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Compute $\sin 6^\circ \sin 42^\circ \sin 66^\circ \sin 78^\circ.$ | \frac{1}{16} | 6/8 |
Circles $C_1$ and $C_2$ are externally tangent, and they are both internally tangent to circle $C_3.$ The radii of $C_1$ and $C_2$ are 4 and 10, respectively, and the centers of the three circles are all collinear. A chord of $C_3$ is also a common external tangent of $C_1$ and $C_2.$ Given that the length of the chord is $\frac{m\sqrt{n}}p$ where $m,n,$ and $p$ are positive integers, $m$ and $p$ are relatively prime, and $n$ is not divisible by the square of any prime, find $m+n+p.$
| 405 | 2/8 |
Given the nine-sided regular polygon $A_1A_2A_3A_4A_5A_6A_7A_8A_9$, how many distinct equilateral triangles in the plane of the polygon have at least two vertices in the set $\{A_1, A_2, \ldots A_9\}$? | 66 | 6/8 |
In an increasing sequence of four positive integers, the first three terms form an arithmetic progression, the last three terms form a geometric progression, and the first and fourth terms differ by $30$. Find the sum of the four terms. | 129 | 7/8 |
For positive real numbers $a,$ $b,$ $c,$ and $d,$ compute the maximum value of
\[\frac{abcd(a + b + c + d)}{(a + b)^2 (c + d)^2}.\] | \frac{1}{4} | 1/8 |
Given integer \( n \geq 3 \), define \(\mathcal{A} = \left\{\{\mathrm{z}_{1}, \mathrm{z}_{2}, \ldots, \mathrm{z}_{n}\} \mid \mathrm{z}_{\mathrm{i}} \in \mathbb{C}, |\mathrm{z}_{\mathrm{i}}| = 1, 1 \leq \mathrm{i} \leq n\right\}\). Determine the value of \(\min_{\mathrm{A} \in \mathcal{A}}\left\{\max_{\substack{\mathrm{u} \in \mathbb{C} \\|\mathrm{u}|=1}}\left\{\prod_{\mathrm{z} \in \mathrm{A}}|\mathrm{u}-\mathrm{z}|\right\}\right\}\) and find all \(\mathrm{A} \in \mathcal{A}\) that attain this minimum value in the maximum expression. | 2 | 6/8 |
Let $K$ be a closed plane curve such that the distance between any two points of $K$ is always less than $1.$ Show that $K$ lies in a circle of radius $\frac{1}{\sqrt{3}}.$ | \frac{1}{\sqrt{3}} | 7/8 |
Given that among 11 balls, 2 are radioactive, and for any group of balls, one test can determine whether there are any radioactive balls in the group (but not how many radioactive balls there are). Prove that if fewer than 7 tests are performed, it is not guaranteed that the pair of radioactive balls can be identified. However, with 7 tests, it is always possible to identify them. | 7 | 5/8 |
Find the smallest positive integer \( n \) such that in any coloring of the \( n \) vertices of a regular \( n \)-gon with three colors (red, yellow, and blue), there exist four vertices of the same color forming an isosceles trapezoid. (An isosceles trapezoid is a convex quadrilateral with one pair of parallel sides and the other pair of non-parallel sides being equal in length.) | 17 | 1/8 |
Simplify the expression and then evaluate: $(a-2b)(a^2+2ab+4b^2)-a(a-5b)(a+3b)$, where $a=-1$ and $b=1$. | -21 | 7/8 |
Let \( a_{1}, a_{2}, \cdots, a_{100} \) be a permutation of \( 1, 2, \cdots, 100 \). For \( 1 \leqslant i \leqslant 100 \), let \( b_{i} = a_{1} + a_{2} + \cdots + a_{i} \) and \( r_{i} \) be the remainder when \( b_{i} \) is divided by 100. Prove that \( r_{1}, r_{2}, \cdots, r_{100} \) take at least 11 different values.
(Canadian Mathematical Olympiad Training Problem, 1991) | 11 | 2/8 |
Compute $139+27+23+11$. | 200 | 7/8 |
Xiao Qian, Xiao Lu, and Xiao Dai are guessing a natural number between 1 and 99. Here are their statements:
Xiao Qian says: "It is a perfect square, and it is less than 5."
