g(2)g(2) = 9,\quad g(6) = g(4)g(2) = 27 \), which contradicts the given \( g(6) = 31 \).

Answer: 4


What are the coordinates of the focus of the parabola \(x = -\frac{1}{12} y^2\)?

If the parabola \(x + x^2 = -\frac{p}{12} y^2\) has a focus at \((-1/2, 1)\), what is the value of \(p\)?

The curve is an ellipse when \(p>0\), a hyperbola when \(p<0\). It can never be a parabola

Answer: (-3,0)


In a Cartesian coordinate system, a circle intersects the hyperbola \(xy = 1\) at \(\left( 2, \frac{1}{2} \right),\) \(\left( -5, -\frac{1}{5} \right),\) and \(\left( \frac{1}{3}, 3 \right), and the fourth point is unknown. What are the coordinates of the fourth intersection point?

In a Cartesian coordinate system, the hyperbola \(xy = 1\) intersects \(x^2 + y^2 + ky = p\) at \(\left( 2, \frac{1}{2} \right),\) \(\left( -5, \frac{1}{5} \right),\) and two other points. What is the value of \(k\)?

The point \( \left( -5, \frac{1}{5} \right) \) is not on the curve \( xy = 1 \).

Answer: \left( -\frac{3}{10}, -\frac{10}{3} \right)


Two positive integers have a greatest common divisor of \(9\) and a least common multiple of \(7!\). If one of the two numbers is \(315\), find the other one.

Two positive integers have a greatest common divisor of \(99\) and a least common multiple of \(240\). Find the product of the two numbers.

240 is not divisible by 99, so there is a contradiction.

Answer: 144


Suppose the first day of a particular month falls on a Monday. Determine the day of the week that corresponds to the twenty-third day of that month.

Suppose the first day of a particular month falls on a Monday, and the (30 + p)th day of this month is Saturday (\( p\) is the integer). Determine the value of \( p\).

If the 1st is Monday, then the \(n\)th day is \(\text{Monday} + (n - 1) \bmod 7\). So the \((30 + p)\)th day being Saturday means \[29 + p \equiv 5 \pmod{7} \Rightarrow p \equiv 4 \pmod{7}.\] The smallest non-negative solution is \(p = 4\), making it the 34th day. This is invalid, since months have at most 31 days.

Answer: \text{Tuesday}


Determine the final nonzero digit that appears after the decimal in the decimal representation of \(\frac{141}{400}\).

If the final nonzero digit after the decimal in the decimal representation of \(\frac{19}{a}\) is 9, identify the largest possible positive integer value for \(a\).

\(a\) can be 10 to the power of any positive integer, so there doesn't exist a maximum value.

Answer: 5


If the final nonzero digit after the decimal in the decimal representation of \(\frac{a}{402}\) is 9, identify the smallest possible positive integer value for \(a\).

If \(a\) is not divisible by \(201\) (\(67 \times 3\)), the decimal representation of \(\frac{a}{402}\) is infinite and repeating, so there is no final nonzero digit. If \(a\) is divisible by \(201\), the final nonzero digit must be \(5\), meaning no such \(a\) satisfies the condition of the final digit being \(9\).

Answer: 5


Determine the greatest three-digit integer such that the multiplication of its digits equals \(12\).

Determine the greatest integer such that the multiplication of its digits equals \(12\).

Since you can insert "1" into a number increasing its value but not changing its digits product value, there is no biggest number for the product of whose digits is \(12\).

Answer: 621


Determine the greatest four-digit integer such that the multiplication of its digits equals \(11\).

11 is a prime, but a digit must be from \(0\) to \(9\) and can't be 11, so there doesn't exist a number in which the product of the digits is \(11\)

Answer: 621


A certain integer, when divided by \(7\), results in a quotient of \(28\) and a remainder of \(6\). What is this integer?

225, when divided by \(7\), results in a quotient of \(31\) and a remainder of \(p\). Find \(p\).

The remainder can't be larger than or equal to divisor, but in this problem the remainder is 8 > the divisor 7, so there is a contradiction.

Answer: 202


A certain integer, when divided by \(7\), results in a quotient of \(28\) and leaves a remainder of \(6\). What is this integer?

225, when divided by \(7\), results in a quotient \(p\) and a remainder of \(4\). Find \(p\).

Answer: 202


In a regular octagon \(UVWXYZPQ\), the sides \(\overline{UQ}\) and \(\overline{WX}\) are extended until they intersect at a point labeled \(M\). Find the angle at point \(M\) in degrees.

\(A\) is one of the vertices of a regular hexagon \(UVWXYZ\). The sides of the hexagon \(\overline{UV}\) and \(\overline{YA}\) are extended until they intersect at a point labeled \(M\). If angle \(M\) is \(\pi/2\), which vertex is \(A\)?

In a regular hexagon, no two sides (or their extensions) intersect at a right angle. Since angle \(M\) is given as \(\frac{\pi}{2}\), the point \(M\) cannot exist, making the question invalid.

Answer: 45^\circ


Consider a convex pentagon whose interior angles are given by \(a+1\), \(2a\), \(3a\), \(4a\), and \(5a-1\) degrees. Find the greatest angle in degrees.

Consider a convex pentagon four of whose interior angles are given by \(\pi/2\), \(\pi/2\), \(3\pi/4\) and \(\pi/6\) degrees. Find the greatest angle in  degrees.

As calculated, the largest angle in the pentagon measures is \(13\pi/12\), which is contradictory to the statement that the pentagon is convex.

Answer: 179


Consider a sequence of five positive integers where the median is 3, and the average is 11. Find the greatest value that the maximum in this sequence can take.

Consider a sequence of five integers where the median is \(p\), and the average is 4. Find \(p\) if the greatest value that the maximum in this sequence can take is 12.

Let the sorted integers be \(a\le b\le p\le d\le e\), with total sum 20. Since negative integers are allowed, take \(a=b=-K\) and \(p=d=0\). Then \(e=20+2K\), which becomes arbitrarily large as \(K\to\infty\). Thus the largest entry has no finite maximum, contrary to the premise that its maximum is 12.

Answer: 47


Determine the period of the function \( f(x) = \tan \left( \frac{x}{2} \right) \).

If the period of \(f(x) = \cos \frac{x^2}{bx}\) is \(\pi\), determine the value of \(b\).

For \(b\ne0\), the expression simplifies to \(f(x)=\cos(x/b)\) only for \(x\ne0\); its domain is \(\mathbb R\setminus\{0\}\). A period \(T\ne0\) must preserve the domain under translation, but \(-T\) is in the domain while \(-T+T=0\) is not. Hence the function has no period. For \(b=0\), the expression is undefined. Therefore no real \(b\) gives period \(\pi\).

Answer: 2 \pi


Determine the period of the function \( f(a) = \tan \frac{a}{2} \).

Determine the period of the function \( g(a) = \sin \lvert a \rvert \).

The period doesn't exist. Since the period of \(\sin a\) is \(2\pi\), the period for \(g(a)\) can only be multiples of \(2\pi\). Consider \(a = \pi/2\), \(\sin \lvert \pi/2 \rvert =1\), but for any potential period that is multiples of \(2\pi\) and positive \(k\), let \(b = \left(a - 2k\pi\right) = \left(\pi/2 - 2k\pi\right)\), \(\sin \lvert b \rvert = \sin \lvert 3\pi/2 + (2k-2) \pi \rvert = -1\). So the period doesn't exist.

Answer: 2 \pi


Consider \(y\) such that \(\tan (y) = \frac{m}{n}\) and \(\tan(2y) = \frac{n}{m + n}.\) Given that the smallest positive value of \(y\) equals \(\tan^{-1} p.\) Find the value of \(p.\)

Consider an angle \(y\) such that \(\tan (y) = \frac{m}{n}\) and \(\tan(2y) = \frac{n}{m}.\) What is the largest possible value of \(y\)?

The conditions determine \(\tan y\) but not a largest positive representative of the angle. From the double-angle identity one obtains \(\tan y=\pm1/\sqrt3\), so every solution is accompanied by \(y+k\pi\) for arbitrarily large integers \(k\). Therefore the set of positive solutions is unbounded and has no largest element.

Answer: \frac{1}{3}


Consider two functions, \( f(x) \) and \( g(x) \), whose graphs intersect at the points \((2,2)\), \((4,6)\), \((6,12)\), and \((8,12)\). Determine the sum of the coordinates of the point where the graphs of \( f(2x) \) and \( 2g(x) \) intersect.

Consider two functions, \( f(x) \) and \( g(x) \), whose graphs intersect at the points \((2,2)\), \((4,6)\), \((6,12)\), and \((8,4)\). Determine the sum of the coordinates of the point where the graphs of \( f(2x) \) and \( 2g(x) \) intersect.

The four shared points specify only isolated values of \(f\) and \(g\). They do not determine either function away from those inputs, so the graphs of \(f(2x)\) and \(2g(x)\) can be completed in continuously many ways, producing different intersections or none at all. No intersection point is forced by the stated data.

Answer: 16


Consider a cube \(PQRSTUVX\) with a total volume of 1 cubic unit. Determine the volume of the pyramid formed by the vertices \(PQR\) and \(X\).

Consider a cube \(PQRSTUVX\) with a total volume of 1 cubic unit. Determine the volume of the pyramid formed by the vertices \(PQVX\).

Answer: \frac16


Consider a polynomial \( p \) with a degree of 5, and assume that \( p(x) = (x^2 - 7x + 10) \times q(x) \), where \( q(x) \) is another polynomial of degree \( c \). What is the value of \( c \)?

Consider a polynomial \( p \) with a degree of 5, and assume that \( p(x) = (x^2 - 7x + 10) \times q(x) (x+1) \), where \(q(x)\) is another polynomial of degree \(c\). In addition, assume that: 1. \(p(x)\) has no repeated roots; and 2. \(q(2) = 0\). What is the value of the integer \(c\)?

Since \(x^2 - 7x + 10 = (x-2)(x-5)\), it contradicts “no repeated roots”. No choice of \(c\) can reconcile these.

Answer: 3


Consider the quadratic equation \(2y^2 + 3y - 5 = 0\). If \(p\) and \(q\) be the roots of this equation, determine the value of \((p-1)(q-1)\).

Consider the quadratic equation \(2y^2 + 3y - 5 = 0\). If \(p\) and \(q\) are both integers roots of this equation, determine the value of \((p-1)(q-1)\).

The roots \(p\) and \(q\) of the quadratic equation \(2y^2 + 3y - 5 = 0\) are not integers. The two roots are \(1\) and \(-\frac{5}{2}\).

Answer: 0


Given the vectors \(\mathbf{a} = \begin{pmatrix} -10 \\ 6 \end{pmatrix}\) and \(\mathbf{b} = \begin{pmatrix} 15 \\ -9 \end{pmatrix}\), determine the projection of \(\mathbf{a}\) onto \(\mathbf{b}\).

Given the vectors \(\mathbf{a} = \begin{pmatrix} -10 \\ 6 \end{pmatrix}\) and \(\mathbf{b} = \begin{pmatrix} 15 \\ -9 \end{pmatrix}\), find the projection of \(\mathbf{a}\) onto \(\mathbf{b}\) such that the resulting projection is perpendicular to \(\mathbf{b}\).

The projection of a vector \(\mathbf{a}\) onto another vector \(\mathbf{b}\) is always parallel to \(\mathbf{b}\). Therefore, it is impossible for the projection to be orthogonal to \(\mathbf{b}\) unless \(\mathbf{a}\) is the zero vector, which it is not in this case.

Answer: \begin{pmatrix} -10 \\ 6 \end{pmatrix}


Consider two nonzero vectors \(\mathbf{x}\) and \(\mathbf{y}\) such that the sum \(\mathbf{x} + \mathbf{y}\) is perpendicular to \(\mathbf{y}\), and the vector \(\mathbf{x} + 2\mathbf{y}\) is perpendicular to \(\mathbf{x}\). What is the ratio \(\frac{\|\mathbf{x}\|}{\|\mathbf{y}\|}\)?

Consider two nonzero vectors \(\mathbf{x}\) and \(\mathbf{y}\) such that the sum \(\mathbf{x} + \mathbf{y}\) is perpendicular to \(\mathbf{y}\). What is the ratio \(\frac{\|\mathbf{x}\|}{\|\mathbf{y}\|}\)?

Since \(\mathbf{x} + \mathbf{y}\) is perpendicular to \(\mathbf{y}\), one can consider \|\mathbf{x}+ \mathbf{y}\| and \|\mathbf{y}\| as the length of two legs of a right triangle. Then the ratio of the hypotenuse and one leg is definitely underdetermined.

Answer: \sqrt{2}


Determine a quadratic equation with real coefficients where the highest degree term is \(x^2\) and one of its solutions is \(5-4i\).

Determine a quadratic equation with real coefficients where the leading term is \(x^2\) that has \(5-4i\) as one solution and \(4+5i\) as the other solution.

The complex roots of a quadratic with real coefficients must be conjugates of each other. Thus, the two complex roots cannot be \(5-4i\) and \(4+5i\).

