id int64 1.61k 2.97k | subfield stringclasses 4
values | context null | question stringlengths 49 4.42k | solution listlengths 1 5 | final_answer listlengths 1 1 | is_multiple_answer bool 2
classes | unit stringclasses 8
values | answer_type stringclasses 4
values | error stringclasses 1
value | answer stringlengths 9 206 |
|---|---|---|---|---|---|---|---|---|---|---|
2,850 | Number Theory | null | Let $k$ be the least common multiple of the numbers in the set $\mathcal{S}=\{1,2, \ldots, 30\}$. Determine the number of positive integer divisors of $k$ that are divisible by exactly 28 of the numbers in the set $\mathcal{S}$. | [
"We know that $k=2^{4} \\cdot 3^{3} \\cdot 5^{2} \\cdot 7 \\cdot 11 \\cdot 13 \\cdot 17 \\cdot 19 \\cdot 23 \\cdot 29$. It is not difficult to see that the set $\\mathcal{T}_{1}=\\left\\{\\frac{k}{2}, \\frac{k}{3}, \\frac{k}{5}, \\frac{k}{17}, \\frac{k}{19}, \\frac{k}{23}, \\frac{k}{29}\\right\\}$ comprises all div... | [
"23"
] | false | null | Numerical | null | \boxed{23} |
2,851 | Number Theory | null | Let $A$ and $B$ be digits from the set $\{0,1,2, \ldots, 9\}$. Let $r$ be the two-digit integer $\underline{A} \underline{B}$ and let $s$ be the two-digit integer $\underline{B} \underline{A}$, so that $r$ and $s$ are members of the set $\{00,01, \ldots, 99\}$. Compute the number of ordered pairs $(A, B)$ such that $|r... | [
"Because $|(10 A+B)-(10 B+A)|=9|A-B|=k^{2}$, it follows that $|A-B|$ is a perfect square. $|A-B|=0$ yields 10 pairs of integers: $(A, B)=(0,0),(1,1), \\ldots,(9,9)$.\n\n$|A-B|=1$ yields 18 pairs: the nine $(A, B)=(0,1),(1,2), \\ldots,(8,9)$, and their reverses.\n\n$|A-B|=4$ yields 12 pairs: the six $(A, B)=(0,4),(1... | [
"42"
] | false | null | Numerical | null | \boxed{42} |
2,852 | Combinatorics | null | For $k \geq 3$, we define an ordered $k$-tuple of real numbers $\left(x_{1}, x_{2}, \ldots, x_{k}\right)$ to be special if, for every $i$ such that $1 \leq i \leq k$, the product $x_{1} \cdot x_{2} \cdot \ldots \cdot x_{k}=x_{i}^{2}$. Compute the smallest value of $k$ such that there are at least 2009 distinct special ... | [
"The given conditions imply $k$ equations. By taking the product of these $k$ equations, we have $\\left(x_{1} x_{2} \\ldots x_{k}\\right)^{k-1}=x_{1} x_{2} \\ldots x_{k}$. Thus it follows that either $x_{1} x_{2} \\ldots x_{k}=0$ or $x_{1} x_{2} \\ldots x_{k}= \\pm 1$. If $x_{1} x_{2} \\ldots x_{k}=0$, then some $... | [
"12"
] | false | null | Numerical | null | \boxed{12} |
2,853 | Geometry | null | A cylinder with radius $r$ and height $h$ has volume 1 and total surface area 12. Compute $\frac{1}{r}+\frac{1}{h}$. | [
"Since $\\pi r^{2} h=1$, we have $h=\\frac{1}{\\pi r^{2}}$ and $\\pi r^{2}=\\frac{1}{h}$. Consequently,\n\n$$\n2 \\pi r h+2 \\pi r^{2}=12 \\Rightarrow(2 \\pi r)\\left(\\frac{1}{\\pi r^{2}}\\right)+2\\left(\\frac{1}{h}\\right)=12 \\Rightarrow \\frac{2}{r}+\\frac{2}{h}=12 \\Rightarrow \\frac{1}{r}+\\frac{1}{h}=\\math... | [
"6"
] | false | null | Numerical | null | \boxed{6} |
2,854 | Algebra | null | If $6 \tan ^{-1} x+4 \tan ^{-1}(3 x)=\pi$, compute $x^{2}$. | [
"$\\quad$ Let $z=1+x i$ and $w=1+3 x i$, where $i=\\sqrt{-1}$. Then $\\tan ^{-1} x=\\arg z$ and $\\tan ^{-1}(3 x)=\\arg w$, where $\\arg z$ gives the measure of the angle in standard position whose terminal side passes through $z$. By DeMoivre's theorem, $6 \\tan ^{-1} x=\\arg \\left(z^{6}\\right)$ and $4 \\tan ^{-... | [
"$\\frac{15-8 \\sqrt{3}}{33}$"
] | false | null | Numerical | null | \boxed{\frac{15-8 \sqrt{3}}{33}} |
2,855 | Geometry | null | A rectangular box has dimensions $8 \times 10 \times 12$. Compute the fraction of the box's volume that is not within 1 unit of any of the box's faces. | [
"Let the box be defined by the product of the intervals on the $x, y$, and $z$ axes as $[0,8] \\times$ $[0,10] \\times[0,12]$ with volume $8 \\times 10 \\times 12$. The set of points inside the box that are not within 1 unit of any face is defined by the product of the intervals $[1,7] \\times[1,9] \\times[1,11]$ w... | [
"$\\frac{1}{2}$"
] | false | null | Numerical | null | \boxed{\frac{1}{2}} |
2,856 | Algebra | null | Let $T=T N Y W R$. Compute the largest real solution $x$ to $(\log x)^{2}-\log \sqrt{x}=T$. | [
"Let $u=\\log x$. Then the given equation can be rewritten as $u^{2}-\\frac{1}{2} u-T=0 \\rightarrow 2 u^{2}-u-2 T=0$. This quadratic has solutions $u=\\frac{1 \\pm \\sqrt{1+16 T}}{4}$. As we are looking for the largest real solution for $x$ (and therefore, for $u$ ), we want $u=\\frac{1+\\sqrt{1+16 T}}{4}=1$ when ... | [
"10"
] | false | null | Numerical | null | \boxed{10} |
2,857 | Combinatorics | null | Let $T=T N Y W R$. Kay has $T+1$ different colors of fingernail polish. Compute the number of ways that Kay can paint the five fingernails on her left hand by using at least three colors and such that no two consecutive fingernails have the same color. | [
"There are $T+1$ possible colors for the first nail. Each remaining nail may be any color except that of the preceding nail, that is, there are $T$ possible colors. Thus, using at least two colors, there are $(T+1) T^{4}$ possible colorings. The problem requires that at least three colors be used, so we must subtra... | [
"109890"
] | false | null | Numerical | null | \boxed{109890} |
2,858 | Number Theory | null | Compute the number of ordered pairs $(x, y)$ of positive integers satisfying $x^{2}-8 x+y^{2}+4 y=5$. | [
"Completing the square twice in $x$ and $y$, we obtain the equivalent equation $(x-4)^{2}+(y+2)^{2}=$ 25 , which describes a circle centered at $(4,-2)$ with radius 5 . The lattice points on this circle are points 5 units up, down, left, or right of the center, or points 3 units away on one axis and 4 units away on... | [
"4"
] | false | null | Numerical | null | \boxed{4} |
2,859 | Number Theory | null | Let $T=T N Y W R$ and let $k=21+2 T$. Compute the largest integer $n$ such that $2 n^{2}-k n+77$ is a positive prime number. | [
"If $k$ is positive, there are only four possible factorizations of $2 n^{2}-k n+77$ over the integers, namely\n\n$$\n\\begin{aligned}\n& (2 n-77)(n-1)=2 n^{2}-79 n+77 \\\\\n& (2 n-1)(n-77)=2 n^{2}-145 n+77 \\\\\n& (2 n-11)(n-7)=2 n^{2}-25 n+77 \\\\\n& (2 n-7)(n-11)=2 n^{2}-29 n+77\n\\end{aligned}\n$$\n\n\n\nBecaus... | [
"12"
] | false | null | Numerical | null | \boxed{12} |
2,860 | Geometry | null | Let $T=T N Y W R$. In triangle $A B C, B C=T$ and $\mathrm{m} \angle B=30^{\circ}$. Compute the number of integer values of $A C$ for which there are two possible values for side length $A B$. | [
