problem stringlengths 10 5.15k | answer stringlengths 0 1.23k |
|---|---|
Let $f(n)$ be the number of distinct prime divisors of $n$ less than 6. Compute $$\sum_{n=1}^{2020} f(n)^{2}$$ | 3431 |
OKRA is a trapezoid with OK parallel to RA. If OK = 12 and RA is a positive integer, how many integer values can be taken on by the length of the segment in the trapezoid, parallel to OK, through the intersection of the diagonals? | 10 |
How many integers satisfy $-4 \le 2x+1 \le 6$? | 5 |
Consider a large square divided into a grid of \(5 \times 5\) smaller squares, each with side length \(1\) unit. A shaded region within the large square is formed by connecting the centers of four smaller squares, creating a smaller square inside. Calculate the ratio of the area of the shaded smaller square to the area of the large square. | \frac{2}{25} |
Let $p(x)$ be a monic polynomial of degree 4, such that $p(1) = 17,$ $p(2) = 34,$ and $p(3) = 51.$ Find $p(0) + p(4).$ | 92 |
Let $b(n)$ be the number of digits in the base -4 representation of $n$. Evaluate $\sum_{i=1}^{2013} b(i)$. | 12345 |
Given that $a$, $b$, and $c$ are the roots of the equation $x^3-3x^2+mx+24=0$, and that $-a$ and $-b$ are the roots of the equation $x^2+nx-6=0$, then the value of $n$ is | -1 |
A salon has the same amount of customers every day. Each customer needs 1 can of hairspray during the styling and is also given 1 can of hairspray to take home. The salon also buys an extra 5 cans of hairspray each day to ensure there is never a shortage. If the salon buys 33 cans of hairspray every day, how many customers do they have each day? | Removing the surplus hairspray shows the salon needs 33 total cans of hairspray – 5 surplus cans of hairspray = <<33-5=28>>28 cans of hairspray for each customer.
Each customer needs 1 can of hairspray for styling + 1 can of hairspray to take home = <<1+1=2>>2 cans of hairspray.
So the salon has a total of 28 cans of hairspray / 2 cans of hairspray per customer = <<28/2=14>>14 customers.
#### 14 |
What is the sum of the first 1234 terms of the sequence where the number of 2s between consecutive 1s increases by 1 each time? | 2419 |
Everyday at school, Jo climbs a flight of $6$ stairs. Jo can take the stairs $1$, $2$, or $3$ at a time. For example, Jo could climb $3$, then $1$, then $2$. In how many ways can Jo climb the stairs? | 24 |
Define the operation $\S$ as follows: $a\,\S\, b=3a+5b$. What is the value of $7\,\S\,2$? | 31 |
Zain has 10 more of each coin than Emerie. If Emerie has six quarters, seven dimes, and five nickels, how many coins does Zain have? | If Emerie has six quarters, Zain has 6+10 = <<6+10=16>>16 quarters.
At the same time, Zain has 7+10 = <<7+10=17>>17 dimes, ten more than Emerie.
The total number of quarters and dimes Zain has is 17+16 = <<17+16=33>>33 coins.
Zain also has 10 more nickels than Emerie, a total of 10+5 = <<10+5=15>>15 nickels.
In total, Zain has 33+15 = <<33+15=48>>48 coins.
#### 48 |
Points \( E, F, M \) are located on the sides \( AB, BC, \) and \( AC \) of triangle \( ABC \), respectively. The segment \( AE \) is one third of side \( AB \), the segment \( BF \) is one sixth of side \( BC \), and the segment \( AM \) is two fifths of side \( AC \). Find the ratio of the area of triangle \( EFM \) to the area of triangle \( ABC \). | 23/90 |
Let \(a\) be a positive real number. Find the value of \(a\) such that the definite integral
\[
\int_{a}^{a^2} \frac{\mathrm{d} x}{x+\sqrt{x}}
\]
achieves its smallest possible value. | 3 - 2\sqrt{2} |
Let \( m \) and \( n \) be positive integers satisfying
\[ m n^{2} + 876 = 4 m n + 217 n. \]
Find the sum of all possible values of \( m \). | 93 |
Simplify completely: $$\sqrt[3]{30^3+40^3+50^3}$$. | 60 |
Find all positive real numbers $\lambda$ such that for all integers $n\geq 2$ and all positive real numbers $a_1,a_2,\cdots,a_n$ with $a_1+a_2+\cdots+a_n=n$, the following inequality holds:
$\sum_{i=1}^n\frac{1}{a_i}-\lambda\prod_{i=1}^{n}\frac{1}{a_i}\leq n-\lambda$. | \lambda \geq e |
Charley bought 30 pencils. She lost 6 pencils while moving to school, and of course, also lost 1/3 of the remaining pencils because she wasn't very good at keeping track of pencils. How many pencils does she currently have? | If Charley bought 30 pencils and lost 6 while moving, she remained with 30-6=<<30-6=24>>24 pencils.