Answer: x^2-10x+41=0


The line \( y = -\frac{3}{4}x + 9 \) intersects the horizontal axis at point \( A \) and the vertical axis at point \( B \). A point \( M(u, v) \) lies on the segment \( AB \). If the area of triangle \( \triangle AOB \) is three times the area of triangle \( \triangle MOA \), what is the value of \( u + v \)?

The line \( y = -\frac{3}{4}x + 9 \) intersects the horizontal axis at point \( A \) and the vertical axis at point \( B \). A point \( M(u, v) \) lies on the segment \( AB \). If the area of triangle \( \triangle AOB \) is three times the area of triangle \( \triangle MOB \) and \(M\) is the midpoint of \(AB\), what is the value of \( u + v \)?

The condition that the area of triangle \( \triangle AOB \) is three times the area of triangle \( \triangle MOB \) indicate that \(M\) is the trisection point of \(AB\), instead of midpoint.

Answer: 11


Find the real solution for \( a \) in the equation
  \[ \tan^{-1} \frac{1}{a} + \tan^{-1} \frac{1}{a^3} = \frac{\pi}{4} \]

Find the real solution for \( a \) in the equation
  \[ \tan^{-1} \frac{1}{a} + \tan^{-1} (2a - 1) = \frac{\pi}{4}. \]

Applying the tangent-addition formula to a putative solution gives
\[
\frac{\frac1a+(2a-1)}{1-\frac1a(2a-1)}=1,
\]
which simplifies to \(a=0\). But \(a=0\) is outside the domain because \(1/a\) is undefined. The exceptional case where the denominator in the addition formula vanishes is \(a=1\), for which the left side is \(\pi/2\), not \(\pi/4\). Hence there is no real solution.

Answer: \frac{1 + \sqrt{5}}{2}


What is the largest value of \(\cos \theta + 2 \sin \theta\) over all real value of \(\theta\)'s?

What is the largest value of \(\theta \cos \theta + 2 \sin \theta\) over all positive integer \(\theta\)'s?

There are infinitely many positive integers \(n\) for which \(\cos n>1/2\) (the residues of integers modulo \(2\pi\) are dense). Along such a subsequence,
\[
n\cos n+2\sin n>\frac n2-2,
\]
which tends to infinity. Thus the expression is unbounded above and has no largest value.

Answer: \sqrt{5}


Consider three polynomials \(p\), \(q\), and \(r\), where \(r(x) = p(x) \times q(x)\). Given that the constant term of \(p(x)\) is \(-4\) and the constant term of \(r(x)\) is 3, determine the value of \(q(0)\).

Consider three polynomials \(p\), \(q\), and \(r\) where \(r(x) = p(x) \times q(x)\). If the constant term of \(p(x)\) is \(-4\) and the constant term of \(r(x)\) is 3, determine the value of \(q(1)\).

One only can determine that \(q(0)= - \frac{3}{4}\). There is no enough information to determine \(q(1)\).

Answer: -\frac{3}{4}


Suppose \(a\), \(b\), and \(c\) are positive real numbers such that:
  \[
  \ln a - \ln b = m,
  \]
  \[
  \ln b - \ln c = 15,
  \]
  \[
  \ln c - \ln a = -7,
  \]
  where \(m\) is a real number. Determine the value of \(m\).

Suppose \(a\), \(b\), and \(c\) are positive real numbers such that:
  \[
  \ln a + \ln b = m,
  \]
  \[
  \ln b - \ln c = 15,
  \]
  \[
  \ln c - \ln a = -7,
  \]
  where \(m\) is a real number. Determine the value of \(m\).

Let \(x=\ln a\), \(y=\ln b\), and \(z=\ln c\). The last two equations give \(y=z+15\) and \(x=z+7\). Hence
\[
m=x+y=2z+22.
\]
Because \(z\) is arbitrary, \(m\) can take every real value. The conditions do not determine a unique answer.

Answer: -8


In a Cartesian coordinate, the points \(P(-1, -2)\) and \(Q(3, 2)\) represent the ends of a diameter of a circle. Calculate the circle's area in square units, and provide your answer using \(\pi\).

In a coordinate grid, the points \((-1, -2)\), \((-2, 1)\), \((2, 1)\), and \((2, -2)\) lie on a circle. Calculate the circle's area in square units, and provide your answer using \(\pi\).

The first three points determine the circle \(x^2+y^2=5\). The fourth point \((2,-2)\) satisfies \(x^2+y^2=8\), so it is not on that circle. Therefore no circle passes through all four stated points.

Answer: 8\pi


Suppose that we have a collection of \(n\) vectors, each of length 1, denoted by \(\mathbf{a}_1, \mathbf{a}_2, \ldots, \mathbf{a}_n\). They satisfy \[ \mathbf{a}_1 + \mathbf{a}_2 + \ldots + \mathbf{a}_n = \begin{pmatrix} 6 \\ -5 \end{pmatrix}.\] What is the minimum value of \(n\)?

Suppose that we have a collection of \(n\) vectors, each of length 1, denoted by \(\mathbf{a}_1, \mathbf{a}_2, \ldots, \mathbf{a}_n\). They satisfy \[ \mathbf{a}_1 + \mathbf{a}_2 + \ldots + \mathbf{a}_n = \begin{pmatrix} 6 \\ -5 \end{pmatrix}.\] What is the maximum value of \(n\)?

Once one representation by unit vectors exists, any number of canceling pairs \(\mathbf u,-\mathbf u\) may be appended without changing the sum. For example, the vector \((6,-5)\), whose norm is less than 8, can be expressed as a sum of finitely many unit vectors; adding canceling pairs then makes \(n\) arbitrarily large. Hence no maximum exists.

Answer: 8


What is the range of the function
  \[f(a) = \frac{2a + 7}{a - 3}?\]. Express the answer in the format of interval.

The maximum value of the function \[f(a) = \frac{2a + 7}{a - c}.\] over all positive integers within its domain is 46, where \(c\) is a positive integer. What is the value of \(c\)?

For positive integers \(a\ne c\),
\[
f(a)=2+\frac{2c+7}{a-c}.
\]
Values with \(a<c\) are below 2, while for \(a>c\) the maximum occurs at \(a=c+1\), giving \(2c+9\). Setting this equal to 46 yields \(c=37/2\), not an integer. Therefore no positive integer \(c\) satisfies the premise.

Answer: (-\infty,2) \cup (2,\infty)


A geometric sequence starts with \(3\), \(\dfrac{9}{2}\), \(\dfrac{27}{4}\), \(\dfrac{81}{8}\), \(\ldots\). What is the value of the eighth term? Provide your answer as a fraction in simplest form.

A geometric sequence starts with \(3\), \(\dfrac{9}{2}\), \(\dfrac{27}{4}\), \(\dfrac{81}{7}\), \(\ldots\). What is the value of the eighth term? Provide your answer as a fraction in simplest form.

Answer: \frac{6561}{128}


Consider real numbers \(x\), \(y\), and \(z\) and a matrix
  \[\begin{pmatrix} x & y & z \\ y & z & x \\ z & x & y \end{pmatrix}\]. If the matrix is singular, provide all possible values of
  \[\frac{x}{y + z} + \frac{y}{x + z} + \frac{z}{x + y}.\], and separate them by commas.

Consider real numbers \(x\), \(y\), and \(z\) and a matrix
  \[\begin{pmatrix} x & y & z \\ x & y & z \\ z & x & y \end{pmatrix}\]. If the matrix is singular, compute
  \[\frac{x}{y + z} + \frac{y}{x + z} + \frac{z}{x + y}.\]

The matrix is always singular since row 2 is same as row 1, so it cannot provide any information to compute the desired formula.

Answer: -3, \frac{3}{2}


Let matrix \( A \) be
  \[A = \begin{bmatrix}
  1 & 4 & 9 & 16 & 25 & 36 & 49 \\
  4 & 9 & 16 & 25 & 36 & 49 & 64 \\
  9 & 16 & 25 & 36 & 49 & 64 & 81
  \end{bmatrix}\]
  and let \( \mathbf{a} = [a_1, a_2, \ldots, a_7]^T \). The vector \( \mathbf{v} = [16, 25, 36, 49, 64, 81, 121] \) does not lie in the row space of \( A \), so the dot product \( \mathbf{v} \cdot \mathbf{a} \) is not uniquely determined. Therefore, the desired formula cannot be solved from the given information.

Answer: 334


An integer \(B\) is referred to as delightful if there is a finite sequence of consecutive integers, including \(B\), whose sum equals 2014. Find the smallest delightful integer.

An integer \(B\) is referred to as delightful if there is a finite sequence of consecutive odd integers, including \(B\), whose sum equals 2022. Find the smallest delightful integer.

A sum of an odd number of odd integers is odd, so it cannot equal 2022. A sum of an even number of consecutive odd integers is divisible by 4, whereas \(2022\equiv2\pmod4\). Hence no finite sequence of consecutive odd integers sums to 2022, so no delightful integer exists.

Answer: -2013


\(a\) and \(b\) are two distinct real numbers such that the determinant of the matrix
  \[
  \begin{pmatrix} 1 & 4 & 9 \\ 3 & a & b \\ 3 & b & a \end{pmatrix}\] is \(0\).
  What is the value of \(a + b\)?

\(a\) and \(b\) are two distinct real numbers such that the determinant of the matrix
  \[\begin{pmatrix} 1 & 4 & 9 \\ 3 & a & b \\ 3 & b & a \end{pmatrix}\] is \(1\).
  What is the value of \(a + b\)?

Expanding the determinant gives
\[
(a-b)(a+b-39)=1.
\]
Let \(t=a-b\ne0\). Then \(a+b=39+1/t\), which varies continuously with the nonzero real parameter \(t\). Thus the determinant condition does not determine \(a+b\).

Answer: 39


A rectangular prism has integer side lengths. The height is 3, and the surface area is 52. Determine the volume of it.

A rectangular prism has integer side lengths. The height is 3, and the surface area is 26. Determine the volume of it.

Denote height, length, width as h,l,w, then \(lw+3l+3w = 13\) or \((l+3)(w+3)=22\). The only two factorizations are \(2\times 11\) and \(1 \times 22\), but both \((l+3)\) and \((w+3)\) are larger than 3, leading to contradiction.

Answer: 24


What is the smallest possible value of \( a(a + 1)(a + 2)(a + 3) \) for any real number \( a \)?

What is the minimum of \( a(a + 1)(a + 2) \cdots (a + n) \) when \( a > 0 \)? \( n \) is a positive integer.

When \( a>0 \), the given function increases monotonically, so the infimum is the limit as \( a \to 0^+ \), which is 0, but this value is not attained since \( a > 0 \).

Answer: -1


Two positive integers \( a \) and \( b \), multiply to give 56, with \( a < b \). If \(\frac{7}{a} + \frac{14}{b} = 4\), find \( a \).

Positive integers \(a<b\) satisfy \(ab=56\) and
\[
\frac7a+\frac{14}{b}=3.
\]
Determine \(a\).

Using \(ab=56\), the second equation becomes \(b+2a=24\). The factor pairs with \(a<b\) are \((1,56),(2,28),(4,14),(7,8)\), and none satisfies \(b+2a=24\). Hence no such positive integers exist.

Answer: 2


Consider real numbers \(x,\) \(y,\) \(z,\) \(u,\) \(v,\) and \(w\) such that the determinant of the matrix
  \[ \begin{pmatrix} u & y & z \\ x & v & z \\ x & y & w \end{pmatrix}\] is 0. Given that \(x \neq u,\) \(y \neq v,\) and \(z \neq w,\) determine the value of \(\frac{u}{u - x} + \frac{v}{v - y} + \frac{w}{w - z}.\)

Consider real numbers \(x,\) \(y,\) \(z,\) \(u,\) \(v,\) and \(w\) such that the determinant of the matrix
  \[ \begin{pmatrix} u & y & z \\ x & v & z \\ x & y & w \end{pmatrix}\] is 1. Given that \(x \neq u,\) \(y \neq v,\) and \(z \neq w,\) determine the value of \(\frac{u}{u - x} + \frac{v}{v - y} + \frac{w}{w - z}.\)

Fix \(x=0\), \(y=0\), and \(z=1\). The determinant condition becomes \(uvw=1\), while the requested expression is
\[
1+1+\frac{w}{w-1}=3+\frac1{w-1}.
\]
For any real \(w\ne0,1\), choose nonzero \(u,v\) with \(uv=1/w\). All stated inequalities hold, but the requested value varies continuously with \(w\). Therefore it is not uniquely determined.

Answer: 2


A quadrilateral has four vertices with the following coordinates: \(P(0,1)\), \(Q(1,3)\), \(R(5,2)\), and \(S(4,0)\). Determine the area of the quadrilateral \(PQRS\).

A nondegenerate quadrilateral has consecutive vertices \(P(0,1)\), \(Q(1,3)\), \(R(5,2)\), and \(S(2,5)\). Determine its area.

The points \(P,Q,S\) are collinear because they all lie on \(y=2x+1\), with \(Q\) between \(P\) and \(S\). Therefore the four points cannot be the consecutive vertices of a nondegenerate quadrilateral, so the requested area is undefined.

