source stringclasses 7
values | question stringlengths 3 1.02k | ground_truth stringlengths 1 202 |
|---|---|---|
Open-Reasoner-Zero/orz_math_57k_collection | Find the number of five-digit positive integers, $n$, that satisfy the following conditions:
(a) the number $n$ is divisible by $5,$
(b) the first and last digits of $n$ are equal, and
(c) the sum of the digits of $n$ is divisible by $5.$ | 200 |
Open-Reasoner-Zero/orz_math_57k_collection | Freddy the frog is jumping around the coordinate plane searching for a river, which lies on the horizontal line $y = 24$. A fence is located at the horizontal line $y = 0$. On each jump Freddy randomly chooses a direction parallel to one of the coordinate axes and moves one unit in that direction. When he is at a point... | 273 |
Open-Reasoner-Zero/orz_math_57k_collection | A rational number written in base eight is $\underline{ab} . \underline{cd}$, where all digits are nonzero. The same number in base twelve is $\underline{bb} . \underline{ba}$. Find the base-ten number $\underline{abc}$. | 321 |
Open-Reasoner-Zero/orz_math_57k_collection | Find the number of functions $f(x)$ from $\{1, 2, 3, 4, 5\}$ to $\{1, 2, 3, 4, 5\}$ that satisfy $f(f(x)) = f(f(f(x)))$ for all $x$ in $\{1, 2, 3, 4, 5\}$. | 756 |
Open-Reasoner-Zero/orz_math_57k_collection | Find the smallest integer $k$ for which the conditions
(1) $a_1,a_2,a_3\cdots$ is a nondecreasing sequence of positive integers
(2) $a_n=a_{n-1}+a_{n-2}$ for all $n>2$
(3) $a_9=k$
are satisfied by more than one sequence. | 748 |
Open-Reasoner-Zero/orz_math_57k_collection | Find the least positive integer $N$ such that the set of $1000$ consecutive integers beginning with $1000\cdot N$ contains no square of an integer. | 282 |
Open-Reasoner-Zero/orz_math_57k_collection | Find the sum of all positive integers $n$ such that $\sqrt{n^2+85n+2017}$ is an integer. | 195 |
Open-Reasoner-Zero/orz_math_57k_collection | Let $a, b, c,$ and $d$ be real numbers that satisfy the system of equations
\begin{align*} a + b &= -3, \\ ab + bc + ca &= -4, \\ abc + bcd + cda + dab &= 14, \\ abcd &= 30. \end{align*}
There exist relatively prime positive integers $m$ and $n$ such that
\[a^2 + b^2 + c^2 + d^2 = \frac{m}{n}.\]Find $m + n$. | 145 |
Open-Reasoner-Zero/orz_math_57k_collection | What is the largest number of towns that can meet the following criteria. Each pair is directly linked by just one of air, bus or train. At least one pair is linked by air, at least one pair by bus and at least one pair by train. No town has an air link, a bus link and a train link. No three towns, $A, B, C$ are such t... | 4 |
Open-Reasoner-Zero/orz_math_57k_collection | A game uses a deck of $n$ different cards, where $n$ is an integer and $n \geq 6.$ The number of possible sets of 6 cards that can be drawn from the deck is 6 times the number of possible sets of 3 cards that can be drawn. Find $n.$ | 13 |
Open-Reasoner-Zero/orz_math_57k_collection | In the accompanying figure, the outer square $S$ has side length $40$. A second square $S'$ of side length $15$ is constructed inside $S$ with the same center as $S$ and with sides parallel to those of $S$. From each midpoint of a side of $S$, segments are drawn to the two closest vertices of $S'$. The result is a four... | 750 |
Open-Reasoner-Zero/orz_math_57k_collection | The solutions to the system of equations
$\log_{225}x+\log_{64}y=4$