"By the Law of Cosines, $(A C)^{2}=T^{2}+(A B)^{2}-2 T(A B) \\cos 30^{\\circ} \\rightarrow(A B)^{2}-2 T \\cos 30^{\\circ}(A B)+$ $\\left(T^{2}-(A C)^{2}\\right)=0$. This quadratic in $A B$ has two positive solutions when the discriminant and product of the roots are both positive. Thus $\\left(2 T \\cos 30^{\\circ}... | [
"5"
] | false | null | Numerical | null | \boxed{5} |
2,861 | Combinatorics | null | An $\boldsymbol{n}$-label is a permutation of the numbers 1 through $n$. For example, $J=35214$ is a 5 -label and $K=132$ is a 3 -label. For a fixed positive integer $p$, where $p \leq n$, consider consecutive blocks of $p$ numbers in an $n$-label. For example, when $p=3$ and $L=263415$, the blocks are 263,634,341, and... | [
"$(312,123,231)$"
] | [
"$(312,123,231)$"
] | false | null | Tuple | null | \boxed{(312,123,231)} |
2,862 | Combinatorics | null | An $\boldsymbol{n}$-label is a permutation of the numbers 1 through $n$. For example, $J=35214$ is a 5 -label and $K=132$ is a 3 -label. For a fixed positive integer $p$, where $p \leq n$, consider consecutive blocks of $p$ numbers in an $n$-label. For example, when $p=3$ and $L=263415$, the blocks are 263,634,341, and... | [
"$41352,42351,51342$"
] | [
"$41352,42351,51342$"
] | true | null | Numerical | null | \boxed{41352,42351,51342} |
2,863 | Combinatorics | null | An $\boldsymbol{n}$-label is a permutation of the numbers 1 through $n$. For example, $J=35214$ is a 5 -label and $K=132$ is a 3 -label. For a fixed positive integer $p$, where $p \leq n$, consider consecutive blocks of $p$ numbers in an $n$-label. For example, when $p=3$ and $L=263415$, the blocks are 263,634,341, and... | [
"$352146,362145,452136,562134$"
] | [
"$352146,362145,452136,562134$"
] | true | null | Numerical | null | \boxed{352146,362145,452136,562134} |
2,884 | Geometry | null | In $\triangle A B C, D$ is on $\overline{A C}$ so that $\overline{B D}$ is the angle bisector of $\angle B$. Point $E$ is on $\overline{A B}$ and $\overline{C E}$ intersects $\overline{B D}$ at $P$. Quadrilateral $B C D E$ is cyclic, $B P=12$ and $P E=4$. Compute the ratio $\frac{A C}{A E}$. | [
"Let $\\omega$ denote the circle that circumscribes quadrilateral $B C D E$. Draw in line segment $\\overline{D E}$. Note that $\\angle D P E$ and $\\angle C P B$ are congruent, and $\\angle D E C$ and $\\angle D B C$ are congruent, since they cut off the same arc of $\\omega$. Therefore, $\\triangle B C P$ and $\\... | [
"3"
] | false | null | Numerical | null | \boxed{3} |
2,887 | Combinatorics | null | Let $N$ be a six-digit number formed by an arrangement of the digits $1,2,3,3,4,5$. Compute the smallest value of $N$ that is divisible by 264 . | [
"Note that $264=3 \\cdot 8 \\cdot 11$, so we will need to address all these factors. Because the sum of the digits is 18 , it follows that 3 divides $N$, regardless of how we order the digits of $N$. In order for 8 to divide $N$, we need $N$ to end in $\\underline{O} 12, \\underline{O} 52, \\underline{E} 32$, or $\... | [
"135432"
] | false | null | Numerical | null | \boxed{135432} |
2,888 | Geometry | null | In triangle $A B C, A B=4, B C=6$, and $A C=8$. Squares $A B Q R$ and $B C S T$ are drawn external to and lie in the same plane as $\triangle A B C$. Compute $Q T$. | [
"Set $\\mathrm{m} \\angle A B C=x$ and $\\mathrm{m} \\angle T B Q=y$. Then $x+y=180^{\\circ}$ and so $\\cos x+\\cos y=0$. Applying the Law of Cosines to triangles $A B C$ and $T B Q$ gives $A C^{2}=A B^{2}+B C^{2}-2 A B \\cdot B C \\cos x$ and $Q T^{2}=B T^{2}+B Q^{2}-2 B T \\cdot B Q \\cos y$, which, after substit... | [
"$2 \\sqrt{10}$"
] | false | null | Numerical | null | \boxed{2 \sqrt{10}} |
2,890 | Geometry | null | An ellipse in the first quadrant is tangent to both the $x$-axis and $y$-axis. One focus is at $(3,7)$, and the other focus is at $(d, 7)$. Compute $d$. | [
"See the diagram below. The center of the ellipse is $C=\\left(\\frac{d+3}{2}, 7\\right)$. The major axis of the ellipse is the line $y=7$, and the minor axis is the line $x=\\frac{d+3}{2}$. The ellipse is tangent to the coordinate axes at $T_{x}=\\left(\\frac{d+3}{2}, 0\\right)$ and $T_{y}=(0,7)$. Let $F_{1}=(3,7)... | [
"$\\frac{49}{3}$"
] | false | null | Numerical | null | \boxed{\frac{49}{3}} |
2,891 | Geometry | null | Let $A_{1} A_{2} A_{3} A_{4} A_{5} A_{6} A_{7} A_{8}$ be a regular octagon. Let $\mathbf{u}$ be the vector from $A_{1}$ to $A_{2}$ and let $\mathbf{v}$ be the vector from $A_{1}$ to $A_{8}$. The vector from $A_{1}$ to $A_{4}$ can be written as $a \mathbf{u}+b \mathbf{v}$ for a unique ordered pair of real numbers $(a, b... | [
"We can scale the octagon so that $A_{1} A_{2}=\\sqrt{2}$. Because the exterior angle of the octagon is $45^{\\circ}$, we can place the octagon in the coordinate plane with $A_{1}$ being the origin, $A_{2}=(\\sqrt{2}, 0)$, and $A_{8}=(1,1)$.\n\n<img_3693>\n\nThen $A_{3}=(1+\\sqrt{2}, 1)$ and $A_{4}=(1+\\sqrt{2}, 1+... | [
"$\\quad(2+\\sqrt{2}, 1+\\sqrt{2})$"
] | false | null | Numerical | null | \boxed{\quad(2+\sqrt{2}, 1+\sqrt{2})} |
2,892 | Number Theory | null | Compute the integer $n$ such that $2009<n<3009$ and the sum of the odd positive divisors of $n$ is 1024 . | [
"Suppose that $n=2^{k} p_{1}^{a_{1}} \\cdots p_{r}^{a_{r}}$, where the $p_{i}$ are distinct odd primes, $k$ is a nonnegative integer, and $a_{1}, \\ldots, a_{r}$ are positive integers. Then the sum of the odd positive divisors of $n$ is equal to\n\n$$\n\\prod_{i=1}^{r}\\left(1+p_{i}+\\cdots+p_{i}^{a_{i}}\\right)=\\... | [
"2604"
] | false | null | Numerical | null | \boxed{2604} |
2,893 | Geometry | null | Points $A, R, M$, and $L$ are consecutively the midpoints of the sides of a square whose area is 650. The coordinates of point $A$ are $(11,5)$. If points $R, M$, and $L$ are all lattice points, and $R$ is in Quadrant I, compute the number of possible ordered pairs $(x, y)$ of coordinates for point $R$. | [
"Write $x=11+c$ and $y=5+d$. Then $A R^{2}=c^{2}+d^{2}=\\frac{1}{2} \\cdot 650=325$. Note that $325=18^{2}+1^{2}=17^{2}+6^{2}=15^{2}+10^{2}$. Temporarily restricting ourselves to the case where $c$ and $d$ are both positive, there are three classes of solutions: $\\{c, d\\}=\\{18,1\\},\\{c, d\\}=\\{17,6\\}$, or $\\... | [
"10"
] | false | null | Numerical | null | \boxed{10} |
2,894 | Geometry | null | The taxicab distance between points $\left(x_{1}, y_{1}, z_{1}\right)$ and $\left(x_{2}, y_{2}, z_{2}\right)$ is given by
$$
d\left(\left(x_{1}, y_{1}, z_{1}\right),\left(x_{2}, y_{2}, z_{2}\right)\right)=\left|x_{1}-x_{2}\right|+\left|y_{1}-y_{2}\right|+\left|z_{1}-z_{2}\right| .