Xiao Lu says: "It is less than 7, and it is a two-digit number."
Xiao Dai says: "The first part of Xiao Qian's statement is correct, but the second part is wrong."
If one of them tells the truth in both statements, one of them tells a lie in both statements, and one of them tells one truth and one lie, then what is the number? (Note: A perfect square is a number that can be expressed as the square of an integer, for example, 4=2 squared, 81=9 squared, so 4 and 9 are considered perfect squares.) | 9 | 4/8 |
Given \(\cos \left(\theta+\frac{\pi}{4}\right)<3\left(\sin ^{5} \theta-\cos ^{5} \theta\right)\) where \(\theta \in[-\pi, \pi]\). Determine the range of values for \(\theta\). | [-\pi,-\frac{3\pi}{4})\cup(\frac{\pi}{4},\pi] | 2/8 |
Find the number of real solutions of the equation
\[\frac{x}{50} = \cos x.\] | 31 | 1/8 |
For some constants \( c \) and \( d \), let \[ g(x) = \left\{
\begin{array}{cl}
cx + d & \text{if } x < 3, \\
10 - 2x & \text{if } x \ge 3.
\end{array}
\right.\] The function \( g \) has the property that \( g(g(x)) = x \) for all \( x \). What is \( c + d \)? | \frac{9}{2} | 1/8 |
Petya created all the numbers that can be made from the digits 2, 0, 1, 8 (each digit can be used no more than once). Find their total sum. | 78331 | 1/8 |
Simplify first, then evaluate: $(1- \frac {2}{x+1})÷ \frac {x^{2}-x}{x^{2}-1}$, where $x=-2$. | \frac {3}{2} | 7/8 |
Evaluate \[ \sum_{k=1}^\infty \frac{(-1)^{k-1}}{k} \sum_{n=0}^\infty \frac{1}{k2^n + 1}. \] | 1 | 1/8 |
Let \( f(x) = x^2 + px + q \). It is known that the inequality \( |f(x)| > \frac{1}{2} \) has no solutions on the interval \([1, 3]\). Find \( \underbrace{f(f(\ldots f}_{2017}\left(\frac{3+\sqrt{7}}{2}\right)) \ldots) \). If necessary, round your answer to two decimal places. | 0.18 | 2/8 |
Find the positive integers $n$ that are not divisible by $3$ if the number $2^{n^2-10}+2133$ is a perfect cube.
<details><summary>Note</summary><span style="color:#BF0000">The wording of this problem is perhaps not the best English. As far as I am aware, just solve the diophantine equation $x^3=2^{n^2-10}+2133$ where $x,n \in \mathbb{N}$ and $3\nmid n$ .</span></details> | 4 | 3/8 |
(a) A bag contains 11 balls numbered from 1 to 11, 7 of which are black and 4 of which are gold. Julio removes two balls as described above. What is the probability that both balls are black?
(b) For some integer \( g \geq 2 \), a second bag contains 6 black balls and \( g \) gold balls; the balls are numbered from 1 to \( g+6 \). Julio removes two balls as described above. The probability that both balls are black is \( \frac{1}{8} \). Determine the value of \( g \).
(c) For some integer \( x \geq 2 \), a third bag contains \( 2x \) black balls and \( x \) gold balls; the balls are numbered from 1 to \( 3x \). Julio removes two balls as described above. The probability that both balls are black is \( \frac{7}{16} \). Determine the value of \( x \).