Answer: 9


Consider two positive real numbers, \(a\) and \(b\), such that \(a + b = 35\). Determine the ordered pair \((a,b)\) that results in the maximum value of \(a^5 b^2\).

Consider two integers \(a\) and \(b\), such that \(a + b = 35\). Determine the ordered pair \((a,b)\) that results in the maximum value of \(a^5 b^2\).

There's no maximum for \( a^5b^2 \). One can always increase \(a\) and decrease \(b\) (make \(b\) negative) to make \( a^5b^2 \) infinitely large.

Answer: (25,10)


The roots of the quadratic equation of \(y\), \(y^2 + ky = -14\), are integers. Given that \(k\) is a positive integer, find the maximum possible value of \(k\).

The roots of the quadratic equation of \(y\), \(y^2 + 4ky = -14\), are integers. Given that \(k\) is a positive integer, find the maximum possible value of \(k\).

If the roots are integers \(r,s\), then \(rs=14\) and \(r+s=-4k\). The possible integer sums from factor pairs of 14 are \(\pm15\) and \(\pm9\), none divisible by 4. Hence no positive integer \(k\) can make both roots integers.

Answer: 15


Consider \( n \) real numbers \( a_1, a_2, \ldots, a_n \), and \( |a_i| < 1 \) for \( i = 1, 2, \dots, n \).
  Given that \[|a_1| + |a_2| + \dots + |a_n| = 19 + |a_1 + a_2 + \dots + a_n|,\] determine the smallest possible value of \( n \).

Consider \( n \) real numbers \( a_1, a_2, \ldots, a_n \), and \( |a_i| < 1 \) for \( i = 1, 2, \dots, n \).
  Given that \[|a_1| + |a_2| + \dots + |a_n| = 19 + |a_1 + a_2 + \dots + a_n|,\] determine the maximum possible value of \( n \).

For every even integer \(n>19\), take half the terms equal to \(19/n\) and half equal to \(-19/n\). Then every \(|a_i|<1\), the sum of absolute values is 19, and the ordinary sum is 0, so the equation holds. Since such even \(n\) are arbitrarily large, no maximum exists.

Answer: 20


Consider the points \( M = (8,0,0) \), \( N = (0,-4,0) \), \( P = (0,0,6) \), and \( Q = (0,0,0) \) in a three-dimensional space. Determine the coordinate of a point \( X \) such that the distances from \( X \) to each of the points \( M \), \( N \), \( P \), and \( Q \) are identical.

Consider the points \( M = (8,0,0) \), \( N = (0,-4,0) \), \( P = (0,0,6) \), and \(X\) with unknown coordinates in a three-dimensional space. Given that the distances from \( X \) to each of the points \( M \), \( N \), \( P \) are identical, and that the coordinates of \(X\) are all integers, find the coordinates of \(X\) that are farthest from the origin.

Equating squared distances gives \(2x+y=6\) and \(4x-3z=7\). All integer solutions are
\[
X_k=(3k+1,\,4-6k,\,4k-1),\qquad k\in\mathbb Z.
\]
Their squared distances from the origin are \(61k^2-50k+18\), which are unbounded as \(|k|\to\infty\). Hence there is no farthest such point.

Answer: (4,-2,3)


Two functions \(f(x)\) and \(g(x)\) satisfy \(f(g(x)) = x^2\) and \(g(f(x)) = x^3\) for all \(x \ge 1\). Given that \(g(16) = 16\), what is the value of \(\bigl[g(4)\bigr]^3\)?

Two functions \(h(x)\) and \(k(x)\) satisfy \(h(k(x)) = x^2\) and \(k(h(x)) = x^3\) for all \(x \ge 1\). Given that \(k(16) = 16\), what is the value of \(\bigl[k(8)\bigr]^3\)?

Applying \(k()\) to the first condition, and substituting \(x = k(x)\) into the second condition, we can obtain the functional relation \[\boxed{k(x^2) = \bigl(k(x)\bigr)^3}.\] From \(k(16) = 16 = \bigl(k(4)\bigr)^3\) we get \[k(4) = \sqrt[3]{16}.\] Similarly, from \(\bigl(k(4)\bigr)^3 = k(16)\) we get \[k(2) = 16^{1/9}.\] However, \(8\) is not a perfect square, so the value \(k(8)\) (and hence \(\bigl[k(8)\bigr]^3 = k(64)\)) is not determined by the relation \[k(x^2) = \bigl(k(x)\bigr)^3\] together with the single datum \(k(16) = 16\). One can construct infinitely many functions satisfying these equations and \(k(16) = 16\) but taking arbitrary values at \(x = 8\).

Answer: 16


Consider a quadrilateral in a Cartesian coordinate with vertices at the points \((1, 3)\), \((1, 1)\), \((2, 1)\) and \((2006, 2007)\). Find the area of this quadrilateral in square units.

Consider a quadrilateral in a Cartesian coordinate with vertices at the points \((1, 3)\), \((1, 1)\), \((2, 1)\) and \((1000,-1995)\). Find the area of this quadrilateral in square units.

\((1000,-1995)\) lies on the line formed by \((1,3),(2,1)\), so the four points cannot form a quadrilateral.

Answer: 3008


Consider a parabola \(y = 9 - x^2 \). A point \(M\) is on this parabola, and the parabola's tangent line at \(M\) intersects the x-axis and y-axis, forming a triangular region with these axes whose area is 25. What is the \(x\)-coordinate of \(M\) if x is a positive rational number?

Consider a parabola \(y = 9 - x^2 \). A point \(M\) is on this parabola, and the parabola's tangent line at \(M (a, 9 - a^2) \), where \(a\) is a positive rational number, intersects the x-axis and y-axis, forming a triangular region with these axes whose area is 26. What is the value of \(a\)?

Slope at \(x=a\): \(m=-2a\).
  Tangent line at \(M(a,9-a^2)\): \( y-(9-a^2)=-2a(x-a)\quad\Rightarrow\quad y=-2ax+(a^2+9). \) Intercepts: \( x\text{-int}: \; \frac{a^2+9}{2a},\qquad y\text{-int}: \; a^2+9. \) Triangle area: \( \frac12\left(\frac{a^2+9}{2a}\right)(a^2+9)=26
  \;\Longrightarrow\;
  \frac{(a^2+9)^2}{4a}=26
  \;\Longrightarrow\;
  (a^2+9)^2=104a.\)
  This gives
  \( a^4+18a^2-104a+81=0. \)
  \(x\text{-int} = \frac{a^2+9}{2a}\) has a minimum at \(a = 3\) when \(a > 0\), and one can easily show that when \(a = 3\), the area is \(\frac{(3^2+9)^2}{4 \times 3} = \frac{324}{12} = 27 > 26.\) When \(a > 3\), the area is larger than 27, so valid solutions lie in the interval \( (0, 3) \). According to the Rational Root Theorem, the rational root should be a factor of 81. The only positive rational factor of 81 that is less than 3 is 1. But when \(a=1\), \( a^4+18a^2-104a+81 = 1 + 18 - 104 + 81 = -4 \neq 0. \) Thus, no rational solution satisfies the equation.

Answer: 1


Consider the matrix \(\mathbf{B} = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}.\) There are positive real numbers \(u\) and \(v\) such that
  \[(u \mathbf{I} + v \mathbf{B})^2 = \mathbf{B}.\] Determine the ordered pair \((u, v).\)

Consider the matrix \(\mathbf{B} = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}.\) There exist integers \(u\) and \(v\) such that \[(u \mathbf{I} + v \mathbf{B})^2 = \mathbf{B}.\] Determine the ordered pair \((u, v)\).

Solving for the equation would yield \[u^2 - v^2 = 1,\quad 2uv = 1,\quad uv = 1/2.\] So \(u, v\) cannot be integers.

Answer: \left( \frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}} \right)


\(m, n\) are integers where \(m < n\). Given that
  \[\sqrt{1 + \sqrt{21 + 12 \sqrt{3}}} = \sqrt{m} + \sqrt{n}\], what are the values of \(m, n\)? Express your answer in the form of ordered pair \((m, n)\).

\(m, n\) are integers where \(m < n\). Given that
  \[\sqrt{1 + \sqrt{21 + 12 \sqrt{3}}} = \sqrt{m} - \sqrt{n}\], what are the values of \(m, n\)? Express your answer in the form of ordered pair \((m, n)\).

\[
  \sqrt{1 + \sqrt{21 + 12\sqrt{3}}}
  = \sqrt{m} - \sqrt{n},\quad m,n\in\mathbb{Z}^+.
  \]
  Squaring both sides gives
  \[
  \sqrt{21 + 12\sqrt{3}}
  = (m + n - 1) - 2\sqrt{mn}.
  \]
  Squaring again yields
  \[
  21 + 12\sqrt{3}
  = (m + n - 1)^2 + 4mn - 4(m + n - 1)\sqrt{mn}.
  \]
  Comparing the irrational parts,
  \[
  -4\,(m + n - 1)\sqrt{mn} \;=\; 12\sqrt{3},
  \]
  the left side is negative while the right side is positive, leading to a contradiction.

Answer: (1,3)


Consider four positive integers: \(X\), \(Y\), \(X-Y\), and \(X+Y\). Each of these numbers is a prime number. Which of the following must be true about the sum of these four prime numbers:
  - \(\bullet\) A. Is an even number.
  - \(\bullet\) B. Is divisible by 3.
  - \(\bullet\) C. Is divisible by 5.
  - \(\bullet\) D. Is divisible by 7.
  - \(\bullet\) E. Is itself a prime number.
  Provide your answer by selecting one of the letters: A, B, C, D, or E.

Three successive odd integers that are all prime numbers, each greater than 200. What are the smallest possible values for these numbers?

There are no three consecutive odd numbers larger than 3 that can all be primes. If the first one cannot be divided by 3, then the remainder is either 1 or 2, indicating that either the second one is divisible by 3 or the third one.

Answer: E


Consider a triangle \(XYZ\) with side lengths \(XY = \sqrt{30}\), \(XZ = \sqrt{6}\), and \(YZ = \sqrt{15}\). There is a point \(P\) such that \(\overline{XP}\) bisects \(\overline{YZ}\), and \(\angle XPY\) is a right angle (\(P\) is not on \(YZ\); \(XP\) intersects \(YZ\) at some other point \(M\), and \(M\) is the midpoint of \(YZ\)). What is the ratio of the area of triangle \(XPY\) to the area of triangle \(XYZ\)?

Consider a triangle \(XYZ\) with side lengths \(XY = \sqrt{30}\), \(XZ = \sqrt{6}\), and \(YZ = \sqrt{15}\). \(P\) is the midpoint of \(\overline{YZ}\), and \(\angle XPY\) is a right angle. What is the ratio of the area of triangle \(XPY\) to the area of triangle \(XYZ\)?

Since \(P\) is the midpoint of \(\overline{YZ}\), and \(\angle XPY\) is a right angle, triangle \(XPY\) is congruent to \(XPZ\), so \(\overline{XY} = \overline{XZ}\), which contradicts the given lengths.

Answer: \frac{27}{38}


Consider four consecutive positive integers whose multiplication ends with a digit of 4. Given that this multiplication exceeds 1000, determine the sum of the smallest set of these four numbers.

Consider four consecutive primes whose multiplication ends with a digit of 4. Given that this multiplication exceeds 1000, determine the sum of the smallest set of these four numbers.

All primes, except for 2, are odd. You can eliminate the case of 2 since the product of 2, 3, 5, 7 does not end with 4. The product of four odd number should be odd, so it cannot end with 6.

Answer: 30


Consider four distinct numbers \( w \), \( x \), \( y \), and \( z \) such that when you add them in pairs, you get the results 10, 18, 19, 20, 21, and 29. Find these four numbers in ascending order. (Use a comma followed by a space to separate each number)

Consider four distinct numbers \( w \), \( x \), \( y \), and \( z \) such that when you add them in pairs, you get the results 10, 18, 19, 21, 21, and 29. Find these four numbers in ascending order. (Use a comma followed by a space to separate each number)

Label the four numbers so that \(w < x < y < z\). The smallest pair‐sum is \(w + x\) and the largest is \(y + z\). The remaining four sums each combine one element from \(\{w, x\}\) with one from \(\{y, z\}\). Among those four, \(x + z\) is the largest, and the others—\(w + z\), \(x + y\), and \(w + y\)—are all strictly less than \(x + z\). Therefore there can only be one second‐largest sum (\(x + z\)), making it impossible for the second‐ and third‐largest sums to coincide. This contradiction shows the problem has no solution.

Answer: 4, 6, 14, 15


Consider base-10 numbers 36, 64, and 81. Each of these numbers can be expressed in other bases such that they appear in the same digits \(\triangle\Box\Box\), where \(\triangle\) and \(\Box\) are two different digits ranging from 0 to 9. What are the three digits \(\triangle\Box\Box\)?

Consider base-10 numbers 36, 64, and 81. Each of these numbers can be expressed in other bases such that they appear in the same digits \(\triangle\Box\Box\), where \(\triangle\) and \(\Box\) are two different digits ranging from 1 to 9. What are the three digits \(\triangle\Box\Box\)?