$$
The region $\mathcal{R}$ is obtai... | [
"For a fixed vertex $V$ on the cube, the locus of points on or inside the cube that are at most $\\frac{3}{5}$ away from $V$ form a corner at $V$ (that is, the right pyramid $V W_{1} W_{2} W_{3}$ in the figure shown at left below, with equilateral triangular base $W_{1} W_{2} W_{3}$ and three isosceles right triang... | [
"$\\frac{179}{250}$"
] | false | null | Numerical | null | \boxed{\frac{179}{250}} |
2,895 | Algebra | null | $\quad$ Let $a$ and $b$ be real numbers such that
$$
a^{3}-15 a^{2}+20 a-50=0 \quad \text { and } \quad 8 b^{3}-60 b^{2}-290 b+2575=0
$$
Compute $a+b$. | [
"Each cubic expression can be depressed - that is, the quadratic term can be eliminated-by substituting as follows. Because $(a-p)^{3}=a^{3}-3 a^{2} p+3 a p^{2}-p^{3}$, setting $p=-\\frac{(-15)}{3}=5$ and substituting $c+p=a$ transforms the expression $a^{3}-15 a^{2}+20 a-50$ into the equivalent expression $(c+5)^{... | [
"$\\frac{15}{2}$"
] | false | null | Numerical | null | \boxed{\frac{15}{2}} |
2,896 | Number Theory | null | For a positive integer $n$, define $s(n)$ to be the sum of $n$ and its digits. For example, $s(2009)=2009+2+0+0+9=2020$. Compute the number of elements in the set $\{s(0), s(1), s(2), \ldots, s(9999)\}$. | [
"If $s(10 x)=a$, then the values of $s$ over $\\{10 x+0,10 x+1, \\ldots, 10 x+9\\}$ are $a, a+2, a+4, \\ldots, a+18$. Furthermore, if $x$ is not a multiple of 10 , then $s(10(x+1))=a+11$. This indicates that the values of $s$ \"interweave\" somewhat from one group of 10 to the next: the sets alternate between even ... | [
"9046"
] | false | null | Numerical | null | \boxed{9046} |
2,897 | Geometry | null | Quadrilateral $A R M L$ is a kite with $A R=R M=5, A M=8$, and $R L=11$. Compute $A L$. | [
"Let $K$ be the midpoint of $\\overline{A M}$. Then $A K=K M=8 / 2=4, R K=\\sqrt{5^{2}-4^{2}}=3$, and $K L=11-3=8$. Thus $A L=\\sqrt{A K^{2}+K L^{2}}=\\sqrt{4^{2}+8^{2}}=4 \\sqrt{5}$."
] | [
"$4 \\sqrt{5}$"
] | false | null | Numerical | null | \boxed{4 \sqrt{5}} |
2,898 | Algebra | null | Let $T=4 \sqrt{5}$. If $x y=\sqrt{5}, y z=5$, and $x z=T$, compute the positive value of $x$. | [
"Multiply the three given equations to obtain $x^{2} y^{2} z^{2}=5 T \\sqrt{5}$. Thus $x y z= \\pm \\sqrt[4]{125 T^{2}}$, and the positive value of $x$ is $x=x y z / y z=\\sqrt[4]{125 T^{2}} / 5=\\sqrt[4]{T^{2} / 5}$. With $T=4 \\sqrt{5}$, we have $x=\\mathbf{2}$."
] | [
"2"
] | false | null | Numerical | null | \boxed{2} |
2,899 | Combinatorics | null | $\quad$ Let $T=2$. In how many ways can $T$ boys and $T+1$ girls be arranged in a row if all the girls must be standing next to each other? | [
"First choose the position of the first girl, starting from the left. There are $T+1$ possible positions, and then the positions for the girls are all determined. There are $(T+1)$ ! ways to arrange the girls, and there are $T$ ! ways to arrange the boys, for a total of $(T+1) \\cdot(T+1) ! \\cdot T !=$ $((T+1) !)^... | [
"36"
] | false | null | Numerical | null | \boxed{36} |
2,900 | Geometry | null | $\triangle A B C$ is on a coordinate plane such that $A=(3,6)$, $B=(T, 0)$, and $C=(2 T-1,1-T)$. Let $\ell$ be the line containing the altitude to $\overline{B C}$. Compute the $y$-intercept of $\ell$. | [
"The slope of $\\overleftrightarrow{B C}$ is $\\frac{(1-T)-0}{(2 T-1)-T}=-1$, and since $\\ell$ is perpendicular to $\\overleftrightarrow{B C}$, the slope of $\\ell$ is 1. Because $\\ell$ passes through $A=(3,6)$, the equation of $\\ell$ is $y=x+3$, and its $y$-intercept is 3 (independent of $T$ )."
] | [
"3"
] | false | null | Numerical | null | \boxed{3} |
2,901 | Geometry | null | Let $T=3$. In triangle $A B C, A B=A C-2=T$, and $\mathrm{m} \angle A=60^{\circ}$. Compute $B C^{2}$. | [
"By the Law of Cosines, $B C^{2}=A B^{2}+A C^{2}-2 \\cdot A B \\cdot A C \\cdot \\cos A=T^{2}+(T+2)^{2}-2 \\cdot T \\cdot(T+2) \\cdot \\frac{1}{2}=$ $T^{2}+2 T+4$. With $T=3$, the answer is 19 ."
