Unnamed: 0
int64 0
56.9k
| problem
stringlengths 16
7.44k
| ground_truth
stringlengths 1
942
| solved_percentage
float64 0
100
|
|---|---|---|---|
1,400 |
Alice is bored in class, so she thinks of a positive integer. Every second after that, she subtracts from her current number its smallest prime divisor, possibly itself. After 2022 seconds, she realizes that her number is prime. Find the sum of all possible values of her initial number.
|
8093
| 42.96875 |
1,401 |
For $k\ge 1$ , define $a_k=2^k$ . Let $$ S=\sum_{k=1}^{\infty}\cos^{-1}\left(\frac{2a_k^2-6a_k+5}{\sqrt{(a_k^2-4a_k+5)(4a_k^2-8a_k+5)}}\right). $$ Compute $\lfloor 100S\rfloor$ .
|
157
| 49.21875 |
1,402 |
A non-negative integer $n$ is said to be *squaredigital* if it equals the square of the sum of its digits. Find all non-negative integers which are squaredigital.
|
0, 1, 81
| 73.4375 |
1,403 |
In a regular hexagon $ABCDEF$ of side length $8$ and center $K$ , points $W$ and $U$ are chosen on $\overline{AB}$ and $\overline{CD}$ respectively such that $\overline{KW} = 7$ and $\angle WKU = 120^{\circ}$ . Find the area of pentagon $WBCUK$ .
*Proposed by Bradley Guo*
|
32\sqrt{3}
| 0 |
1,404 |
A circle is inscribed in a regular octagon with area $2024$ . A second regular octagon is inscribed in the circle, and its area can be expressed as $a + b\sqrt{c}$ , where $a, b, c$ are integers and $c$ is square-free. Compute $a + b + c$ .
|
1520
| 14.0625 |
1,405 |
If $\displaystyle\prod_{i=6}^{2021} (1-\tan^2((2^i)^\circ))$ can be written in the form $a^b$ for positive integers $a,b$ with $a$ squarefree, find $a+b$ .
*Proposed by Deyuan Li and Andrew Milas*
|
2018
| 35.9375 |
1,406 |
$5.$ Let x be a real number with $0<x<1$ and let $0.c_1c_2c_3...$ be the decimal expansion of x.Denote by $B(x)$ the set of all subsequences of $c_1c_2c_3$ that consist of 6 consecutive digits.
For instance , $B(\frac{1}{22})={045454,454545,545454}$ Find the minimum number of elements of $B(x)$ as $x$ varies among all irrational numbers with $0<x<1$
|
7
| 0 |
1,407 |
It is known that $\int_1^2x^{-1}\arctan (1+x)\ dx = q\pi\ln(2)$ for some rational number $q.$ Determine $q.$ Here, $0\leq\arctan(x)<\frac{\pi}{2}$ for $0\leq x <\infty.$
|
q = \frac{3}{8}
| 0 |
1,408 |
Find the number of $4$ -digit numbers (in base $10$ ) having non-zero digits and which are divisible by $4$ but not by $8$ .
|
729
| 75 |
1,409 |
Rhombus $PQRS$ is inscribed in rectangle $ABCD$ so that vertices $P$ , $Q$ , $R$ , and $S$ are interior points on sides $\overline{AB}$ , $\overline{BC}$ , $\overline{CD}$ , and $\overline{DA}$ , respectively. It is given that $PB=15$ , $BQ=20$ , $PR=30$ , and $QS=40$ . Let $m/n$ , in lowest terms, denote the perimeter of $ABCD$ . Find $m+n$ .
|
677
| 0 |
1,410 |
Suppose $x$ is a positive real number such that $\{x\}, [x]$ and $x$ are in a geometric progression. Find the least positive integer $n$ such that $x^n > 100$ . (Here $[x]$ denotes the integer part of $x$ and $\{x\} = x - [x]$ .)
|
10
| 39.84375 |
1,411 |
Let $L$ be the number formed by $2022$ digits equal to $1$ , that is, $L=1111\dots 111$ .
Compute the sum of the digits of the number $9L^2+2L$ .
|
4044
| 95.3125 |
1,412 |
A tree has 10 pounds of apples at dawn. Every afternoon, a bird comes and eats x pounds of apples. Overnight, the amount of food on the tree increases by 10%. What is the maximum value of x such that the bird can sustain itself indefinitely on the tree without the tree running out of food?
|
\frac{10}{11}
| 34.375 |
1,413 |
Square $ABCD$ has side length $13$ , and points $E$ and $F$ are exterior to the square such that $BE=DF=5$ and $AE=CF=12$ . Find $EF^{2}$ .