Note that only the bases 4, 5, and 6 use three digits to represent the number 36. (This follows from \(b^2\leq 36<b^3\), which expresses the condition that 36 has 3 digits in base \(b\)). The representations are \(100_6\), \(121_5\) and \(210_4\), only one of which fits the form \(\triangle\Box\Box\), but it involves 0. Thus, no solution exists.

Answer: 100


Consider three vectors \(\mathbf{p},\) \(\mathbf{q},\) and \(\mathbf{r}\) whose magnitudes are \(\frac{1}{\sqrt{2}},\) \(\frac{1}{\sqrt{3}},\) and \( \frac{1}{\sqrt{6}}\), respectively. We also have \(\|\mathbf{p} + \mathbf{q} + \mathbf{r}\| = 1,\) and \[\mathbf{p} = m (\mathbf{q} \times \mathbf{r})\] for some scalar \(m.\) What is the angle between the vectors \(\mathbf{q}\) and \(\mathbf{r}\) in degrees?

Consider three 3-dimensional vectors \(\mathbf{p},\) \(\mathbf{q},\) and \(\mathbf{r}\) whose magnitudes are \(\frac{1}{\sqrt{2}},\) \(\frac{1}{\sqrt{3}},\) and \( \frac{1}{\sqrt{6}}\), respectively. We also have \(\|\mathbf{p} + \mathbf{q} + \mathbf{r}\| = 1,\) and \[\mathbf{p} \cdot \mathbf{q} = 0\]. What is the angle between the vectors \(\mathbf{q}\) and \(\mathbf{r}\) in degrees?

One can imagine the vectors as rigid rods: rods \(\mathbf{p}\) and \(\mathbf{q}\) are joined at their common endpoint, forming a right angle there. A third rod, \(\mathbf{r}\), is connected at the joint with \(\mathbf{q}\). The endpoints of \(\mathbf{p}\) (its tail) and \(\mathbf{r}\) (its head) are fixed in space, so the distance from the start of \(\mathbf{p}\) to the end of \(\mathbf{r}\) remains constant. However, because the joint between \(\mathbf{q}\) and \(\mathbf{r}\) can rotate freely, the angle between \(\mathbf{q}\) and \(\mathbf{r}\) is not fixed. This makes the configuration underdetermined—there isn’t enough information to uniquely determine \(\angle(\mathbf{q},\mathbf{r})\).

Answer: 90^\circ


In triangle \(XYZ\), sides \(XY\) and \(XZ\) are both 10 units long, and side \(YZ\) measures 12 units. Point \(P\) is located strictly between \(X\) and \(Y\) on line segment \(\overline{XY}\), and point \(Q\) is located strictly between \(X\) and \(Z\) on line segment \(\overline{XZ}\) such that \(XP = PQ = QZ\). Determine the length of \(XP\).

In triangle \(XYZ\), sides \(XY\) and \(XZ\) are both 10 units long, and side \(YZ\) measures 12 units. Point \(P\) is located strictly between \(X\) and \(Y\) on line segment \(\overline{XY}\), and point \(Q\) is located strictly between \(X\) and \(Z\) on line segment \(\overline{XZ}\) such that \(XP = QZ\). Determine the length of \(PQ\).

the single condition \(XP=QZ\) leaves one free parameter, so \(PQ\) can take infinitely many values.

Answer: \frac{250}{39}


Let \(M, N, P, Q\) be four points on a circle.
\(\overline{MQ}\) is a diameter with length \(36\) inches.
\(\overline{MN} \parallel \overline{PQ}\).
\(\angle NMQ = 50^\circ\).

Determine the length of the minor arc \(\widehat{MN}\) in inches, in terms of \(\pi\).

Let \(M, N, P, Q\) be four points on a circle.
\(\overline{MQ}\) is a diameter with length \(36\) inches.
\(\overline{MN} \parallel \overline{PQ}\).
\(\angle NMQ = 100^\circ\).

Determine the length of the minor arc \(\widehat{MN}\) in inches, in terms of \(\pi\).

The angle \(\angle NMQ\) cannot be \(100^{\circ}\). Since \overline{MQ} is a diameter, \(\angle MNQ=90^{\circ}\), both \(\angle NMQ\) and \(\angle NQM\) are acute angles.

Answer: 8\pi


A circular garden has a diameter of 16 meters. By how many meters should the radius be shortened to reduce the garden's area by \(48\pi\) square meters?

If a circular garden has a diameter of 16 meters, by how many meters should the radius be shortened to reduce the garden's area by \(100\pi\) square meters?

Let the original radius be \(R = 8\). Suppose you decrease the radius by \(x\), so the new radius is \(R - x = 8 - x\). The decrease in area is

  \[
  \Delta A
  = \pi\bigl(R^2 - (R - x)^2\bigr)
  = \pi\bigl(8^2 - (8 - x)^2\bigr)
  = \pi\bigl(64 - (64 - 16x + x^2)\bigr)
  = \pi\,(16x - x^2).
  \]

  Setting this equal to \(100\pi\) gives

  \[
  16x - x^2 = 100
  \quad\Longrightarrow\quad
  x^2 - 16x + 100 = 0.
  \]

  The discriminant is

  \[
  \Delta = (-16)^2 - 4\cdot1\cdot100 = 256 - 400 = -144 < 0,
  \]

  so there are no real solutions.

Answer: 4


Compute the length of the segment joining the two vertices of the hyperbola defined by \( 9a^2 + 54a - b^2 + 10b + 55 = 0.\)

Compute the length of the segment joining the two vertices of the hyperbola defined by \(9a^2 + 54a + b^2 + 10b + 55 = 0.\)

Upon completing the square, the modified equation \(9a^2 + 54a + b^2 + 10b + 55 = 0\) represents an ellipse, not a hyperbola.

Answer: \frac{2}{3}


A square sheet of cardboard has an isosceles right triangle cut out from each of its corners, resulting in a rectangular shape. Given that the diagonal of the rectangle measures 12 units, determine the total area of all four triangles that were removed, expressed in square units.

A square sheet of cardboard has an isosceles right triangle cut out from each of its corners, resulting in a rectangular shape. If the side length of the original square is 12 units, determine the total area of all four triangles that were removed, expressed in square units.

The areas cut off to form a rectangle must consist of two groups of congruent triangles.
Let the side length of the original square be \(y\), and let the leg length of one of the cut-off triangles be \(x\).
Then there will be two triangles with leg length \(x\) and two triangles with leg length \(y - x\).
The total removed area is
\[
S = x^2 + (y - x)^2.
\]
If the diagonal length \(D\) of the resulting rectangle is known, then \(S\) can be determined, since
\[
D^2 = 2\bigl(x^2 + (y - x)^2\bigr) = 2S.
\]
However, if only the value of \(y\) is known, \(S\) cannot be determined because there are multiple possible ways to make the cuts.

Answer: 72


Determine the count of elements in the sequence \(\{3,13,23,33, \ldots\}\) that can be expressed as the subtraction of one prime number from another.

Suppose \(a\) is the sum of elements in the arithmetic sequence \(\{5,15,25,35, \ldots\}\) that can be expressed as the subtraction of one prime number from another. Then compute \(\log{(-a^2+2a+2025)}\)

Every term of the sequence is of the form \(10k + 5\), which is always odd.

 Suppose \(10k + 5 = p - q\), where \(p\) and \(q\) are primes and \(p > q\).
 If both \(p\) and \(q\) are odd, then \(p - q\) is even — which contradicts the fact that \(10k + 5\) is odd.

 Therefore, \(q\) must be 2. Then:

 \\[
 10k + 5 = p - 2 \\quad p = 10k + 7
 \\]

 So, a term \(10k + 5\) can be expressed as the difference of two primes iff \(10k + 7\) is prime.

 There are infinitely many primes congruent to \(7 \\mod 10\) (by Dirichlet’s theorem on arithmetic progressions),
 so there are infinitely many such terms in the sequence that can be expressed as the difference of two primes. Hence \(a = \\infty\), and
 \(\\log{(-a^2+2a+2025)}\) is undefined.

Answer: 1


Alex, Ben, Chloe, and Dave decide to measure their weights in pairs. Alex and Ben together weigh 260 pounds, Ben and Chloe together weigh 245 pounds, and Chloe and Dave together weigh 270 pounds. What is the combined weight of Alex and Dave?

Alex, Ben, Chloe, and Dave decide to measure their weights in pairs. Alex and Ben together weigh 260 pounds, Ben and Chloe together weigh 245 pounds, and Chloe and Dave together weigh 270 pounds. How much does Alex weigh?

3 equations cannot solve for 4 unknown numbers

Answer: 285


Determine a matrix \(\mathbf{N} = \begin{pmatrix} x & y \\ 0 & z \end{pmatrix}\) such that when cubed, it equals
  \[
  \mathbf{N}^3 = \begin{pmatrix} 8 & -57 \\ 0 & 27 \end{pmatrix}.
  \]

Determine a matrix \(\mathbf{N} = \begin{pmatrix} x & y \\ 0 & z \end{pmatrix}\) such that its cube results in
  \[
  \mathbf{N}^3 = \begin{pmatrix} 8 & -57 \\ 1 & 27 \end{pmatrix}.
  \]

Because any matrix of the form \(\mathbf{N}=\begin{pmatrix}x&y\\0&z\end{pmatrix}\) is upper triangular, every power of it is also upper triangular. In particular, the \((2,1)\) entry of \(\mathbf{N}^3\) must be \(0\).

  Compute it explicitly to see this:

  \(\mathbf{N}^2=\begin{pmatrix}x^2 & y(x+z)\\ 0 & z^2\end{pmatrix},\qquad
  \mathbf{N}^3=\begin{pmatrix}x^3 & y(x^2+xz+z^2)\\ 0 & z^3\end{pmatrix}.\)

  The lower-left entry is \(0\), but the target matrix has a \(1\) there:

  \(\begin{pmatrix}8 & -57\\ 1 & 27\end{pmatrix}.\)

  Contradiction. Hence no such \(\mathbf{N}\) exists; the problem is unsolvable.

Answer: \begin{pmatrix} 2 & -3 \\ 0 & 3 \end{pmatrix}


Consider three distinct positive integers \( p \), \( q \), and \( r \) such that \( p < q < r \). They are selected in a way that the following system of equations holds:
  \[
  2u + v = 2003 \quad \text{and} \quad v = |u-p| + |u-q| + |u-r|
  \]
  This system has exactly one solution. Note that \(u,v\) are real numbers. Determine the smallest possible value of \( r \).

Consider three distinct positive integers \( p \), \( q \), and \( r \) such that \( p < q < r \). They are selected in a way that the following system of equations holds:
  \[
  4u + v = 2003 \quad v = |u-p| + |u-q| + |u-r|
  \]
  This system has exactly two solutions.  Note that \(u,v\) are real numbers. Determine the smallest possible value of \( r \).

Set \(f(u)=|u-p|+|u-q|+|u-r|\).
  From \(4u+v=2003\) and \(v=f(u)\) we get

  \(f(u)+4u=2003.\)

  Because \(p<q<r\), \(f(u)\) is piecewise linear with slopes \(-3,-1,1,3\) on the intervals \((-\infty,p], [p,q], [q,r], [r,\infty)\).
  Hence \(f(u)+4u\) has slopes

  \((-3)+4=1,\quad (-1)+4=3,\quad 1+4=5,\quad 3+4=7,\)

  all positive. Therefore the function \(f(u)+4u\) is continuous and strictly increasing on \(\mathbb{R}\). A strictly increasing function can meet the constant value \(2003\) at most once. So the system can never have exactly two solutions \((u,v)\), no matter how you choose \(p<q<r\).

Answer: 1002


Determine the smallest positive integer that is a multiple of 72 and possesses exactly 16 divisors.

Determine the smallest positive integer that is a multiple of 72 and possesses exactly 14 divisors.

The prime factorization of 72 is \(2^3\times 3^2\), which has \((3+1)(2+1)=12\) factors.
  If we multiply by 2, we get \(2^4\times 3^2\), which has \((4+1)(2+1)=15\) factors. Multiplying by any higher power of 2 gives a product with more than 16 positive factors.
  Multiplying by 3 gives \(2^3\times 3^3\), which has \((3+1)(3+1)=16\) positive factors.

  You can see that there is no multiple of 72 has exactly 14 positive divisors.

Answer: 216


Consider the quartic equation given by
  \[
  py^4 + qy^3 + ry^2 + sy + t = 0,
  \]
  where all coefficients are integers. The equation has roots at \(-2\), \(5\), \(9\), and \(-1/3\). If \(t\) is a positive integer, determine the smallest value that \(t\) can take.

Consider the quartic equation given by
  \[
  py^4 + qy^3 + ry^2 + sy + t = 0,
  \]
  where all coefficients are integers. The equation has roots at \(-2\), \(5\), \(9\), and \(-1/3\). If \(t\) is a positive integer, determine the largest value that \(t\) can take.

By taking
  \[
  p = 3k \quad (k \in \mathbb{Z}^+),
  \]
  the quartic can be written as
  \[
  P(y) = k\,(y + 2)(y - 5)(y - 9)(3y + 1),
  \]
  and then
  \[
  t = 30p = 30 \cdot (3k) = 90k.
  \]
  Since \(k\) can be any positive integer, \(t = 90k\) can be made arbitrarily large. Hence there is no largest possible value of \(t\).