Because she isn't good at keeping track of her pencils, Charley lost 1/3*24 = <<1/3*24=8>>8 more pencils.
She currently has 24-8 = <<24-8=16>>16 pencils.
#### 16 |
Let \[f(x) = \left\{
\begin{array}{cl}
\sqrt{x} &\text{ if }x>4, \\
x^2 &\text{ if }x \le 4.
\end{array}
\right.\]Find $f(f(f(2)))$. | 4 |
Six pepperoni circles will exactly fit across the diameter of a $12$-inch pizza when placed. If a total of $24$ circles of pepperoni are placed on this pizza without overlap, what fraction of the pizza is covered by pepperoni? | \frac{2}{3} |
In the diagram, $PQRS$ is a trapezoid with an area of $12.$ $RS$ is twice the length of $PQ.$ What is the area of $\triangle PQS?$
[asy]
draw((0,0)--(1,4)--(7,4)--(12,0)--cycle);
draw((7,4)--(0,0));
label("$S$",(0,0),W);
label("$P$",(1,4),NW);
label("$Q$",(7,4),NE);
label("$R$",(12,0),E);
[/asy] | 4 |
From the set of integers $\{1,2,3,\dots,2009\}$, choose $k$ pairs $\{a_i,b_i\}$ with $a_i<b_i$ so that no two pairs have a common element. Suppose that all the sums $a_i+b_i$ are distinct and less than or equal to $2009$. Find the maximum possible value of $k$. | 803 |
In the array of 13 squares shown below, 8 squares are colored red, and the remaining 5 squares are colored blue. If one of all possible such colorings is chosen at random, the probability that the chosen colored array appears the same when rotated 90 degrees around the central square is $\frac{1}{n}$ , where n is a positive integer. Find n.
[asy] draw((0,0)--(1,0)--(1,1)--(0,1)--(0,0)); draw((2,0)--(2,2)--(3,2)--(3,0)--(3,1)--(2,1)--(4,1)--(4,0)--(2,0)); draw((1,2)--(1,4)--(0,4)--(0,2)--(0,3)--(1,3)--(-1,3)--(-1,2)--(1,2)); draw((-1,1)--(-3,1)--(-3,0)--(-1,0)--(-2,0)--(-2,1)--(-2,-1)--(-1,-1)--(-1,1)); draw((0,-1)--(0,-3)--(1,-3)--(1,-1)--(1,-2)--(0,-2)--(2,-2)--(2,-1)--(0,-1)); size(100);[/asy] | 429 |
How many ordered pairs of real numbers $(x,y)$ satisfy the following system of equations? \[\left\{ \begin{aligned} x+3y&=3 \\ \left| |x| - |y| \right| &= 1 \end{aligned}\right.\] | 3 |
Let \( F_{1} \) and \( F_{2} \) be the left and right foci of the hyperbola \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 \) (where \( a > 0 \) and \( b > 0 \)). There exists a point \( P \) on the right branch of the hyperbola such that \( \left( \overrightarrow{OP} + \overrightarrow{OF_{2}} \right) \cdot \overrightarrow{PF_{2}} = 0 \), where \( O \) is the origin. Additionally, \( \left| \overrightarrow{PF_{1}} \right| = \sqrt{3} \left| \overrightarrow{PF_{2}} \right| \). Determine the eccentricity of the hyperbola. | \sqrt{3} + 1 |
The equation
\[(x - \sqrt[3]{13})(x - \sqrt[3]{53})(x - \sqrt[3]{103}) = \frac{1}{3}\]has three distinct solutions $r,$ $s,$ and $t.$ Calculate the value of $r^3 + s^3 + t^3.$ | 170 |
Central High School is competing against Northern High School in a backgammon match. Each school has three players, and the contest rules require that each player play two games against each of the other school's players. The match takes place in six rounds, with three games played simultaneously in each round. In how many different ways can the match be scheduled? | 900 |
There are 250 books inside a library. On Tuesday, 120 books are taken out to be read by children. On Wednesday, 35 books are returned. On Thursday, another 15 books are withdrawn from the library. How many books are now in the library? | On Tuesday, 120 books were taken out 250 - 120 = <<250-120=130>>130 books
On Wednesday, 35 books were returned 130 + 35 = <<130+35=165>>165 books
On Thursday, 15 books were borrowed 165 - 15 = 150 books
#### 150 |
Given the function $$f(x)=\sin^{2}x+ \sqrt {3}\sin x\cos x+2\cos^{2}x，x∈R$$.