[asy]
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draw(B--C--F--D--C^^D--A--E--B--A, linewidth(0.7));
dot(A^^B^^C^^D^^E^^F);
pair point=(13,13);
label(" $A$ ", A, dir(point--A));
label(" $B$ ", B, dir(point--B));
label(" $C$ ", C, dir(point--C));
label(" $D$ ", D, dir(point--D));
label(" $E$ ", E, dir(point--E));
label(" $F$ ", F, dir(point--F));[/asy]
|
578
| 71.09375 |
1,414 |
Let $ n $ $(n\geq2)$ be an integer. Find the greatest possible value of the expression $$ E=\frac{a_1}{1+a_1^2}+\frac{a_2}{1+a_2^2}+\ldots+\frac{a_n}{1+a_n^2} $$ if the positive real numbers $a_1,a_2,\ldots,a_n$ satisfy $a_1+a_2+\ldots+a_n=\frac{n}{2}.$ What are the values of $a_1,a_2,\ldots,a_n$ when the greatest value is achieved?
|
\frac{2n}{5}
| 88.28125 |
1,415 |
Find the positive integer $n$ such that the least common multiple of $n$ and $n - 30$ is $n + 1320$ .
|
165
| 96.09375 |
1,416 |
There are $10001$ students at an university. Some students join together to form several clubs (a student may belong to different clubs). Some clubs join together to form several societies (a club may belong to different societies). There are a total of $k$ societies. Suppose that the following conditions hold:
*i.)* Each pair of students are in exactly one club.
*ii.)* For each student and each society, the student is in exactly one club of the society.
*iii.)* Each club has an odd number of students. In addition, a club with ${2m+1}$ students ( $m$ is a positive integer) is
in exactly $m$ societies.
Find all possible values of $k$ .
*Proposed by Guihua Gong, Puerto Rico*
|
5000
| 35.15625 |
1,417 |
Find the sum of all the digits in the decimal representations of all the positive integers less than $1000.$
|
13500
| 93.75 |
1,418 |
The equation $ ax^3\plus{}bx^2\plus{}cx\plus{}d\equal{}0$ has three distinct solutions. How many distinct solutions does the following equation have:
$ 4(ax^3\plus{}bx^2\plus{}cx\plus{}d)(3ax\plus{}b)\equal{}(3ax^2\plus{}2bx\plus{}c)^2?$
|
2
| 14.84375 |
1,419 |
Let $u$ be a real number. On the coordinate plane, consider two parabolas $C_1: y=-x^2+1,\ C_2: y=(x-u)^2+u$ .
The range of $u$ such that $C_1$ and $C_2$ have at least one point of intersection is expressed by $a\leq u\leq b$ for some real numbers $a,\ b$ .
(1) Find the values of $a,\ b$ .
(2) When $u$ satisfies $a\leq u\leq b$ , let $P_1(x_1,\ y_1),\ P_2(x_2,\ y_2)$ be the point of intersections of $C_1$ and $C_2$ .
Note : if the point of intersection is just one, then we regard the intersection as $P_1=P_2$ .
Express $2|x_1y_2-x_2y_1|$ in terms of $u$ .
(3) Let $f(u)$ be the expression obtained in (2), evaluate $I=\int_ a^b f(u)du.$
|
I = \frac{21\pi}{8}
| 1.5625 |
1,420 |
Find $$ \inf_{\substack{ n\ge 1 a_1,\ldots ,a_n >0 a_1+\cdots +a_n <\pi }} \left( \sum_{j=1}^n a_j\cos \left( a_1+a_2+\cdots +a_j \right)\right) . $$
|
-\pi
| 52.34375 |
1,421 |
Kevin colors three distinct squares in a $3\times 3$ grid red. Given that there exist two uncolored squares such that coloring one of them would create a horizontal or vertical red line, find the number of ways he could have colored the original three squares.
|
36
| 0.78125 |
1,422 |
Let $f$ be a function such that $f (x + y) = f (x) + f (y)$ for all $x,y \in R$ and $f (1) = 100$ . Calculate $\sum_{k = 1}^{10}f (k!)$ .
|
403791300
| 100 |
1,423 |
Given the following system of equations: $$ \begin{cases} R I +G +SP = 50 R I +T + M = 63 G +T +SP = 25 SP + M = 13 M +R I = 48 N = 1 \end{cases} $$
Find the value of L that makes $LMT +SPR I NG = 2023$ true.
|
\frac{341}{40}
| 0 |
1,424 |
Find the greatest real $k$ such that, for every tetrahedron $ABCD$ of volume $V$ , the product of areas of faces $ABC,ABD$ and $ACD$ is at least $kV^2$ .
|
\frac{9}{2}
| 2.34375 |
1,425 |
What is the minimum number of planes determined by $6$ points in space which are not all coplanar, and among which no three are collinear?
|
11
| 0 |
1,426 |
Two distinct points $A$ and $B$ are chosen at random from 15 points equally spaced around a circle centered at $O$ such that each pair of points $A$ and $B$ has the same probability of being chosen. The probability that the perpendicular bisectors of $OA$ and $OB$ intersect strictly inside the circle can be expressed in the form $\frac{m}{n}$ , where $m,n$ are relatively prime positive integers. Find $m+n$ .