$\log_{x}225-\log_{y}64=1$
are $(x_1,y_1)$ and $(x_2,y_2)$. Find $\log_{30}\left(x_1y_1x_2y_2\right)$. | 12 |
Open-Reasoner-Zero/orz_math_57k_collection | A frog is positioned at the origin of the coordinate plane. From the point $(x, y)$, the frog can jump to any of the points $(x + 1, y)$, $(x + 2, y)$, $(x, y + 1)$, or $(x, y + 2)$. Find the number of distinct sequences of jumps in which the frog begins at $(0, 0)$ and ends at $(4, 4)$. | 556 |
Open-Reasoner-Zero/orz_math_57k_collection | Find the positive integer $n\,$ for which
\[\lfloor\log_2{1}\rfloor+\lfloor\log_2{2}\rfloor+\lfloor\log_2{3}\rfloor+\cdots+\lfloor\log_2{n}\rfloor=1994\]
(For real $x\,$, $\lfloor x\rfloor\,$ is the greatest integer $\le x.\,$) | 312 |
Open-Reasoner-Zero/orz_math_57k_collection | There is a prime number $p$ such that $16p+1$ is the cube of a positive integer. Find $p$. | 307 |
Open-Reasoner-Zero/orz_math_57k_collection | Call a three-term strictly increasing arithmetic sequence of integers special if the sum of the squares of the three terms equals the product of the middle term and the square of the common difference. Find the sum of the third terms of all special sequences. | 31 |
Open-Reasoner-Zero/orz_math_57k_collection | Rectangle $ABCD$ has side lengths $AB=84$ and $AD=42$. Point $M$ is the midpoint of $\overline{AD}$, point $N$ is the trisection point of $\overline{AB}$ closer to $A$, and point $O$ is the intersection of $\overline{CM}$ and $\overline{DN}$. Point $P$ lies on the quadrilateral $BCON$, and $\overline{BP}$ bisects the a... | 546 |
Open-Reasoner-Zero/orz_math_57k_collection | Let
$S_1=\{(x,y)|\log_{10}(1+x^2+y^2)\le 1+\log_{10}(x+y)\}$
and
$S_2=\{(x,y)|\log_{10}(2+x^2+y^2)\le 2+\log_{10}(x+y)\}$.
What is the ratio of the area of $S_2$ to the area of $S_1$?
$\mathrm{(A) \ } 98\qquad \mathrm{(B) \ } 99\qquad \mathrm{(C) \ } 100\qquad \mathrm{(D) \ } 101\qquad \mathrm{(E) \ } 102$ | 102 |
Open-Reasoner-Zero/orz_math_57k_collection | Determine $3x_4+2x_5$ if $x_1$, $x_2$, $x_3$, $x_4$, and $x_5$ satisfy the system of equations below.
$2x_1+x_2+x_3+x_4+x_5=6$
$x_1+2x_2+x_3+x_4+x_5=12$
$x_1+x_2+2x_3+x_4+x_5=24$
$x_1+x_2+x_3+2x_4+x_5=48$
$x_1+x_2+x_3+x_4+2x_5=96$ | 181 |
Open-Reasoner-Zero/orz_math_57k_collection | Find the least positive integer $n$ such that when $3^n$ is written in base $143$, its two right-most digits in base $143$ are $01$. | 195 |
Open-Reasoner-Zero/orz_math_57k_collection | Let $a,b,c,d$ be real numbers such that $b-d \ge 5$ and all zeros $x_1, x_2, x_3,$ and $x_4$ of the polynomial $P(x)=x^4+ax^3+bx^2+cx+d$ are real. Find the smallest value the product $(x_1^2+1)(x_2^2+1)(x_3^2+1)(x_4^2+1)$ can take. | 16 |
Open-Reasoner-Zero/orz_math_57k_collection | For every subset $T$ of $U = \{ 1,2,3,\ldots,18 \}$, let $s(T)$ be the sum of the elements of $T$, with $s(\emptyset)$ defined to be $0$. If $T$ is chosen at random among all subsets of $U$, the probability that $s(T)$ is divisible by $3$ is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find ... | 683 |
Open-Reasoner-Zero/orz_math_57k_collection | An ordered pair $( b , c )$ of integers, each of which has absolute value less than or equal to five, is chosen at random, with each
such ordered pair having an equal likelihood of being chosen. What is the probability that the equation $x^ 2 + bx + c = 0$ will
not have distinct positive real roots?