(I) Find the smallest positive period and the interval where the function is monotonically increasing;
(II) Find the maximum value of the function on the interval $$[- \frac {π}{3}， \frac {π}{12}]$$. | \frac { \sqrt {3}+3}{2} |
The maximum value of the function $f(x) = 8\sin x - \tan x$, defined on $\left(0, \frac{\pi}{2}\right)$, is $\_\_\_\_\_\_\_\_\_\_\_\_$. | 3\sqrt{3} |
Xiaoli decides which subject among history, geography, or politics to review during tonight's self-study session based on the outcome of a mathematical game. The rules of the game are as follows: in the Cartesian coordinate system, starting from the origin $O$, and then ending at points $P_{1}(-1,0)$, $P_{2}(-1,1)$, $P_{3}(0,1)$, $P_{4}(1,1)$, $P_{5}(1,0)$, to form $5$ vectors. By randomly selecting any two vectors and calculating the dot product $y$ of these two vectors, if $y > 0$, she will review history; if $y=0$, she will review geography; if $y < 0$, she will review politics.
$(1)$ List all possible values of $y$;
$(2)$ Calculate the probability of Xiaoli reviewing history and the probability of reviewing geography. | \dfrac{3}{10} |
Evaluate
\[\begin{vmatrix} y + 1 & y & y \\ y & y + 1 & y \\ y & y & y + 1 \end{vmatrix}.\] | 3y + 1 |
A positive integer has exactly 8 divisors. The sum of its smallest 3 divisors is 15. Additionally, for this four-digit number, one prime factor minus five times another prime factor is equal to two times the third prime factor. What is this number? | 1221 |
How many times will a clock strike over the course of 12 hours if it chimes on the half-hours as well? | 90 |
In a certain school, 3 teachers are chosen from a group of 6 to give support teaching in 3 remote areas, with each area receiving one teacher. There are restrictions such that teacher A and teacher B cannot go together, and teacher A can only go with teacher C or not go at all. How many different dispatch plans are there? | 42 |
On the sides $AB$ and $CD$ of rectangle $ABCD$, points $E$ and $F$ are marked such that $AFCE$ forms a rhombus. It is known that $AB = 16$ and $BC = 12$. Find $EF$. | 15 |
There are five gifts priced at 2 yuan, 5 yuan, 8 yuan, 11 yuan, and 14 yuan, and five boxes priced at 1 yuan, 3 yuan, 5 yuan, 7 yuan, and 9 yuan. Each gift is paired with one box. How many different total prices are possible? | 19 |
In a conference room, 40 chairs with a capacity of 2 people each were arranged in rows in preparation for the board meeting of a company, whose number of members was the same as the chairs' capacity. If 2/5 of the chairs were not occupied, and the rest each had two people, calculate the number of board members who did attend the meeting. | The total capacity of the 40 chairs was 40*2=<<40*2=80>>80 people.
If 2/5 of the chairs were unoccupied, 2/5*80=<<2/5*80=32>>32 people missed the board meeting since the number of members was the same as the chair's capacity.