(d) For some integer \( r \geq 3 \), a fourth bag contains 10 black balls, 18 gold balls, and \( r \) red balls; the balls are numbered from 1 to \( r+28 \). This time, Julio draws three balls one after another. The probability that two of these three balls are black and one of these three balls is gold is at least \( \frac{1}{3000} \). What is the largest possible value of \( r \)? | 217 | 6/8 |
Let \( x, y, z, u, v \in \mathbf{R}_{+} \). Determine the maximum value of \( f = \frac{xy + yz + zu + uv}{2x^2 + y^2 + 2z^2 + u^2 + 2v^2} \). | 1/2 | 1/8 |
Let $A = (0,0)$ and $B = (b,2)$ be points on the coordinate plane. Let $ABCDEF$ be a convex equilateral hexagon such that $\angle FAB = 120^\circ,$ $\overline{AB}\parallel \overline{DE},$ $\overline{BC}\parallel \overline{EF,}$ $\overline{CD}\parallel \overline{FA},$ and the y-coordinates of its vertices are distinct elements of the set $\{0,2,4,6,8,10\}.$ The area of the hexagon can be written in the form $m\sqrt {n},$ where $m$ and $n$ are positive integers and n is not divisible by the square of any prime. Find $m + n.$
| 51 | 1/8 |
There are six identical red balls and three identical green balls in a pail. Four of these balls are selected at random and then these four balls are arranged in a line in some order. Find the number of different-looking arrangements of the selected balls. | 15 | 5/8 |
Given an annual interest rate of $i$, compounded annually, how much principal is required to withdraw $1 at the end of the first year, $4 at the end of the second year, $\cdots$, and $n^{2}$ at the end of the $n$-th year indefinitely? | \frac{(1+i)(2+i)}{i^3} | 1/8 |
In the convex quadrilateral \(ABCD\), the diagonals \(AC\) and \(BD\) are equal. Moreover, \(\angle BAC = \angle ADB\) and \(\angle CAD + \angle ADC = \angle ABD\). Find the angle \(BAD\). | 60 | 2/8 |
The radius of the Earth measures approximately $6378 \mathrm{~km}$ at the Equator. Suppose that a wire is adjusted exactly over the Equator.
Next, suppose that the length of the wire is increased by $1 \mathrm{~m}$, so that the wire and the Equator are concentric circles around the Earth. Can a standing man, an ant, or an elephant pass underneath this wire? | 0.159 | 2/8 |
A math competition consists of three problems, each of which receives an integer score from 0 to 7. For any two competitors, it is known that there is at most one problem in which they received the same score. Find the maximum number of competitors in this competition. | 64 | 3/8 |
Let $T$ be the set of all ordered triples of integers $(b_1, b_2, b_3)$ with $1 \leq b_1, b_2, b_3 \leq 20$. Each ordered triple in $T$ generates a sequence according to the rule $b_n = b_{n-1} \cdot |b_{n-2} - b_{n-3}|$ for all $n \geq 4$. Find the number of such sequences for which $b_n = 0$ for some $n$. | 780 | 1/8 |
Given the symbol $R_k$ represents an integer whose base-ten representation is a sequence of $k$ ones, find the number of zeros in the quotient $Q=R_{24}/R_4$. | 15 | 5/8 |
How many lattice points $(v, w, x, y, z)$ does a $5$ -sphere centered on the origin, with radius $3$ , contain on its surface or in its interior? | 1343 | 1/8 |
A pyramid has a square base with side of length 1 and has lateral faces that are equilateral triangles. A cube is placed within the pyramid so that one face is on the base of the pyramid and its opposite face has all its edges on the lateral faces of the pyramid. What is the volume of this cube? | 5\sqrt{2} - 7 | 5/8 |
Given that $x > 0$ and $y > 0$, and $\frac{4}{x} + \frac{3}{y} = 1$.
(I) Find the minimum value of $xy$ and the values of $x$ and $y$ when the minimum value is obtained.
(II) Find the minimum value of $x + y$ and the values of $x$ and $y$ when the minimum value is obtained. | 7 + 4\sqrt{3} | 6/8 |
In a circular circus arena with a radius of 10 meters, a lion runs. Moving along a broken line, the lion runs 30 kilometers.
Prove that the sum of all the angles by which the lion turns is at least 2998 radians. | 2998 | 4/8 |
Let \( a, b, c, d \in \{-1, 0, 1\} \). If the ordered array \((a, b, c, d)\) satisfies that \(a+b\), \(c+d\), \(a+c\), and \(b+d\) are all different, then \((a, b, c, d)\) is called a "good array." What is the probability that a randomly selected ordered array \((a, b, c, d)\) is a good array? | \frac{16}{81} | 1/8 |
Given an acute triangle \( \triangle ABC \) with angles \( A, B, C \) opposite to sides \( a, b, c \) respectively, and the condition \( \sin A = 2a \sin B \). The circumradius of \( \triangle ABC \) is \( \frac{\sqrt{3}}{6} \). Determine the range of values for the perimeter of \( \triangle ABC \). | (\frac{1+\sqrt{3}}{2},\frac{3}{2}] | 1/8 |
How many ordered triples of integers $(a,b,c)$ satisfy $|a+b|+c = 19$ and $ab+|c| = 97$?