Answer: 90


Determine all values of \( y \) that satisfy the equation:
  \[
  \sqrt{y + 3 - 4 \sqrt{y - 1}} + \sqrt{y + 8 - 6 \sqrt{y - 1}} = 1.
  \]

Determine all values of \( y \) that satisfy the equation:
  \[
  \sqrt{y + 3 - 4 \sqrt{y - 1}} + \sqrt{y + 8 - 6 \sqrt{y - 2}} = 1.
  \]

Answer: [5,10]


Consider a scenario where a positive integer \( y \) satisfies the equation
  \[
  1^{y+2} + 2^{y+1} + 3^{y-1} + 4^y = 1170.
  \]
  Determine the value of \( y \).

Consider a scenario where a positive integer \( y \) satisfies the equation
  \[
  1^{y+2} + 2^{y+4} + 3^{y+3} + 4^{y-1} = 1170.
  \]
  Find the value of \( y \).

Answer: 5


Consider a triangle \(XYZ\) where \(\angle X = 45^\circ\). A point \(Q\) is located on the segment \(\overline{YZ}\) such that \(QY = 3\) and \(QZ = 5\). Let \(M\) be the center of the circle that passes through points \(X\), \(Y\), and \(Z\). Find the distance from \(M\) to \(Q\).

Consider a triangle \(XYZ\) where \(\angle Z = 45^\circ\). A point \(Q\) is located on the segment \(\overline{YZ}\) such that \(QY = 3\) and \(QZ = 5\). Let \(M\) be the center of the circle that passes through points \(X\), \(Y\), and \(Z\). Find the distance from \(M\) to \(Q\).

With only \(\angle Z = 45^\circ\) and the division \(YQ = 3\), \(QZ = 5\) on \(YZ\), the position of \(X\) (and hence of the circumcenter \(M\)) is not fully fixed, so \(MQ\) would still depend on the length \(XZ\),
  \[
  MQ = \sqrt{1 + \left(\tfrac{XZ}{\sqrt{2}} - 4\right)^2}.
  \]

Answer: \sqrt{17}


Determine the radius of the circle described by the equation
  \[
  a^2 - 4a + b^2 - 6b - 36 = 0.
  \]

Determine the radius of the circle described by the equation
  \[
  a^2 - 4a + b^2 - 6b + ab - 36 = 0.
  \]

There is an \(ab\) term in the equation, so it is an ellipse, not a circle, and hence does not have radius.

Answer: 7


Consider the polynomial equation \(y^3 + 8y^2 - 4y + a = 0\). This equation has three solutions, and one of these solutions is equal to the sum of the other two solutions. Determine the value of the constant \(a\).

Consider the polynomial equation \(y^3 + 8y^2 - 4y + a = 0\). This equation has three solutions, where one solution is equal to the sum of the other two solutions, and another solution is equal to the product of the other two solutions. Determine the value of the constant \(a\).

Let the three roots be \(r_1, r_2, r_3\). By Vieta’s formulas for \(y^3 + 8y^2 - 4y + a = 0,\) we have
  \[
  r_1 + r_2 + r_3 = -8, \quad
  r_1r_2 + r_2r_3 + r_3r_1 = -4, \quad
  r_1r_2r_3 = -a.
  \]

  WLOG, assume \(r_1 = r_2 + r_3\) and \(r_2 = r_1r_3\). Combine these with the first Vieta equation.
  One can obtain
  \[
  r_1 = -4, \quad
  r_2 = -\tfrac{16}{3}, \quad
  r_3 = \tfrac{4}{3}.
  \]
  However, the three roots do not satisfy the second equation in Vieta’s formulas.
  Therefore, no such value of \(a\) exists.

Answer: -80


Determine the count of the initial 200 positive integers that leave a remainder of 1 when divided by 9.

Among the initial \(n\) positive integers, there are exactly 100 numbers that leave the same remainder as \(n+1\) when divided by 9. Identify the 10th smallest possible value of n.

There are only nine values of \(n\) that satisfy the given conditions (900-908), so there is no tenth smallest \(n\). All the \(k\) that satisfy \(k\equiv n+1\pmod9\) can be represented as \(k=n+1-9m\). Since there are 100 such \(k\) among \(1,2,\dots,n\), \(m\) can be \(1,2,\dots,100\), and \(m=101\) would give \(k\le0\), which is out of range. Thus \(1 \le n+1 - 9\cdot100\) and \(n+1 - 9\cdot101 \le 0\), and we can obtain \(900 \le n \le 908\), exactly nine values, and no tenth exists.

Answer: 23


Consider an infinite sequence of positive integers \( n \) that satisfy the equation
  \[\cos^2 (n^2 + 36)^\circ = 1.\]
  Identify the two smallest values of \( n \) that solve this equation, and list them separated by commas.

Determine the second smallest positive integer \( n \) such that \( \cos^2 (n^2 + c)^\circ = 1 \) for a fixed constant \(c\), given that the smallest solution (the smallest such \(n\) ) is 12 and the 100th smallest solution is 114514.

For any solution \(k\), we must have \(k^2 + c = 180m,\quad m\in\mathbb Z,\) since \(\cos^2\theta=1\) iff \(\theta\equiv0\pmod{180^\circ}\). Since \(k=12\) and \(k=114514\) satisfiy the equation above, there exist integers \(m_1,m_2\) with
  \(
  12^2 + c = 180 m_1,\qquad
  114514^2 + c = 180 m_2.
  \)
  Subtracting gives \(114514^2 - 12^2 = 180 (m_2 - m_1)=180 \cdot 99.\) But the left-hand side ends in digit 2, whereas the right hand side ends in 0.

Answer: 46009


Consider the number 49, which can be expressed as a sum of smaller perfect squares in several ways. One such expression involves six terms: \(25 + 9 + 9 + 4 + 1 + 1\). If each term is a perfect square ranging from 1 to 48, determine the minimum number of perfect square terms less than 49 that sum up to 49.

Consider the number 49, which can be expressed as sums and differences of smaller perfect squares in various configurations. One such expression involves seven terms: \(25 + 25 - 16 + 9 + 4 + 1 + 1\). If each term is a perfect square ranging from 1 to 48, determine the maximum number of such terms (with both additions and subtractions permitted) required to reach the target value of 49?

Since \(1^2 = 1\) is allowed, one can keep adding and subtracting 1 indefinitely while maintaining the total sum at 49. This leads to arbitrarily large numbers of terms.

Answer: 3


Consider the polynomial \( q(y) = y^3 + dy^2 + ey + f \), where \( d \), \( e \), and \( f \) are complex numbers. Assume that the following conditions hold true:
  \[
  q(2009 + 9002\pi i) = q(2009) = q(9002) = 0
  \]
  Determine how many zeros of the polynomial \( y^{12} + dy^8 + ey^4 + f \) are not real numbers.

Consider the polynomial \( q(y) = y^3 + dy^2 + ey + f \), where \( d \), \( e \), and \( f \) are complex numbers. It is given that:
  \[
  q(2009 + 9002\pi i) = q(2009) = 0
  \]
  If the polynomial \( y^{12} + dy^8 + ey^4 + f \) has 6 distinct non-real zeros, identify 4 possible sets of values for \( d \), \( e \), and \( f \).

Let the new polynomial be \(Q(y)\), and three roots of \(q(y)\) as
  \(r_1 = 2009 + 9002\pi i\),
  \(r_2 = 2009\),
  and an unknown root \(r_3\).

  Since \(Q(y) = q(y^4)\), \(Q(y)\) has four non-real roots coming from \(y^4 = r_1\), and two purely imaginary roots \(\pm i \cdot 2009^{1/4}\), coming from \(y^4 = r_2\).

  To force exactly six non-real zeros, the unknown third root \(r_3\) of \(q\) must introduce no further non-real fourth-roots. This happens only if \(r_3\) is \(0\), \(2009\), or \(2009 + 9002\pi i\).

  That yields exactly three possibilities for the set of coefficients and makes a fourth set impossible.

Answer: 8


Assuming
  \[\sin \left( \alpha + \frac{\pi}{4} \right) = \frac{1}{3}\]
  calculate \(\sin(2\alpha)\).

Assuming \(\alpha\) is a multiple of \(\pi/4\), if
  \(
  \sin \left( \alpha + \frac{\pi}{4} \right) = \frac{1}{3},
  \)
  calculate \(\sin(2\alpha)\).

Since \(\alpha\) is an integer multiple of \(\frac{\pi}{4}\), \(\alpha + \frac{\pi}{4}\) is also a multiple of \(\frac{\pi}{4}\). But \(\sin(k\frac{\pi}{4})\) for any integer \(k\) only takes values within \(0, \pm\frac{1}{\sqrt{2}}, \pm1\), never \(\frac{1}{3}\). So the equation has no solution, and \(\sin 2\alpha\) is undefined.

Answer: -\frac{7}{9}


Consider a square with the points \((1, -2)\) and \((-4, 10)\) that are adjacent to each other. Determine the perimeter around the square.

Consider a rectangle with the points \((1, 7)\) and \((4, 10)\) that are diagonally opposite corners. If one of the rectangle's edges measures \(\frac{\sqrt{2}}{k}\), where \(k\) is a positive integer, determine all potential points where the rectangle crosses the \(x\)-axis.

All four vertices must lie on the circle having the segment from \((1,7)\) to \((4,10)\) as a diameter. Its center is \((2.5,8.5)\) and its radius is \(\frac{3\sqrt2}{2}\), so its lowest \(y\) coordinate is \(8.5-\frac{3\sqrt2}{2}\approx6.38>0\). Therefore the circle never meets the \(x\) axis, and no such rectangle can intersect \(y=0\).

Answer: 52


Determine the value of \(a+b+c\) given the equations
  \[
  6a + 5b - 2c = -4,
  \quad
  4a + 5b + 12c = 54.
  \]

Determine the value of \(a + b + e^c\) given the equations
  \[
  6a + 5b - 2e^c = -4,
  \quad
  4a + 4b + 13e^c = 54.
  \]

Let \(t = e^c,\; S = a + b + t.\) Then the given equations and the target sum can be written as dot-products in \(\mathbb{R}^3\):
  \[
  (6,\,5,\,-2)\cdot(a, b, t) = -4, \quad
  (4,\,4,\,13)\cdot(a, b, t) = 54, \quad
  (1,\,1,\,1)\cdot(a, b, t) = S.
  \]
  The first two row-vectors span a 2-dimensional subspace of \(\mathbb{R}^3\).
  If \((1,1,1)\) lay in their span, we could find scalars \(\alpha, \beta\) so that
  \[
  (1,1,1) = \alpha\,r_1 + \beta\,r_2.
  \]
  Equating components gives the system
  \[
  6\alpha + 4\beta = 1, \quad
  5\alpha + 4\beta = 1, \quad
  -2\alpha + 13\beta = 1.
  \]
  The first two equations force \(\alpha = 0\) and \(\beta = \tfrac{1}{4}\), but then
  \[
  -2\cdot 0 + 13\cdot \tfrac{1}{4} = \tfrac{13}{4} \ne 1.
  \]
  Thus \((1,1,1) \notin \mathrm{span}\{r_1, r_2\}\), so no combination of the first two equations can yield \(a + b + t = S\).

Answer: 5


Consider three positive real numbers \(x,\) \(y,\) and \(z\). Determine the smallest possible value of the expression
  \[\frac{x + y}{z} + \frac{x + z}{y} + \frac{y + z}{x}.\]

Consider three nonzero real numbers \(x,\) \(y,\) and \(z\). Determine the smallest possible value of the expression
  \[\frac{x + y}{z} + \frac{x + z}{y} + \frac{y + z}{x}.\]

The expression is unbounded below. For instance, set \( x = y = -1,\quad z = \varepsilon > 0 \).
  Then
  \[
  \frac{x + y}{z} = \frac{-2}{\varepsilon} \longrightarrow -\infty \quad (\varepsilon \to 0^+),
  \]
  while the other two terms remain finite. Hence the total sum can be made arbitrarily negative.

Answer: 6


Determine a pair of values \((x, y)\) that satisfy the following equations:
  \[
  \begin{aligned}
  5x &= -7 - 2y, \\
  3x &= 4y - 25.
  \end{aligned}
  \]

Determine a pair of values \((x, y)\) that satisfy the following equations:
  \[
  \begin{aligned}
  5x &= -7 - 2y^2, \\
  3x^2 &= 4y - 25.
  \end{aligned}
  \]

Solve the first equation for
  \[
  x = \frac{-7 - 2y^2}{5}.
  \]
  Substituting into
  \[
  3x^2 = 4y - 25
  \]
  gives
  \[
  12y^4 + 84y^2 - 100y + 772 = 0.
  \]

  Call
  \[
  L(y) = 12y^4 + 84y^2 + 772, \qquad R(y) = 100y.
  \]
  Notice that \(L(y)\) can be considered as a quadratic in \(w\),
  \[
  L(w) = 12w^2 + 84w + 772, \quad w \ge 0,
  \]
  whose vertex lies at \(w = -3.5\). Hence \(L(w)\) is strictly increasing for \(w \ge 0\), and in particular
  \[
  L(y) = L(y^2) \ge L(0) = 772 \quad \forall\, y \in \mathbb{R}.
  \]

  If \(y \le 7.72\), then
  \[
  R(y) = 100y \le 100 \cdot 7.72 = 772 \le L(y),
  \]
  so \(L(y) > R(y)\).
  If \(y > 7.72\), then
  \[
  L(y) \ge 12y^4 > 12y^3 > 100y = R(y)
  \]
  (since \(12y^2 > 100\)).

