problem
stringlengths 10
7.44k
| answer
stringlengths 1
270
| difficulty
stringclasses 8
values |
|---|---|---|
Let $E$ be the intersection point of the diagonals of a convex quadrilateral $ABCD$. Denote $F_{1}$, $F_{2}$, and $F_{3}$ as the areas of $\triangle ABE$, $\triangle CDE$, and the quadrilateral $ABCD$, respectively. Prove that $\sqrt{F_{1}} + \sqrt{F_{2}} \leq \sqrt{F}$ and determine when equality holds. | \sqrt{F_1}+\sqrt{F_2}\le\sqrt{F} | 7/8 |
Given an ellipse with the equation $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 (a > b > 0)$ and an eccentricity of $\frac{1}{2}$. $F\_1$ and $F\_2$ are the left and right foci of the ellipse, respectively. A line $l$ passing through $F\_2$ intersects the ellipse at points $A$ and $B$. The perimeter of $\triangle F\_1AB$ is $8$.
(I) Find the equation of the ellipse.
(II) If the slope of line $l$ is $0$, and its perpendicular bisector intersects the $y$-axis at $Q$, find the range of the $y$-coordinate of $Q$.
(III) Determine if there exists a point $M(m, 0)$ on the $x$-axis such that the $x$-axis bisects $\angle AMB$. If it exists, find the value of $m$; otherwise, explain the reason. | m = 4 | 2/8 |
Let $a, b, c$ be the sides of triangle $\triangle ABC$, and let $S$ be its area. Prove that
\[a^{2} + b^{2} + c^{2} \geq 4 \sqrt{3} S + (a - b)^{2} + (b - c)^{2} + (c - a)^{2},\]
with equality if and only if $a = b = c$. | ^2+b^2+^2\ge4\sqrt{3}S+(b)^2+()^2+()^2 | 3/8 |
Let \(p\) and \(q\) be relatively prime positive integers such that \(\dfrac pq = \dfrac1{2^1} + \dfrac2{4^2} + \dfrac3{2^3} + \dfrac4{4^4} + \dfrac5{2^5} + \dfrac6{4^6} + \cdots\), where the numerators always increase by 1, and the denominators alternate between powers of 2 and 4, with exponents also increasing by 1 for each subsequent term. Compute \(p+q\). | 169 | 7/8 |
Given a positive integer \( n \), there are \( 3n \) numbers that satisfy:
\[
0 \leq a_{1} \leq a_{2} \leq \cdots \leq a_{3n},
\]
and
\[
\left(\sum_{i=1}^{3n} a_{i}\right)^{3} \geq k_{n}\left(\sum_{i=1}^{n} a_{i} a_{n+i} a_{2n+i}\right) \text{ always holds. }
\]
Find the best possible value of \( k_{n} \) (expressed in terms of \( n \)). | 27n^2 | 1/8 |
Find the largest integer $ n$ satisfying the following conditions:
(i) $ n^2$ can be expressed as the difference of two consecutive cubes;
(ii) $ 2n\plus{}79$ is a perfect square. | 181 | 2/8 |
For each positive integer $n$ let $a_n$ be the least positive integer multiple of $23$ such that $a_n \equiv 1 \pmod{2^n}.$ Find the number of positive integers $n$ less than or equal to $1000$ that satisfy $a_n = a_{n+1}.$ | 363 | 1/8 |
In triangle \( ABC \), median \( AM \) is drawn. Circle \(\omega\) passes through point \( A \), touches line \( BC \) at point \( M \), and intersects sides \( AB \) and \( AC \) at points \( D \) and \( E \) respectively. On arc \( AD \), not containing point \( E \), a point \( F \) is chosen such that \(\angle BFE = 72^{\circ}\). It turns out that \(\angle DEF = \angle ABC\). Find the angle \(\angle CME\). | 36 | 1/8 |
Let $n$ be a positive integer, $n\geq 2$ . For each $t\in \mathbb{R}$ , $t\neq k\pi$ , $k\in\mathbb{Z}$ , we consider the numbers
\[ x_n(t) = \sum_{k=1}^n k(n-k)\cos{(tk)} \textrm{ and } y_n(t) = \sum_{k=1}^n k(n-k)\sin{(tk)}. \]
Prove that if $x_n(t) = y_n(t) =0$ if and only if $\tan {\frac {nt}2} = n \tan {\frac t2}$ .