$\textbf{(A)}\ 0\qquad\textbf{(B)}\ 4\qquad\textbf{(C)}\ 6\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 12$ | \textbf{(E)}\12 | 1/8 |
Let \( P(x) \) be the monic polynomial with rational coefficients of minimal degree such that \( \frac{1}{\sqrt{2}}, \frac{1}{\sqrt{3}}, \ldots, \frac{1}{\sqrt{1000}} \) are roots of \( P \). What is the sum of the coefficients of \( P \)? | \frac{1}{16000} | 2/8 |
The roots of the equation $2\sqrt{x} + 2x^{-\frac{1}{2}} = 5$ can be found by solving: | 4x^2-17x+4 = 0 | 1/8 |
Let $x$ be Walter's age in 1996. If Walter was one-third as old as his grandmother in 1996, then his grandmother's age in 1996 is $3x$. The sum of their birth years is $3864$, and the sum of their ages in 1996 plus $x$ and $3x$ is $1996+1997$. Express Walter's age at the end of 2001 in terms of $x$. | 37 | 1/8 |
Let $[x]$ denote the greatest integer less than or equal to $x$. When $0 \leqslant x \leqslant 10$, find the number of distinct integers represented by the function $f(x) = [x] + [2x] + [3x] + [4x]$. | 61 | 2/8 |
Two differentiable real functions \( f(x) \) and \( g(x) \) satisfy
\[ \frac{f^{\prime}(x)}{g^{\prime}(x)} = e^{f(x) - g(x)} \]
for all \( x \), and \( f(0) = g(2003) = 1 \). Find the largest constant \( c \) such that \( f(2003) > c \) for all such functions \( f, g \). | 1 - \ln 2 | 4/8 |
There are already $N$ people seated around a circular table with 60 chairs. What is the smallest possible value of $N$ such that the next person to sit down will have to sit next to someone? | 20 | 6/8 |
Three households, A, B, and C, plan to subscribe to newspapers. There are 7 different newspapers available. Each household subscribes to exactly 3 different newspapers, and it is known that each pair of households subscribes to exactly one newspaper in common. How many different subscription methods are there for the three households in total? | 5670 | 1/8 |
There are 21 nonzero numbers. For each pair of these numbers, their sum and product are calculated. It turns out that half of all the sums are positive and half are negative. What is the maximum possible number of positive products? | 120 | 4/8 |
In $\triangle ABC$, let $BC = a$, $CA = b$, and $AB = c$. If $2a^2 + b^2 + c^2 = 4$, find the maximum area of $\triangle ABC$. | \frac{\sqrt{5}}{5} | 4/8 |
Vasya wrote a note on a piece of paper, folded it in quarters, and wrote "MAME" on top. He then unfolded the note, added something more, folded it again randomly along the crease lines (not necessarily as before), and left it on the table with a random side facing up. Find the probability that the inscription "MAME" remains on top. | 1/8 | 2/8 |
In the rectangular coordinate system $xOy$, the ellipse $\Gamma: \frac{x^{2}}{3} + y^{2} = 1$ has its left vertex at $P$. Points $A$ and $B$ are two moving points on $\Gamma$. Determine the range of values for $\overrightarrow{P A} \cdot \overrightarrow{P B}$. | [-\frac{1}{4},12] | 3/8 |
The real function \( f \) has the property that, whenever \( a, b, n \) are positive integers such that \( a+b=2^{n} \), the equation \( f(a)+f(b)=n^{2} \) holds. What is \( f(2002) \)? | 96 | 4/8 |
Let $n$ be a positive integer. A frog starts on the number line at $0$ . Suppose it makes a finite sequence of hops, subject to two conditions:
- The frog visits only points in $\{1, 2, \dots, 2^n-1\}$ , each at most once.