*Ray Li.*
|
11
| 0.78125 |
1,427 |
The circle centered at point $A$ with radius $19$ and the circle centered at point $B$ with radius $32$ are both internally tangent to a circle centered at point $C$ with radius $100$ such that point $C$ lies on segment $\overline{AB}$ . Point $M$ is on the circle centered at $A$ and point $N$ is on the circle centered at $B$ such that line $MN$ is a common internal tangent of those two circles. Find the distance $MN$ .
|
140
| 57.8125 |
1,428 |
Find the smallest positive integer $n$ such that \[2^{1989}\; \vert \; m^{n}-1\] for all odd positive integers $m>1$ .
|
2^{1987}
| 83.59375 |
1,429 |
Let $ p>2 $ be a prime number. For any permutation $ \pi = ( \pi(1) , \pi(2) , \cdots , \pi(p) ) $ of the set $ S = \{ 1, 2, \cdots , p \} $ , let $ f( \pi ) $ denote the number of multiples of $ p $ among the following $ p $ numbers:
\[ \pi(1) , \pi(1) + \pi(2) , \cdots , \pi(1) + \pi(2) + \cdots + \pi(p) \]
Determine the average value of $ f( \pi) $ taken over all permutations $ \pi $ of $ S $ .
|
2 - \frac{1}{p}
| 10.15625 |
1,430 |
A $3 \times 3$ grid of unit cells is given. A *snake of length $k$* is an animal which occupies an ordered $k$ -tuple of cells in this grid, say $(s_1, \dots, s_k)$ . These cells must be pairwise distinct, and $s_i$ and $s_{i+1}$ must share a side for $i = 1, \dots, k-1$ . After being placed in a finite $n \times n$ grid, if the snake is currently occupying $(s_1, \dots, s_k)$ and $s$ is an unoccupied cell sharing a side with $s_1$ , the snake can *move* to occupy $(s, s_1, \dots, s_{k-1})$ instead. The snake has *turned around* if it occupied $(s_1, s_2, \dots, s_k)$ at the beginning, but after a finite number of moves occupies $(s_k, s_{k-1}, \dots, s_1)$ instead.
Find the largest integer $k$ such that one can place some snake of length $k$ in a $3 \times 3$ grid which can turn around.
|
5
| 13.28125 |
1,431 |
Two circumferences of radius $1$ that do not intersect, $c_1$ and $c_2$ , are placed inside an angle whose vertex is $O$ . $c_1$ is tangent to one of the rays of the angle, while $c_2$ is tangent to the other ray. One of the common internal tangents of $c_1$ and $c_2$ passes through $O$ , and the other one intersects the rays of the angle at points $A$ and $B$ , with $AO=BO$ . Find the distance of point $A$ to the line $OB$ .
|
2
| 56.25 |
1,432 |
Suppose that $2^{2n+1}+ 2^{n}+1=x^{k}$ , where $k\geq2$ and $n$ are positive integers. Find all possible values of $n$ .
|
n = 4
| 0 |
1,433 |
Find the number of ordered triples of positive integers $(a, b, c)$ such that $abc$ divides $(ab + 1)(bc + 1)(ca + 1)$ .
|
16
| 50.78125 |
1,434 |
Determine the smallest possible value of $$ |2^m - 181^n|, $$ where $m$ and $n$ are positive integers.
|
7
| 89.0625 |
1,435 |
A teacher suggests four possible books for students to read. Each of six students selects one of the four books. How many ways can these selections be made if each of the books is read by at least one student?
|
1560
| 98.4375 |
1,436 |
In the octagon below all sides have the length $1$ and all angles are equal.
