Unnamed: 0
int64 0
56.9k
| problem
stringlengths 16
7.44k
| ground_truth
stringlengths 1
942
| solved_percentage
float64 0
100
|
|---|---|---|---|
1,000 |
Let $ M(n )\equal{}\{\minus{}1,\minus{}2,\ldots,\minus{}n\}$ . For every non-empty subset of $ M(n )$ we consider the product of its elements. How big is the sum over all these products?
|
-1
| 46.875 |
1,001 |
$ P(x)$ is a quadratic trinomial. What maximum number of terms equal to the sum of the two preceding terms can occur in the sequence $ P(1)$ , $ P(2)$ , $ P(3)$ , $ \dots?$
*Proposed by A. Golovanov*
|
2
| 17.1875 |
1,002 |
Let $n$ be a positive integer and consider an arrangement of $2n$ blocks in a straight line, where $n$ of them are red and the rest blue. A swap refers to choosing two consecutive blocks and then swapping their positions. Let $A$ be the minimum number of swaps needed to make the first $n$ blocks all red and $B$ be the minimum number of swaps needed to make the first $n$ blocks all blue. Show that $A+B$ is independent of the starting arrangement and determine its value.
|
n^2
| 68.75 |
1,003 |
A word is an ordered, non-empty sequence of letters, such as $word$ or $wrod$ . How many distinct $3$ -letter words can be made from a subset of the letters $c, o, m, b, o$ , where each letter in the list is used no more than the number of times it appears?
|
33
| 75 |
1,004 |
We call the polynomial $P (x)$ simple if the coefficient of each of its members belongs to the set $\{-1, 0, 1\}$ .
Let $n$ be a positive integer, $n> 1$ . Find the smallest possible number of terms with a non-zero coefficient in a simple $n$ -th degree polynomial with all values at integer places are divisible by $n$ .
|
2
| 65.625 |
1,005 |
Define $ a \circledast b = a + b-2ab $ . Calculate the value of $$ A=\left( ...\left(\left(\frac{1}{2014}\circledast \frac{2}{2014}\right)\circledast\frac{3}{2014}\right)...\right)\circledast\frac{2013}{2014} $$
|
\frac{1}{2}
| 87.5 |
1,006 |
For $n \geq 3$ , let $(b_0, b_1,..., b_{n-1}) = (1, 1, 1, 0, ..., 0).$ Let $C_n =
(c_{i, j})$ the $n \times n$ matrix defined by $c_{i, j} = b _{(j -i) \mod n}$ . Show
that $\det (C_n) = 3$ if $n$ is not a multiple of 3 and $\det (C_n) = 0$ if $n$
is a multiple of 3.
|
0
| 3.90625 |
1,007 |
In parallelogram $ABCD$ , $AC=10$ and $BD=28$ . The points $K$ and $L$ in the plane of $ABCD$ move in such a way that $AK=BD$ and $BL=AC$ . Let $M$ and $N$ be the midpoints of $CK$ and $DL$ , respectively. What is the maximum walue of $\cot^2 (\tfrac{\angle BMD}{2})+\tan^2(\tfrac{\angle ANC}{2})$ ?
|
2
| 49.21875 |
1,008 |
We have a calculator with two buttons that displays and integer $x$ . Pressing the first button replaces $x$ by $\lfloor \frac{x}{2} \rfloor$ , and pressing the second button replaces $x$ by $4x+1$ . Initially, the calculator displays $0$ . How many integers less than or equal to $2014$ can be achieved through a sequence of arbitrary button presses? (It is permitted for the number displayed to exceed 2014 during the sequence. Here, $\lfloor y \rfloor$ denotes the greatest integer less than or equal to the real number $y$ ).
|
233
| 0 |
1,009 |
A jacket was originally priced $\textdollar 100$ . The price was reduced by $10\%$ three times and increased by $10\%$ four times in some order. To the nearest cent, what was the final price?
|
106.73
| 60.9375 |
1,010 |
Find all non-negative integer numbers $n$ for which there exists integers $a$ and $b$ such that $n^2=a+b$ and $n^3=a^2+b^2.$
|
0, 1, 2
| 82.03125 |
1,011 |
There are 1000 rooms in a row along a long corridor. Initially the first room contains 1000 people and the remaining rooms are empty. Each minute, the following happens: for each room containing more than one person, someone in that room decides it is too crowded and moves to the next room. All these movements are simultaneous (so nobody moves more than once within a minute). After one hour, how many different
rooms will have people in them?