$\textbf{(A) }\frac... | \frac{111}{121} |
Open-Reasoner-Zero/orz_math_57k_collection | Ms. Math's kindergarten class has $16$ registered students. The classroom has a very large number, $N$, of play blocks which satisfies the conditions:
(a) If $16$, $15$, or $14$ students are present in the class, then in each case all the blocks can be distributed in equal numbers to each student, and
(b) There are thr... | 148 |
Open-Reasoner-Zero/orz_math_57k_collection | Let $S$ be a list of positive integers--not necessarily distinct--in which the number $68$ appears. The average (arithmetic mean) of the numbers in $S$ is $56$. However, if $68$ is removed, the average of the remaining numbers drops to $55$. What is the largest number that can appear in $S$? | 649 |
Open-Reasoner-Zero/orz_math_57k_collection | Let $S$ be the set of all polynomials of the form $z^3 + az^2 + bz + c$, where $a$, $b$, and $c$ are integers. Find the number of polynomials in $S$ such that each of its roots $z$ satisfies either $|z| = 20$ or $|z| = 13$. | 540 |
Open-Reasoner-Zero/orz_math_57k_collection | Let $L$ be the line with slope $\frac{5}{12}$ that contains the point $A=(24,-1)$, and let $M$ be the line perpendicular to line $L$ that contains the point $B=(5,6)$. The original coordinate axes are erased, and line $L$ is made the $x$-axis and line $M$ the $y$-axis. In the new coordinate system, point $A$ is on th... | 31 |
Open-Reasoner-Zero/orz_math_57k_collection | A point $P$ is outside a circle and is $13$ inches from the center. A secant from $P$ cuts the circle at $Q$ and $R$
so that the external segment of the secant $PQ$ is $9$ inches and $QR$ is $7$ inches. The radius of the circle is: | 5 |
Open-Reasoner-Zero/orz_math_57k_collection | A box contains chips, each of which is red, white, or blue. The number of blue chips is at least half the number of white chips, and at most one third the number of red chips. The number which are white or blue is at least $55$. The minimum number of red chips is
$\textbf{(A) }24\qquad \textbf{(B) }33\qquad \textbf{(C)... | 57 |
Open-Reasoner-Zero/orz_math_57k_collection | Bernardo and Silvia play the following game. An integer between $0$ and $999$ inclusive is selected and given to Bernardo. Whenever Bernardo receives a number, he doubles it and passes the result to Silvia. Whenever Silvia receives a number, she adds $50$ to it and passes the result to Bernardo. The winner is the last ... | 7 |
Open-Reasoner-Zero/orz_math_57k_collection | A computer screen shows a $98 \times 98$ chessboard, colored in the usual way. One can select with a mouse any rectangle with sides on the lines of the chessboard and click the mouse button: as a result, the colors in the selected rectangle switch (black becomes white, white becomes black). Find, with proof, the minim... | 98 |
Open-Reasoner-Zero/orz_math_57k_collection | Find the number of rational numbers $r$, $0<r<1$, such that when $r$ is written as a fraction in lowest terms, the numerator and the denominator have a sum of 1000. | 200 |
Open-Reasoner-Zero/orz_math_57k_collection | What is the largest even integer that cannot be written as the sum of two odd composite numbers? | 38 |
Open-Reasoner-Zero/orz_math_57k_collection | Let $S$ be the set of positive integers $N$ with the property that the last four digits of $N$ are $2020,$ and when the last four digits are removed, the result is a divisor of $N.$ For example, $42,020$ is in $S$ because $4$ is a divisor of $42,020.$ Find the sum of all the digits of all the numbers in $S.$ For exampl... | 93 |
Open-Reasoner-Zero/orz_math_57k_collection | Suppose that $(a_1,b_1),$ $(a_2,b_2),$ $\dots,$ $(a_{100},b_{100})$ are distinct ordered pairs of nonnegative integers. Let $N$ denote the number of pairs of integers $(i,j)$ satisfying $1\leq i<j\leq 100$ and $|a_ib_j-a_jb_i|=1$. Determine the largest possible value of $N$ over all possible choices of the $100$ ordere... | 197 |
Open-Reasoner-Zero/orz_math_57k_collection | Let $n$ be the least positive integer for which $149^n-2^n$ is divisible by $3^3\cdot5^5\cdot7^7.$ Find the number of positive integer divisors of $n.$ | 270 |
Open-Reasoner-Zero/orz_math_57k_collection | Let $S\,$ be a set with six elements. In how many different ways can one select two not necessarily distinct subsets of $S\,$ so that the union of the two subsets is $S\,$? The order of selection does not matter; for example, the pair of subsets $\{a, c\},\{b, c, d, e, f\}$ represents the same selection as the pair $\... | 365 |
Open-Reasoner-Zero/orz_math_57k_collection | Two real numbers $a$ and $b$ are chosen independently and uniformly at random from the interval $(0, 75)$. Let $O$ and $P$ be two points on the plane with $OP = 200$. Let $Q$ and $R$ be on the same side of line $OP$ such that the degree measures of $\angle POQ$ and $\angle POR$ are $a$ and $b$ respectively, and $\angle... | 41 |