Determine the distance between the corners $A$ and $B$ .
|
1 + \sqrt{2}
| 17.1875 |
1,437 |
Find all 4-digit numbers $\overline{abcd}$ that are multiples of $11$ , such that the 2-digit number $\overline{ac}$ is a multiple of $7$ and $a + b + c + d = d^2$ .
|
3454
| 98.4375 |
1,438 |
Ninety-eight apples who always lie and one banana who always tells the truth are randomly arranged along a line. The first fruit says "One of the first forty fruit is the banana!'' The last fruit responds "No, one of the $\emph{last}$ forty fruit is the banana!'' The fruit in the middle yells "I'm the banana!'' In how many positions could the banana be?
|
21
| 0 |
1,439 |
Let $n \geq 3$ be a positive integer. Find the smallest positive real $k$ , satisfying the following condition: if $G$ is a connected graph with $n$ vertices and $m$ edges, then it is always possible to delete at most $k(m-\lfloor \frac{n} {2} \rfloor)$ edges, so that the resulting graph has a proper vertex coloring with two colors.
|
k = \frac{1}{2}
| 0 |
1,440 |
You are given a positive integer $n$ . What is the largest possible number of numbers that can be chosen from the set $\{1, 2, \ldots, 2n\}$ so that there are no two chosen numbers $x > y$ for which $x - y = (x, y)$ ?
Here $(x, y)$ denotes the greatest common divisor of $x, y$ .
*Proposed by Anton Trygub*
|
n
| 95.3125 |
1,441 |
A cake has a shape of triangle with sides $19,20$ and $21$ . It is allowed to cut it it with a line into two pieces and put them on a round plate such that pieces don't overlap each other and don't stick out of the plate. What is the minimal diameter of the plate?
|
21
| 31.25 |
1,442 |
Let $d$ be the greatest common divisor of $2^{30^{10}}-2$ and $2^{30^{45}}-2$ . Find the remainder when $d$ is divided by $2013$ .
|
2012
| 81.25 |
1,443 |
Find all pairs of integers $(a,b)$ so that each of the two cubic polynomials $$ x^3+ax+b \, \, \text{and} \, \, x^3+bx+a $$ has all the roots to be integers.
*Proposed by Prithwijit De and Sutanay Bhattacharya*
|
(0, 0)
| 54.6875 |
1,444 |
The diagram below shows an $8$ x $7$ rectangle with a 3-4-5 right triangle drawn in each corner. The lower two triangles have their sides of length 4 along the bottom edge of the rectangle, while the upper two
triangles have their sides of length 3 along the top edge of the rectangle. A circle is tangent to the hypotenuse of each triangle. The diameter of the circle is $\frac{m}{n}$ , where m and n are relatively prime positive integers. Find m + n.
For diagram go to http://www.purplecomet.org/welcome/practice, go to the 2015 middle school contest questions, and then go to #20
|
47
| 0 |
1,445 |
Let $a$ and $b$ be positive integers such that $a>b$ and the difference between $a^2+b$ and $a+b^2$ is prime. Compute all possible pairs $(a,b)$ .
|
(2, 1)
| 97.65625 |
1,446 |
For positive integer $n,$ let $s(n)$ be the sum of the digits of n when n is expressed in base ten. For example, $s(2022) = 2 + 0 + 2 + 2 = 6.$ Find the sum of the two solutions to the equation $n - 3s(n) = 2022.$
|
4107
| 78.90625 |
1,447 |
In a simple graph with 300 vertices no two vertices of the same degree are adjacent (boo hoo hoo).
What is the maximal possible number of edges in such a graph?
|
42550
| 0 |
1,448 |
David is taking a true/false exam with $9$ questions. Unfortunately, he doesn’t know the answer to any of the questions, but he does know that exactly $5$ of the answers are True. In accordance with this, David guesses the answers to all $9$ questions, making sure that exactly $5$ of his answers are True. What is the probability he answers at least $5$ questions correctly?
|
\frac{9}{14}
| 0 |
1,449 |
Let $p$ , $q$ , $r$ , and $s$ be 4 distinct primes such that $p+q+r+s$ is prime, and the numbers $p^2+qr$ and $p^2+qs$ are both perfect squares. What is the value of $p+q+r+s$ ?
|
23
| 48.4375 |
1,450 |
Square $SEAN$ has side length $2$ and a quarter-circle of radius $1$ around $E$ is cut out. Find the radius of the largest circle that can be inscribed in the remaining figure.
|
5 - 3\sqrt{2}
| 2.34375 |
1,451 |
For arbiterary integers $n,$ find the continuous function $f(x)$ which satisfies the following equation. \[\lim_{h\rightarrow 0}\frac{1}{h}\int_{x-nh}^{x+nh}f(t) dt=2f(nx).\] Note that $x$ can range all real numbers and $f(1)=1.$
|
f(x) = x
| 85.9375 |
1,452 |
Define $ n!!$ to be $ n(n\minus{}2)(n\minus{}4)\ldots3\cdot1$ for $ n$ odd and $ n(n\minus{}2)(n\minus{}4)\ldots4\cdot2$ for $ n$ even. When $ \displaystyle \sum_{i\equal{}1}^{2009} \frac{(2i\minus{}1)!!}{(2i)!!}$ is expressed as a fraction in lowest terms, its denominator is $ 2^ab$ with $ b$ odd. Find $ \displaystyle \frac{ab}{10}$ .
|
401
| 50.78125 |
1,453 |
Find (with proof) all natural numbers $n$ such that, for some natural numbers $a$ and $b$ , $a\ne b$ , the digits in the decimal representations of the two numbers $n^a+1$ and $n^b+1$ are in reverse order.
|
n = 3
| 0 |
1,454 |
In $\triangle ABC$ points $M,O$ are midpoint of $AB$ and circumcenter. It is true, that $OM=R-r$ . Bisector of external $\angle A$ intersect $BC$ at $D$ and bisector of external $\angle C$ intersect $AB$ at $E$ .