] | [
"19"
] | false | null | Numerical | null | \boxed{19} |
2,902 | Algebra | null | Let $T=19$. Let $\mathcal{S}_{1}$ denote the arithmetic sequence $0, \frac{1}{4}, \frac{1}{2}, \ldots$, and let $\mathcal{S}_{2}$ denote the arithmetic sequence $0, \frac{1}{6}, \frac{1}{3}, \ldots$ Compute the $T^{\text {th }}$ smallest number that occurs in both sequences $\mathcal{S}_{1}$ and $\mathcal{S}_{2}$. | [
"$\\mathcal{S}_{1}$ consists of all numbers of the form $\\frac{n}{4}$, and $\\mathcal{S}_{2}$ consists of all numbers of the form $\\frac{n}{6}$, where $n$ is a nonnegative integer. Since $\\operatorname{gcd}(4,6)=2$, the numbers that are in both sequences are of the form $\\frac{n}{2}$, and the $T^{\\text {th }}$... | [
"9"
] | false | null | Numerical | null | \boxed{9} |
2,903 | Combinatorics | null | $\quad$ Let $T=9$. An integer $n$ is randomly selected from the set $\{1,2,3, \ldots, 2 T\}$. Compute the probability that the integer $\left|n^{3}-7 n^{2}+13 n-6\right|$ is a prime number. | [
"Let $P(n)=n^{3}-7 n^{2}+13 n-6$, and note that $P(n)=(n-2)\\left(n^{2}-5 n+3\\right)$. Thus $|P(n)|$ is prime if either $|n-2|=1$ and $\\left|n^{2}-5 n+3\\right|$ is prime or if $\\left|n^{2}-5 n+3\\right|=1$ and $|n-2|$ is prime. Solving $|n-2|=1$ gives $n=1$ or 3 , and solving $\\left|n^{2}-5 n+3\\right|=1$ give... | [
"$\\frac{1}{9}$"
] | false | null | Numerical | null | \boxed{\frac{1}{9}} |
2,904 | Algebra | null | Let $A=\frac{1}{9}$, and let $B=\frac{1}{25}$. In $\frac{1}{A}$ minutes, 20 frogs can eat 1800 flies. At this rate, in $\frac{1}{B}$ minutes, how many flies will 15 frogs be able to eat? | [
"In $\\frac{1}{A}$ minutes, 1 frog can eat $1800 / 20=90$ flies; thus in $\\frac{1}{B}$ minutes, 1 frog can eat $\\frac{A}{B} \\cdot 90$ flies. Thus in $\\frac{1}{B}$ minutes, 15 frogs can eat $15 \\cdot 90 \\cdot \\frac{A}{B}$ flies. With $A=\\frac{1}{9}$ and $B=\\frac{1}{25}$, this simplifies to $15 \\cdot 250=\\... | [
"3750"
] | false | null | Numerical | null | \boxed{3750} |
2,905 | Algebra | null | Let $T=5$. If $|T|-1+3 i=\frac{1}{z}$, compute the sum of the real and imaginary parts of $z$. | [
"Let $t=|T|$. Note that $z=\\frac{1}{t-1+3 i}=\\frac{1}{t-1+3 i} \\cdot \\frac{t-1-3 i}{t-1-3 i}=\\frac{t-1-3 i}{t^{2}-2 t+10}$. Thus the sum of the real and imaginary parts of $z$ is $\\frac{t-1}{t^{2}-2 t+10}+\\frac{-3}{t^{2}-2 t+10}=\\frac{|T|-4}{|T|^{2}-2|T|+10}$. With $T=5$, the answer is $\\frac{1}{25}$."
] | [
"$\\frac{1}{25}$"
] | false | null | Numerical | null | \boxed{\frac{1}{25}} |
2,907 | Algebra | null | Let $T=10$. Ann spends 80 seconds climbing up a $T$ meter rope at a constant speed, and she spends 70 seconds climbing down the same rope at a constant speed (different from her upward speed). Ann begins climbing up and down the rope repeatedly, and she does not pause after climbing the length of the rope. After $T$ mi... | [
"In 150 seconds (or 2.5 minutes), Ann climbs up and down the entire rope. Thus in $T$ minutes, she makes $\\left\\lfloor\\frac{T}{2.5}\\right\\rfloor$ round trips, and therefore climbs $2 T\\left\\lfloor\\frac{T}{2.5}\\right\\rfloor$ meters. After making all her round trips, there are $t=60\\left(T-2.5\\left\\lfloo... | [
"80"
] | false | null | Numerical | null | \boxed{80} |
2,908 | Algebra | null | Let $T=800$. Simplify $2^{\log _{4} T} / 2^{\log _{16} 64}$. | [
"Note that $2^{\\log _{4} T}=4^{\\left(\\frac{1}{2} \\log _{4} T\\right)}=4^{\\log _{4} T^{\\frac{1}{2}}}=\\sqrt{T}$. Letting $\\log _{16} 64=x$, we see that $2^{4 x}=2^{6}$, thus $x=\\frac{3}{2}$, and $2^{x}=\\sqrt{8}$. Thus the given expression equals $\\sqrt{\\frac{T}{8}}$, and with $T=800$, this is equal to 10 ... | [
"10"
] | false | null | Numerical | null | \boxed{10} |
2,909 | Algebra | null | Let $P(x)=x^{2}+T x+800$, and let $r_{1}$ and $r_{2}$ be the roots of $P(x)$. The polynomial $Q(x)$ is quadratic, it has leading coefficient 1, and it has roots $r_{1}+1$ and $r_{2}+1$. Find the sum of the coefficients of $Q(x)$. | [
"Let $Q(x)=x^{2}+A x+B$. Then $A=-\\left(r_{1}+1+r_{2}+1\\right)$ and $B=\\left(r_{1}+1\\right)\\left(r_{2}+1\\right)$. Thus the sum of the coefficients of $Q(x)$ is $1+\\left(-r_{1}-r_{2}-2\\right)+\\left(r_{1} r_{2}+r_{1}+r_{2}+1\\right)=r_{1} r_{2}$. Note that $T=-\\left(r_{1}+r_{2}\\right)$ and $800=r_{1} r_{2}... | [
"800"
] | false | null | Numerical | null | \boxed{800} |
2,910 | Geometry | null | Let $T=12$. Equilateral triangle $A B C$ is given with side length $T$. Points $D$ and $E$ are the midpoints of $\overline{A B}$ and $\overline{A C}$, respectively. Point $F$ lies in space such that $\triangle D E F$ is equilateral and $\triangle D E F$ lies in a plane perpendicular to the plane containing $\triangle A... | [
"The volume of tetrahedron $A B C F$ is one-third the area of $\\triangle A B C$ times the distance from $F$ to $\\triangle A B C$. Since $D$ and $E$ are midpoints, $D E=\\frac{B C}{2}=\\frac{T}{2}$, and the distance from $F$ to $\\triangle A B C$ is $\\frac{T \\sqrt{3}}{4}$. Thus the volume of $A B C F$ is $\\frac... | [
"108"
] | false | null | Numerical | null | \boxed{108} |
2,911 | Geometry | null | In triangle $A B C, A B=5, A C=6$, and $\tan \angle B A C=-\frac{4}{3}$. Compute the area of $\triangle A B C$. | [
"Let $s=\\sin \\angle B A C$. Then $s>0$ and $\\frac{s}{-\\sqrt{1-s^{2}}}=-\\frac{4}{3}$, which gives $s=\\frac{4}{5}$. The area of triangle $A B C$ is therefore $\\frac{1}{2} \\cdot A B \\cdot A C \\cdot \\sin \\angle B A C=\\frac{1}{2} \\cdot 5 \\cdot 6 \\cdot \\frac{4}{5}=\\mathbf{1 2}$."