Open-Reasoner-Zero/orz_math_57k_collection | For positive integers $n$ and $k$, let $f(n, k)$ be the remainder when $n$ is divided by $k$, and for $n > 1$ let $F(n) = \max_{\substack{1\le k\le \frac{n}{2}}} f(n, k)$. Find the remainder when $\sum\limits_{n=20}^{100} F(n)$ is divided by $1000$. | 512 |
Open-Reasoner-Zero/orz_math_57k_collection | Circles $\omega_1$ and $\omega_2$ with radii $961$ and $625$, respectively, intersect at distinct points $A$ and $B$. A third circle $\omega$ is externally tangent to both $\omega_1$ and $\omega_2$. Suppose line $AB$ intersects $\omega$ at two points $P$ and $Q$ such that the measure of minor arc $\widehat{PQ}$ is $120... | 672 |
Open-Reasoner-Zero/orz_math_57k_collection | While watching a show, Ayako, Billy, Carlos, Dahlia, Ehuang, and Frank sat in that order in a row of six chairs. During the break, they went to the kitchen for a snack. When they came back, they sat on those six chairs in such a way that if two of them sat next to each other before the break, then they did not sit next... | 90 |
Open-Reasoner-Zero/orz_math_57k_collection | Two distinct, real, infinite geometric series each have a sum of $1$ and have the same second term. The third term of one of the series is $1/8$, and the second term of both series can be written in the form $\frac{\sqrt{m}-n}p$, where $m$, $n$, and $p$ are positive integers and $m$ is not divisible by the square of an... | 518 |
Open-Reasoner-Zero/orz_math_57k_collection | For each integer $n\geq3$, let $f(n)$ be the number of $3$-element subsets of the vertices of the regular $n$-gon that are the vertices of an isosceles triangle (including equilateral triangles). Find the sum of all values of $n$ such that $f(n+1)=f(n)+78$. | 245 |
Open-Reasoner-Zero/orz_math_57k_collection | The vertices of a regular nonagon (9-sided polygon) are to be labeled with the digits 1 through 9 in such a way that the sum of the numbers on every three consecutive vertices is a multiple of 3. Two acceptable arrangements are considered to be indistinguishable if one can be obtained from the other by rotating the no... | 144 |
Open-Reasoner-Zero/orz_math_57k_collection | In rectangle $ABCD$, $AB = 12$ and $BC = 10$. Points $E$ and $F$ lie inside rectangle $ABCD$ so that $BE = 9$, $DF = 8$, $\overline{BE} \parallel \overline{DF}$, $\overline{EF} \parallel \overline{AB}$, and line $BE$ intersects segment $\overline{AD}$. The length $EF$ can be expressed in the form $m \sqrt{n} - p$, wh... | 36 |
Open-Reasoner-Zero/orz_math_57k_collection | Consider arrangements of the $9$ numbers $1, 2, 3, \dots, 9$ in a $3 \times 3$ array. For each such arrangement, let $a_1$, $a_2$, and $a_3$ be the medians of the numbers in rows $1$, $2$, and $3$ respectively, and let $m$ be the median of $\{a_1, a_2, a_3\}$. Let $Q$ be the number of arrangements for which $m = 5$. Fi... | 360 |
Open-Reasoner-Zero/orz_math_57k_collection | Sides $AB,~ BC, ~CD$ and $DA$, respectively, of convex quadrilateral $ABCD$ are extended past $B,~ C ,~ D$ and $A$ to points $B',~C',~ D'$ and $A'$. Also, $AB = BB' = 6,~ BC = CC' = 7, ~CD = DD' = 8$ and $DA = AA' = 9$; and the area of $ABCD$ is $10$. The area of $A 'B 'C'D'$ is | 50 |
Open-Reasoner-Zero/orz_math_57k_collection | A triangle has vertices $A(0,0)$, $B(12,0)$, and $C(8,10)$. The probability that a randomly chosen point inside the triangle is closer to vertex $B$ than to either vertex $A$ or vertex $C$ can be written as $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$. | 409 |
Open-Reasoner-Zero/orz_math_57k_collection | Let the sum of a set of numbers be the sum of its elements. Let $S$ be a set of positive integers, none greater than 15. Suppose no two disjoint subsets of $S$ have the same sum. What is the largest sum a set $S$ with these properties can have? | 61 |
Open-Reasoner-Zero/orz_math_57k_collection | A convex quadrilateral has area $30$ and side lengths $5, 6, 9,$ and $7,$ in that order. Denote by $\theta$ the measure of the acute angle formed by the diagonals of the quadrilateral. Then $\tan \theta$ can be written in the form $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$. | 47 |
Open-Reasoner-Zero/orz_math_57k_collection | For a permutation $p = (a_1,a_2,\ldots,a_9)$ of the digits $1,2,\ldots,9$, let $s(p)$ denote the sum of the three $3$-digit numbers $a_1a_2a_3$, $a_4a_5a_6$, and $a_7a_8a_9$. Let $m$ be the minimum value of $s(p)$ subject to the condition that the units digit of $s(p)$ is $0$. Let $n$ denote the number of permutations ... | 162 |
Open-Reasoner-Zero/orz_math_57k_collection | Define a T-grid to be a $3\times3$ matrix which satisfies the following two properties:
Exactly five of the entries are $1$'s, and the remaining four entries are $0$'s.