Find possible values of $\angle CED$ *D. Shiryaev*
|
90^\circ
| 83.59375 |
1,455 |
Let $S$ be the set of all real values of $x$ with $0 < x < \pi/2$ such that $\sin x$ , $\cos x$ , and $\tan x$ form the side lengths (in some order) of a right triangle. Compute the sum of $\tan^2 x$ over all $x$ in $S$ .
|
\sqrt{2}
| 39.0625 |
1,456 |
Let $n$ be a positive integer. Positive numbers $a$ , $b$ , $c$ satisfy $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=1$ . Find the greatest possible value of $$ E(a,b,c)=\frac{a^{n}}{a^{2n+1}+b^{2n} \cdot c + b \cdot c^{2n}}+\frac{b^{n}}{b^{2n+1}+c^{2n} \cdot a + c \cdot a^{2n}}+\frac{c^{n}}{c^{2n+1}+a^{2n} \cdot b + a \cdot b^{2n}} $$
|
\frac{1}{3^{n+1}}
| 82.8125 |
1,457 |
Find the remainder when $(1^2+1)(2^2+1)(3^2+1)\dots(42^2+1)$ is divided by $43$ . Your answer should be an integer between $0$ and $42$ .
|
4
| 88.28125 |
1,458 |
What is the greatest number of integers that can be selected from a set of $2015$ consecutive numbers so that no sum of any two selected numbers is divisible by their difference?
|
672
| 41.40625 |
1,459 |
A frog is standing in a center of a $3 \times 3$ grid of lilypads. Each minute, the frog chooses a square that shares exactly one side with their current square uniformly at random, and jumps onto the lilypad on their chosen square. The frog stops jumping once it reaches a lilypad on a corner of the grid. What is the expected number of times the frog jumps?
*2021 CCA Math Bonanza Lightning Round #3.2*
|
3
| 14.84375 |
1,460 |
Find all sequences of positive integers $\{a_n\}_{n=1}^{\infty}$ , for which $a_4=4$ and
\[\frac{1}{a_1a_2a_3}+\frac{1}{a_2a_3a_4}+\cdots+\frac{1}{a_na_{n+1}a_{n+2}}=\frac{(n+3)a_n}{4a_{n+1}a_{n+2}}\]
for all natural $n \geq 2$ .
*Peter Boyvalenkov*
|
a_n = n
| 53.90625 |
1,461 |
An isosceles triangle has angles of $50^\circ,x^\circ,$ and $y^\circ$ . Find the maximum possible value of $x-y$ .
[i]Proposed by Nathan Ramesh
|
30^\circ
| 78.125 |
1,462 |
A game is played on an ${n \times n}$ chessboard. At the beginning there are ${99}$ stones on each square. Two players ${A}$ and ${B}$ take turns, where in each turn the player chooses either a row or a column and removes one stone from each square in the chosen row or column. They are only allowed to choose a row or a column, if it has least one stone on each square. The first player who cannot move, looses the game. Player ${A}$ takes the first turn. Determine all n for which player ${A}$ has a winning strategy.
|
n
| 80.46875 |
1,463 |
Let $ABCD$ be a rhombus with angle $\angle A = 60^\circ$ . Let $E$ be a point, different from $D$ , on the line $AD$ . The lines $CE$ and $AB$ intersect at $F$ . The lines $DF$ and $BE$ intersect at $M$ . Determine the angle $\angle BMD$ as a function of the position of $E$ on $AD.$
|
120^\circ
| 1.5625 |
1,464 |
For each positive integer $k$ , let $d(k)$ be the number of positive divisors of $k$ and $\sigma(k)$ be the sum of positive divisors of $k$ . Let $\mathbb N$ be the set of all positive integers. Find all functions $f: \mathbb{N} \to \mathbb N$ such that \begin{align*}
f(d(n+1)) &= d(f(n)+1)\quad \text{and}
f(\sigma(n+1)) &= \sigma(f(n)+1)
\end{align*}
for all positive integers $n$ .