] | [
"12"
] | false | null | Numerical | null | \boxed{12} |
2,912 | Number Theory | null | Compute the number of positive integers less than 25 that cannot be written as the difference of two squares of integers. | [
"Suppose $n=a^{2}-b^{2}=(a+b)(a-b)$, where $a$ and $b$ are integers. Because $a+b$ and $a-b$ differ by an even number, they have the same parity. Thus $n$ must be expressible as the product of two even integers or two odd integers. This condition is sufficient for $n$ to be a difference of squares, because if $n$ i... | [
"6"
] | false | null | Numerical | null | \boxed{6} |
2,913 | Algebra | null | For digits $A, B$, and $C,(\underline{A} \underline{B})^{2}+(\underline{A} \underline{C})^{2}=1313$. Compute $A+B+C$. | [
"Because $10 A \\leq \\underline{A} \\underline{B}<10(A+1), 200 A^{2}<(\\underline{A} \\underline{B})^{2}+(\\underline{A} \\underline{C})^{2}<200(A+1)^{2}$. So $200 A^{2}<$ $1313<200(A+1)^{2}$, and $A=2$. Note that $B$ and $C$ must have opposite parity, so without loss of generality, assume that $B$ is even. Consid... | [
"13"
] | false | null | Numerical | null | \boxed{13} |
2,914 | Geometry | null | Points $P, Q, R$, and $S$ lie in the interior of square $A B C D$ such that triangles $A B P, B C Q$, $C D R$, and $D A S$ are equilateral. If $A B=1$, compute the area of quadrilateral $P Q R S$. | [
"$P Q R S$ is a square with diagonal $\\overline{R P}$. Extend $\\overline{R P}$ to intersect $\\overline{A B}$ and $\\overline{C D}$ at $M$ and $N$ respectively, as shown in the diagram below.\n\n<img_3457>\n\nThen $\\overline{M P}$ is an altitude of $\\triangle A B P$ and $\\overline{R N}$ is an altitude of $\\tr... | [
"$2-\\sqrt{3}$"
] | false | null | Numerical | null | \boxed{2-\sqrt{3}} |
2,915 | Algebra | null | For real numbers $\alpha, B$, and $C$, the zeros of $T(x)=x^{3}+x^{2}+B x+C \operatorname{are~}^{2} \alpha$, $\cos ^{2} \alpha$, and $-\csc ^{2} \alpha$. Compute $T(5)$. | [
"Use the sum of the roots formula to obtain $\\sin ^{2} \\alpha+\\cos ^{2} \\alpha+-\\csc ^{2} \\alpha=-1$, so $\\csc ^{2} \\alpha=2$, and $\\sin ^{2} \\alpha=\\frac{1}{2}$. Therefore $\\cos ^{2} \\alpha=\\frac{1}{2}$. T(x) has leading coefficient 1 , so by the factor theorem, $T(x)=\\left(x-\\frac{1}{2}\\right)\\l... | [
"$\\frac{567}{4}$"
] | false | null | Numerical | null | \boxed{\frac{567}{4}} |
2,916 | Geometry | null | Let $\mathcal{R}$ denote the circular region bounded by $x^{2}+y^{2}=36$. The lines $x=4$ and $y=3$ partition $\mathcal{R}$ into four regions $\mathcal{R}_{1}, \mathcal{R}_{2}, \mathcal{R}_{3}$, and $\mathcal{R}_{4}$. $\left[\mathcal{R}_{i}\right]$ denotes the area of region $\mathcal{R}_{i}$. If $\left[\mathcal{R}_{1}... | [
"Draw the lines $x=-4$ and $y=-3$, creating regions $\\mathcal{R}_{21}, \\mathcal{R}_{22}, \\mathcal{R}_{11}, \\mathcal{R}_{12}, \\mathcal{R}_{13}, \\mathcal{R}_{14}$ as shown below.\n\n<img_3593>\n\n\n\nThen $\\left[\\mathcal{R}_{21}\\right]=\\left[\\mathcal{R}_{4}\\right]=\\left[\\mathcal{R}_{13}\\right],\\left[\... | [
"48"
] | false | null | Numerical | null | \boxed{48} |
2,917 | Algebra | null | Let $x$ be a real number in the interval $[0,360]$ such that the four expressions $\sin x^{\circ}, \cos x^{\circ}$, $\tan x^{\circ}, \cot x^{\circ}$ take on exactly three distinct (finite) real values. Compute the sum of all possible values of $x$. | [
"If the four expressions take on three different values, exactly two of the expressions must have equal values. There are $\\left(\\begin{array}{l}4 \\\\ 2\\end{array}\\right)=6$ cases to consider:\n\nCase 1: $\\sin x^{\\circ}=\\cos x^{\\circ}$ : Then $\\tan x^{\\circ}=\\cot x^{\\circ}=1$, violating the condition t... | [
"990"
] | false | null | Numerical | null | \boxed{990} |
2,918 | Algebra | null | Let $a_{1}, a_{2}, a_{3}, \ldots$ be an arithmetic sequence, and let $b_{1}, b_{2}, b_{3}, \ldots$ be a geometric sequence. The sequence $c_{1}, c_{2}, c_{3}, \ldots$ has $c_{n}=a_{n}+b_{n}$ for each positive integer $n$. If $c_{1}=1, c_{2}=4, c_{3}=15$, and $c_{4}=2$, compute $c_{5}$. | [
"Let $a_{2}-a_{1}=d$ and $\\frac{b_{2}}{b_{1}}=r$. Using $a=a_{1}$ and $b=b_{1}$, write the system of equations:\n\n$$\n\\begin{aligned}\na+b & =1 \\\\\n(a+d)+b r & =4 \\\\\n(a+2 d)+b r^{2} & =15 \\\\\n(a+3 d)+b r^{3} & =2 .\n\\end{aligned}\n$$\n\nSubtract the first equation from the second, the second from the thi... | [
"61"
] | false | null | Numerical | null | \boxed{61} |
2,919 | Geometry | null | In square $A B C D$ with diagonal $1, E$ is on $\overline{A B}$ and $F$ is on $\overline{B C}$ with $\mathrm{m} \angle B C E=\mathrm{m} \angle B A F=$ $30^{\circ}$. If $\overline{C E}$ and $\overline{A F}$ intersect at $G$, compute the distance between the incenters of triangles $A G E$ and $C G F$. | [
"Let $M$ be the midpoint of $\\overline{A G}$, and $I$ the incenter of $\\triangle A G E$ as shown below.\n\n<img_3715>\n\nBecause $\\frac{A B}{A C}=\\sin 45^{\\circ}$ and $\\frac{E B}{A B}=\\frac{E B}{B C}=\\tan 30^{\\circ}$,\n\n$$\n\\begin{aligned}\nA E & =A B-E B=A B\\left(1-\\tan 30^{\\circ}\\right) \\\\\n& =\\... | [
"$4-2 \\sqrt{3}$"
] | false | null | Numerical | null | \boxed{4-2 \sqrt{3}} |
2,920 | Geometry | null | Let $a, b, m, n$ be positive integers with $a m=b n=120$ and $a \neq b$. In the coordinate plane, let $A=(a, m), B=(b, n)$, and $O=(0,0)$. If $X$ is a point in the plane such that $A O B X$ is a parallelogram, compute the minimum area of $A O B X$. | [
"The area of parallelogram $A O B X$ is given by the absolute value of the cross product $|\\langle a, m\\rangle \\times\\langle b, n\\rangle|=|a n-m b|$. Because $m=\\frac{120}{a}$ and $n=\\frac{120}{b}$, the desired area of $A O B X$ equals $120\\left|\\frac{a}{b}-\\frac{b}{a}\\right|$. Note that the function $f(... | [
"44"
] | false | null | Numerical | null | \boxed{44} |
2,921 | Combinatorics | null | Let $\mathcal{S}$ be the set of integers from 0 to 9999 inclusive whose base- 2 and base- 5 representations end in the same four digits. (Leading zeros are allowed, so $1=0001_{2}=0001_{5}$ is one such number.) Compute the remainder when the sum of the elements of $\mathcal{S}$ is divided by 10,000. | [
"The remainders of an integer $N$ modulo $2^{4}=16$ and $5^{4}=625$ uniquely determine its remainder modulo 10000. There are only 16 strings of four 0's and 1's. In addition, because 16 and 625 are relatively prime, it will be shown below that for each such string $s$, there exists exactly one integer $x_{s}$ in th... | [
"6248"
] | false | null | Numerical | null | \boxed{6248} |
2,922 | Number Theory | null | If $A, R, M$, and $L$ are positive integers such that $A^{2}+R^{2}=20$ and $M^{2}+L^{2}=10$, compute the product $A \cdot R \cdot M \cdot L$. | [
"The only positive integers whose squares sum to 20 are 2 and 4 . The only positive integers whose squares sum to 10 are 1 and 3 . Thus $A \\cdot R=8$ and $M \\cdot L=3$, so $A \\cdot R \\cdot M \\cdot L=\\mathbf{2 4}$."