*Constantin Buse* | \tan(\frac{nt}{2})=n\tan(\frac{}{2}) | 1/8 |
Let $\Omega$ be a unit circle and $A$ be a point on $\Omega$ . An angle $0 < \theta < 180^\circ$ is chosen uniformly at random, and $\Omega$ is rotated $\theta$ degrees clockwise about $A$ . What is the expected area swept by this rotation? | 2\pi | 1/8 |
Given the function \( f(x) = x^2 + x + \sqrt{3} \), if for all positive numbers \( a, b, c \), the inequality \( f\left(\frac{a+b+c}{3} - \sqrt[3]{abc}\right) \geq f\left(\lambda \left(\frac{a+b}{2} - \sqrt{ab}\right)\right) \) always holds, find the maximum value of the positive number \( \lambda \). | \frac{2}{3} | 4/8 |
Given that the random variable $X$ follows a normal distribution $N(2,\sigma^{2})$, and its normal distribution density curve is the graph of the function $f(x)$, and $\int_{0}^{2} f(x)dx=\dfrac{1}{3}$, calculate $P(x > 4)$. | \dfrac{1}{3} | 1/8 |
In an acute-angled triangle \( ABC \), the angle bisector \( AN \), the altitude \( BH \), and the line perpendicular to side \( AB \) passing through its midpoint intersect at a single point. Find angle \( BAC \). | 60 | 6/8 |
$x$ is a positive rational number, and $(x)$ represents the number of prime numbers less than or equal to $x$. For instance, $(5)=3$, meaning there are three prime numbers (2, 3, and 5) less than or equal to 5. Thus, $(x)$ defines an operation on $x$. Find the value of $((20) \times (1) + (7))$. | 2 | 1/8 |
Person A and person B start from locations A and B respectively, walking towards each other at speeds of 65 meters per minute and 55 meters per minute simultaneously. They meet after 10 minutes. What is the distance between A and B in meters? Also, what is the distance of the meeting point from the midpoint between A and B in meters? | 50 | 6/8 |
A dormitory is installing a shower room for 100 students. How many shower heads are economical if the boiler preheating takes 3 minutes per shower head, and it also needs to be heated during the shower? Each group is allocated 12 minutes for showering. | 20 | 1/8 |
Given a positive integer \( n \) and a sequence of real numbers \( a_{1}, a_{2}, \cdots, a_{n} \) such that for each \( m \leq n \), it holds that \( \left|\sum_{k=1}^{m} \frac{a_{k}}{k}\right| \leq 1 \), find the maximum value of \( \left|\sum_{k=1}^{n} a_{k}\right| \). | 2n-1 | 5/8 |
Let \( S = \{ 1, 2, \cdots, 2005 \} \). Find the smallest number \( n \) such that in any subset of \( n \) pairwise coprime numbers from \( S \), there is at least one prime number. | 16 | 5/8 |
Let $P_n$ be the number of permutations $\pi$ of $\{1,2,\dots,n\}$ such that \[|i-j|=1\text{ implies }|\pi(i)-\pi(j)|\le 2\] for all $i,j$ in $\{1,2,\dots,n\}.$ Show that for $n\ge 2,$ the quantity \[P_{n+5}-P_{n+4}-P_{n+3}+P_n\] does not depend on $n,$ and find its value. | 4 | 1/8 |
An isosceles triangle with a base \( a \) and a base angle \( \alpha \) is inscribed in a circle. Additionally, a second circle is constructed, which is tangent to both of the triangle's legs and the first circle. Find the radius of the second circle. | \frac{}{2\sin\alpha(1+\cos\alpha)} | 1/8 |
A theater box office sells $2n$ tickets at 5 cents each, with each person limited to buying one ticket. Initially, the box office has no money, and among the first $2n$ people in line, half have 5 cent coins while the other half only have one dollar bills. How many different ways can these $2n$ ticket buyers line up so that the box office does not experience any difficulty in providing change? | \frac{1}{n+1}\binom{2n}{n} | 1/8 |
A point \( C \) is taken on the segment \( AB \). A line passing through point \( C \) intersects the circles with diameters \( AC \) and \( BC \) at points \( K \) and \( L \), and also intersects the circle with diameter \( AB \) at points \( M \) and \( N \). Prove that \( KM = LN \). | KM=LN | 3/8 |
Find the number of ordered quadruples of positive integers \((a, b, c, d)\) such that \(a, b, c\), and \(d\) are all (not necessarily distinct) factors of 30 and \(a b c d > 900\). | 1940 | 1/8 |
In a $3 \times 3$ grid (each cell is a $1 \times 1$ square), place two identical chess pieces, with at most one piece per cell. There are ___ different ways to arrange the pieces (if two arrangements can overlap by rotation, they are considered the same arrangement). | 10 | 4/8 |
Parallelogram $ABCD$ is given such that $\angle ABC$ equals $30^o$ . Let $X$ be the foot of the perpendicular from $A$ onto $BC$ , and $Y$ the foot of the perpendicular from $C$ to $AB$ . If $AX = 20$ and $CY = 22$ , find the area of the parallelogram.