Among the eight rows, columns, and long diagonals (the long diagonals are $\{a_{13},a_{22},a_{31}\}$ and $\{a_{11},a_{22},a_{33}\}$, no more than one... | 68 |
Open-Reasoner-Zero/orz_math_57k_collection | An $m\times n\times p$ rectangular box has half the volume of an $(m + 2)\times(n + 2)\times(p + 2)$ rectangular box, where $m, n,$ and $p$ are integers, and $m\le n\le p.$ What is the largest possible value of $p$? | 130 |
Open-Reasoner-Zero/orz_math_57k_collection | A frog begins at $P_0 = (0,0)$ and makes a sequence of jumps according to the following rule: from $P_n = (x_n, y_n),$ the frog jumps to $P_{n+1},$ which may be any of the points $(x_n + 7, y_n + 2),$ $(x_n + 2, y_n + 7),$ $(x_n - 5, y_n - 10),$ or $(x_n - 10, y_n - 5).$ There are $M$ points $(x, y)$ with $|x| + |y| \l... | 373 |
Open-Reasoner-Zero/orz_math_57k_collection | Segments $\overline{AB}, \overline{AC},$ and $\overline{AD}$ are edges of a cube and $\overline{AG}$ is a diagonal through the center of the cube. Point $P$ satisfies $BP=60\sqrt{10}$, $CP=60\sqrt{5}$, $DP=120\sqrt{2}$, and $GP=36\sqrt{7}$. Find $AP.$ | 192 |
Open-Reasoner-Zero/orz_math_57k_collection | Suppose one of the eight lettered identical squares is included with the four squares in the T-shaped figure outlined. How many of the resulting figures can be folded into a topless cubical box?
$\text{(A)}\ 2 \qquad \text{(B)}\ 3 \qquad \text{(C)}\ 4 \qquad \text{(D)}\ 5 \qquad \text{(E)}\ 6$ | 6 |
Open-Reasoner-Zero/orz_math_57k_collection | Octagon $ABCDEFGH$ with side lengths $AB = CD = EF = GH = 10$ and $BC = DE = FG = HA = 11$ is formed by removing 6-8-10 triangles from the corners of a $23$ $\times$ $27$ rectangle with side $\overline{AH}$ on a short side of the rectangle, as shown. Let $J$ be the midpoint of $\overline{AH}$, and partition the octagon... | 184 |
Open-Reasoner-Zero/orz_math_57k_collection | Let $SP_1P_2P_3EP_4P_5$ be a heptagon. A frog starts jumping at vertex $S$. From any vertex of the heptagon except $E$, the frog may jump to either of the two adjacent vertices. When it reaches vertex $E$, the frog stops and stays there. Find the number of distinct sequences of jumps of no more than $12$ jumps that end... | 351 |
Open-Reasoner-Zero/orz_math_57k_collection | Circle $C_0$ has radius $1$, and the point $A_0$ is a point on the circle. Circle $C_1$ has radius $r<1$ and is internally tangent to $C_0$ at point $A_0$. Point $A_1$ lies on circle $C_1$ so that $A_1$ is located $90^{\circ}$ counterclockwise from $A_0$ on $C_1$. Circle $C_2$ has radius $r^2$ and is internally tangent... | 110 |
Open-Reasoner-Zero/orz_math_57k_collection | The area of the region bounded by the graph of\[x^2+y^2 = 3|x-y| + 3|x+y|\]is $m+n\pi$, where $m$ and $n$ are integers. What is $m + n$?