The number of board members who attended the meeting was 80-32=<<80-32=48>>48
#### 48 |
In a finite sequence of real numbers, the sum of any 7 consecutive terms is negative and the sum of any 11 consecutive terms is positive. How many terms can such a sequence have at most? | 16 |
Chad sandwiches 2 crackers with a scoop of peanut butter. He has 5 of these crackers a night before bed. A box of crackers has 4 sleeves with each sleeve holding 28 crackers. How many nights will 5 boxes of crackers last him? | Chad uses 2 crackers for each "sandwich" and has 5 sandwiches a night so he eats 2*5 = <<2*5=10>>10 crackers a night
The box has 4 sleeves of crackers and each sleeve has 28 crackers for a total of 4*28 = <<4*28=112>>112 crackers
1 box has 112 crackers so 5 boxes will have 112*5 = <<112*5=560>>560 crackers
He eats 10 crackers a night and 5 boxes have 560 crackers so they will last him 560//10 = <<560//10=56>>56 nights
#### 56 |
Grandma wants to order 5 personalized backpacks for each of her grandchildren's first days of school. The backpacks are 20% off of $20.00 and having their names monogrammed on the back pack will cost $12.00 each. How much will the backpacks cost? | She needs 5 backpacks that are $20.00 each so that's 5*20 = $<<5*20=100.00>>100.00
The backpacks are currently 20% off so that's 100*.20 = $<<100*.20=20.00>>20.00 off
The backpacks costs $100.00 but are $20.00 off so they will cost $100-$20 = $<<100-20=80.00>>80.00
She wants each of the 5 backpacks to have her grandchild's name monogramed on them which costs $12.00 each so that's 5*$12 = $<<5*12=60.00>>60.00
The backpacks cost $80.00 and the monogramming costs $60.00 so together they will cost $80+$60 = $<<80+60=140.00>>140.00
#### 140 |
Given an integer $k\geq 2$, determine all functions $f$ from the positive integers into themselves such that $f(x_1)!+f(x_2)!+\cdots f(x_k)!$ is divisibe by $x_1!+x_2!+\cdots x_k!$ for all positive integers $x_1,x_2,\cdots x_k$.
$Albania$ | f(n) = n |
On square $ABCD$, points $E,F,G$, and $H$ lie on sides $\overline{AB},\overline{BC},\overline{CD},$ and $\overline{DA},$ respectively, so that $\overline{EG} \perp \overline{FH}$ and $EG=FH = 34$. Segments $\overline{EG}$ and $\overline{FH}$ intersect at a point $P$, and the areas of the quadrilaterals $AEPH, BFPE, CGPF,$ and $DHPG$ are in the ratio $269:275:405:411.$ Find the area of square $ABCD$.
[asy] pair A = (0,sqrt(850)); pair B = (0,0); pair C = (sqrt(850),0); pair D = (sqrt(850),sqrt(850)); draw(A--B--C--D--cycle); dotfactor = 3; dot("$A$",A,dir(135)); dot("$B$",B,dir(215)); dot("$C$",C,dir(305)); dot("$D$",D,dir(45)); pair H = ((2sqrt(850)-sqrt(306))/6,sqrt(850)); pair F = ((2sqrt(850)+sqrt(306)+7)/6,0); dot("$H$",H,dir(90)); dot("$F$",F,dir(270)); draw(H--F); pair E = (0,(sqrt(850)-6)/2); pair G = (sqrt(850),(sqrt(850)+sqrt(100))/2); dot("$E$",E,dir(180)); dot("$G$",G,dir(0)); draw(E--G); pair P = extension(H,F,E,G); dot("$P$",P,dir(60)); label("$w$", intersectionpoint( A--P, E--H )); label("$x$", intersectionpoint( B--P, E--F )); label("$y$", intersectionpoint( C--P, G--F )); label("$z$", intersectionpoint( D--P, G--H ));[/asy] | 850 |
Given a parabola with vertex \( V \) and a focus \( F \), and points \( B \) and \( C \) on the parabola such that \( BF=25 \), \( BV=24 \), and \( CV=20 \), determine the sum of all possible values of the length \( FV \). | \frac{50}{3} |
5 points in a plane are situated so that no two of the lines joining a pair of points are coincident, parallel, or perpendicular. Through each point, lines are drawn perpendicular to each of the lines through two of the other 4 points. Determine the maximum number of intersections these perpendiculars can have. | 315 |
Suppose a regular tetrahedron \( P-ABCD \) has all edges equal in length. Using \(ABCD\) as one face, construct a cube \(ABCD-EFGH\) on the other side of the regular tetrahedron. Determine the cosine of the angle between the skew lines \( PA \) and \( CF \). | \frac{2 + \sqrt{2}}{4} |
What is the base five product of the numbers $132_{5}$ and $12_{5}$? | 2114_5 |
5/8 of shoppers at All Goods Available store prefer to avoid the check-out line on weekends and instead go through the express lane. If the number of shoppers in the store is 480, calculate the number of shoppers who pay at the check-out lane. | At the store, 5/8 of shoppers prefer to avoid the check-out line, a total of 5/8*480 = <<5/8*480=300>>300
The total number of shoppers at the store is 480; those who prefer the check-out line is 480-300 = <<480-300=180>>180
#### 180 |
Mr. Willson worked on making his furniture for 3/4 an hour on Monday. On Tuesday, he worked for half an hour. Then he worked for 2/3 an hour on Wednesday and 5/6 of an hour on Thursday. If he worked for 75 minutes on Friday, how many hours in all did he work from Monday to Friday? | Since there are 60 minutes in an hour, then Mr. Willson worked for 60 minutes x 3/4 = <<60*3/4=45>>45 minutes on Monday.