| 880 | 7/8 |
Let \\(n\\) be a positive integer, and \\(f(n) = 1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n}\\). It is calculated that \\(f(2) = \frac{3}{2}\\), \\(f(4) > 2\\), \\(f(8) > \frac{5}{2}\\), and \\(f(16) > 3\\). Observing the results above, according to the pattern, it can be inferred that \\(f(128) > \_\_\_\_\_\_\_\_． | \frac{9}{2} | 7/8 |
Given the function $f(x)=2\ln x - ax^2 + 3$,
(1) Discuss the monotonicity of the function $y=f(x)$;
(2) If there exist real numbers $m, n \in [1, 5]$ such that $f(m)=f(n)$ holds when $n-m \geq 2$, find the maximum value of the real number $a$. | \frac{\ln 3}{4} | 5/8 |
Consider $ \triangle ABC$ and points $ M \in (AB)$ , $ N \in (BC)$ , $ P \in (CA)$ , $ R \in (MN)$ , $ S \in (NP)$ , $ T \in (PM)$ such that $ \frac {AM}{MB} \equal{} \frac {BN}{NC} \equal{} \frac {CP}{PA} \equal{} k$ and $ \frac {MR}{RN} \equal{} \frac {NS}{SP} \equal{} \frac {PT}{TN} \equal{} 1 \minus{} k$ for some $ k \in (0, 1)$ . Prove that $ \triangle STR \sim \triangle ABC$ and, furthermore, determine $ k$ for which the minimum of $ [STR]$ is attained. | \frac{1}{2} | 2/8 |
Given the sample 7, 8, 9, x, y has an average of 8, and xy=60, then the standard deviation of this sample is \_\_\_\_\_\_. | \sqrt{2} | 3/8 |
A pedestrian is moving in a straight line towards a crosswalk at a constant speed of 3.6 km/h. Initially, the distance from the pedestrian to the crosswalk is 40 meters. The length of the crosswalk is 6 meters. What distance from the crosswalk will the pedestrian be after two minutes? | 74 | 4/8 |
The figure shows a square in the interior of a regular hexagon. The square and regular hexagon share a common side. What is the degree measure of $\angle ABC$? [asy]
size(150);
pair A, B, C, D, E, F, G, H;
A=(0,.866);
B=(.5,1.732);
C=(1.5,1.732);
D=(2,.866);
E=(1.5,0);
F=(.5,0);
G=(.5,1);
H=(1.5,1);
draw(A--B);
draw(B--C);
draw(C--D);
draw(D--E);
draw(E--F);
draw(F--A);
draw(F--G);
draw(G--H);
draw(H--E);
draw(D--H);
label("A", C, N);
label("B", D, E);
label("C", H, N);
[/asy] | 45 | 3/8 |
In a regular 1976-gon, the midpoints of all sides and the midpoints of all diagonals are marked. What is the maximum number of marked points that lie on one circle? | 1976 | 1/8 |
Given two points A and B on a number line, their distance is 2, and the distance between point A and the origin O is 3. Then, the sum of all possible distances between point B and the origin O equals to . | 12 | 4/8 |
Circles \(\omega_{1}, \omega_{2},\) and \(\omega_{3}\) are centered at \(M, N,\) and \(O\), respectively. The points of tangency between \(\omega_{2}\) and \(\omega_{3}\), \(\omega_{3}\) and \(\omega_{1}\), and \(\omega_{1}\) and \(\omega_{2}\) are \(A, B,\) and \(C\), respectively. Line \(MO\) intersects \(\omega_{3}\) and \(\omega_{1}\) again at \(P\) and \(Q\), respectively, and line \(AP\) intersects \(\omega_{2}\) again at \(R\). Given that \(ABC\) is an equilateral triangle of side length 1, compute the area of \(PQR\). | 2\sqrt{3} | 1/8 |
Find the smallest natural number \( N \) that is divisible by \( p \), ends with \( p \), and has a digit sum equal to \( p \), given that \( p \) is a prime number and \( 2p+1 \) is a cube of a natural number. | 11713 | 4/8 |
\( M \) is the midpoint of side \( BC \) of triangle \( ABC \). Let \( r_1 \) and \( r_2 \) be the radii of the circles inscribed in triangles \( ABM \) and \( ACM \), respectively. Prove that \( r_1 < 2r_2 \). | r_1<2r_2 | 5/8 |
Let $p(x)$ and $q(x)$ be two cubic polynomials such that $p(0)=-24$ , $q(0)=30$ , and \[p(q(x))=q(p(x))\] for all real numbers $x$ . Find the ordered pair $(p(3),q(6))$ . | (3,-24) | 1/8 |
Alice starts with the number 0. She can apply 100 operations on her number. In each operation, she can either add 1 to her number or square her number. After applying all operations, her score is the minimum distance from her number to any perfect square. What is the maximum score she can attain? | 94 | 1/8 |
Let $\triangle{PQR}$ be a right triangle with $PQ = 90$, $PR = 120$, and $QR = 150$. Let $C_{1}$ be the inscribed circle. Construct $\overline{ST}$ with $S$ on $\overline{PR}$ and $T$ on $\overline{QR}$, such that $\overline{ST}$ is perpendicular to $\overline{PR}$ and tangent to $C_{1}$. Construct $\overline{UV}$ with $U$ on $\overline{PQ}$ and $V$ on $\overline{QR}$ such that $\overline{UV}$ is perpendicular to $\overline{PQ}$ and tangent to $C_{1}$. Let $C_{2}$ be the inscribed circle of $\triangle{RST}$ and $C_{3}$ the inscribed circle of $\triangle{QUV}$. The distance between the centers of $C_{2}$ and $C_{3}$ can be written as $\sqrt {10n}$. What is $n$?