$\textbf{(A)} ~18\qquad\textbf{(B)} ~27\qquad\textbf{(C)} ~36\qquad\textbf{(D)} ~45\qquad\textbf{(E)} ~54$ | 54 |
Open-Reasoner-Zero/orz_math_57k_collection | A circular disc with diameter $D$ is placed on an $8\times 8$ checkerboard with width $D$ so that the centers coincide. The number of checkerboard squares which are completely covered by the disc is
$\textbf{(A) }48\qquad \textbf{(B) }44\qquad \textbf{(C) }40\qquad \textbf{(D) }36\qquad \textbf{(E) }32$ | 32 |
Open-Reasoner-Zero/orz_math_57k_collection | If $0 < x < \frac{\pi}{2}$ and $\frac{\sin x}{1 + \cos x} = \frac{1}{3}$ , what is $\frac{\sin 2x}{1 + \cos 2x}$ ? | \frac{3}{4} |
Open-Reasoner-Zero/orz_math_57k_collection | Find the sum of all four-digit integers whose digits are a rearrangement of the digits $1$ , $2$ , $3$ , $4$ , such as $1234$ , $1432$ , or $3124$ . | 66660 |
Open-Reasoner-Zero/orz_math_57k_collection | Let $x_1, x_2,... , x_{84}$ be the roots of the equation $x^{84} + 7x - 6 = 0$ . Compute $\sum_{k=1}^{84} \frac{x_k}{x_k-1}$ . | \frac{77}{2} |
Open-Reasoner-Zero/orz_math_57k_collection | Juan wrote a natural number and Maria added a digit $ 1$ to the left and a digit $ 1$ to the right. Maria's number exceeds to the number of Juan in $14789$ . Find the number of Juan. | 532 |
Open-Reasoner-Zero/orz_math_57k_collection | A point $(x,y)$ in the first quadrant lies on a line with intercepts $(a,0)$ and $(0,b)$ , with $a,b > 0$ . Rectangle $M$ has vertices $(0,0)$ , $(x,0)$ , $(x,y)$ , and $(0,y)$ , while rectangle $N$ has vertices $(x,y)$ , $(x,b)$ , $(a,b)$ , and $(a,y)$ . What is the ratio of the area of $M$ to that... | 1 |
Open-Reasoner-Zero/orz_math_57k_collection | Floyd looked at a standard $12$ hour analogue clock at $2\!:\!36$ . When Floyd next looked at the clock, the angles through which the hour hand and minute hand of the clock had moved added to $247$ degrees. How many minutes after $3\!:\!00$ was that? | 14 |
Open-Reasoner-Zero/orz_math_57k_collection | Let $z_1$ and $z_2$ be the zeros of the polynomial $f(x) = x^2 + 6x + 11$ . Compute $(1 + z^2_1z_2)(1 + z_1z_2^2)$ . | 1266 |
Open-Reasoner-Zero/orz_math_57k_collection | Jerry's favorite number is $97$ . He knows all kinds of interesting facts about $97$ :
- $97$ is the largest two-digit prime.
- Reversing the order of its digits results in another prime.
- There is only one way in which $97$ can be written as a difference of two perfect squares.
- There is only one way in whic... | 96 |
Open-Reasoner-Zero/orz_math_57k_collection | Two circles $\mathcal{C}_1$ and $\mathcal{C}_2$ with centers $(1, 1)$ and $(4, 5)$ and radii $r_1 < r_2$ , respectively, are drawn on the coordinate plane. The product of the slopes of the two common external tangents of $\mathcal{C}_1$ and $\mathcal{C}_2$ is $3$ . If the value of $(r_2 - r_1)^2$ can be... | 13 |
Open-Reasoner-Zero/orz_math_57k_collection | Tina writes four letters to her friends Silas, Jessica, Katie, and Lekan. She prepares an envelope for Silas, an envelope for Jessica, an envelope for Katie, and an envelope for Lekan. However, she puts each letter into a random envelope. What is the probability that no one receives the letter they are supposed to rece... | \frac{3}{8} |
Open-Reasoner-Zero/orz_math_57k_collection | What is the smallest possible natural number $n$ for which the equation $x^2 -nx + 2014 = 0$ has integer roots? | 91 |
Open-Reasoner-Zero/orz_math_57k_collection | Compute the number of positive integers less than or equal to $10000$ which are relatively prime to $2014$ . | 4648 |
Open-Reasoner-Zero/orz_math_57k_collection | Find the remainder when the number of positive divisors of the value $$ (3^{2020}+3^{2021})(3^{2021}+3^{2022})(3^{2022}+3^{2023})(3^{2023}+3^{2024}) $$ is divided by $1000$ .