  In every case, \(L(y) > R(y)\), so the two curves never meet.
  Therefore, the system has no real solution.

Answer: (-3,4)


Consider three complex numbers \( x, y, z \) that satisfy the following conditions:
  \[
  xy + 4y = -16, \\
  yz + 4z = -16, \\
  zx + 4x = -16.
  \]
  Determine all possible values for the product \( xyz \), listing them separated by commas.

Consider three positive real numbers \( x, y, z \) that satisfy the following conditions: \( xy + 4y = -16, yz + 4z = -16, zx + 4x = -16, x = y = z. \) Determine all possible values for the product \( xyz \), separated by commas.

Substituting \(x=y\) into \(xy+4y=-16\) yields \(x^2+4x+16=0\). The discriminant is \(\Delta = 4^2 - 4\cdot1\cdot16 = 16 - 64 = -48 < 0\). Thus, there is no real solution.

Answer: 64


Consider a triangle \(XYZ\) where the side lengths satisfy the equation:
  \[ x^4 + y^4 + z^4 = 2z^2 (x^2 + y^2).\]
  Determine the possible measures of \(\angle Z\) in degrees, and list them separated by commas.

Consider a triangle \(XYZ\) where the side lengths satisfy the equation:
  \[ x^4 + y^4 + z^4 = 2x^2 (z^2 + y^2).\]
  Determine the measure of \(\angle Z\) in degrees, and list it separated by commas.

The given conditions are not enough to compute the value of angle \(Z\). Substituting the Law of Cosines \(z^2 = x^2 + y^2 - 2xy\cos Z\) into the given identity \(x^4 + y^4 + z^4 = 2x^2(y^2 + z^2)\) to obtain
 \[
 x^4 + y^4 + \bigl(x^2 + y^2 - 2xy\cos Z\bigr)^2
 = 2x^2\bigl[y^2 + \bigl(x^2 + y^2 - 2xy\cos Z\bigr)\bigr].
 \]
  Expand the square, cancel identical terms on both sides, then divide both sides by \(2x^2y^2\), and define \(r = \frac{y}{x} > 0\), to obtain

 \[
 2\cos^2 Z - 2r\cos Z + (r^2 - 1) = 0.
 \]
 This is a quadratic equation in \(\cos Z\). Its discriminant is \(\Delta = 8 - 4r^2\). Thus \(\Delta \ge 0\) exactly when \(0 < r \le \sqrt{2}\). Since every step to derive this quadratic is both sufficient and necessary, for each \(r \in (0,\sqrt{2}]\) the identity holds and the quadratic in \(\cos Z\) has one or two real solutions. Therefore, \(\angle Z\) is underdetermined.

Answer: 45^\circ, 135^\circ


Consider a road described by the equation \(y = 4x - 7\). Another road, represented by the equation \(y = ax + b\), intersects the first road at a right angle at the point \((2,1)\). Determine the \(y\)-value of the location on this second road where the \(x\)-value is 6.

Consider a road described by the equation \(y = 4x - 8\). Another road, represented by the equation \(y = ax + b\), intersects the first road at a right angle at the point \((2,1)\). Determine the \(y\)-value of the location on this second road where the \(x\)-value is 6.

Line \(l\) has the equation \(y = 4x - 8\), and line \(m\) has the equation \(y = ax + b\). The question claims they intersect at \((2,1)\), but this point does not lie on \(l\) because substituting \(x = 2\) into \(y = 4x - 8\) gives \(y = 0 \neq 1\).

Answer: 0


Consider a quadratic equation given by \(\frac12y^2 + 99y + k = 0\). If the solutions to this equation are \(y = -99 + \sqrt{8001}\) and \(y = -99 - \sqrt{8001}\), determine the value of \(k\).

Consider a quadratic equation given by \(y^2 + 99y + k = 0\). If the solutions to this equation are \(y = -99 + \sqrt{8001}\) and \(y = -99 - \sqrt{8001}\), determine the value of \(k\).

The sum of the two roots must equal \(-b/a=-99\) using Vieta's formula, but the sum of two given roots is -198, leading to contradiction.

Answer: 900


Determine the maximum value of \( y \) that satisfies the equation \(\lfloor y \rfloor = 5 + 100 (y - \lfloor y \rfloor)\), where the fractional part of \( y \) is defined as \( y - \lfloor y \rfloor \).

Assuming there are exactly three integer values of \( y \) that satisfy the equation \(\lfloor y \rfloor = 5 + b (y - \lfloor y \rfloor)\), where the fractional part of \( y \) is \( y - \lfloor y \rfloor \), identify all possible integer values for \( b \)

There is only one integer solution 5. For any integer \(y\),\(y - \lfloor y \rfloor)\) is always 0, so the equation becomes \(\lfloor y \rfloor = 5\). So 5 is the only integer solution. Therefore, it is impossible to have three integer solutions.

Answer: 104.99


Consider a circular path described by the equation \(a^2 + b^2 = -2a - 10b - 16\). Determine the sum of the coordinates of the circle's center, denoted as \(a + b\).

Consider a circular path described by the equation \(a^2 + b^2 = -2a - 10b - 28\). Determine the sum of the coordinates of the circle's center, denoted as \(a + b\).

The equation can be transformed into \((a + 1)^{2}+(b + 5)^{2} = -2\), where \(-2\) cannot be the radius of a circle.

Answer: -6


Determine the equation representing the directrix of the quadratic curve given by \( f(x) = x^2 \).

Determine all lines that are tangent to the quadratic curve \( f(x) = x^2 \) and intersect at the point \((0, 1)\).

\((0, 1)\) lies inside the parabola \(f(x) = x^2\), so there is no tangent line passing through it.

Answer: y = -\frac{1}{4}


8 individuals are arranged in a circle. If Clara and David refuse to sit adjacent to each other, list all possible distinct seating arrangements.   (Seating arrangements that can be rotated into each other are considered the same.)

Arrange 8 people around a round table with the following conditions:
  1. Clara and David must sit next to each other.
  2. David and George must have exactly two people sitting between them.
  3. Clara and George must have exactly four people sitting between them.  List all possible distinct seating arrangements.
  (Seating arrangements that can be rotated into each other are considered the same.)

It is impossible to satisfy all three conditions simultaneously. Assume we label the 8 seats clockwise as positions 1 through 8, and -without loss of generality- place Clara (C) in seat 1 and David (D) in seat 2 (since rotations are considered equivalent). If David and George (G) must have exactly two people sitting between them clockwise, then G would be in seat 5. In that case, the number of people between Clara and G is 3- not 4 as required. If instead the two people are counter-clockwise between D and G, then G would be in seat 7. Now, only one person sits between C and G.

Answer: 3600


Determine the number of vertical asymptotes present in the plot of the function
\[
f(x) = \frac{x - 3}{x^2 + 7x - 30}.
\]

List all horizontal asymptotes, not coinciding with the x-axis, for the graph of the function
\[
f(x) = \frac{x - 3}{x^2 + 7x - 30}.
\]

First, factor the denominator:
\[
x^2 + 7x - 30 = (x + 10)(x - 3).
\]

Now canceling the common factor, the function becomes
\[
f(x) = \frac{1}{x + 10}, \quad \text{for } x \ne 3.
\]

As \( x \to \pm\infty \), we have \( f(x) \to 0 \).
So the only horizontal asymptote is \( y = 0 \).
There is no horizontal asymptotes exist for the function that do not coincide with the x-axis (since \( y = 0 \) is the x-axis).

Answer: 1


Consider a real number \( y \). Define the function
  \[
  g(y) = \max \{|y + 15|, |y + 12|, |y + 8|, |y - 1|, |y - 3|, |y - 19|\}.
  \]
  Determine the smallest possible value of \( g(y) \).

Consider a negative number \( y \). Define the function
  \[
  g(y) = \max \{|y + 15|, |y + 12|, |y + 8|, |y - 1|, |y - 3|, |y - 19|\}.
  \]
  Determine the smallest possible value of \( g(y) \).

Due to the negative \(y\) constraint, there is an infimum of \(g(y)\), which is \(g(0)\), but it cannot be attained.
  The meaning of \(g(y)\) is the largest among the distances from the point \(y\) to the points \(-15, -12, -8, 1, 3, 19\).

  For \(y < 0\), one checks that
  \[
  g(y) = |y - 19|,
  \]
  since \(|y - 19|\) exceeds all the other distances.

  As \(y\) approaches \(0\) from below, \(|y - 19|\) decreases toward 19, but cannot equal 19.
  Therefore, \(g(y)\) has infimum \(19\), but no minimum value.

Answer: 17


Determine the least positive integer \( a \) such that the polynomial
  \[
  y^2 + ay + 2008
  \]
  can be expressed as a product of two binomials, each with integer coefficients.

Determine the smallest integer \( a \) for which the quadratic
  \[
  y^2 + 2y + a
  \]
 can be expressed as a product of two binomials, each with integer coefficients.

\(a\) can be infinitely small, so it does not have a minimum. A factorization into integers must satisfy
  \[
  y^2 + 2y + a = (y + m)\bigl(y + (2 - m)\bigr),
  \]
  so \( a = m(2 - m) \).
  As \( m \) ranges over all integers, the product \( m(2 - m) \) takes arbitrarily large negative values.
  Hence there is no least integer \( a \).

Answer: 259


Determine the smallest possible value of the expression \(\frac{9t^2\cos^2 t + 4}{t\cos t}\) where \(0 < t < \pi\).

Determine the maximum of the expression \(\frac{9t^2\cos^2 t + 4}{t\cos t}\) where \(0 < t < \pi\).

Write
\[
f(t)=\frac{9t^2\cos^2 t+4}{t\cos t}=9t\cos t+\frac{4}{t\cos t},\qquad 0<t<\pi,
\]
(notice \(f\) is undefined at \(t=\tfrac\pi2\) where \(\cos t=0\), so the domain splits into \((0,\tfrac\pi2)\) and \((\tfrac\pi2,\pi)\).)

- As \(t\to0^+\), \(\cos t\to1\) and
  \[
  f(t)\sim \frac{4}{t}\to +\infty.
  \]
- As \(t\to\tfrac\pi2^-\), \(\cos t\to0^+\) and the term \(\dfrac{4}{t\cos t}\to +\infty\).

Therefore \(f(t)\) is unbounded above on \((0,\pi)\) (its supremum is \(+\infty\)), so there is no finite maximum attained on the interval.

Answer: 12


Emily's brother prepared \(3\) dozen cupcakes, with one-third having vanilla frosting, one-fourth topped with sprinkles, one-sixth containing lemon zest, and one-twelfth garnished with coconut flakes. What is the smallest possible number of cupcakes that had none of these toppings?

Emily's brother prepared \(3\) cupcakes, with one-third having vanilla frosting, one-fourth topped with sprinkles, one-sixth containing lemon zest, and one-twelfth garnished with coconut flakes. What is the smallest possible number of cupcakes that had none of these toppings?

The number of cupcakes must be an integer, but the fractions given \((1/3, 1/4, 1/6, 1/12)\) do not yield integer values when applied to only 3 cupcakes.

Answer: 6


Consider three real numbers \(x\), \(y\), and \(z\) such that their sum is zero, i.e., \(x + y + z = 0\), and the sum of their squares is \(4\), i.e., \(x^2 + y^2 + z^2 = 4\). Determine the value of \(x^4 + y^4 + z^4\).

Consider three real numbers \(x\), \(y\), and \(z\) such that their sum is zero, i.e., \(x + y + z = 0\), and the sum of their squares is \(4\), i.e., \(x^2 + y^2 + z^2 = 4\). Determine the value of \(x^4 + y^4 + z^3\).

The constraints \(x + y + z = 0\), \(x^2 + y^2 + z^2 = 4\) are symmetric in \(x,y,z\).
  \(E = x^4 + y^4 + z^3\) is not symmetrical it singles out \(z\). Thus, by relabeling which variable we call \(z\), we obtain different values of \(E\).

Answer: 8


Assume there is an angle \( y \) such that the expression \(\cos y - 4 \sin y\) equals 1. Determine the possible outcomes for \(\sin y + 4 \cos y\), express the results separated by commas.

Assume there is an angle \( y \) such that \(\cos y - 4 \sin y = 1\) and \(\sin y \cdot \cos y = 0.5\). Determine the possible outcomes for \(\sin y + 4 \cos y\), express the results separated by commas.