|
61
| 55.46875 |
1,012 |
Let $a_1 = 2,$ and for $n\ge 1,$ let $a_{n+1} = 2a_n + 1.$ Find the smallest value of an $a_n$ that is not a prime number.
|
95
| 100 |
1,013 |
Given a triangle $ABC$ with integer side lengths, where $BD$ is an angle bisector of $\angle ABC$ , $AD=4$ , $DC=6$ , and $D$ is on $AC$ , compute the minimum possible perimeter of $\triangle ABC$ .
|
25
| 97.65625 |
1,014 |
For real numbers $a$ and $b$ , define $$ f(a,b) = \sqrt{a^2+b^2+26a+86b+2018}. $$ Find the smallest possible value of the expression $$ f(a, b) + f (a,-b) + f(-a, b) + f (-a, -b). $$
|
4 \sqrt{2018}
| 45.3125 |
1,015 |
In isosceles right-angled triangle $ABC$ , $CA = CB = 1$ . $P$ is an arbitrary point on the sides of $ABC$ . Find the maximum of $PA \cdot PB \cdot PC$ .
|
\frac{\sqrt{2}}{4}
| 32.8125 |
1,016 |
Suppose $X$ is a random variable that takes on only nonnegative integer values, with $E[X]=1,$ $E[X^2]=2,$ and $E[X^3]=5.$ (Here $E[Y]$ denotes the expectation of the random variable $Y.$ ) Determine the smallest possible value of the probability of the event $X=0.$
|
\frac{1}{3}
| 42.1875 |
1,017 |
Find all triples $(p, q, r)$ of prime numbers for which $4q - 1$ is a prime number and $$ \frac{p + q}{p + r} = r - p $$ holds.
*(Walther Janous)*
|
(2, 3, 3)
| 92.1875 |
1,018 |
The map below shows an east/west road connecting the towns of Acorn, Centerville, and Midland, and a north/south road from Centerville to Drake. The distances from Acorn to Centerville, from Centerville to Midland, and from Centerville to Drake are each 60 kilometers. At noon Aaron starts at Acorn and bicycles east at 17 kilometers per hour, Michael starts at Midland and bicycles west at 7 kilometers per hour, and David starts at Drake and bicycles at a constant rate in a straight line across an open field. All three bicyclists arrive at exactly the same time at a point along the road from Centerville to Midland. Find the number of kilometers that David bicycles.
[center][/center]
|
65
| 33.59375 |
1,019 |
Suppose that $a$ and $b$ are natural numbers such that \[p=\frac{b}{4}\sqrt{\frac{2a-b}{2a+b}}\] is a prime number. What is the maximum possible value of $p$ ?
|
5
| 45.3125 |
1,020 |
Tim and Allen are playing a match of *tenus*. In a match of *tenus*, the two players play a series of games, each of which is won by one of the two players. The match ends when one player has won exactly two more games than the other player, at which point the player who has won more games wins the match. In odd-numbered games, Tim wins with probability $3/4$ , and in the even-numbered games, Allen wins with probability $3/4$ . What is the expected number of games in a match?
|
\frac{16}{3}
| 1.5625 |
1,021 |
In a right triangle, the altitude from a vertex to the hypotenuse splits the hypotenuse into two segments of lengths $a$ and $b$ . If the right triangle has area $T$ and is inscribed in a circle of area $C$ , find $ab$ in terms of $T$ and $C$ .
|
\frac{\pi T^2}{C}
| 51.5625 |
1,022 |
The gure below shows a large square divided into $9$ congruent smaller squares. A shaded square bounded by some of the diagonals of those smaller squares has area $14$ . Find the area of the large square.
|
63
| 40.625 |
1,023 |
If $a_0$ is a positive real number, consider the sequence $\{a_n\}$ defined by:
\[ a_{n+1} = \frac{a^2_n - 1}{n+1}, n \geq 0. \]
Show that there exist a real number $a > 0$ such that:
**i.)** for all $a_0 \geq a,$ the sequence $\{a_n\} \rightarrow \infty,$
**ii.)** for all $a_0 < a,$ the sequence $\{a_n\} \rightarrow 0.$
|
a = 2
| 0 |
1,024 |
Find all values of the positive integer $k$ that has the property:
There are no positive integers $a,b$ such that the expression $A(k,a,b)=\frac{a+b}{a^2+k^2b^2-k^2ab}$ is a composite positive number.
|
k = 1
| 0 |
1,025 |
Let $y=f(x)$ be a function of the graph of broken line connected by points $(-1,\ 0),\ (0,\ 1),\ (1,\ 4)$ in the $x$ - $y$ plane.