| 725 | 7/8 |
Given set $A = \{1, 2, 3\}$, and functions $f$ and $g$ are from set $A$ to set $A$. Find the number of function pairs $(f, g)$ such that the intersection of the images of $f$ and $g$ is empty. | 42 | 3/8 |
In the store "Third is not superfluous," there is a promotion: if a customer presents three items at the checkout, the cheapest one of them is free. Ivan wants to buy 11 items costing $100, 200, 300, ..., 1100 rubles. What is the minimum amount of money he can spend to buy these items? | 4800 | 2/8 |
Let $ABC$ be an isosceles triangle with $AB = AC = 4$ and $BC = 5$ . Two circles centered at $B$ and $C$ each have radius $2$ , and the line through the midpoint of $\overline{BC}$ perpendicular to $\overline{AC}$ intersects the two circles in four different points. If the greatest possible distance between any two of those four points can be expressed as $\frac{\sqrt{a}+b\sqrt{c}}{d}$ for positive integers $a$ , $b$ , $c$ , and $d$ with gcd $(b, d) = 1$ and $a$ and $c$ each not divisible by the square of any prime, find $a + b + c + d$ . | 451 | 2/8 |
From the vertex \( A \) of triangle \( ABC \), the perpendiculars \( AM \) and \( AN \) are dropped onto the bisectors of the external angles at vertices \( B \) and \( C \) respectively. Prove that the segment \( MN \) is equal to the semiperimeter of triangle \( ABC \). | \frac{}{2} | 1/8 |
In a right square prism \( P-ABCD \) with side edges and base edges both equal to 4, determine the total length of all the curves formed on its surface by points that are 3 units away from vertex \( P \). | 6\pi | 1/8 |
The 200-digit number \( M \) is composed of 200 ones. What is the sum of the digits of the product \( M \times 2013 \)? | 1200 | 2/8 |
Define $f\left(n\right)=\textrm{LCM}\left(1,2,\ldots,n\right)$ . Determine the smallest positive integer $a$ such that $f\left(a\right)=f\left(a+2\right)$ .
*2017 CCA Math Bonanza Lightning Round #2.4* | 13 | 5/8 |
A unit cube has vertices $P_1,P_2,P_3,P_4,P_1',P_2',P_3',$ and $P_4'$. Vertices $P_2$, $P_3$, and $P_4$ are adjacent to $P_1$, and for $1\le i\le 4,$ vertices $P_i$ and $P_i'$ are opposite to each other. A regular octahedron has one vertex in each of the segments $\overline{P_1P_2}$, $\overline{P_1P_3}$, $\overline{P_1P_4}$, $\overline{P_1'P_2'}$, $\overline{P_1'P_3'}$, and $\overline{P_1'P_4'}$. Find the side length of the octahedron.
[asy]
import three;
size(5cm);
triple eye = (-4, -8, 3);
currentprojection = perspective(eye);
triple[] P = {(1, -1, -1), (-1, -1, -1), (-1, 1, -1), (-1, -1, 1), (1, -1, -1)}; // P[0] = P[4] for convenience
triple[] Pp = {-P[0], -P[1], -P[2], -P[3], -P[4]};
// draw octahedron
triple pt(int k){ return (3*P[k] + P[1])/4; }
triple ptp(int k){ return (3*Pp[k] + Pp[1])/4; }
draw(pt(2)--pt(3)--pt(4)--cycle, gray(0.6));
draw(ptp(2)--pt(3)--ptp(4)--cycle, gray(0.6));
draw(ptp(2)--pt(4), gray(0.6));
draw(pt(2)--ptp(4), gray(0.6));
draw(pt(4)--ptp(3)--pt(2), gray(0.6) + linetype("4 4"));
draw(ptp(4)--ptp(3)--ptp(2), gray(0.6) + linetype("4 4"));
// draw cube
for(int i = 0; i < 4; ++i){
draw(P[1]--P[i]); draw(Pp[1]--Pp[i]);
for(int j = 0; j < 4; ++j){
if(i == 1 || j == 1 || i == j) continue;
draw(P[i]--Pp[j]); draw(Pp[i]--P[j]);
}
dot(P[i]); dot(Pp[i]);
dot(pt(i)); dot(ptp(i));
}
label("$P_1$", P[1], dir(P[1]));
label("$P_2$", P[2], dir(P[2]));
label("$P_3$", P[3], dir(-45));
label("$P_4$", P[4], dir(P[4]));
label("$P'_1$", Pp[1], dir(Pp[1]));
label("$P'_2$", Pp[2], dir(Pp[2]));
label("$P'_3$", Pp[3], dir(-100));
label("$P'_4$", Pp[4], dir(Pp[4]));
[/asy] | \frac{3 \sqrt{2}}{4} | 3/8 |
Fill the numbers $1, 2, \cdots, 9$ into a $3 \times 3$ grid such that each cell contains one number, each row's numbers increase from left to right, and each column's numbers decrease from top to bottom. How many different ways are there to achieve this?