He worked for 60 minutes/2 = <<60/2=30>>30 minutes on Tuesday.
He worked for 60 minutes x 2/3 = <<60*2/3=40>>40 minutes on Wednesday.
He worked for 60 minutes x 5/6 = <<60*5/6=50>>50 minutes on Thursday.
So Mr. Willson worked for 45 + 30 + 40 + 50 + 75 = <<240=240>>240 minutes.
In hours, it is equal to 240/60 = <<240/60=4>>4 hours.
#### 4 |
Given that 2 students exercised 0 days, 4 students exercised 1 day, 5 students exercised 2 days, 3 students exercised 4 days, 7 students exercised 5 days, and 2 students exercised 6 days, calculate the average number of days exercised last week by the students in Ms. Brown's class. | 3.17 |
Evaluate
\[\log_{10}(\tan 1^{\circ})+\log_{10}(\tan 2^{\circ})+\log_{10}(\tan 3^{\circ})+\cdots+\log_{10}(\tan 88^{\circ})+\log_{10}(\tan 89^{\circ}).\] | 0 |
Let $a$ and $b$ be nonzero real numbers such that $\tfrac{1}{3a}+\tfrac{1}{b}=2011$ and $\tfrac{1}{a}+\tfrac{1}{3b}=1$ . What is the quotient when $a+b$ is divided by $ab$ ? | 1509 |
For integers $a$, $b$, $c$, and $d$, $(x^2+ax+b)(x^2+cx+d)=x^4+x^3-2x^2+17x-5$. What is the value of $a+b+c+d$? | 5 |
**Q8.** Given a triangle $ABC$ and $2$ point $K \in AB, \; N \in BC$ such that $BK=2AK, \; CN=2BN$ and $Q$ is the common point of $AN$ and $CK$ . Compute $\dfrac{ S_{ \triangle ABC}}{S_{\triangle BCQ}}.$ | 7/4 |
Let $\alpha$ be a real number. Determine all polynomials $P$ with real coefficients such that $$P(2x+\alpha)\leq (x^{20}+x^{19})P(x)$$ holds for all real numbers $x$. | P(x)\equiv 0 |
What fraction of the pizza is left for Wally if Jovin takes $\frac{1}{3}$ of the pizza, Anna takes $\frac{1}{6}$ of the pizza, and Olivia takes $\frac{1}{4}$ of the pizza? | \frac{1}{4} |
Convert the point $( -5, 0, -8 )$ in rectangular coordinates to cylindrical coordinates. Enter your answer in the form $(r,\theta,z),$ where $r > 0$ and $0 \le \theta < 2 \pi.$ | (5,\pi,-8) |
$O$ is the origin, and $F$ is the focus of the parabola $C:y^{2}=4x$. A line passing through $F$ intersects $C$ at points $A$ and $B$, and $\overrightarrow{FA}=2\overrightarrow{BF}$. Find the area of $\triangle OAB$. | \dfrac{3\sqrt{2}}{2} |
What is the least possible value of the expression (x+1)(x+2)(x+3)(x+4) + 2021 where x is a real number? | 2020 |
In triangle $XYZ$, $\angle Y = 90^\circ$, $YZ = 4$, and $XY = 5$. What is $\tan X$? | \frac{4}{3} |
John Smith buys 3 cakes for $12 each and splits the cost with his brother. How much did he pay? | The cakes cost 3*12=$<<3*12=36>>36
So he paid 36/2=$<<36/2=18>>18
#### 18 |
The product of the digits of 3214 is 24. How many distinct four-digit positive integers are such that the product of their digits equals 12? | 36 |
How many factors of 8000 are perfect squares? | 8 |
You are given the digits $0$, $1$, $2$, $3$, $4$, $5$. Form a four-digit number with no repeating digits.