Find the minimum value of $\int_{-1}^1 \{f(x)-(a|x|+b)\}^2dx.$ *2010 Tohoku University entrance exam/Economics, 2nd exam*
|
\frac{8}{3}
| 12.5 |
1,026 |
Let $ P_1$ be a regular $ r$ -gon and $ P_2$ be a regular $ s$ -gon $ (r\geq s\geq 3)$ such that each interior angle of $ P_1$ is $ \frac {59}{58}$ as large as each interior angle of $ P_2$ . What's the largest possible value of $ s$ ?
|
117
| 21.875 |
1,027 |
Determine the smallest integer $n > 1$ with the property that $n^2(n - 1)$ is divisible by 2009.
|
42
| 48.4375 |
1,028 |
Let $A$ be the set $\{k^{19}-k: 1<k<20, k\in N\}$ . Let $G$ be the GCD of all elements of $A$ .
Then the value of $G$ is?
|
798
| 0 |
1,029 |
On January $20$ , $2018$ , Sally notices that her $7$ children have ages which sum to a perfect square: their ages are $1$ , $3$ , $5$ , $7$ , $9$ , $11$ , and $13$ , with $1+3+5+7+9+11+13=49$ . Let $N$ be the age of the youngest child the next year the sum of the $7$ children's ages is a perfect square on January $20$ th, and let $P$ be that perfect square. Find $N+P$ .
*2018 CCA Math Bonanza Lightning Round #2.3*
|
218
| 35.15625 |
1,030 |
Nine tiles are numbered $1, 2, 3, \ldots, 9,$ respectively. Each of three players randomly selects and keeps three of the tile, and sums those three values. The probability that all three players obtain an odd sum is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$
|
17
| 42.1875 |
1,031 |
A company has five directors. The regulations of the company require that any majority (three or more) of the directors should be able to open its strongroom, but any minority (two or less) should not be able to do so. The strongroom is equipped with ten locks, so that it can only be opened when keys to all ten locks are available. Find all positive integers $n$ such that it is possible to give each of the directors a set of keys to $n$ different locks, according to the requirements and regulations of the company.
|
n = 6
| 0 |
1,032 |
Circle $o$ contains the circles $m$ , $p$ and $r$ , such that they are tangent to $o$ internally and any two of them are tangent between themselves. The radii of the circles $m$ and $p$ are equal to $x$ . The circle $r$ has radius $1$ and passes through the center of the circle $o$ . Find the value of $x$ .
|
\frac{8}{9}
| 32.03125 |
1,033 |
Medians $AM$ and $BE$ of a triangle $ABC$ intersect at $O$ . The points $O, M, E, C$ lie on one circle. Find the length of $AB$ if $BE = AM =3$ .
|
2\sqrt{3}
| 35.9375 |
1,034 |
In a $2 \times 8$ squared board, you want to color each square red or blue in such a way that on each $2 \times 2$ sub-board there are at least $3$ boxes painted blue. In how many ways can this coloring be done?
Note. A $2 \times 2$ board is a square made up of $4$ squares that have a common vertex.
|
341
| 0 |
1,035 |
$ f: \mathbb N \times \mathbb Z \rightarrow \mathbb Z$ satisfy the given conditions
$ a)$ $ f(0,0)\equal{}1$ , $ f(0,1)\equal{}1$ ,
$ b)$ $ \forall k \notin \left\{0,1\right\}$ $ f(0,k)\equal{}0$ and
$ c)$ $ \forall n \geq 1$ and $ k$ , $ f(n,k)\equal{}f(n\minus{}1,k)\plus{}f(n\minus{}1,k\minus{}2n)$
find the sum $ \displaystyle\sum_{k\equal{}0}^{\binom{2009}{2}}f(2008,k)$
|
2^{2008}
| 85.15625 |
1,036 |
What is the largest possible value of the expression $$ gcd \,\,\, (n^2 + 3, (n + 1)^2 + 3 ) $$ for naturals $n$ ?