|
f(n) = n
| 97.65625 |
1,465 |
16 students took part in a competition. All problems were multiple choice style. Each problem had four choices. It was said that any two students had at most one answer in common, find the maximum number of problems.
|
5
| 35.9375 |
1,466 |
Jane is 25 years old. Dick is older than Jane. In $n$ years, where $n$ is a positive integer, Dick's age and Jane's age will both be two-digit number and will have the property that Jane's age is obtained by interchanging the digits of Dick's age. Let $d$ be Dick's present age. How many ordered pairs of positive integers $(d,n)$ are possible?
|
25
| 77.34375 |
1,467 |
How many ordered triplets $(a, b, c)$ of positive integers such that $30a + 50b + 70c \leq 343$ .
|
30
| 98.4375 |
1,468 |
Find the $2019$ th strictly positive integer $n$ such that $\binom{2n}{n}$ is not divisible by $5$ .
|
37805
| 12.5 |
1,469 |
The positive integers are colored with black and white such that:
- There exists a bijection from the black numbers to the white numbers,
- The sum of three black numbers is a black number, and
- The sum of three white numbers is a white number.
Find the number of possible colorings that satisfies the above conditions.
|
2
| 58.59375 |
1,470 |
Find all ordered pairs $(a,b)$ of positive integers that satisfy $a>b$ and the equation $(a-b)^{ab}=a^bb^a$ .
|
(4, 2)
| 93.75 |
1,471 |
Compute the number of subsets $S$ of $\{0,1,\dots,14\}$ with the property that for each $n=0,1,\dots,
6$ , either $n$ is in $S$ or both of $2n+1$ and $2n+2$ are in $S$ .
*Proposed by Evan Chen*
|
2306
| 10.15625 |
1,472 |
In the following alpha-numeric puzzle, each letter represents a different non-zero digit. What are all possible values for $b+e+h$ ?
$ \begin{tabular}{cccc} &a&b&c &d&e&f + & g&h&i \hline 1&6&6&5 \end{tabular}$ *Proposed by Eugene Chen*
|
15
| 25.78125 |
1,473 |
Let $f : N \to R$ be a function, satisfying the following condition:
for every integer $n > 1$ , there exists a prime divisor $p$ of $n$ such that $f(n) = f \big(\frac{n}{p}\big)-f(p)$ .
If $f(2^{2007}) + f(3^{2008}) + f(5^{2009}) = 2006$ , determine the value of $f(2007^2) + f(2008^3) + f(2009^5)$
|
9
| 2.34375 |
1,474 |
Find the smallest value of the expression $|253^m - 40^n|$ over all pairs of positive integers $(m, n)$ .
*Proposed by Oleksii Masalitin*
|
9
| 84.375 |
1,475 |
Let *Revolution* $(x) = x^3 +Ux^2 +Sx + A$ , where $U$ , $S$ , and $A$ are all integers and $U +S + A +1 = 1773$ . Given that *Revolution* has exactly two distinct nonzero integer roots $G$ and $B$ , find the minimum value of $|GB|$ .
*Proposed by Jacob Xu*
<details><summary>Solution</summary>*Solution.* $\boxed{392}$ Notice that $U + S + A + 1$ is just *Revolution* $(1)$ so *Revolution* $(1) = 1773$ . Since $G$ and $B$ are integer roots we write *Revolution* $(X) = (X-G)^2(X-B)$ without loss of generality. So Revolution $(1) = (1-G)^2(1-B) = 1773$ . $1773$ can be factored as $32 \cdot 197$ , so to minimize $|GB|$ we set $1-G = 3$ and $1-B = 197$ . We get that $G = -2$ and $B = -196$ so $|GB| = \boxed{392}$ .</details>
|
392
| 91.40625 |
1,476 |
Determine the maximum possible real value of the number $ k$ , such that
\[ (a \plus{} b \plus{} c)\left (\frac {1}{a \plus{} b} \plus{} \frac {1}{c \plus{} b} \plus{} \frac {1}{a \plus{} c} \minus{} k \right )\ge k\]
for all real numbers $ a,b,c\ge 0$ with $ a \plus{} b \plus{} c \equal{} ab \plus{} bc \plus{} ca$ .
|
1
| 7.03125 |
1,477 |
Determine the values of the real parameter $a$ , such that the equation
\[\sin 2x\sin 4x-\sin x\sin 3x=a\]
has a unique solution in the interval $[0,\pi)$ .
|
a = 1
| 0 |
1,478 |
Suppose $P (x)$ is a polynomial with real coefficients such that $P (t) = P (1)t^2 + P (P (1))t + P (P (P (1)))$ for all real numbers $t$ . Compute the largest possible value of $P(P(P(P(1))))$ .