*Proposed by pog* | 783 |
Open-Reasoner-Zero/orz_math_57k_collection | Suppose $xy-5x+2y=30$ , where $x$ and $y$ are positive integers. Find the sum of all possible values of $x$ | 31 |
Open-Reasoner-Zero/orz_math_57k_collection | Paul fi lls in a $7\times7$ grid with the numbers $1$ through $49$ in a random arrangement. He then erases his work and does the same thing again, to obtain two diff erent random arrangements of the numbers in the grid. What is the expected number of pairs of numbers that occur in either the same row as each othe... | 73.5 |
Open-Reasoner-Zero/orz_math_57k_collection | A hotel packed breakfast for each of three guests. Each breakfast should have consisted of three types of rolls, one each of nut, cheese, and fruit rolls. The preparer wrapped each of the nine rolls and once wrapped, the rolls were indistinguishable from one another. She then randomly put three rolls in a bag for each ... | 79 |
Open-Reasoner-Zero/orz_math_57k_collection | Let $a_1 = 1$ and $a_{n+1} = a_n \cdot p_n$ for $n \geq 1$ where $p_n$ is the $n$ th prime number, starting with $p_1 = 2$ . Let $\tau(x)$ be equal to the number of divisors of $x$ . Find the remainder when $$ \sum_{n=1}^{2020} \sum_{d \mid a_n} \tau (d) $$ is divided by 91 for positive integers $d$ . R... | 40 |
Open-Reasoner-Zero/orz_math_57k_collection | A group of women working together at the same rate can build a wall in $45$ hours. When the work started, all the women did not start working together. They joined the worked over a period of time, one by one, at equal intervals. Once at work, each one stayed till the work was complete. If the first woman worked 5 ti... | 75 |
Open-Reasoner-Zero/orz_math_57k_collection | Farmer Tim is lost in the densely-forested Cartesian plane. Starting from the origin he walks a sinusoidal path in search of home; that is, after $t$ minutes he is at position $(t,\sin t)$ .
Five minutes after he sets out, Alex enters the forest at the origin and sets out in search of Tim. He walks in such a way t... | 3\sqrt{3} |
Open-Reasoner-Zero/orz_math_57k_collection | Let $ a$ , $ b$ , $ c$ , $ x$ , $ y$ , and $ z$ be real numbers that satisfy the three equations
\begin{align*}
13x + by + cz &= 0
ax + 23y + cz &= 0
ax + by + 42z &= 0.
\end{align*}Suppose that $ a \ne 13$ and $ x \ne 0$ . What is the value of
\[ \frac{13}{a - 13} + \frac{23}{b - 23} + \... | -2 |
Open-Reasoner-Zero/orz_math_57k_collection | Let $T = TNFTPP$ . When properly sorted, $T - 35$ math books on a shelf are arranged in alphabetical order from left to right. An eager student checked out and read all of them. Unfortunately, the student did not realize how the books were sorted, and so after finishing ... | 2161 |
Open-Reasoner-Zero/orz_math_57k_collection | Eight football teams play matches against each other in such a way that no two teams meet twice and no three teams play all of the three possible matches.
What is the largest possible number of matches? | 16 |
Open-Reasoner-Zero/orz_math_57k_collection | Let $m,n$ be natural numbers such that $\hspace{2cm} m+3n-5=2LCM(m,n)-11GCD(m,n).$ Find the maximum possible value of $m+n$ . | 70 |
Open-Reasoner-Zero/orz_math_57k_collection | Find value of $$ \frac{1}{1+x+xy}+\frac{1}{1+y+yz}+\frac{1}{1+z+zx} $$ if $x$ , $y$ and $z$ are real numbers usch that $xyz=1$ | 1 |
Open-Reasoner-Zero/orz_math_57k_collection | Suppose that $N$ is a three digit number divisible by $7$ such that upon removing its middle digit, the remaining two digit number is also divisible by $7$ . What is the minimum possible value of $N$ ?
*2019 CCA Math Bonanza Lightning Round #3.1* | 154 |
Open-Reasoner-Zero/orz_math_57k_collection | Garfield and Odie are situated at $(0,0)$ and $(25,0)$ , respectively. Suddenly, Garfield and Odie dash in the direction of the point $(9, 12)$ at speeds of $7$ and $10$ units per minute, respectively. During this chase, the minimum distance between Garfield and Odie can be written as $\frac{m}{\sqrt{n}}$ fo... | 159 |
Open-Reasoner-Zero/orz_math_57k_collection | ABC is a triangle with coordinates A =(2, 6), B =(0, 0), and C =(14, 0).
(a) Let P be the midpoint of AB. Determine the equation of the line perpendicular to AB passing through P.
(b) Let Q be the point on line BC for which PQ is perpendicular to AB. Determine the length of AQ.