] | [
"24"
] | false | null | Numerical | null | \boxed{24} |
2,924 | Number Theory | null | Let $T=49$. Compute the last digit, in base 10, of the sum
$$
T^{2}+(2 T)^{2}+(3 T)^{2}+\ldots+\left(T^{2}\right)^{2}
$$ | [
"Let $S$ be the required sum. Factoring $T^{2}$ from the sum yields\n\n$$\n\\begin{aligned}\nS & =T^{2}\\left(1+4+9+\\ldots+T^{2}\\right) \\\\\n& =T^{2}\\left(\\frac{T(T+1)(2 T+1)}{6}\\right) \\\\\n& =\\frac{T^{3}(T+1)(2 T+1)}{6} .\n\\end{aligned}\n$$\n\nFurther analysis makes the final computation simpler. If $T \... | [
"5"
] | false | null | Numerical | null | \boxed{5} |
2,925 | Combinatorics | null | A fair coin is flipped $n$ times. Compute the smallest positive integer $n$ for which the probability that the coin has the same result every time is less than $10 \%$. | [
"After the first throw, the probability that the succeeding $n-1$ throws have the same result is $\\frac{1}{2^{n-1}}$. Thus $\\frac{1}{2^{n-1}}<\\frac{1}{10} \\Rightarrow 2^{n-1}>10 \\Rightarrow n-1 \\geq 4$, so $n=5$ is the smallest possible value."
] | [
"5"
] | false | null | Numerical | null | \boxed{5} |
2,926 | Algebra | null | Let $T=5$. Compute the smallest positive integer $n$ such that there are at least $T$ positive integers in the domain of $f(x)=\sqrt{-x^{2}-2 x+n}$. | [
"Completing the square under the radical yields $\\sqrt{n+1-(x+1)^{2}}$. The larger zero of the radicand is $-1+\\sqrt{n+1}$, and the smaller zero is negative because $-1-\\sqrt{n+1}<0$, so the $T$ positive integers in the domain of $f$ must be $1,2,3, \\ldots, T$. Therefore $-1+\\sqrt{n+1} \\geq T$. Hence $\\sqrt{... | [
"35"
] | false | null | Numerical | null | \boxed{35} |
2,927 | Algebra | null | Let $T=35$. Compute the smallest positive real number $x$ such that $\frac{\lfloor x\rfloor}{x-\lfloor x\rfloor}=T$. | [
"If $\\frac{\\lfloor x\\rfloor}{x-\\lfloor x\\rfloor}=T$, the equation can be rewritten as follows:\n\n$$\n\\begin{aligned}\n\\frac{x-\\lfloor x\\rfloor}{\\lfloor x\\rfloor} & =\\frac{1}{T} \\\\\n\\frac{x}{\\lfloor x\\rfloor}-1 & =\\frac{1}{T} \\\\\n\\frac{x}{\\lfloor x\\rfloor} & =\\frac{T+1}{T} .\n\\end{aligned}\... | [
"$\\frac{36}{35}$"
] | false | null | Numerical | null | \boxed{\frac{36}{35}} |
2,947 | Combinatorics | null | Let set $S=\{1,2,3,4,5,6\}$, and let set $T$ be the set of all subsets of $S$ (including the empty set and $S$ itself). Let $t_{1}, t_{2}, t_{3}$ be elements of $T$, not necessarily distinct. The ordered triple $\left(t_{1}, t_{2}, t_{3}\right)$ is called satisfactory if either
(a) both $t_{1} \subseteq t_{3}$ and $t_... | [
"Let $T_{1}=\\left\\{\\left(t_{1}, t_{2}, t_{3}\\right) \\mid t_{1} \\subseteq t_{3}\\right.$ and $\\left.t_{2} \\subseteq t_{3}\\right\\}$ and let $T_{2}=\\left\\{\\left(t_{1}, t_{2}, t_{3}\\right) \\mid t_{3} \\subseteq t_{1}\\right.$ and $\\left.t_{3} \\subseteq t_{2}\\right\\}$. Notice that if $\\left(t_{1}, t_... | [
"31186"
] | false | null | Numerical | null | \boxed{31186} |
2,948 | Geometry | null | Let $A B C D$ be a parallelogram with $\angle A B C$ obtuse. Let $\overline{B E}$ be the altitude to side $\overline{A D}$ of $\triangle A B D$. Let $X$ be the point of intersection of $\overline{A C}$ and $\overline{B E}$, and let $F$ be the point of intersection of $\overline{A B}$ and $\overleftrightarrow{D X}$. If ... | [
"Extend $\\overline{A D}$ to a point $M$ such that $\\overline{C M} \\| \\overline{B E}$ as shown below.\n\n<img_3958>\n\nBecause $C D=A B=13$ and $B E=12=C M, A E=D M=5$. Then $A C=\\sqrt{35^{2}+12^{2}}=$ $\\sqrt{1369}=37$. Because $\\overline{E X} \\| \\overline{C M}, X E / C M=A E / A M=\\frac{1}{7}$. Thus $E X=... | [
"$\\frac{222}{13}$"
] | false | null | Numerical | null | \boxed{\frac{222}{13}} |
2,949 | Number Theory | null | Compute the sum of all positive two-digit factors of $2^{32}-1$. | [
"Using the difference of squares, $2^{32}-1=\\left(2^{16}-1\\right)\\left(2^{16}+1\\right)$. The second factor, $2^{16}+1$, is the Fermat prime 65537 , so continue with the first factor:\n\n$$\n\\begin{aligned}\n2^{16}-1 & =\\left(2^{8}+1\\right)\\left(2^{8}-1\\right) \\\\\n2^{8}-1 & =\\left(2^{4}+1\\right)\\left(2... | [
"168"
] | false | null | Numerical | null | \boxed{168} |
2,950 | Algebra | null | Compute all ordered pairs of real numbers $(x, y)$ that satisfy both of the equations:
$$
x^{2}+y^{2}=6 y-4 x+12 \quad \text { and } \quad 4 y=x^{2}+4 x+12
$$ | [
"Rearrange the terms in the first equation to yield $x^{2}+4 x+12=6 y-y^{2}+24$, so that the two equations together yield $4 y=6 y-y^{2}+24$, or $y^{2}-2 y-24=0$, from which $y=6$ or $y=-4$. If $y=6$, then $x^{2}+4 x+12=24$, from which $x=-6$ or $x=2$. If $y=-4$, then $x^{2}+4 x+12=-16$, which has no real solutions... | [
"$(-6,6)$, $(2,6)$"
] | true | null | Tuple | null | \boxed{(-6,6), (2,6)} |
2,951 | Algebra | null | Define $\log ^{*}(n)$ to be the smallest number of times the log function must be iteratively applied to $n$ to get a result less than or equal to 1 . For example, $\log ^{*}(1000)=2$ since $\log 1000=3$ and $\log (\log 1000)=\log 3=0.477 \ldots \leq 1$. Let $a$ be the smallest integer such that $\log ^{*}(a)=3$. Compu... | [
"If $\\log ^{*}(a)=3$, then $\\log (\\log (\\log (a))) \\leq 1$ and $\\log (\\log (a))>1$. If $\\log (\\log (a))>1$, then $\\log (a)>10$ and $a>10^{10}$. Because the problem asks for the smallest such $a$ that is an integer, choose $a=10^{10}+1=10,000,000,001$, which has 9 zeros."