- The length of each hop is in $\{2^0, 2^1, 2^2, \dots\}$ . (The hops may be either direction, left or right.)
Let $S$ be the sum of the (positive) lengths of all hops in the sequence. What is the maximum possible value of $S$ ?
*Ashwin Sah* | \frac{4^n-1}{3} | 1/8 |
Let $[x]$ denote the greatest integer less than or equal to the real number $x$. Calculate the value of the sum
$$
\left[\frac{n+1}{2}\right]+\left[\frac{n+2}{2^{2}}\right]+\ldots+\left[\frac{n+2^{k}}{2^{k+1}}\right]+\ldots
$$
for every positive integer $n$, and prove the correctness of the obtained result. | n | 3/8 |
Let $\mathbb{N}_{\geqslant 1}$ be the set of positive integers.
Find all functions $f \colon \mathbb{N}_{\geqslant 1} \to \mathbb{N}_{\geqslant 1}$ such that, for all positive integers $m$ and $n$ :
\[\mathrm{GCD}\left(f(m),n\right) + \mathrm{LCM}\left(m,f(n)\right) =
\mathrm{GCD}\left(m,f(n)\right) + \mathrm{LCM}\left(f(m),n\right).\]
Note: if $a$ and $b$ are positive integers, $\mathrm{GCD}(a,b)$ is the largest positive integer that divides both $a$ and $b$ , and $\mathrm{LCM}(a,b)$ is the smallest positive integer that is a multiple of both $a$ and $b$ . | f(n)=n | 4/8 |
Let $z=z(x,y)$ be implicit function with two variables from $2sin(x+2y-3z)=x+2y-3z$ . Find $\frac{\partial z}{\partial x}+\frac{\partial z}{\partial y}$ . | 1 | 7/8 |
Petya and Vasya came up with ten polynomials of the fifth degree. Then Vasya began to sequentially name consecutive natural numbers (starting from some number), and Petya substituted each named number into one of the polynomials of his choice and wrote the obtained values on the board from left to right. It turned out that the numbers written on the board form an arithmetic progression (exactly in that order). What is the maximum number of numbers that Vasya could name? | 50 | 4/8 |
Let $ n, m$ be positive integers of different parity, and $ n > m$ . Find all integers $ x$ such that $ \frac {x^{2^n} \minus{} 1}{x^{2^m} \minus{} 1}$ is a perfect square. | 0 | 6/8 |
In the diagram below, $AB = AC = 115,$ $AD = 38,$ and $CF = 77.$ Compute $\frac{[CEF]}{[DBE]}.$
[asy]
unitsize(0.025 cm);
pair A, B, C, D, E, F;
B = (0,0);
C = (80,0);
A = intersectionpoint(arc(B,115,0,180),arc(C,115,0,180));
D = interp(A,B,38/115);
F = interp(A,C,(115 + 77)/115);
E = extension(B,C,D,F);
draw(C--B--A--F--D);
label("$A$", A, N);
label("$B$", B, SW);
label("$C$", C, NE);
label("$D$", D, W);
label("$E$", E, SW);
label("$F$", F, SE);
[/asy] | \frac{19}{96} | 2/8 |
Xiao Ming arranges chess pieces in a two-layer hollow square array (the diagram shows the top left part of the array). After arranging the inner layer, 60 chess pieces are left. After arranging the outer layer, 32 chess pieces are left. How many chess pieces does Xiao Ming have in total? | 80 | 1/8 |
There are 30 students in a class going to watch a movie. Their student numbers are $1, 2, \cdots, 30$; they have movie tickets numbered 1 to 30. The movie tickets need to be distributed according to the following rule: For any two students A and B, if A's student number can be divided by B's student number, then A's movie ticket number should also be divisible by B's movie ticket number. How many different ways are there to distribute the movie tickets? | 48 | 1/8 |
There are 1000 lights and 1000 switches. Each switch simultaneously controls all lights whose numbers are multiples of the switch's number. Initially, all lights are on. Now, if switches numbered 2, 3, and 5 are pulled, how many lights will remain on? | 499 | 7/8 |
Determine the integer \( m \), \( -180 \leq m \leq 180 \), such that \(\sin m^\circ = \sin 945^\circ.\) | -135 | 3/8 |
A sequence is defined recursively as follows: \( t_{1} = 1 \), and for \( n > 1 \):
- If \( n \) is even, \( t_{n} = 1 + t_{\frac{n}{2}} \).