(A) 12
(B) 24
(C) 42
(D) 48 | 42 | 1/8 |
Six chairs sit in a row. Six people randomly seat themselves in the chairs. Each person randomly chooses either to set their feet on the floor, to cross their legs to the right, or to cross their legs to the left. There is only a problem if two people sitting next to each other have the person on the right crossing their legs to the left and the person on the left crossing their legs to the right. The probability that this will **not** happen is given by $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$ . | 1106 | 6/8 |
In a two-story house that is inhabited in both floors as well as on the ground floor, 35 people live above someone and 45 people live below someone. One third of all the people living in the house live on the first floor.
How many people live in the house in total? | 60 | 6/8 |
A secret agent is trying to decipher a passcode. So far, he has obtained the following information:
- It is a four-digit number.
- It is not divisible by seven.
- The digit in the tens place is the sum of the digit in the units place and the digit in the hundreds place.
- The number formed by the first two digits of the code (in this order) is fifteen times the last digit of the code.
- The first and last digits of the code (in this order) form a prime number.
Does the agent have enough information to decipher the code? Justify your conclusion. | 4583 | 7/8 |
In Mr. Jacob's class, $12$ of the $20$ students received a 'B' on the latest exam. If the same proportion of students received a 'B' in Mrs. Cecilia's latest exam, and Mrs. Cecilia originally had $30$ students, but $6$ were absent during the exam, how many students present for Mrs. Cecilia’s exam received a 'B'? | 14 | 5/8 |
Let $ABCD$ be a unit square. For any interior points $M,N$ such that the line $MN$ does not contain a vertex of the square, we denote by $s(M,N)$ the least area of the triangles having their vertices in the set of points $\{ A,B,C,D,M,N\}$ . Find the least number $k$ such that $s(M,N)\le k$ , for all points $M,N$ .
*Dinu Șerbănescu* | 1/8 | 1/8 |
In the Cartesian coordinate system $(xOy)$, the parametric equations of curve $C_{1}$ are given by $\begin{cases}x=2t-1 \\ y=-4t-2\end{cases}$ $(t$ is the parameter$)$, and in the polar coordinate system with the coordinate origin $O$ as the pole and the positive half of the $x$-axis as the polar axis, the polar equation of curve $C_{2}$ is $\rho= \frac{2}{1-\cos \theta}$.
(1) Write the Cartesian equation of curve $C_{2}$;
(2) Let $M_{1}$ be a point on curve $C_{1}$, and $M_{2}$ be a point on curve $C_{2}$. Find the minimum value of $|M_{1}M_{2}|$. | \frac{3 \sqrt{5}}{10} | 7/8 |
Shanille O'Keal shoots free throws on a basketball court. She hits the first and misses the second, and thereafter the probability that she hits the next shot is equal to the proportion of shots she has hit so far. What is the probability she hits exactly $50$ of her first $100$ shots? | \frac{1}{99} | 3/8 |
There exist constants $b_1, b_2, b_3, b_4, b_5, b_6, b_7$ such that
\[
\cos^7 \theta = b_1 \cos \theta + b_2 \cos 2 \theta + b_3 \cos 3 \theta + b_4 \cos 4 \theta + b_5 \cos 5 \theta + b_6 \cos 6 \theta + b_7 \cos 7 \theta
\]
for all angles $\theta$. Find $b_1^2 + b_2^2 + b_3^2 + b_4^2 + b_5^2 + b_6^2 + b_7^2$. | \frac{429}{1024} | 5/8 |
The sequence $\mathrm{Az}\left(a_{n}\right)$ is defined as follows:
$$
a_{0}=a_{1}=1, \quad a_{n+1}=a_{n}+\frac{1}{a_{n-1}}
$$
Show that $a_{n} \geq \sqrt{n}$. | a_n\ge\sqrt{n} | 7/8 |
Let $\omega$ be a circle, and let $ABCD$ be a quadrilateral inscribed in $\omega$. Suppose that $BD$ and $AC$ intersect at a point $E$. The tangent to $\omega$ at $B$ meets line $AC$ at a point $F$, so that $C$ lies between $E$ and $F$. Given that $AE=6, EC=4, BE=2$, and $BF=12$, find $DA$. | 2 \sqrt{42} | 5/8 |
Find the largest integer $n$ satisfying the following conditions:
(i) $n^2$ can be expressed as the difference of two consecutive cubes;
(ii) $2n + 79$ is a perfect square.
| 181 | 4/8 |
Given a square with four vertices and its center, find the probability that the distance between any two of these five points is less than the side length of the square. | \frac{2}{5} | 6/8 |
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 $100$ ordered pairs. | 197 | 2/8 |
Compute $\arccos(\cos 9).$ All functions are in radians. | 9 - 2\pi | 7/8 |
In the diagram, there are more than three triangles. If each triangle has the same probability of being selected, what is the probability that a selected triangle has all or part of its interior shaded? Express your answer as a common fraction.