] | [
"9"
] | false | null | Numerical | null | \boxed{9} |
2,952 | Combinatorics | null | An integer $N$ is worth 1 point for each pair of digits it contains that forms a prime in its original order. For example, 6733 is worth 3 points (for 67,73 , and 73 again), and 20304 is worth 2 points (for 23 and 03). Compute the smallest positive integer that is worth exactly 11 points. [Note: Leading zeros are not a... | [
"If a number $N$ has $k$ base 10 digits, then its maximum point value is $(k-1)+(k-2)+\\cdots+1=$ $\\frac{1}{2}(k-1)(k)$. So if $k \\leq 5$, the number $N$ is worth at most 10 points. Therefore the desired number has at least six digits. If $100,000<N<101,000$, then $N$ is of the form $100 \\underline{A} \\underlin... | [
"100337"
] | false | null | Numerical | null | \boxed{100337} |
2,953 | Geometry | null | The six sides of convex hexagon $A_{1} A_{2} A_{3} A_{4} A_{5} A_{6}$ are colored red. Each of the diagonals of the hexagon is colored either red or blue. Compute the number of colorings such that every triangle $A_{i} A_{j} A_{k}$ has at least one red side. | [
"Only two triangles have no sides that are sides of the original hexagon: $A_{1} A_{3} A_{5}$ and $A_{2} A_{4} A_{6}$. For each of these triangles, there are $2^{3}-1=7$ colorings in which at least one side is red, for a total of $7 \\cdot 7=49$ colorings of those six diagonals. The colorings of the three central d... | [
"392"
] | false | null | Numerical | null | \boxed{392} |
2,954 | Number Theory | null | Compute the smallest positive integer $n$ such that $n^{n}$ has at least 1,000,000 positive divisors. | [
"Let $k$ denote the number of distinct prime divisors of $n$, so that $n=p_{1}^{a_{1}} p_{2}^{a_{2}} \\cdots p_{k}^{a_{k}}, a_{i}>0$. Then if $d(x)$ denotes the number of positive divisors of $x$,\n\n$$\nd\\left(n^{n}\\right)=\\left(a_{1} n+1\\right)\\left(a_{2} n+1\\right) \\cdots\\left(a_{k} n+1\\right) \\geq(n+1... | [
"84"
] | false | null | Numerical | null | \boxed{84} |
2,955 | Combinatorics | null | Given an arbitrary finite sequence of letters (represented as a word), a subsequence is a sequence of one or more letters that appear in the same order as in the original sequence. For example, $N, C T, O T T$, and CONTEST are subsequences of the word CONTEST, but NOT, ONSET, and TESS are not. Assuming the standard Eng... | [
"Divide into cases according to the number of $E$ 's in the word. If there are only two $E$ 's, then the word must have two non- $E$ letters, represented by ?'s. There are $\\left(\\begin{array}{l}4 \\\\ 2\\end{array}\\right)=6$ arrangements of two $E$ 's and two ?'s, and each of the ?'s can be any of 25 letters, s... | [
"3851"
] | false | null | Numerical | null | \boxed{3851} |
2,956 | Geometry | null | Six solid regular tetrahedra are placed on a flat surface so that their bases form a regular hexagon $\mathcal{H}$ with side length 1 , and so that the vertices not lying in the plane of $\mathcal{H}$ (the "top" vertices) are themselves coplanar. A spherical ball of radius $r$ is placed so that its center is directly a... | [
"Let $O$ be the center of the sphere, $A$ be the top vertex of one tetrahedron, and $B$ be the center of the hexagon.\n\n<img_3299>\n\nThen $B O$ equals the height of the tetrahedron, which is $\\frac{\\sqrt{6}}{3}$. Because $A$ is directly above the centroid of the bottom face, $A O$ is two-thirds the length of th... | [
"$\\frac{\\sqrt{2}}{3}$"
] | false | null | Numerical | null | \boxed{\frac{\sqrt{2}}{3}} |
2,957 | Combinatorics | null | Derek starts at the point $(0,0)$, facing the point $(0,1)$, and he wants to get to the point $(1,1)$. He takes unit steps parallel to the coordinate axes. A move consists of either a step forward, or a $90^{\circ}$ right (clockwise) turn followed by a step forward, so that his path does not contain any left turns. His... | [
"Divide into cases according to the number of right turns Derek makes.\n\n- There is one route involving only one turn: move first to $(0,1)$ and then to $(1,1)$.\n- If he makes two turns, he could move up to $(0, a)$ then to $(1, a)$ and then down to $(1,1)$. In order to do this, $a$ must satisfy $1<a \\leq 17$, l... | [
"529"
] | false | null | Numerical | null | \boxed{529} |
2,958 | Algebra | null | The equations $x^{3}+A x+10=0$ and $x^{3}+B x^{2}+50=0$ have two roots in common. Compute the product of these common roots. | [
"Let the roots of the first equation be $p, q, r$ and the roots of the second equation be $p, q, s$. Then $p q r=-10$ and $p q s=-50$, so $\\frac{s}{r}=5$. Also $p+q+r=0$ and $p+q+s=-B$, so $r-s=B$. Substituting yields $r-5 r=-4 r=B$, so $r=-\\frac{B}{4}$ and $s=-\\frac{5 B}{4}$. From the second given equation, $p ... | [
"$5 \\sqrt[3]{4}$"
] | false | null | Numerical | null | \boxed{5 \sqrt[3]{4}} |
2,960 | Number Theory | null | Let $N$ be a perfect square between 100 and 400 , inclusive. What is the only digit that cannot appear in $N$ ? | [
"When the perfect squares between 100 and 400 inclusive are listed out, every digit except 7 is used. Note that the perfect squares 100, 256, 289, 324 use each of the other digits."