- If \( n \) is odd, \( t_{n} = \frac{1}{t_{n-1}} \).
Given that \( t_{n} = \frac{19}{87} \), find the sum of the digits of \( n \).
(From the 38th American High School Mathematics Examination, 1987) | 15 | 5/8 |
How many ordered triples of integers $(a,b,c)$ satisfy $|a+b|+c = 19$ and $ab+|c| = 97$? | 12 | 6/8 |
$12$ knights are sitting at a round table. Every knight is an enemy with two of the adjacent knights but with none of the others. $5$ knights are to be chosen to save the princess, with no enemies in the group. How many ways are there for the choice? | 36 | 7/8 |
Four steel balls, each with a radius of 1, are completely packed into a container in the shape of a regular tetrahedron. Find the minimum height of this regular tetrahedron. | 2+\frac{2 \sqrt{6}}{3} | 1/8 |
Let $x=\frac{\sum\limits_{n=1}^{44} \cos n^\circ}{\sum\limits_{n=1}^{44} \sin n^\circ}$. What is the greatest integer that does not exceed $100x$?
| 241 | 7/8 |
Equilateral triangles $ABC$ and $A_{1}B_{1}C_{1}$ with a side length of 12 are inscribed in a circle $S$ such that point $A$ lies on the arc $B_{1}C_{1}$, and point $B$ lies on the arc $A_{1}B_{1}$. Find $AA_{1}^{2} + BB_{1}^{2} + CC_{1}^{2}$. | 288 | 4/8 |
Little Wang has three ballpoint pens of the same style but different colors. Each pen has a cap that matches its color. Normally, Wang keeps the pen and cap of the same color together, but sometimes he mixes and matches the pens and caps. If Wang randomly pairs the pens and caps, calculate the probability that he will mismatch the colors of two pairs. | \frac{1}{2} | 7/8 |
\[
\sum_{k=1}^{70} \frac{k}{x-k} \geq \frac{5}{4}
\]
is a union of disjoint intervals the sum of whose lengths is 1988. | 1988 | 3/8 |
$A, B, C, D$ are points along a circle, in that order. $AC$ intersects $BD$ at $X$ . If $BC=6$ , $BX=4$ , $XD=5$ , and $AC=11$ , fi nd $AB$ | 6 | 1/8 |
Find the largest real number $\lambda$ such that
\[a_1^2 + \cdots + a_{2019}^2 \ge a_1a_2 + a_2a_3 + \cdots + a_{1008}a_{1009} + \lambda a_{1009}a_{1010} + \lambda a_{1010}a_{1011} + a_{1011}a_{1012} + \cdots + a_{2018}a_{2019}\]
for all real numbers $a_1, \ldots, a_{2019}$ . The coefficients on the right-hand side are $1$ for all terms except $a_{1009}a_{1010}$ and $a_{1010}a_{1011}$ , which have coefficient $\lambda$ . | 3/2 | 1/8 |
The volume of a bottle of kvass is 1.5 liters. The first person drank half of the bottle, the second drank a third of what was left after the first, the third drank a quarter of what was left after the previous ones, and so on, with the fourteenth person drinking one-fifteenth of what remained. How much kvass is left in the bottle? | 0.1\, | 1/8 |
Evaluate $\frac{7}{3} + \frac{11}{5} + \frac{19}{9} + \frac{37}{17} - 8$. | \frac{628}{765} | 7/8 |
The natural number \(a\) is divisible by 35 and has 75 different divisors, including 1 and \(a\). Find the smallest such \(a\). | 490000 | 3/8 |
For n real numbers $a_{1},\, a_{2},\, \ldots\, , a_{n},$ let $d$ denote the difference between the greatest and smallest of them and $S = \sum_{i<j}\left |a_i-a_j \right|.$ Prove that \[(n-1)d\le S\le\frac{n^{2}}{4}d\] and find when each equality holds. | (n-1)\leS\le\frac{n^2}{4} | 6/8 |
Given an isosceles triangle \(XYZ\) with \(XY = YZ\) and an angle at the vertex equal to \(96^{\circ}\). Point \(O\) is located inside triangle \(XYZ\) such that \(\angle OZX = 30^{\circ}\) and \(\angle OXZ = 18^{\circ}\). Find the measure of angle \(\angle YOX\). | 78 | 1/8 |