[asy]
draw((0,0)--(1,0)--(0,1)--(0,0)--cycle,linewidth(1));
draw((0,0)--(.5,0)--(.5,.5)--(0,0)--cycle,linewidth(1));
label("A",(0,1),NW);
label("B",(.5,.5),NE);
label("C",(1,0),SE);
label("D",(.5,0),S);
label("E",(0,0),SW);
filldraw((.5,0)--(1,0)--(.5,.5)--(.5,0)--cycle,gray,black);[/asy] | \frac{3}{5} | 1/8 |
Three people are sitting in a row of eight seats. If there must be empty seats on both sides of each person, then the number of different seating arrangements is. | 24 | 3/8 |
The vertex of the parabola $y = x^2 - 8x + c$ will be a point on the $x$-axis if the value of $c$ is:
$\textbf{(A)}\ - 16 \qquad \textbf{(B) }\ - 4 \qquad \textbf{(C) }\ 4 \qquad \textbf{(D) }\ 8 \qquad \textbf{(E) }\ 16$ | \textbf{(E)}\16 | 1/8 |
Show that if the numbers \( a_{1}, a_{2}, \ldots \) are not all zeros and satisfy the relation \( a_{n+2} = \left|a_{n+1}\right| - a_{n} \), then from some point on they are periodic and the smallest period is 9. | 9 | 2/8 |
On the clock tower of the train station, there is an electronic timing clock. On the boundary of the circular clock face, there is a small color light at each minute mark. At 9:37:20 PM, there are ____ small color lights within the acute angle formed between the minute hand and the hour hand. | 11 | 5/8 |
Triangle $ABC$ has sides $\overline{AB}$, $\overline{BC}$, and $\overline{CA}$ of length 43, 13, and 48, respectively. Let $\omega$ be the circle circumscribed around $\triangle ABC$ and let $D$ be the intersection of $\omega$ and the perpendicular bisector of $\overline{AC}$ that is not on the same side of $\overline{AC}$ as $B$. The length of $\overline{AD}$ can be expressed as $m\sqrt{n}$, where $m$ and $n$ are positive integers and $n$ is not divisible by the square of any prime. Find the greatest integer less than or equal to $m + \sqrt{n}$.
| 12 | 7/8 |
For which real \( k \) do we have \( \cosh x \leq \exp(k x^2) \) for all real \( x \)? | \frac{1}{2} | 6/8 |
If positive numbers \(a, b, c\) and constant \(k\) satisfy the inequality \(\frac{k a b c}{a+b+c} \leq (a+b)^2 + (a+b+4c)^2\), find the maximum value of the constant \(k\). | 100 | 6/8 |
Given six balls numbered 1, 2, 3, 4, 5, 6 and boxes A, B, C, D, each to be filled with one ball, with the conditions that ball 2 cannot be placed in box B and ball 4 cannot be placed in box D, determine the number of different ways to place the balls into the boxes. | 252 | 4/8 |
For every odd number $p>1$ we have:
$\textbf{(A)}\ (p-1)^{\frac{1}{2}(p-1)}-1 \; \text{is divisible by} \; p-2\qquad \textbf{(B)}\ (p-1)^{\frac{1}{2}(p-1)}+1 \; \text{is divisible by} \; p\\ \textbf{(C)}\ (p-1)^{\frac{1}{2}(p-1)} \; \text{is divisible by} \; p\qquad \textbf{(D)}\ (p-1)^{\frac{1}{2}(p-1)}+1 \; \text{is divisible by} \; p+1\\ \textbf{(E)}\ (p-1)^{\frac{1}{2}(p-1)}-1 \; \text{is divisible by} \; p-1$ | \textbf{(A)} | 1/8 |
In a tournament, any two players play against each other. Each player earns one point for a victory, 1/2 for a draw, and 0 points for a loss. Let \( S \) be the set of the 10 lowest scores. We know that each player obtained half of their score by playing against players in \( S \).
a) What is the sum of the scores of the players in \( S \)?
b) Determine how many participants are in the tournament.