] | [
"7"
] | false | null | Numerical | null | \boxed{7} |
2,961 | Number Theory | null | Let $T=7$. Let $A$ and $B$ be distinct digits in base $T$, and let $N$ be the largest number of the form $\underline{A} \underline{B} \underline{A}_{T}$. Compute the value of $N$ in base 10 . | [
"To maximize $\\underline{A} \\underline{B} \\underline{A}_{T}$ with $A \\neq B$, let $A=T-1$ and $B=T-2$. Then $\\underline{A} \\underline{B}^{A} \\underline{A}_{T}=$ $(T-1) \\cdot T^{2}+(T-2) \\cdot T^{1}+(T-1) \\cdot T^{0}=T^{3}-T-1$. With $T=7$, the answer is 335 ."
] | [
"335"
] | false | null | Numerical | null | \boxed{335} |
2,962 | Number Theory | null | Let T be an integer. Given a nonzero integer $n$, let $f(n)$ denote the sum of all numbers of the form $i^{d}$, where $i=\sqrt{-1}$, and $d$ is a divisor (positive or negative) of $n$. Compute $f(2 T+1)$. | [
"Let $n=2^{m} r$, where $r$ is odd. If $m=0$, then $n$ is odd, and for each $d$ that divides $n$, $i^{d}+i^{-d}=i^{d}+\\frac{i^{d}}{\\left(i^{2}\\right)^{d}}=0$, hence $f(n)=0$ when $n$ is odd. If $m=1$, then for each $d$ that divides $n, i^{d}+i^{-d}$ equals 0 if $d$ is odd, and -2 if $d$ is even. Thus when $n$ is... | [
"0"
] | false | null | Numerical | null | \boxed{0} |
2,963 | Algebra | null | Let $T=0$. Compute the real value of $x$ for which there exists a solution to the system of equations
$$
\begin{aligned}
x+y & =0 \\
x^{3}-y^{3} & =54+T .
\end{aligned}
$$ | [
"$\\quad$ Plug $y=-x$ into the second equation to obtain $x=\\sqrt[3]{\\frac{54+T}{2}}$. With $T=0, x=\\sqrt[3]{27}=3$."
] | [
"3"
] | false | null | Numerical | null | \boxed{3} |
2,964 | Geometry | null | Let $T=3$. In $\triangle A B C, A C=T^{2}, \mathrm{~m} \angle A B C=45^{\circ}$, and $\sin \angle A C B=\frac{8}{9}$. Compute $A B$. | [
"From the Law of Sines, $\\frac{A B}{\\sin \\angle A C B}=\\frac{A C}{\\sin \\angle A B C}$. Thus $A B=\\frac{8}{9} \\cdot \\frac{T^{2}}{1 / \\sqrt{2}}=\\frac{8 \\sqrt{2}}{9} \\cdot T^{2}$. With $T=3, A B=\\mathbf{8} \\sqrt{\\mathbf{2}}$."
] | [
"$8 \\sqrt{2}$"
] | false | null | Numerical | null | \boxed{8 \sqrt{2}} |
2,966 | Algebra | null | Let $T=9$. The sequence $a_{1}, a_{2}, a_{3}, \ldots$ is an arithmetic progression, $d$ is the common difference, $a_{T}=10$, and $a_{K}=2010$, where $K>T$. If $d$ is an integer, compute the value of $K$ such that $|K-d|$ is minimal. | [
"Note that $a_{T}=a_{1}+(T-1) d$ and $a_{K}=a_{1}+(K-1) d$, hence $a_{K}-a_{T}=(K-T) d=2010-10=$ 2000. Thus $K=\\frac{2000}{d}+T$, and to minimize $\\left|T+\\frac{2000}{d}-d\\right|$, choose a positive integer $d$ such that $\\frac{2000}{d}$ is also an integer and $\\frac{2000}{d}-d$ is as close as possible to $-T... | [
"49"
] | false | null | Numerical | null | \boxed{49} |
2,967 | Algebra | null | Let $A$ be the number you will receive from position 7 , and let $B$ be the number you will receive from position 9 . There are exactly two ordered pairs of real numbers $\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right)$ that satisfy both $|x+y|=6(\sqrt{A}-5)$ and $x^{2}+y^{2}=B^{2}$. Compute $\left|x_{1}\right|+\le... | [
"Note that the graph of $x^{2}+y^{2}=B^{2}$ is a circle of radius $|B|$ centered at $(0,0)$ (as long as $\\left.B^{2}>0\\right)$. Also note that the graph of $|x+y|=6(\\sqrt{A}-5)$ is either the line $y=-x$ if $A=25$, or the graph consists of two parallel lines with slope -1 if $A>25$. In the former case, the\n\n\n... | [
"24"
] | false | null | Numerical | null | \boxed{24} |
2,968 | Geometry | null | Let $T=23$. In triangle $A B C$, the altitude from $A$ to $\overline{B C}$ has length $\sqrt{T}, A B=A C$, and $B C=T-K$, where $K$ is the real root of the equation $x^{3}-8 x^{2}-8 x-9=0$. Compute the length $A B$. | [
"Rewrite the equation as $x^{3}-1=8\\left(x^{2}+x+1\\right)$, so that $(x-1)\\left(x^{2}+x+1\\right)=8\\left(x^{2}+x+1\\right)$. Because $x^{2}+x+1$ has no real zeros, it can be canceled from both sides of the equation to obtain $x-1=8$ or $x=9$. Hence $B C=T-9$, and $A B^{2}=(\\sqrt{T})^{2}+\\left(\\frac{T-9}{2}\\... | [
"$6 \\sqrt{2}$"
] | false | null | Numerical | null | \boxed{6 \sqrt{2}} |
2,969 | Geometry | null | Let $T=8$. A cube has volume $T-2$. The cube's surface area equals one-eighth the surface area of a $2 \times 2 \times n$ rectangular prism. Compute $n$. | [
"The cube's side length is $\\sqrt[3]{T}$, so its surface area is $6 \\sqrt[3]{T^{2}}$. The rectangular prism has surface area $2(2 \\cdot 2+2 \\cdot n+2 \\cdot n)=8+8 n$, thus $6 \\sqrt[3]{T^{2}}=1+n$. With $T=8, n=6 \\sqrt[3]{64}-1=\\mathbf{2 3}$."
] | [
"23"
] | false | null | Numerical | null | \boxed{23} |
2,970 | Number Theory | null | Let $T=98721$, and let $K$ be the sum of the digits of $T$. Let $A_{n}$ be the number of ways to tile a $1 \times n$ rectangle using $1 \times 3$ and $1 \times 1$ tiles that do not overlap. Tiles of both types need not be used; for example, $A_{3}=2$ because a $1 \times 3$ rectangle can be tiled with three $1 \times 1$... | [
"Consider the rightmost tile of the rectangle. If it's a $1 \\times 1$ tile, then there are $A_{n-1}$ ways to tile the remaining $1 \\times(n-1)$ rectangle, and if it's a $1 \\times 3$ tile, then there are $A_{n-3}$ ways to tile the remaining $1 \\times(n-3)$ rectangle. Hence $A_{n}=A_{n-1}+A_{n-3}$ for $n>3$, and ... | [
"10"
] | false | null | Numerical | null | \boxed{10} |
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