<details><summary>Click to expand</summary>original wording]Kāda ir izteiksmes LKD (n2 + 3, (n + 1)2 + 3) lielākā iespējamā vērtība naturāliem n?</details>
|
13
| 50 |
1,037 |
Define the *bigness*of a rectangular prism to be the sum of its volume, its surface area, and the lengths of all of its edges. Find the least integer $N$ for which there exists a rectangular prism with integer side lengths and *bigness* $N$ and another one with integer side lengths and *bigness* $N + 1$ .
|
55
| 80.46875 |
1,038 |
Four mathletes and two coaches sit at a circular table. How many distinct arrangements are there of these six people if the two coaches sit opposite each other?
|
24
| 76.5625 |
1,039 |
Find all positive integers $x$ such that $2x+1$ is a perfect square but none of the integers $2x+2, 2x+3, \ldots, 3x+2$ are perfect squares.
|
4
| 67.96875 |
1,040 |
Given a $15\times 15$ chessboard. We draw a closed broken line without self-intersections such that every edge of the broken line is a segment joining the centers of two adjacent cells of the chessboard. If this broken line is symmetric with respect to a diagonal of the chessboard, then show that the length of the broken line is $\leq 200$ .
|
200
| 85.15625 |
1,041 |
We define a sequence $a_n$ so that $a_0=1$ and
\[a_{n+1} = \begin{cases} \displaystyle \frac{a_n}2 & \textrm { if } a_n \equiv 0 \pmod 2, a_n + d & \textrm{ otherwise. } \end{cases} \]
for all postive integers $n$ .
Find all positive integers $d$ such that there is some positive integer $i$ for which $a_i=1$ .
|
d
| 10.9375 |
1,042 |
Find the least value of $k$ such that for all $a,b,c,d \in \mathbb{R}$ the inequality
\[ \begin{array} c \sqrt{(a^2+1)(b^2+1)(c^2+1)} +\sqrt{(b^2+1)(c^2+1)(d^2+1)} +\sqrt{(c^2+1)(d^2+1)(a^2+1)} +\sqrt{(d^2+1)(a^2+1)(b^2+1)} \ \ge 2( ab+bc+cd+da+ac+bd)-k \end{array}\]
holds.
|
4
| 29.6875 |
1,043 |
The temperatures $ f^\circ \text{F}$ and $ c^\circ \text{C}$ are equal when $ f \equal{} \frac {9}{5}c \plus{} 32$ . What temperature is the same in both $ ^\circ \text{F}$ and $ ^\circ \text{C}$ ?
|
-40
| 99.21875 |
1,044 |
Find the smallest positive value of $36^k - 5^m$ , where $k$ and $m$ are positive integers.
|
11
| 97.65625 |
1,045 |
A positive integer $n$ between $1$ and $N=2007^{2007}$ inclusive is selected at random. If $a$ and $b$ are natural numbers such that $a/b$ is the probability that $N$ and $n^3-36n$ are relatively prime, find the value of $a+b$ .
|
1109
| 25 |
1,046 |
A tetrahedron is a polyhedron composed of four triangular faces. Faces $ABC$ and $BCD$ of a tetrahedron $ABCD$ meet at an angle of $\pi/6$ . The area of triangle $\triangle ABC$ is $120$ . The area of triangle $\triangle BCD$ is $80$ , and $BC = 10$ . What is the volume of the tetrahedron? We call the volume of a tetrahedron as one-third the area of it's base times it's height.
|
320
| 92.1875 |
1,047 |
For each positive natural number $n$ let $d (n)$ be the number of its divisors including $1$ and $n$ .