(I) How many different four-digit numbers can be formed?
(II) How many of these four-digit numbers have a tens digit that is larger than both the units digit and the hundreds digit? | 100 |
Given the function $y=4^{x}-6\times2^{x}+8$, find the minimum value of the function and the value of $x$ when the minimum value is obtained. | -1 |
During the Easter egg hunt, Kevin found 5 eggs, Bonnie found 13 eggs, George found 9 and Cheryl found 56. How many more eggs did Cheryl find than the other three children found? | We know that Kevin found 5, Bonnie found 13 and George found 9 so 5+13+9 = <<5+13+9=27>>27
Cheryl found 56 eggs while the others found 27 eggs so 56-27 = <<56-27=29>>29 more eggs
#### 29 |
Consider equilateral triangle $ABC$ with side length $1$ . Suppose that a point $P$ in the plane of the triangle satisfies \[2AP=3BP=3CP=\kappa\] for some constant $\kappa$ . Compute the sum of all possible values of $\kappa$ .
*2018 CCA Math Bonanza Lightning Round #3.4* | \frac{18\sqrt{3}}{5} |
Chandra now has five bowls and five glasses, and each expands to a new set of colors: red, blue, yellow, green, and purple. However, she dislikes pairing the same colors; thus, a bowl and glass of the same color cannot be paired together like a red bowl with a red glass. How many acceptable combinations can Chandra make when choosing a bowl and a glass? | 44 |
The expression $\dfrac{\sqrt[4]{7}}{\sqrt[3]{7}}$ equals 7 raised to what power? | -\frac{1}{12} |
Let \[f(x) =
\begin{cases}
x^2+2 &\text{if } x<n, \\
2x+5 &\text{if }x\ge{n}.
\end{cases}
\]If the graph $y=f(x)$ is continuous, find the sum of all possible values of $n$. | 2 |
You, your friend, and two strangers are sitting at a table. A standard $52$ -card deck is randomly dealt into $4$ piles of $13$ cards each, and each person at the table takes a pile. You look through your hand and see that you have one ace. Compute the probability that your friend’s hand contains the three remaining aces. | 22/703 |
Given that $\sec x - \tan x = \frac{5}{4},$ find all possible values of $\sin x.$ | \frac{1}{4} |
In triangle \(ABC\), the sides opposite to angles \(A, B,\) and \(C\) are denoted by \(a, b,\) and \(c\) respectively. Given that \(c = 10\) and \(\frac{\cos A}{\cos B} = \frac{b}{a} = \frac{4}{3}\). Point \(P\) is a moving point on the incircle of triangle \(ABC\), and \(d\) is the sum of the squares of the distances from \(P\) to vertices \(A, B,\) and \(C\). Find \(d_{\min} + d_{\max}\). | 160 |
How many distinct three-digit positive integers have only odd digits? | 125 |
Let $c_i$ denote the $i$ th composite integer so that $\{c_i\}=4,6,8,9,...$ Compute
\[\prod_{i=1}^{\infty} \dfrac{c^{2}_{i}}{c_{i}^{2}-1}\]
(Hint: $\textstyle\sum^\infty_{n=1} \tfrac{1}{n^2}=\tfrac{\pi^2}{6}$ ) | \frac{12}{\pi^2} |
In the Cartesian coordinate system $xOy$, it is known that the circle $C: x^{2} + y^{2} + 8x - m + 1 = 0$ intersects with the line $x + \sqrt{2}y + 1 = 0$ at points $A$ and $B$. If $\triangle ABC$ is an equilateral triangle, then the value of the real number $m$ is. | -11 |
Square $EFGH$ is inside the square $ABCD$ so that each side of $EFGH$ can be extended to pass through a vertex of $ABCD$. Square $ABCD$ has side length $\sqrt {50}$ and $BE = 1$. What is the area of the inner square $EFGH$? | 36 |
In the equation $\frac{1}{m} + \frac{1}{n} = \frac{1}{4}$, where $m$ and $n$ are positive integers, determine the sum of all possible values for $n$. | 51 |
In the rectangle below, line segment $MN$ separates the rectangle into $2$ sections. What is the largest number of sections into which the rectangle can be separated when $4$ line segments (including $MN$) are drawn through the rectangle? [asy]
size(3cm,3cm);
pair A,B,C,D,M,N;