On a sunny day, 3000 people, including children, boarded a cruise ship. Two-fifths of the people were women, and a third were men. If 25% of the women and 15% of the men were wearing sunglasses, and there were also 180 children on board with 10% wearing sunglasses, how many people in total were wearing sunglasses? | 530 | 1/8 |
Suppose two distinct integers are chosen from between 1 and 29, inclusive. What is the probability that their product is neither a multiple of 2 nor 3? | \dfrac{45}{406} | 7/8 |
Find all \( a_{0} \in \mathbb{R} \) such that the sequence defined by
\[ a_{n+1} = 2^{n} - 3a_{n}, \quad n = 0, 1, 2, \cdots \]
is increasing. | \frac{1}{5} | 4/8 |
Alice, Bob, and Carol repeatedly take turns tossing a die. Alice begins; Bob always follows Alice; Carol always follows Bob; and Alice always follows Carol. Find the probability that Carol will be the first one to toss a six. (The probability of obtaining a six on any toss is $\frac{1}{6}$, independent of the outcome of any other toss.)
$\textbf{(A) } \frac{1}{3} \textbf{(B) } \frac{2}{9} \textbf{(C) } \frac{5}{18} \textbf{(D) } \frac{25}{91} \textbf{(E) } \frac{36}{91}$ | \textbf{(D)}\frac{25}{91} | 1/8 |
A dog and a cat simultaneously grab a sausage from different ends with their teeth. If the dog bites off its piece and runs away, the cat will get 300 grams more than the dog. If the cat bites off its piece and runs away, the dog will get 500 grams more than the cat. How much sausage is left if both bite off their pieces and run away? | 400 | 7/8 |
If each face of a tetrahedron is not an isosceles triangle, then what is the minimum number of edges of different lengths?
(A) 3
(B) 4
(C) 5
(D) 6 | 3 | 1/8 |
Given a set $S$ consisting of two undefined elements "pib" and "maa", and the four postulates: $P_1$: Every pib is a collection of maas, $P_2$: Any two distinct pibs have one and only one maa in common, $P_3$: Every maa belongs to two and only two pibs, $P_4$: There are exactly four pibs. Consider the three theorems: $T_1$: There are exactly six maas, $T_2$: There are exactly three maas in each pib, $T_3$: For each maa there is exactly one other maa not in the same pid with it. The theorems which are deducible from the postulates are:
$\textbf{(A)}\ T_3 \; \text{only}\qquad \textbf{(B)}\ T_2 \; \text{and} \; T_3 \; \text{only} \qquad \textbf{(C)}\ T_1 \; \text{and} \; T_2 \; \text{only}\\ \textbf{(D)}\ T_1 \; \text{and} \; T_3 \; \text{only}\qquad \textbf{(E)}\ \text{all}$ | \textbf{(E)}\ | 1/8 |
A contest has six problems worth seven points each. On any given problem, a contestant can score either 0, 1, or 7 points. How many possible total scores can a contestant achieve over all six problems? | 28 | 7/8 |
On each side of a unit square, an equilateral triangle of side length 1 is constructed. On each new side of each equilateral triangle, another equilateral triangle of side length 1 is constructed. The interiors of the square and the 12 triangles have no points in common. Let $R$ be the region formed by the union of the square and all the triangles, and $S$ be the smallest convex polygon that contains $R$. What is the area of the region that is inside $S$ but outside $R$? | 1 | 1/8 |
Let $ABC$ be an equilateral triangle and a point M inside the triangle such that $MA^2 = MB^2 +MC^2$ . Draw an equilateral triangle $ACD$ where $D \ne B$ . Let the point $N$ inside $\vartriangle ACD$ such that $AMN$ is an equilateral triangle. Determine $\angle BMC$ .