For which positive natural numbers $n$ , for every divisor $t$ of $n$ , that $d (t)$ is a divisor of $d (n)$ ?
|
n
| 0 |
1,048 |
In triangle $ABC$ with $AB=8$ and $AC=10$ , the incenter $I$ is reflected across side $AB$ to point $X$ and across side $AC$ to point $Y$ . Given that segment $XY$ bisects $AI$ , compute $BC^2$ . (The incenter is the center of the inscribed circle of triangle .)
|
84
| 0.78125 |
1,049 |
On Monday a school library was attended by 5 students, on Tuesday, by 6, on Wednesday, by 4, on Thursday, by 8, and on Friday, by 7. None of them have attended the library two days running. What is the least possible number of students who visited the library during a week?
|
15
| 7.8125 |
1,050 |
Determine all three-digit numbers $N$ having the property that $N$ is divisible by 11, and $\dfrac{N}{11}$ is equal to the sum of the squares of the digits of $N$ .
|
803
| 13.28125 |
1,051 |
Michael, David, Evan, Isabella, and Justin compete in the NIMO Super Bowl, a round-robin cereal-eating tournament. Each pair of competitors plays exactly one game, in which each competitor has an equal chance of winning (and there are no ties). The probability that none of the five players wins all of his/her games is $\tfrac{m}{n}$ for relatively prime positive integers $m$ , $n$ . Compute $100m + n$ .
*Proposed by Evan Chen*
|
1116
| 3.90625 |
1,052 |
Let $a, b, c$ , and $d$ be real numbers such that $a^2 + b^2 + c^2 + d^2 = 3a + 8b + 24c + 37d = 2018$ . Evaluate $3b + 8c + 24d + 37a$ .
|
1215
| 6.25 |
1,053 |
Find all primes that can be written both as a sum of two primes and as a difference of two primes.
|
5
| 92.96875 |
1,054 |
Given triangle $ ABC$ of area 1. Let $ BM$ be the perpendicular from $ B$ to the bisector of angle $ C$ . Determine the area of triangle $ AMC$ .
|
\frac{1}{2}
| 98.4375 |
1,055 |
Let $(a_n)$ be defined by $a_1=a_2=1$ and $a_n=a_{n-1}+a_{n-2}$ for $n>2$ . Compute the sum $\frac{a_1}2+\frac{a_2}{2^2}+\frac{a_3}{2^3}+\ldots$ .
|
2
| 64.0625 |
1,056 |
Points $P_1, P_2, P_3,$ and $P_4$ are $(0,0), (10, 20), (5, 15),$ and $(12, -6)$ , respectively. For what point $P \in \mathbb{R}^2$ is the sum of the distances from $P$ to the other $4$ points minimal?
|
(6, 12)
| 0.78125 |
1,057 |
Let $S$ be the set of all lattice points $(x, y)$ in the plane satisfying $|x|+|y|\le 10$ . Let $P_1,P_2,\ldots,P_{2013}$ be a sequence of 2013 (not necessarily distinct) points such that for every point $Q$ in $S$ , there exists at least one index $i$ such that $1\le i\le 2013$ and $P_i = Q$ . Suppose that the minimum possible value of $|P_1P_2|+|P_2P_3|+\cdots+|P_{2012}P_{2013}|$ can be expressed in the form $a+b\sqrt{c}$ , where $a,b,c$ are positive integers and $c$ is not divisible by the square of any prime. Find $a+b+c$ . (A *lattice point* is a point with all integer coordinates.)
<details><summary>Clarifications</summary>
- $k = 2013$ , i.e. the problem should read, ``... there exists at least one index $i$ such that $1\le i\le 2013$ ...''. An earlier version of the test read $1 \le i \le k$ .
</details>
*Anderson Wang*
|
222
| 3.125 |
1,058 |
Consider all the positive integers $N$ with the property that all of the divisors of $N$ can be written as $p-2$ for some prime number $p$ . Then, there exists an integer $m$ such that $m$ is the maximum possible number of divisors of all
numbers $N$ with such property. Find the sum of all possible values of $N$ such that $N$ has $m$ divisors.
*Proposed by **FedeX333X***
|
135
| 25 |
1,059 |
In the diagram below, three squares are inscribed in right triangles. Their areas are $A$ , $M$ , and $N$ , as indicated in the diagram. If $M = 5$ and $N = 12$ , then $A$ can be expressed as $a + b\sqrt{c}$ , where $a$ , $b$ , and $c$ are positive integers and $c$ is not divisible by the square of any prime. Compute $a + b + c$ .