|
\frac{1}{9}
| 10.9375 |
1,479 |
Let $ f:\mathbb{Z}_{>0}\rightarrow\mathbb{R} $ be a function such that for all $n > 1$ there is a prime divisor $p$ of $n$ such that \[ f(n)=f\left(\frac{n}{p}\right)-f(p). \]
Furthermore, it is given that $ f(2^{2014})+f(3^{2015})+f(5^{2016})=2013 $ . Determine $ f(2014^2)+f(2015^3)+f(2016^5) $ .
|
\frac{49}{3}
| 0 |
1,480 |
For a positive integer $n$ not divisible by $211$ , let $f(n)$ denote the smallest positive integer $k$ such that $n^k - 1$ is divisible by $211$ . Find the remainder when $$ \sum_{n=1}^{210} nf(n) $$ is divided by $211$ .
*Proposed by ApraTrip*
|
48
| 2.34375 |
1,481 |
Isosceles triangle $\triangle{ABC}$ has $\angle{ABC}=\angle{ACB}=72^\circ$ and $BC=1$ . If the angle bisector of $\angle{ABC}$ meets $AC$ at $D$ , what is the positive difference between the perimeters of $\triangle{ABD}$ and $\triangle{BCD}$ ?
*2019 CCA Math Bonanza Tiebreaker Round #2*
|
1
| 47.65625 |
1,482 |
One hundred and one of the squares of an $n\times n$ table are colored blue. It is known that there exists a unique way to cut the table to rectangles along boundaries of its squares with the following property: every rectangle contains exactly one blue square. Find the smallest possible $n$ .
|
101
| 63.28125 |
1,483 |
Let $ABCD$ be a parallelogram such that $\angle BAD = 60^{\circ}.$ Let $K$ and $L$ be the midpoints of $BC$ and $CD,$ respectively. Assuming that $ABKL$ is a cyclic quadrilateral, find $\angle ABD.$
|
75^\circ
| 0.78125 |
1,484 |
Anumber of schools took part in a tennis tournament. No two players from the same school played against each other. Every two players from different schools played exactly one match against each other. A match between two boys or between two girls was called a *single* and that between a boy and a girl was called a *mixed single*. The total number of boys differed from the total number of girls by at most 1. The total number of singles differed from the total number of mixed singles by at most 1. At most how many schools were represented by an odd number of players?
|
3
| 35.15625 |
1,485 |
Given an integer $n\ge 2$ , compute $\sum_{\sigma} \textrm{sgn}(\sigma) n^{\ell(\sigma)}$ , where all $n$ -element permutations are considered, and where $\ell(\sigma)$ is the number of disjoint cycles in the standard decomposition of $\sigma$ .
|
n!
| 1.5625 |
1,486 |
Let $\triangle ABC$ be a triangle with $AB=85$ , $BC=125$ , $CA=140$ , and incircle $\omega$ . Let $D$ , $E$ , $F$ be the points of tangency of $\omega$ with $\overline{BC}$ , $\overline{CA}$ , $\overline{AB}$ respectively, and furthermore denote by $X$ , $Y$ , and $Z$ the incenters of $\triangle AEF$ , $\triangle BFD$ , and $\triangle CDE$ , also respectively. Find the circumradius of $\triangle XYZ$ .
*Proposed by David Altizio*
|
30
| 10.15625 |
1,487 |
Let $ S \equal{} \{1,2,3,\cdots ,280\}$ . Find the smallest integer $ n$ such that each $ n$ -element subset of $ S$ contains five numbers which are pairwise relatively prime.
|
217
| 3.125 |
1,488 |
Find all naturals $k$ such that $3^k+5^k$ is the power of a natural number with the exponent $\ge 2$ .
|
k = 1
| 0 |
1,489 |
Triangle $ABC$ has right angle at $B$ , and contains a point $P$ for which $PA = 10$ , $PB = 6$ , and $\angle APB = \angle BPC = \angle CPA$ . Find $PC$ .
[asy]
pair A=(0,5), B=origin, C=(12,0), D=rotate(-60)*C, F=rotate(60)*A, P=intersectionpoint(A--D, C--F);
draw(A--P--B--A--C--B^^C--P);
dot(A^^B^^C^^P);
pair point=P;
label(" $A$ ", A, dir(point--A));
label(" $B$ ", B, dir(point--B));
label(" $C$ ", C, dir(point--C));
label(" $P$ ", P, NE);[/asy]
|
33
| 16.40625 |
1,490 |
Determine all functions $ f$ mapping the set of positive integers to the set of non-negative integers satisfying the following conditions:
(1) $ f(mn) \equal{} f(m)\plus{}f(n)$ ,
(2) $ f(2008) \equal{} 0$ , and
(3) $ f(n) \equal{} 0$ for all $ n \equiv 39\pmod {2008}$ .
|
f(n) = 0
| 84.375 |
1,491 |
There are $27$ cards, each has some amount of ( $1$ or $2$ or $3$ ) shapes (a circle, a square or a triangle) with some color (white, grey or black) on them. We call a triple of cards a *match* such that all of them have the same amount of shapes or distinct amount of shapes, have the same shape or distinct shapes and have the same color or distinct colors. For instance, three cards shown in the figure are a *match* be cause they have distinct amount of shapes, distinct shapes but the same color of shapes.