A=(0,0);
B=(1.5,0);
C=(1.5,1);
D=(0,1);
draw (A--B--C--D--A);
M=(0.8,0);
N=(1.2,1);
draw(M--N);
label("M",M,S);
label("N",N,NNE);
[/asy] | 11 |
Suppose the state of Georgia uses a license plate format "LLDLLL", and the state of Nebraska uses a format "LLDDDDD". Assuming all 10 digits are equally likely to appear in the numeric positions, and all 26 letters are equally likely to appear in the alpha positions, how many more license plates can Nebraska issue than Georgia? | 21902400 |
Let \( f(n) = 3n^2 - 3n + 1 \). Find the last four digits of \( f(1) + f(2) + \cdots + f(2010) \). | 1000 |
Three cats sat on a fence, meowing at the moon. The first cat meowed 3 times per minute. The second cat meowed twice as frequently as the first cat. And the third cat meowed at one-third the frequency of the second cat. What is the combined total number of meows the three cats make in 5 minutes? | The second cat meowed twice as frequently as the three meows per minute from the first cat, for a total of 2*3=<<2*3=6>>6 meows per minute.
The third cat meowed at one-third the frequency of the second cat, for a total of 6/3=<<6/3=2>>2 meows per minute.
Thus, combined, the three cats meow 3+6+2=<<3+6+2=11>>11 times per minute.
In five minutes, the three cats meow 5*11=<<5*11=55>>55 times.
#### 55 |
Let \( y = \cos \frac{2 \pi}{9} + i \sin \frac{2 \pi}{9} \). Compute the value of
\[
(3y + y^3)(3y^3 + y^9)(3y^6 + y^{18})(3y^2 + y^6)(3y^5 + y^{15})(3y^7 + y^{21}).
\] | 112 |
If $x=11$, $y=-8$, and $2x-3z=5y$, what is the value of $z$? | \frac{62}{3} |
If the acute angle $\theta$ satisfies $\sin (\pi \cos \theta) = \cos (\pi \sin \theta)$, then what is $\sin 2\theta$? | \frac{3}{4} |
Light of a blue laser (wavelength $\lambda=475 \, \text{nm}$ ) goes through a narrow slit which has width $d$ . After the light emerges from the slit, it is visible on a screen that is $ \text {2.013 m} $ away from the slit. The distance between the center of the screen and the first minimum band is $ \text {765 mm} $ . Find the width of the slit $d$ , in nanometers.
*(Proposed by Ahaan Rungta)* | 1250 |
In a store, an Uno Giant Family Card costs $12. When Ivan bought ten pieces, he was given a discount of $2 each. How much did Ivan pay in all? | Instead of $12 each, an Uno Giant Family Card costs $12 - $2 = $<<12-2=10>>10 each.
Ivan paid $10 x 10 = $<<10*10=100>>100 for the ten pieces Uno Giant Family Card.
#### 100 |
There are five unmarked envelopes on a table, each with a letter for a different person. If the mail is randomly distributed to these five people, with each person getting one letter, what is the probability that exactly four people get the right letter? | 0 |
It takes 15 mink skins to make a coat. Andy buys 30 minks and each mink has 6 babies, but half the total minks are set free by activists. How many coats can he make? | First find the total number of baby minks: 30 minks * 6 babies/mink = <<30*6=180>>180 minks
Add this to the number of adult minks: 180 minks + 30 minks = <<180+30=210>>210 minks
Then divide this number in half to find how many aren't set free: 210 minks / 2 = <<210/2=105>>105 minks
Then divide the remaining number of minks by the number of minks per coat to find the number of coats: 105 minks / 15 minks/coat = <<105/15=7>>7 coats
#### 7 |
Malcolm works in a company where they normally pack 40 apples in a box, producing 50 full boxes per day. Operations went as normal in one week. But in the next week, they packed 500 fewer apples per day. What's the total number of apples packed in the two weeks? | In the first week, each day they pack 40*50 = <<40*50=2000>>2000 apples
The total number of apples packed in the boxes is 2000*7 = <<2000*7=14000>>14000 apples in the first week.