[asy]
size(250);
defaultpen (linewidth (0.7) + fontsize (10));
pair O = origin, A = (1, 1), B = (4/3, 1/3), C = (2/3, 5/3), P = (3/2, 0), Q = (0,3);
draw (P--O--Q--cycle^^(0, 5/3)--C--(2/3,1)^^(0,1)--A--(1,0)^^(1,1/3)--B--(4/3,0));
label(" $A$ ", (.5,.5));
label(" $M$ ", (7/6, 1/6));
label(" $N$ ", (1/3, 4/3));[/asy]
*Proposed by Aaron Lin*
|
36
| 3.90625 |
1,060 |
Let $\triangle ABC$ have $M$ as the midpoint of $BC$ and let $P$ and $Q$ be the feet of the altitudes from $M$ to $AB$ and $AC$ respectively. Find $\angle BAC$ if $[MPQ]=\frac{1}{4}[ABC]$ and $P$ and $Q$ lie on the segments $AB$ and $AC$ .
|
90^\circ
| 97.65625 |
1,061 |
Find the smallest positive integer $n$ that is divisible by $100$ and has exactly $100$ divisors.
|
162000
| 6.25 |
1,062 |
Say that an integer $n \ge 2$ is *delicious* if there exist $n$ positive integers adding up to 2014 that have distinct remainders when divided by $n$ . What is the smallest delicious integer?
|
4
| 43.75 |
1,063 |
The tangent lines to the circumcircle of triangle ABC passing through vertices $B$ and $C$ intersect at point $F$ . Points $M$ , $L$ and $N$ are the feet of the perpendiculars from vertex $A$ to the lines $FB$ , $FC$ and $BC$ respectively. Show that $AM+AL \geq 2AN$
|
AM + AL \geq 2AN
| 97.65625 |
1,064 |
Euler's formula states that for a convex polyhedron with $V$ vertices, $E$ edges, and $F$ faces, $V - E + F = 2$ . A particular convex polyhedron has 32 faces, each of which is either a triangle or a pentagon. At each of its $V$ vertices, $T$ triangular faces and $P$ pentagonal faces meet. What is the value of $100P + 10T + V$ ?
|
250
| 25 |
1,065 |
Equilateral triangle $ABC$ has point $P$ inside and point $Q$ on side $BC$ , satisfying $PB = PQ = QC$ and $\angle PBC = 20^\circ$ .
Find the measure of $\angle AQP$ .
|
60^\circ
| 0.78125 |
1,066 |
Compute \[\left\lfloor \dfrac{2007!+2004!}{2006!+2005!}\right\rfloor.\] (Note that $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$ .)
|
2006
| 91.40625 |
1,067 |
The real numbers $x$ , $y$ , $z$ , and $t$ satisfy the following equation:
\[2x^2 + 4xy + 3y^2 - 2xz -2 yz + z^2 + 1 = t + \sqrt{y + z - t} \]
Find 100 times the maximum possible value for $t$ .
|
125
| 0.78125 |
1,068 |
Pratyya and Payel have a number each, $n$ and $m$ respectively, where $n>m.$ Everyday, Pratyya multiplies his number by $2$ and then subtracts $2$ from it, and Payel multiplies his number by $2$ and then add $2$ to it. In other words, on the first day their numbers will be $(2n-2)$ and $(2m+2)$ respectively. Find minimum integer $x$ with proof such that if $n-m\geq x,$ then Pratyya's number will be larger than Payel's number everyday.
|
4
| 34.375 |
1,069 |
Consider the addition problem:
\begin{tabular}{ccccc}
&C&A&S&H
+&&&M&E
\hline
O&S&I&D&E
\end{tabular}
where each letter represents a base-ten digit, and $C,M,O \ne 0.$ (Distinct letters are allowed to represent
the same digit.) How many ways are there to assign values to the letters so that the addition problem
is true?
|
0
| 15.625 |
1,070 |
Find the sum of all the integers $N > 1$ with the properties that the each prime factor of $N $ is either $2, 3,$ or $5,$ and $N$ is not divisible by any perfect cube greater than $1.$
|
2820
| 52.34375 |
1,071 |
Given a point $p$ and a line segment $l$ , let $d(p,l)$ be the distance between them. Let $A$ , $B$ , and $C$ be points in the plane such that $AB=6$ , $BC=8$ , $AC=10$ . What is the area of the region in the $(x,y)$ -plane formed by the ordered pairs $(x,y)$ such that there exists a point $P$ inside triangle $ABC$ with $d(P,AB)+x=d(P,BC)+y=d(P,AC)?$
|
\frac{288}{5}
| 0 |
1,072 |
Let $ABC$ be a triangle with $\angle A = 60^o$ . Line $\ell$ intersects segments $AB$ and $AC$ and splits triangle $ABC$ into an equilateral triangle and a quadrilateral. Let $X$ and $Y$ be on $\ell$ such that lines $BX$ and $CY$ are perpendicular to ℓ. Given that $AB = 20$ and $AC = 22$ , compute $XY$ .