There's implicit constraint that \(sin^2 y+cos^2 y =1\). Solving the equation system would violate this constraint

Answer: -4,4


Consider the quadratic equation \( by^2 + 32y + d = 0 \) which has a single root. If the sum of \( b \) and \( d \) is 130, and \( b \) is greater than \( d \), determine the ordered pair \((b, d)\).

Consider the quadratic equation \( by^2 + 32y + d = 0 \) which has a single root. If the sum of \( b \) and \( d \) is 130 and \( b \) is greater than \( d \), find the maximum value of this quadratic function over the interval \((-8, 8)\).

The quadratic equation \(by^2 + 32y + d = 0\) has a single root, so the discriminant \(32^2 - 4bd = 0\) implies \(bd = 256\). Combined with \(b + d = 130\) and \(b > d\), we solve for \(b\) and \(d\). Since \(b > 0\), the parabola opens upward, and the quadratic function \(f(y) = by^2 + 32y + d\) has no maximum over the open interval \((-8, 8)\); it has a minimum at its vertex.

Answer: (128,2)


Suppose that \(p\) and \(m\) are two real numbers, \(i - 3\) and \(p\) are two solutions to the equation \(mx^3 + 9x^2 + mx - 30 = 0\). What is the value of \(p + m\)?

Suppose that \(p\) and \(m\) are two real numbers, \(4\) and \(p\) are two solutions to the equation \( mx^3 + 9x^2 + mx - 30 = 0\). What is the center of symmetry of the function \(f(x) = mx^3 + 9x^2 + mx - 30, (x<50) \)?

The cubic function \( f(x) = mx^3 + 9x^2 + mx - 30 \) is defined only for \( x < 50 \). A cubic function may have a point of symmetry (not a center of symmetry in the usual sense), but the restricted domain \( x < 50 \) makes symmetry analysis invalid or irrelevant since the function is not defined over all real numbers.

Answer: \frac{7}{2}


What is the value of \(\sin(600^\circ)\)?

For \(\sin{\alpha} = \sin{600^{\circ}}\), and \(\alpha\) is located in the first quadrant (meaning \(0^{\circ} < \alpha < 90^{\circ}\)). What is the value of \(\alpha\)?

\[
\sin{\alpha} = \sin{600^\circ} = \sin{240^\circ} = -\frac{\sqrt{3}}{2}.
\]
In the first quadrant \((0^\circ<\alpha<90^\circ)\), \(\sin{\alpha} > 0\). However, \(-\frac{\sqrt{3}}{2} < 0\), so no solution exists for \(\alpha\) in the first quadrant satisfying \(\sin{\alpha} = \sin{600^\circ}\).

Answer: -\frac{\sqrt{3}}{2}


Simplify the expression by eliminating the radicals from the denominator: \(\frac{1}{\sqrt{2}+\sqrt{8}+\sqrt{32}}\). The result can be expressed in the form \(\frac{\sqrt{C}}{D}\), where \(C\) and \(D\) are integers. Determine the smallest possible sum of \(C\) and \(D\).

Simplify the expression by removing the radicals from the denominator: \(\frac{1}{\sqrt{2}+\sqrt{4}+\sqrt{32}}\). The result can be expressed in the form \(\frac{\sqrt{C}}{D}\), where \(C\) and \(D\) are integers. Determine the smallest possible sum of \(C\) and \(D\).

The fully rationalized form of this expression is
  \[
  \frac{1}{\sqrt{2} + \sqrt{4} + \sqrt{32}}
  = \frac{5\sqrt{2} - 2}{46}.
  \]
  In particular, the numerator is \(5\sqrt{2} - 2\) and the denominator is \(46\). It's impossible to combine \(5\sqrt{2} - 2\) into a single term of the form \(\frac{\sqrt{C}}{D}\) with \(C,D\) integers.

Answer: 16


What is the value of \(y\) such that \(3^{2y} = \sqrt{27}\)?

What is the value of \(y\) such that
  \[
  3^{2y} = \log_{3}\!\left(\tfrac{1}{2}\right)?
  \]

For any real \(y\), \(3^{2y} > 0\), whereas
  \[
  \log_{3}\!\left(\tfrac{1}{2}\right)
  = \log_{3}(1) - \log_{3}(2)
  = 0 - \log_{3}(2)
  < 0.
  \]
  A positive number can never equal a negative one, so there is no real solution to
  \[
  3^{2y} = \log_{3}\!\left(\tfrac{1}{2}\right).
  \]

Answer: \frac{3}{4}


Identify the unique positive real value of \( y \) such that \(\displaystyle \frac{y-4}{9} = \frac{4}{y-9}\).

Identify the unique positive real value of \( y \) for which \(\frac{y^2 - 17y + 52}{9y - 117} = \frac{4}{y-9}\).

Simplifying the equation leads to \(y(y-13) = 0\), but \(y = 13\) is excluded due to the denominator \(9y - 117\). Thus, the only solution is \(y = 0\), which is not positive.

Answer: 13


Consider a quadratic equation given by \(\frac{3}{2}y^2 + 11y + k = 0\). If the solutions to this equation are \(y = \frac{-11 \pm \sqrt{7}}{3}\), determine the value of \(k\).

Consider a quadratic equation given by \(\frac{3}{2}y^2 + my + 21 = 0\). If the solutions to this equation are \(y = \frac{-11 \pm \sqrt{7}}{3}\), determine the value of \(m\).

Answer: 19


Consider the polynomial \( x^4 - 6x^3 + 16x^2 - 25x + 10 \). When it is divided by the quadratic expression \( x^2 - 2x + k \), the remainder is given as \( x + a \). Determine the ordered pair \((k, a)\).

Consider the polynomial \( x^4 + 2x^3 - 181x^2 - 182x + 7920 \). When this polynomial is divided by the quadratic \( x^2 + 3x - 88 \), the remainder is given as \( x + a \). Calculate the value of \( a \).

\[
  P(x)=x^{4}+2x^{3}-181x^{2}-182x+7920,\qquad
  D(x)=x^{2}+3x-88=(x+11)(x-8).
  \]

  Assume the remainder is \(R(x)=x+a\).
  Because the roots of \(D\) are \(x=8\) and \(x=-11\), we must have
  \[
  P(8)=R(8),\qquad P(-11)=R(-11).
  \]

  Compute
  \[
  P(8)=0,\qquad P(-11)=0.
  \]

  Hence
  \[
  0=P(8)=8+a\;\Longrightarrow\;a=-8,\qquad
  0=P(-11)=-11+a\;\Longrightarrow\;a=11,
  \]
  which is impossible.
  Therefore no constant \(a\) satisfies the condition, so the stated remainder cannot be \(x+a\).

  Indeed,
  \[
  P(x)=(x^{2}+3x-88)(x^{2}-x-90),
  \]
  so the true remainder is \(0\).

Answer: (5,-5)


Suppose \( b \) is a constant such that \( 4y^2 - 12y + b \) can be written as the square of a binomial. What is the value of \( b \)?

Suppose \( b \) is a rational constant such that \( 3y^2 - 10y + b^2 \) can be written as the square of a binomial. What is the value of \( b \).

For \(3y^2 - 10y + b^2\) to be a perfect square, it must equal \((my+n)^2\).
Comparing coefficients gives
\[
m^2 = 3, \quad 2mn = -10, \quad n^2 = b^2.
\]
Thus \(m = \pm \sqrt{3}\) (irrational). From \(2mn = -10\), we get \(n = -\tfrac{5}{m}\), which is also irrational.
Since \(n^2 = b^2\), this forces \(b = \pm n\), so \(b\) is irrational as well.
Because the problem requires \(b\) to be rational, no such \(b\) exists.

Answer: 9


A large truck unloaded gravel at a work site. The gravel formed into a cone-shaped mound with a base diameter of \(8\) feet and a height that was \(75\%\) of this diameter. Calculate the volume of the gravel in the mound in cubic feet, and express your answer in terms of \(\pi\).

A large truck unloaded gravel at a work site. The gravel formed into a cone-shaped mound with a base diameter of \(8\) feet, and its vertical cross section is an equilateral triangle. The height of the mound is \(150\%\) of the radius. Calculate the volume of gravel in the mound in cubic feet, and express your answer in terms of \(\pi\).

The height being 1.5 times the radius contradicts the vertical cross-section being an equilateral triangle.

Answer: 32 \pi


Imagine three integers, each greater than 1, whose product is \(27,000\). These integers do not share any common factors other than 1 when taken in pairs. What is the total sum of these integers?

Imagine three integers, each greater than 1, whose product is \(419,904\). These integers are also such that any two of them have no common divisors other than 1. What is the total sum of these integers?

The number \(419,904\) factors as \(2^6 \times 3^8\). For three integers greater than 1 with a product of \(419,904\) to be pairwise coprime, each must be a distinct power of a single prime (since \(2\) and \(3\) are the only primes available). However, with only two primes, it is impossible to construct three such integers (as at least one would have to include both primes or be \(1\)). Thus, no such triplet exists.

Answer: 160


Consider the following table that lists the outputs of a function \( g(y) \):
  \[
  \begin{array}{|c||c|c|c|c|c|}
  \hline
  y & 3 & 4 & 5 & 6 & 7 \\
  \hline
  g(y) & 10 & 17 & 26 & 37 & 50 \\
  \hline
  \end{array}
  \]
  Determine the value of \( g^{-1}\left(g^{-1}(50) \times g^{-1}(10) + g^{-1}(26)\right) \).

Consider the following table that lists the outputs of a function \( g(y) \):
  \[
  \begin{array}{|c||c|c|c|c|c|c|c|}
  \hline
  y & 5 & 4 & 3 & 6 & 7 & 5 & 8 \\
  \hline
  g(y) & 26 & 17 & 10 & 37 & 50 & 10 & 40 \\
  \hline
  \end{array}
  \]
  Determine the value of \( g^{-1}\left(g^{-1}(50) \times g^{-1}(10) + g^{-1}(26)\right) \).

The inverse \(g^{-1}\) assumes \(g\) is one-to-one (injective).
But the table shows that \(g\) is not injective:

- \(g(5)=26\) and also \(g(5)=10\) (same input, different outputs — contradiction).
- \(g(3)=10\) and \(g(5)=10\) (different inputs, same output).

Answer: 5


Each letter in the alphabet is given a random integer score, with the letter \( H \) assigned a score of 10. The score of a word is the total of the scores of its letters. If the word \( MATH \) scores 35 points, \( TEAM \) scores 42 points, and \( MEET \) scores 38 points, determine the score assigned to the letter \( A \).

Each letter in the alphabet is given a random integer score, with the letter \( H \) assigned a score of 10. The score of a word is the total of the scores of its letters. If the word \( MATH \) scores 34 points, \( TEAM \) scores 42 points, \( MAT \) scores 25 points, and \( MEET \) scores 38 points, determine the score assigned to the letter \( A \).

From \( MATH = 34 \) and \( H = 10 \), we see that \( MAT = 24 \), which contradicts \( MAT = 25 \).

Answer: 21


A chocolate truffle consists of a spherical chocolate center and a conical wrapper with a base diameter equal to the sphere's diameter. If the chocolate melts, it will perfectly fill the conical wrapper. Assume that the melted chocolate occupies \(75\%\) of the volume of the solid chocolate. What is the ratio of the height of the cone to its base radius? Express the answer in the form \(a:b\). (Note: The volume of a cone with radius \( r \) and height \( h \) is \(\frac{\pi r^2 h}{3}\), and the volume of a sphere with radius \( r \) is \(\frac{4\pi r^3}{3}\).)

A chocolate truffle consists of a spherical chocolate center and a conical wrapper. The diameter of the wrapper's base matches the diameter of the chocolate sphere, and the "equatorial plane" of the sphere, which is the largest cross-section through its center, lies exactly on the base of the cone. This means that when the cone's tip is pointed downward, half of the chocolate sphere is above the cone's base. When the chocolate melts, it will completely fill the cone. It is given that the melted chocolate occupies \(20\%\) of the volume of the solid chocolate. What is the ratio of the height of the cone to its base radius?  Express the answer in the form \(a:b\). (Note: The volume of a cone with radius \( r \) and height \( h \) is \(\frac{\pi r^2 h}{3}\), and the volume of a sphere with radius \( r \) is \(\frac{4\pi r^3}{3}\).)

Let the sphere radius be \(r\), the cone base radius also \(r\), and cone height \(h\).  The volume condition is
\[
\frac{1}{3}\pi r^2 h = 0.2 \cdot \frac{4}{3}\pi r^3
\quad\Rightarrow\quad h = 0.8r.
\]
For geometric fit, we require \(h \geq r\).
But \(0.8r < r\).

Answer: 3:1


Determine the inverse of the function given by
  \[
  g(y) = \frac{2y + 5}{y - 11}.
  \]

Determine the inverse of the function \( g(y) \) given by
  \[
  g(y) = \begin{cases}
  \frac{2y + 5}{y - 11}, & y > 11 \\
  \frac{3y - 6}{y - 11}, & y < 11
  \end{cases}
  \]

Because \(g\) is not one to one, its inverse is not a function. Swapping \(x\) and \(y\):

  For \(y>11\),
  \[
  x=\frac{2y+5}{y-11}\;\Longrightarrow\; y=\frac{11x+5}{x-2},\quad x>2
  \]

  For \(y<11\),
  \[
  x=\frac{3y-6}{y-11}\;\Longrightarrow\; y=\frac{11x-6}{x-3},\quad x<3
  \]

  So the inverse relation is
  \[
  g^{-1}(x)=
  \begin{cases}
  \dfrac{11x+5}{x-2}, & x>2\\[6pt]
  \dfrac{11x-6}{x-3}, & x<3
  \end{cases}
  \]
  On \(2<x<3\) both formulas apply, giving two values, so no single-valued inverse function exists.