Note: Each player plays only once with each opponent. | 25 | 3/8 |
Given that the total amount of money originally owned by Moe, Loki, and Nick was $72, and each of Loki, Moe, and Nick gave Ott$\, 4, determine the fractional part of the group's money that Ott now has. | \frac{1}{6} | 7/8 |
In the Cartesian coordinate system $(xOy)$, the equation of circle $C$ is $((x-4)^{2}+y^{2}=1)$. If there exists at least one point on the line $y=kx-3$ such that a circle with this point as the center and $2$ as the radius intersects with circle $C$, then the maximum value of $k$ is _____. | \frac{24}{7} | 7/8 |
Two circles touch in $M$ , and lie inside a rectangle $ABCD$ . One of them touches the sides $AB$ and $AD$ , and the other one touches $AD,BC,CD$ . The radius of the second circle is four times that of the first circle. Find the ratio in which the common tangent of the circles in $M$ divides $AB$ and $CD$ . | 1:1 | 1/8 |
Two types of anti-inflammatory drugs must be selected from $X_{1}$, $X_{2}$, $X_{3}$, $X_{4}$, $X_{5}$, with the restriction that $X_{1}$ and $X_{2}$ must be used together, and one type of antipyretic drug must be selected from $T_{1}$, $T_{2}$, $T_{3}$, $T_{4}$, with the further restriction that $X_{3}$ and $T_{4}$ cannot be used at the same time. Calculate the number of different test schemes. | 14 | 4/8 |
A semicircle has diameter $XY$ . A square $PQRS$ with side length 12 is inscribed in the semicircle with $P$ and $S$ on the diameter. Square $STUV$ has $T$ on $RS$ , $U$ on the semicircle, and $V$ on $XY$ . What is the area of $STUV$ ? | 36 | 6/8 |
Prove that the reciprocal of the real root of the equation \( x^3 - x + 1 = 0 \) satisfies the equation \( x^5 + x + 1 = 0 \). Find these roots to one decimal place of accuracy. | -0.8 | 4/8 |
Prove that the feet of the perpendiculars dropped from the intersection point of the diagonals of a cyclic quadrilateral to its sides form a quadrilateral into which a circle can be inscribed. Find the radius of this circle if the diagonals of the given cyclic quadrilateral are perpendicular, the radius of the circumscribed circle is $R$, and the distance from its center to the intersection point of the diagonals is $d$. | \frac{R^2-^2}{2R} | 1/8 |
In a competition there are $18$ teams and in each round $18$ teams are divided into $9$ pairs where the $9$ matches are played coincidentally. There are $17$ rounds, so that each pair of teams play each other exactly once. After $n$ rounds, there always exists $4$ teams such that there was exactly one match played between these teams in those $n$ rounds. Find the maximum value of $n$ . | 7 | 2/8 |
Given $2n$ people, some of whom are friends with each other. In this group, each person has exactly $k(k \geqslant 1)$ friends. Find the positive integer $k$ such that the $2n$ people can always be divided into two groups of $n$ people each, and every person in each group has at least one friend in their own group. | 2 | 2/8 |
What is the largest possible number of subsets of the set \(\{1, 2, \ldots, 2n + 1\}\) such that the intersection of any two subsets consists of one or several consecutive integers? | (n+1)^2 | 4/8 |
Let \( k>1 \) be a real number. Calculate:
(a) \( L = \lim _{n \rightarrow \infty} \int_{0}^{1}\left(\frac{k}{\sqrt[n]{x} + k - 1}\right)^{n} \, \mathrm{d}x \).
(b) \( \lim _{n \rightarrow \infty} n\left[L - \int_{0}^{1}\left(\frac{k}{\sqrt[n]{x} + k - 1}\right)^{n} \, \mathrm{d}x\right] \). | \frac{k}{(k-1)^2} | 1/8 |
Suppose $\overline{AB}$ is a segment of unit length in the plane. Let $f(X)$ and $g(X)$ be functions of the plane such that $f$ corresponds to rotation about $A$ $60^\circ$ counterclockwise and $g$ corresponds to rotation about $B$ $90^\circ$ clockwise. Let $P$ be a point with $g(f(P))=P$ ; what is the sum of all possible distances from $P$ to line $AB$ ? | \frac{1 + \sqrt{3}}{2} | 4/8 |
A function $f$ defined on integers such that $f (n) =n + 3$ if $n$ is odd $f (n) = \frac{n}{2}$ if $n$ is even
If $k$ is an odd integer, determine the values for which $f (f (f (k))) = k$ . | 1 | 6/8 |
An ant crawls along the edges of a cube with side length 1 unit. Starting from one of the vertices, in each minute the ant travels from one vertex to an adjacent vertex. After crawling for 7 minutes, the ant is at a distance of \(\sqrt{3}\) units from the starting point. Find the number of possible routes the ant has taken. | 546 | 3/8 |
In the diagram shown, $\overrightarrow{OA}\perp\overrightarrow{OC}$ and $\overrightarrow{OB}\perp\overrightarrow{OD}$. If $\angle{AOD}$ is 3.5 times $\angle{BOC}$, what is $\angle{AOD}$? [asy]