|
21
| 1.5625 |
1,073 |
Let $ I_n\equal{}\int_0^{\sqrt{3}} \frac{1}{1\plus{}x^{n}}\ dx\ (n\equal{}1,\ 2,\ \cdots)$ .
(1) Find $ I_1,\ I_2$ .
(2) Find $ \lim_{n\to\infty} I_n$ .
|
1
| 61.71875 |
1,074 |
The $\textit{arithmetic derivative}$ $D(n)$ of a positive integer $n$ is defined via the following rules:
- $D(1) = 0$ ;
- $D(p)=1$ for all primes $p$ ;
- $D(ab)=D(a)b+aD(b)$ for all positive integers $a$ and $b$ .
Find the sum of all positive integers $n$ below $1000$ satisfying $D(n)=n$ .
|
31
| 66.40625 |
1,075 |
How many ordered triples $(a,b,c)$ of integers with $1\le a\le b\le c\le 60$ satisfy $a\cdot b=c$ ?
|
134
| 100 |
1,076 |
Let $\triangle ABC$ with $AB=AC$ and $BC=14$ be inscribed in a circle $\omega$ . Let $D$ be the point on ray $BC$ such that $CD=6$ . Let the intersection of $AD$ and $\omega$ be $E$ . Given that $AE=7$ , find $AC^2$ .
*Proposed by Ephram Chun and Euhan Kim*
|
105
| 13.28125 |
1,077 |
Find all pairs of real number $x$ and $y$ which simultaneously satisfy the following 2 relations: $x+y+4=\frac{12x+11y}{x^2+y^2}$ $y-x+3=\frac{11x-12y}{x^2+y^2}$
|
(2, 1)
| 2.34375 |
1,078 |
Based on a city's rules, the buildings of a street may not have more than $9$ stories. Moreover, if the number of stories of two buildings is the same, no matter how far they are from each other, there must be a building with a higher number of stories between them. What is the maximum number of buildings that can be built on one side of a street in this city?
|
511
| 0 |
1,079 |
A solid rectangular block is formed by gluing together $N$ congruent 1-cm cubes face to face. When the block is viewed so that three of its faces are visible, exactly 231 of the 1-cm cubes cannot be seen. Find the smallest possible value of $N$ .
|
384
| 93.75 |
1,080 |
Let $ ABC$ be a triangle with circumradius $ R$ , perimeter $ P$ and area $ K$ . Determine the maximum value of: $ \frac{KP}{R^3}$ .
|
\frac{27}{4}
| 93.75 |
1,081 |
Find all real numbers $x, y, z$ that satisfy the following system $$ \sqrt{x^3 - y} = z - 1 $$ $$ \sqrt{y^3 - z} = x - 1 $$ $$ \sqrt{z^3 - x} = y - 1 $$
|
(1, 1, 1)
| 92.96875 |
1,082 |
I have five different pairs of socks. Every day for five days, I pick two socks at random without replacement to wear for the day. Find the probability that I wear matching socks on both the third day and the fifth day.
|
\frac{1}{63}
| 3.90625 |
1,083 |
Define $H_n = 1+\frac{1}{2}+\cdots+\frac{1}{n}$ . Let the sum of all $H_n$ that are terminating in base 10 be $S$ . If $S = m/n$ where m and n are relatively prime positive integers, find $100m+n$ .
*Proposed by Lewis Chen*
|
9920
| 5.46875 |
1,084 |
Compute the sum of all positive integers $a\leq 26$ for which there exist integers $b$ and $c$ such that $a+23b+15c-2$ and $2a+5b+14c-8$ are both multiples of $26$ .
|
31
| 76.5625 |
1,085 |
Let $k$ be the product of every third positive integer from $2$ to $2006$ , that is $k = 2\cdot 5\cdot 8\cdot 11 \cdots 2006$ . Find the number of zeros there are at the right end of the decimal representation for $k$ .
|
168
| 32.03125 |
1,086 |
The five-digit number $12110$ is divisible by the sum of its digits $1 + 2 + 1 + 1 + 0 = 5.$ Find the greatest five-digit number which is divisible by the sum of its digits
|
99972
| 99.21875 |
1,087 |
We delete the four corners of a $8 \times 8$ chessboard. How many ways are there to place eight non-attacking rooks on the remaining squares?