When 500 fewer apples were packed each day in the next week, the total became 2000-500 = <<2000-500=1500>>1500 apples per day.
The total number of apples packed in the boxes that week is 1500*7 = <<1500*7=10500>>10500 apples.
The total is 14000+10500 = <<14000+10500=24500>>24500 apples for the two weeks
#### 24500 |
The complex number $2+i$ and the complex number $\frac{10}{3+i}$ correspond to points $A$ and $B$ on the complex plane, calculate the angle $\angle AOB$. | \frac{\pi}{4} |
A barn with a roof is rectangular in shape, $12$ yd. wide, $15$ yd. long, and $6$ yd. high. Calculate the total area to be painted. | 828 |
On a plane, points are colored in the following way:
1. Choose any positive integer \( m \), and let \( K_{1}, K_{2}, \cdots, K_{m} \) be circles with different non-zero radii such that \( K_{i} \subset K_{j} \) or \( K_{j} \subset K_{i} \) for \( i \neq j \).
2. Points chosen inside the circles are colored differently from the points outside the circles on the plane.
Given that there are 2019 points on the plane such that no three points are collinear, determine the maximum number of different colors possible that satisfy the given conditions. | 2019 |
A lateral face of a regular triangular pyramid $SABC$ is inclined to the base plane $ABC$ at an angle $\alpha = \operatorname{arctg} \frac{3}{4}$. Points $M, N, K$ are midpoints of the sides of the base $ABC$. The triangle $MNK$ serves as the lower base of a rectangular prism. The edges of the upper base of the prism intersect the lateral edges of the pyramid $SABC$ at points $F, P,$ and $R$ respectively. The total surface area of the polyhedron with vertices at points $M, N, K, F, P, R$ is $53 \sqrt{3}$. Find the side length of the triangle $ABC$. | 16 |
When $n$ standard 8-sided dice are rolled, the probability of obtaining a sum of 3000 is greater than zero and is the same as the probability of obtaining a sum of S. Find the smallest possible value of S. | 375 |
If $f(x)$ is a monic quartic polynomial such that $f(-1)=-1$, $f(2)=-4$, $f(-3)=-9$, and $f(4)=-16$, find $f(1)$. | 23 |
Mattis is hosting a badminton tournament for $40$ players on $20$ courts numbered from $1$ to $20$. The players are distributed with $2$ players on each court. In each round a winner is determined on each court. Afterwards, the player who lost on court $1$, and the player who won on court $20$ stay in place. For the remaining $38$ players, the winner on court $i$ moves to court $i + 1$ and the loser moves to court $i - 1$. The tournament continues until every player has played every other player at least once. What is the minimal number of rounds the tournament can last? | 39 |
Given that $[x]$ is the greatest integer less than or equal to $x$, calculate $\sum_{N=1}^{1024}\left[\log _{2} N\right]$. | 8204 |
The sequence \(a_{0}, a_{1}, a_{2}, \cdots, a_{n}\) satisfies \(a_{0}=\sqrt{3}\), \(a_{n+1}=\left\lfloor a_{n}\right\rfloor + \frac{1}{\left\{a_{n}\right\}}\), where \(\left\lfloor a_{n}\right\rfloor\) and \(\left\{a_{n}\right\}\) represent the integer part and fractional part of \(a_{n}\), respectively. Find \(a_{2016}\). | 3024 + \sqrt{3} |
A board game spinner is divided into four regions labeled $A$, $B$, $C$, and $D$. The probability of the arrow stopping on region $A$ is $\frac{3}{8}$, the probability of it stopping in $B$ is $\frac{1}{4}$, and the probability of it stopping in region $C$ is equal to the probability of it stopping in region $D$. What is the probability of the arrow stopping in region $C$? Express your answer as a common fraction. | \frac{3}{16} |
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