unitsize(1.5cm);
defaultpen(linewidth(.7pt)+fontsize(10pt));
dotfactor=4;
pair O=(0,0), A=dir(0), B=dir(50), C=dir(90), D=dir(140);
pair[] dots={O,A,B,C,D};
dot(dots);
draw(O--1.2*D,EndArrow(4));
draw(O--1.2*B,EndArrow(4));
draw(O--1.2*C,EndArrow(4));
draw(O--1.2*A,EndArrow(4));
label("$D$",D,SW);
label("$C$",C,W);
label("$B$",B,E);
label("$A$",A,N);
label("$O$",O,S);
[/asy] | 140\text{ degrees} | 7/8 |
In a class of $ n\geq 4$ some students are friends. In this class any $ n \minus{} 1$ students can be seated in a round table such that every student is sitting next to a friend of him in both sides, but $ n$ students can not be seated in that way. Prove that the minimum value of $ n$ is $ 10$ . | 10 | 7/8 |
Given an angle and a fixed point \( P \) on the angle bisector. The line \( e_{1} \) passing through \( P \) intersects the sides of the angle at segment lengths \( a_{1} \) and \( b_{1} \), while the line \( e_{2} \) passing through \( P \) intersects the sides of the angle at segment lengths \( a_{2} \) and \( b_{2} \). Prove that
$$
\frac{1}{a_{1}}+\frac{1}{b_{1}}=\frac{1}{a_{2}}+\frac{1}{b_{2}}
$$ | \frac{1}{a_{1}}+\frac{1}{b_{1}}=\frac{1}{a_{2}}+\frac{1}{b_{2}} | 4/8 |
In the following diagram, \(ABCD\) is a square, \(BD \parallel CE\) and \(BE = BD\). Let \(\angle E = x^{\circ}\). Find \(x\). | 30 | 6/8 |
Given an isosceles trapezoid $\mathrm{ABCE}$ with bases $\mathrm{BC}$ and $\mathrm{AE}$, where $\mathrm{BC}$ is smaller than $\mathrm{AE}$ with lengths 3 and 4 respectively. The smaller lateral side $\mathrm{AB}$ is equal to $\mathrm{BC}$. Point $\mathrm{D}$ lies on $\mathrm{AE}$ such that $\mathrm{AD}$ : $\mathrm{DE}=3:1$. Point $\mathrm{F}$ lies on $\mathrm{AD}$ such that $\mathrm{AF}$ : $\mathrm{FD}=2:1$. Point $\mathrm{G}$ lies on $\mathrm{BD}$ such that $\mathrm{BG}$ : $\mathrm{GD}=1:2$. Determine the angle measure $\angle \mathrm{CFG}$. | 45 | 1/8 |
Let \( a, b, c \) be positive real numbers such that
\[
\begin{cases}
a^{2} + ab + b^{2} = 25 \\
b^{2} + bc + c^{2} = 49 \\
c^{2} + ca + a^{2} = 64
\end{cases}
\]
Find \( (a+b+c)^{2} \). | 129 | 6/8 |
Determine the minimum of the expression
$$
\frac{2}{|a-b|}+\frac{2}{|b-c|}+\frac{2}{|c-a|}+\frac{5}{\sqrt{ab+bc+ca}}
$$
under the conditions that \(ab + bc + ca > 0\), \(a + b + c = 1\), and \(a, b, c\) are distinct. | 10\sqrt{6} | 6/8 |
Find the largest real number \( m \) such that for all positive numbers \( a, b, \) and \( c \) satisfying \( a + b + c = 1 \),
$$
10\left(a^{3}+b^{3}+c^{3}\right)-m\left(a^{5}+b^{5}+c^{5}\right) \geqslant 1.
$$ | 9 | 3/8 |
For every integer $n > 1$ , the sequence $\left( {{S}_{n}} \right)$ is defined by ${{S}_{n}}=\left\lfloor {{2}^{n}}\underbrace{\sqrt{2+\sqrt{2+...+\sqrt{2}}}}_{n\ radicals} \right\rfloor $ where $\left\lfloor x \right\rfloor$ denotes the floor function of $x$ . Prove that ${{S}_{2001}}=2\,{{S}_{2000}}+1$ .
.
| S_{2001}=2S_{2000}+1 | 3/8 |
On $5\times 5$ squares, we cover the area with several S-Tetrominos (=Z-Tetrominos) along the square so that in every square, there are two or fewer tiles covering that (tiles can be overlap). Find the maximum possible number of squares covered by at least one tile. | 24 | 2/8 |
The locus of the centers of all circles of given radius $a$, in the same plane, passing through a fixed point, is:
$\textbf{(A) }\text{a point}\qquad \textbf{(B) }\text{a straight line}\qquad \textbf{(C) }\text{two straight lines}\qquad \textbf{(D) }\text{a circle}\qquad \textbf{(E) }\text{two circles}$ | \textbf{(D)} | 1/8 |
A farmer sold domestic rabbits. By the end of the market, he sold exactly one-tenth as many rabbits as the price per rabbit in forints. He then distributed the revenue between his two sons. Starting with the older son, the boys alternately received one-hundred forint bills, but at the end, the younger son received only a few ten-forint bills. The father then gave him his pocket knife and said that this made their shares equal in value. How much was the pocket knife worth? | 40 | 3/8 |
A jobber buys an article at $$24$ less $12\frac{1}{2}\%$. He then wishes to sell the article at a gain of $33\frac{1}{3}\%$ of his cost
after allowing a $20\%$ discount on his marked price. At what price, in dollars, should the article be marked?
$\textbf{(A)}\ 25.20 \qquad \textbf{(B)}\ 30.00 \qquad \textbf{(C)}\ 33.60 \qquad \textbf{(D)}\ 40.00 \qquad \textbf{(E)}\ \text{none of these}$ | \textbf{(E)}\ | 1/8 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.