*Proposed by Evan Chen*
|
21600
| 22.65625 |
1,088 |
If $a$ and $b$ are the roots of $x^2 - 2x + 5$ , what is $|a^8 + b^8|$ ?
|
1054
| 67.1875 |
1,089 |
Let $ABCDEF$ be a regular hexagon with side length 10 inscribed in a circle $\omega$ . $X$ , $Y$ , and $Z$ are points on $\omega$ such that $X$ is on minor arc $AB$ , $Y$ is on minor arc $CD$ , and $Z$ is on minor arc $EF$ , where $X$ may coincide with $A$ or $B$ (and similarly for $Y$ and $Z$ ). Compute the square of the smallest possible area of $XYZ$ .
*Proposed by Michael Ren*
|
7500
| 2.34375 |
1,090 |
Find all positive integers $n$ for which both numbers \[1\;\;\!\!\!\!\underbrace{77\ldots 7}_{\text{ $n$ sevens}}\!\!\!\!\quad\text{and}\quad 3\;\; \!\!\!\!\underbrace{77\ldots 7}_{\text{ $n$ sevens}}\] are prime.
|
n = 1
| 0 |
1,091 |
Find all functions $f:\ \mathbb{R}\rightarrow\mathbb{R}$ such that for any real number $x$ the equalities are true: $f\left(x+1\right)=1+f(x)$ and $f\left(x^4-x^2\right)=f^4(x)-f^2(x).$ [source](http://matol.kz/comments/3373/show)
|
f(x) = x
| 99.21875 |
1,092 |
$12$ students need to form five study groups. They will form three study groups with $2$ students each and two study groups with $3$ students each. In how many ways can these groups be formed?
|
138600
| 96.09375 |
1,093 |
An increasing arithmetic sequence with infinitely many terms is determined as follows. A single die is thrown and the number that appears is taken as the first term. The die is thrown again and the second number that appears is taken as the common difference between each pair of consecutive terms. Determine with proof how many of the 36 possible sequences formed in this way contain at least one perfect square.
|
27
| 69.53125 |
1,094 |
(F.Nilov) Given right triangle $ ABC$ with hypothenuse $ AC$ and $ \angle A \equal{} 50^{\circ}$ . Points $ K$ and $ L$ on the cathetus $ BC$ are such that $ \angle KAC \equal{} \angle LAB \equal{} 10^{\circ}$ . Determine the ratio $ CK/LB$ .
|
2
| 4.6875 |
1,095 |
A set $M$ consists of $n$ elements. Find the greatest $k$ for which there is a collection of $k$ subsets of $M$ such that for any subsets $A_{1},...,A_{j}$ from the collection, there is an element belonging to an odd number of them
|
k = n
| 80.46875 |
1,096 |
Let $ n$ be a positive integer and $ [ \ n ] = a.$ Find the largest integer $ n$ such that the following two conditions are satisfied:
$ (1)$ $ n$ is not a perfect square;
$ (2)$ $ a^{3}$ divides $ n^{2}$ .
|
24
| 50.78125 |
1,097 |
Let $k$ be a positive integer with the following property: For every subset $A$ of $\{1,2,\ldots, 25\}$ with $|A|=k$ , we can find distinct elements $x$ and $y$ of $A$ such that $\tfrac23\leq\tfrac xy\leq\tfrac 32$ . Find the smallest possible value of $k$ .
|
7
| 10.15625 |
1,098 |
Two circles with radius $2$ and radius $4$ have a common center at P. Points $A, B,$ and $C$ on the larger circle are the vertices of an equilateral triangle. Point $D$ is the intersection of the smaller circle and the line segment $PB$ . Find the square of the area of triangle $ADC$ .
|
192
| 39.0625 |
1,099 |
A set of $8$ problems was prepared for an examination. Each student was given $3$ of them. No two students received more than one common problem. What is the largest possible number of students?
|
8
| 53.90625 |
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