What is the maximum number of cards that we can choose such that non of the triples make a *match*?
*Proposed by Amin Bahjati*
|
9
| 87.5 |
1,492 |
Find the smallest positive integer $n$ such that if we color in red $n$ arbitrary vertices of the cube , there will be a vertex of the cube which has the three vertices adjacent to it colored in red.
|
5
| 53.125 |
1,493 |
Let ${ a\uparrow\uparrow b = {{{{{a^{a}}^a}^{\dots}}}^{a}}^{a}} $ , where there are $ b $ a's in total. That is $ a\uparrow\uparrow b $ is given by the recurrence \[ a\uparrow\uparrow b = \begin{cases} a & b=1 a^{a\uparrow\uparrow (b-1)} & b\ge2\end{cases} \] What is the remainder of $ 3\uparrow\uparrow( 3\uparrow\uparrow ( 3\uparrow\uparrow 3)) $ when divided by $ 60 $ ?
|
27
| 82.03125 |
1,494 |
If the base of a rectangle is increased by $ 10\%$ and the area is unchanged, then the altitude is decreased by:
$ \textbf{(A)}\ 9\% \qquad\textbf{(B)}\ 10\% \qquad\textbf{(C)}\ 11\% \qquad\textbf{(D)}\ 11\frac {1}{9}\% \qquad\textbf{(E)}\ 9\frac {1}{11}\%$
|
9 \frac{1}{11}\%
| 100 |
1,495 |
Let $a$ be a positive real numbers. Let $t,\ u\ (t<u)$ be the $x$ coordinates of the point of intersections of the curves : $C_1:y=|\cos x|\ (0\leq x\leq \pi),\ C_2:y=a\sin x\ (0\leq x\leq \pi).$ Denote by $S_1$ the area of the part bounded by $C_1,\ C_2$ and $y$ -axis in $0\leq x\leq t$ , and by $S_2$ the area of the part bounded by $C_1,\ C_2$ in $t\leq x\leq u$ . When $a$ moves over all positive real numbers, find the minimum value of $S_1+S_2$ .
|
2\sqrt{2} - 2
| 0 |
1,496 |
Joshua likes to play with numbers and patterns. Joshua's favorite number is $6$ because it is the units digit of his birth year, $1996$ . Part of the reason Joshua likes the number $6$ so much is that the powers of $6$ all have the same units digit as they grow from $6^1$ :
\begin{align*}6^1&=6,6^2&=36,6^3&=216,6^4&=1296,6^5&=7776,6^6&=46656,\vdots\end{align*}
However, not all units digits remain constant when exponentiated in this way. One day Joshua asks Michael if there are simple patterns for the units digits when each one-digit integer is exponentiated in the manner above. Michael responds, "You tell me!" Joshua gives a disappointed look, but then Michael suggests that Joshua play around with some numbers and see what he can discover. "See if you can find the units digit of $2008^{2008}$ ," Michael challenges. After a little while, Joshua finds an answer which Michael confirms is correct. What is Joshua's correct answer (the units digit of $2008^{2008}$ )?
|
6
| 96.875 |
1,497 |
A list of positive integers is called good if the maximum element of the list appears exactly once. A sublist is a list formed by one or more consecutive elements of a list. For example, the list $10,34,34,22,30,22$ the sublist $22,30,22$ is good and $10,34,34,22$ is not. A list is very good if all its sublists are good. Find the minimum value of $k$ such that there exists a very good list of length $2019$ with $k$ different values on it.
|
11
| 0 |
1,498 |
Find all real polynomials $ f$ with $ x,y \in \mathbb{R}$ such that
\[ 2 y f(x \plus{} y) \plus{} (x \minus{} y)(f(x) \plus{} f(y)) \geq 0.
\]
|
f(x) = cx
| 2.34375 |
1,499 |
Let $S = \{1, 2, \ldots, 2016\}$ , and let $f$ be a randomly chosen bijection from $S$ to itself. Let $n$ be the smallest
positive integer such that $f^{(n)}(1) = 1$ , where $f^{(i)}(x) = f(f^{(i-1)}(x))$ . What is the expected value of $n$ ?
|
\frac{2017}{2}
| 20.3125 |
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