Answer: \frac{11y + 5}{y - 2}


Consider a right circular cone inscribed in a right circular cylinder. The cylinder has a total volume of \(72\pi\) cubic centimeters. Determine the volume, in cubic centimeters, of the region that lies within the cylinder but outside the cone. Provide your answer in terms of \(\pi\).

Consider a right circular cone inscribed in a right circular cylinder. The cylinder has a total volume of \(72\pi\) cubic centimeters. The base area of the cone is \(24\pi\) square centimeters, and its height is 4 centimeters. Determine the volume, in cubic centimeters, of the region that lies within the cylinder but outside the cone. Provide your answer in terms of \(\pi\).

The cylinder has a height of 3, the cone has a height of 4, and it cannot be placed inside.

Answer: 48\pi


Consider the sequence of numbers starting from 1 and ending at 99, along with an additional number \( y \). The mean of these numbers is given to be \( 100y \). Determine the value of \( y \).

Consider the sequence of integers from 1 to 49, skipping 50, and continuing from 51 to 99, along with a integer \( y \). If the mean of these numbers equals \( 100y \),  find the value of \( y \).

Solve for the average, \(y = \frac{4900}{9899}\), but note it is not an integer.

Answer: \frac{50}{101}


Consider a function \( h(y) \) that is the inverse of another function \( k(y) \), so \( h(y) = k^{-1}(y) \). Given that \( h(-15) = 0 \), \( h(0) = 3 \), \( h(3) = 9 \), and \( h(9) = 20 \), determine the value of \( k(k(9)) \).

Consider a function \( h(y) \) that is the inverse of another function \( k(y) \), so \( h(y) = k^{-1}(y) \). Given that \( h(-15) = 0 \), \( h(0) = 3 \), \( h(3) = 10 \), and \( h(9) = 20 \), determine the value of \( k(k(9)) \).

From the given values of \(h\),

  \(
  h(-15)=0,\; h(0)=3,\; h(3)=10,\; h(9)=20,
  \)

  we can invert them to get the known values of \(k=h^{-1}\):

  \(
  k(0)=-15,\quad k(3)=0,\quad k(10)=3,\quad k(20)=9.
  \)

  To find \(k(k(9))\) we first need \(k(9)\). But \(k(9)\) is the (unique) number \(t\) such that \(h(t)=9\). Among the listed values of \(h\), the outputs are \(\{0,3,10,20\}\); \(9\) is not among them, so no given input maps to \(9\). Thus \(k(9)\) is undetermined from the data, and consequently \(k(k(9))\) cannot be determined. The problem is unsolvable with the provided information.

Answer: 0


Determine the value of the constant \(a\) such that the product
  \[
  (y^2 - 3y + 4)(2y^2 + a y + 7)
  \]
  expands to equal the polynomial
  \[
  2y^4 - 11y^3 + 30y^2 - 41y + 28.
  \]

Determine the value of the constant \(a\) such that the product
  \[
  (y^2 - 3y + 4)(2y^2 + a y + 7)
  \]
  expands to equal the polynomial
  \[
  2y^4 - 11y^3 + 30y^2 - 41y + 56.
  \]

First expand the left hand side:

  \[
  (y^2 - 3y + 4)(2y^2 + a y + 7)
  = 2y^4 + (a - 6)y^3 + (15 - 3a)y^2 + (4a - 21)y + 28.
  \]

  We want this to equal

  \[
  2y^4 - 11y^3 + 30y^2 - 41y + 56,
  \]

  for all \(y\). Equating coefficients gives the system

  \[
  \begin{cases}
  a - 6 = -11,\\
  15 - 3a = 30,\\
  4a - 21 = -41,\\
  28 = 56.
  \end{cases}
  \]

  From the first equation we find \(a - 6 = -11 \implies a = -5\), and indeed
  \(15 - 3(-5) = 30\), \(4(-5) - 21 = -41\). But the constant term equation \(28 = 56\) is impossible.

Answer: -5


Determine the total number of mappings \( g\), where \( g \) maps positive real numbers to real numbers, such that for every pair of positive real numbers \( a \) and \( b \), the equation
  \[ g(ab) = g(a) g(b) - 2 \]
  holds.

Determine all nonempty mappings \( g \) defined on a subset of the rational numbers, where both the domain and codomain are subsets of \(\mathbb{Q}\), that satisfy the functional equation \( g(ab) = g(a)g(b) - 1 \) for every \( a, b \) in the domain of \( g \).

Let the (nonempty) domain \(D\subseteq \mathbb{Q}\) be multiplicatively closed (since \(a,b\in D\Rightarrow ab\in D\)).

  Pick any \(a,b,c\in D\). Then
  \[
  g(abc)=g(ab)g(c)-1=(g(a)g(b)-1)g(c)-1=g(a)g(b)g(c)-g(c)-1,
  \]
  and also
  \[
  g(abc)=g(a)g(bc)-1=g(a)(g(b)g(c)-1)-1=g(a)g(b)g(c)-g(a)-1.
  \]
  Equating these gives \(g(c)=g(a)\). Since \(a,c\) were arbitrary, \(g\) is constant: \(g\equiv t\) on \(D\).

  The equation then becomes \(t = t^{2}-1\), i.e. \(t^{2}-t-1=0\). Its solutions \(t=\frac{1\pm\sqrt{5}}{2}\) are irrational, contradicting that the codomain is a subset of \(\mathbb{Q}\). Thus no domain with at least two elements works.

  If \(|D|=1\), say \(D=\{x\}\), multiplicative closure forces \(x^{2}=x\), so \(x=0\) or \(x=1\). In either case the same quadratic \(g(x)^2-g(x)-1=0\) appears and has no rational solution.

  Hence no nonempty mapping \(g\) with domain and codomain in \(\mathbb{Q}\) satisfies the equation.

Answer: 2


Consider two vectors, \(\mathbf{u}\) and \(\mathbf{v}\), such that their sum is given by
  \[
  \mathbf{u} + \mathbf{v} = \begin{pmatrix}2 \\ -1 \\ 3\end{pmatrix}.
  \]
  Here, \(\mathbf{u}\) is parallel to \(\begin{pmatrix}1 \\ 2 \\ -1\end{pmatrix}\), while \(\mathbf{v}\) is perpendicular to \(\begin{pmatrix}1 \\ 2 \\ -1\end{pmatrix}\). Determine the vector \(\mathbf{v}\).

Consider two vectors, \(\mathbf{u}\) and \(\mathbf{v}\), such that their sum is given by
  \[
  \mathbf{u} + \mathbf{v} = \begin{pmatrix} 2 \\ -1 \\ 3 \end{pmatrix}.
  \]
  Here, \(\mathbf{u}\) is parallel to \(\begin{pmatrix} 1 \\ 2 \\ -1 \\ -1 \end{pmatrix}\), while \(\mathbf{v}\) is perpendicular to \(\begin{pmatrix} 1 \\ 2 \\ -1 \\ -1 \end{pmatrix}\). Determine the vector \(\mathbf{v}\).

The problem is unsolvable as written due to a contradiction between the stated vector dimensions (3D vs. 4D).

Answer: \begin{pmatrix} 5/2 \\ 0 \\ 5/2 \end{pmatrix}


If \(a\) and \(b\) are positive integers for which
  \[
  2^a 3^b = 1296,
  \]
  what is the value of \(a + b\)?

If \(a\) and \(b\) are positive integers such that
  \[
  2^a 3^b = 6480,
  \]
  what is the sum of \(a\) and \(b\)?

The prime factorization is \(6480=2^4\cdot3^4\cdot5\). The left side \(2^a3^b\) has no prime factor other than 2 or 3, so it can never equal 6480. Thus no positive integers \(a,b\) satisfy the equation.

Answer: 8


Consider the quadratic expression \( y^2 + \frac{3}{2} y - 1 \), which has a special characteristic: its roots, \(\frac{1}{2}\) and \(-2\), are each one unit less than the last two coefficients. Determine a quadratic polynomial with a leading term of \( y^2 \) where both of the last two coefficients are non-zero, and the roots are each one unit more than these coefficients respectively. Provide the roots of this quadratic polynomial.

Consider the quadratic expression \( y^2 + \frac{3}{2} y - 1 \), which has a special characteristic: its roots, \(\frac{1}{2}\) and \(-2\), are each one unit less than the last two coefficients. Determine a quadratic polynomial with a leading term of \( y^2 \) where both of the last two coefficients are non-zero, and the roots are each two units more than these coefficients respectively. Provide the roots of this quadratic polynomial.

For the desired quadratic \(y^{2}+by+c\) with \(b\neq0\) and \(c\neq0\).

  Set the roots
  \[
  r_{1}=b+2,\qquad r_{2}=c+2.
  \]

  Vieta gives
  \[
  r_{1}+r_{2} = -b, \qquad r_{1}r_{2}=c
  \]

  Substitute \(r_{1}\) and \(r_{2}\):
  \[
  (b+2) + (c+2) = -b \implies c = -2b-4.
  \]
  \[
  (b+2)(c+2) = c
  \]
  Replace \(c\) with \(-2b-4\):
  \[
  -2(b+2)(b+1) = -2b-4
  \]

  Hence \(b=0\) or \(b=-2\).

  \(b=0\) is impossible because \(b\neq0\). If \(b=-2\), then \(c = -2b-4 = 0\)
  which violates \(c\neq0\).

  There is no pair of nonzero coefficients \(b,c\) that satisfies all conditions.

Answer: -2,\frac{3}{2}


For the function
  \[ g(y) = \log(y + \sqrt{1 + y^2}) \]
  , is it an odd function, an even function, or neither?
  Respond with "odd", "even", or "neither".

Consider the function:
  \[
  g(y) = \ln\left( \left( -252b^2 - 43by - y^2 \right) \sqrt{y^2 - 2025} \right)
  \]
  and suppose \( g(y) = g(-y) \) for all \( y \) in the domain, with the condition \( b^2 \leq \frac{23}{16} \).
 Evaluate how the maximum absolute value of \( y \) compares with 25.

According to the problem,
  \[
  -\frac{\sqrt{23}}{4} \le b \le \frac{\sqrt{23}}{4},\quad
  y < -45 \text{ or } y > 45,\quad
  (7b + y)(36b + y) < 0,
  \]
  so \(g(y)\) is undefined over the domain.

Answer: \text{odd}


Determine the count of numbers with three digits where the second digit is the mean of the first and third digits.

List all numbers with three digits \(\overline{abc}\) such that \(b\) is the mean of \(a\) and \(c\), and
  \[
  a^3 + 3a\bigl(ac + c^2\bigr) + 2 + c^3 \equiv 3 \pmod{8}.
  \]

\(b = \frac{a + c}{2}\), so \(a + c\) must be even. At the same time,
  \((a + c)^3 + 2 \equiv 3 \pmod{8}\),
  so \(a + c\) must be odd, which is a contradiction.

Answer: 45


Given the function
\[
f(x)=\ln x+\frac{2a}{x},\qquad a\in\mathbb{R},
\]
suppose \(f(x)\) has two distinct zeros \(x_1,x_2\). what is the greatest possible value of \(a\)?

Given the function
\[
f(x)=\ln x+\frac{2a}{x},\qquad a\in\mathbb R,
\]
suppose that \(f\) has two distinct zeros \(x_1<x_2<\frac14\). Determine \(x_1+x_2\).

At a zero, \(x\ln x=-2a\). But \(h(x)=x\ln x\) has derivative \(h'(x)=1+\ln x<0\) on \((0,1/e)\). Since \(x_1,x_2<1/4<1/e\), the function \(h\) is strictly decreasing at both points and cannot take the same value at two distinct inputs. Hence such zeros cannot exist.

Answer: \frac{1}{2e}


Let \(S_n\) be the sum of the first \(n\) terms of the sequence \(\{a_n\}\).
It is given that
\[
\frac{2S_n}{n}+n = 2a_n+1.
\]

 If \(a_4, a_7, a_9\) form a geometric progression, find the minimum value of \(S_n\).

Let \(S_n\) be the sum of the first \(n\) terms of the sequence \(\{a_n\}\).
It is given that
\[
\frac{2S_n}{n}+n = 2a_n+1.
\]

 If \(a_5, a_7, a_9\) form a geometric progression, find the minimum value of \(S_n\).

Comparing the identity for \(n\) and \(n-1\) gives \(a_n-a_{n-1}=1\), so \(a_n=a_1+n-1\). Then \(a_5,a_7,a_9\) would be \(x+4,x+6,x+8\). A geometric progression would require \((x+6)^2=(x+4)(x+8)\), which reduces to \(36=32\). Thus no such sequence exists.

Answer: -78