(c) There is a (unique) circle passin... | 5\sqrt{2} |
Open-Reasoner-Zero/orz_math_57k_collection | On a $3 \times 3$ board the numbers from $1$ to $9$ are written in some order and without repeating. We say that the arrangement obtained is *Isthmian* if the numbers in any two adjacent squares have different parity. Determine the number of different Isthmian arrangements.
Note: Two arrangements are considered ... | 720 |
Open-Reasoner-Zero/orz_math_57k_collection | For positive integers $n$ and $k$ , let $\mho(n,k)$ be the number of distinct prime divisors of $n$ that are at least $k$ . For example, $\mho(90, 3)=2$ , since the only prime factors of $90$ that are at least $3$ are $3$ and $5$ . Find the closest integer to
\[\sum_{n=1}^\infty \sum_{k=1}^\infty \frac{... | 167 |
Open-Reasoner-Zero/orz_math_57k_collection | Find the largest value of the expression $\frac{p}{R}\left( 1- \frac{r}{3R}\right)$ , where $p,R, r$ is, respectively, the perimeter, the radius of the circumscribed circle and the radius of the inscribed circle of a triangle. | \frac{5\sqrt{3}}{2} |
Open-Reasoner-Zero/orz_math_57k_collection | Two toads named Gamakichi and Gamatatsu are sitting at the points $(0,0)$ and $(2,0)$ respectively. Their goal is to reach $(5,5)$ and $(7,5)$ respectively by making one unit jumps in positive $x$ or $y$ direction at a time. How many ways can they do this while ensuring that there is no point on the plane w... | 19152 |
Open-Reasoner-Zero/orz_math_57k_collection | Let $a_1 < a_2 < \cdots < a_k$ denote the sequence of all positive integers between $1$ and $91$ which are relatively prime to $91$ , and set $\omega = e^{2\pi i/91}$ . Define \[S = \prod_{1\leq q < p\leq k}\left(\omega^{a_p} - \omega^{a_q}\right).\] Given that $S$ is a positive integer, compute the number of... | 1054 |
Open-Reasoner-Zero/orz_math_57k_collection | Let $f(x) = x^3 - 3x + b$ and $g(x) = x^2 + bx -3$ , where $b$ is a real number. What is the sum of all possible values of $b$ for which the equations $f(x)$ = 0 and $g(x) = 0$ have a common root? | 0 |
Open-Reasoner-Zero/orz_math_57k_collection | There is a positive integer s such that there are s solutions to the equation $64sin^2(2x)+tan^2(x)+cot^2(x)=46$ in the interval $(0,\frac{\pi}{2})$ all of the form $\frac{m_k}{n_k}\pi$ where $m_k$ and $n_k$ are relatively prime positive integers, for $k = 1, 2, 3, . . . , s$ . Find $(m_1 + n_1) + (m_2 + n_2)... | 100 |
Open-Reasoner-Zero/orz_math_57k_collection | Let $ABC$ be an acute triangle. $PQRS$ is a rectangle with $P$ on $AB$ , $Q$ and $R$ on $BC$ , and $S$ on $AC$ such that $PQRS$ has the largest area among all rectangles $TUVW$ with $T$ on $AB$ , $U$ and $V$ on $BC$ , and $W$ on $AC$ . If $D$ is the point on $BC$ such that $AD\perp B... | 4 |
Open-Reasoner-Zero/orz_math_57k_collection | A binary string is a string consisting of only 0’s and 1’s (for instance, 001010, 101, etc.). What is the probability that a randomly chosen binary string of length 10 has 2 consecutive 0’s? Express your answer as a fraction. | \frac{55}{64} |
Open-Reasoner-Zero/orz_math_57k_collection | For a real number $x,$ let $\lfloor x\rfloor$ denote the greatest integer less than or equal to $x,$ and let $\{x\} = x -\lfloor x\rfloor$ denote the fractional part of $x.$ The sum of all real numbers $\alpha$ that satisfy the equation $$ \alpha^2+\{\alpha\}=21 $$ can be expressed in the form $$ \fra... | 169 |
Open-Reasoner-Zero/orz_math_57k_collection | Forty teams play a tournament in which every team plays every other team exactly once. No ties occur, and each team has a $50 \%$ chance of winning any game it plays. The probability that no two teams win the same number of games is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $\log... | 742 |
Open-Reasoner-Zero/orz_math_57k_collection | Suppose that $(a_n)$ is a sequence of real numbers such that the series $$ \sum_{n=1}^\infty\frac{a_n}n $$ is convergent. Show that the sequence $$ b_n=\frac1n\sum^n_{j=1}a_j $$ is convergent and find its limit. | 0 |
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