| 150 | 1/8 |
From an equal quantity of squares with sides 1, 2, and 3, create a square of the smallest possible size. | 14 | 6/8 |
Given a regular polygon with \(2n + 1\) sides. Consider all triangles that have the vertices of this \(2n + 1\)-sided polygon as their vertices. Find the number of these triangles that include the center of the \(2n + 1\)-sided polygon. | \frac{n(n+1)(2n+1)}{6} | 3/8 |
Let the side lengths of a triangle be integers \( l \), \( m \), and \( n \), with \( l > m > n \). It is given that \( \left\{\frac{3^{l}}{10^{4}}\right\} = \left\{\frac{3^{m}}{10^{4}}\right\} = \left\{\frac{3^{n}}{10^{4}}\right\} \), where \(\{x\} = x - [x]\) and \([x]\) denotes the greatest integer less than or equal to \( x \). Find the minimum perimeter of such a triangle. | 3003 | 3/8 |
Suppose \( m > 0 \). If for any set of positive numbers \( a, b, c \) that satisfy \( a b c \leq \frac{1}{4} \) and \( \frac{1}{a^{2}} + \frac{1}{b^{2}} + \frac{1}{c^{2}} < m \), there exists a triangle with side lengths \( a, b, c \). Determine the maximum value of the real number \( m \) and explain the reasoning. | 9 | 5/8 |
It is known that the numbers EGGPLANT and TOAD are divisible by 3. What is the remainder when the number CLAN is divided by 3? (Letters denote digits, identical letters represent identical digits, different letters represent different digits). | 0 | 1/8 |
A necklace consists of 100 red beads and some number of blue beads. It is known that on any segment of the necklace containing 10 red beads, there are at least 7 blue beads. What is the minimum number of blue beads that can be in this necklace? (The beads in the necklace are arranged cyclically, meaning the last bead is adjacent to the first one.) | 78 | 1/8 |
A six-digit number begins with digit 1 and ends with digit 7. If the digit in the units place is decreased by 1 and moved to the first place, the resulting number is five times the original number. Find this number. | 142857 | 1/8 |
Ricsi, Dénes, and Attila often play ping-pong against each other, where two of them team up on one side. Dénes and Attila win against Ricsi three times as often as they lose; Dénes wins against Ricsi and Attila as often as he loses; finally, Attila wins against Ricsi and Dénes twice as often as he loses. Recently, they spent an entire afternoon playing six matches, playing each combination twice. What is the probability that Ricsi won at least once, either alone or with a partner? | \frac{15}{16} | 3/8 |
If $x$ is a real number and $\lceil x \rceil = 11,$ how many possible values are there for $\lceil x^2 \rceil$? | 21 | 7/8 |
In trapezoid \(ABCD\), \(\overrightarrow{AB} = 2 \overrightarrow{DC}\), \(|\overrightarrow{BC}| = 6\). Point \(P\) is a point in the plane of trapezoid \(ABCD\) and satisfies \(\overrightarrow{AP} + \overrightarrow{BP} + 4 \overrightarrow{DP} = 0\). Additionally, \(\overrightarrow{DA} \cdot \overrightarrow{CB} = |\overrightarrow{DA}| \cdot |\overrightarrow{DP}|\). Point \(Q\) is a variable point on side \(AD\). Find the minimum value of \(|\overrightarrow{PQ}|\). | \frac{4 \sqrt{2}}{3} | 4/8 |
A sphere is centered at a point with integer coordinates and passes through the three points $(2,0,0)$, $(0,4,0),(0,0,6)$, but not the origin $(0,0,0)$. If $r$ is the smallest possible radius of the sphere, compute $r^{2}$. | 51 | 7/8 |
An unpainted cone has radius \( 3 \mathrm{~cm} \) and slant height \( 5 \mathrm{~cm} \). The cone is placed in a container of paint. With the cone's circular base resting flat on the bottom of the container, the depth of the paint in the container is \( 2 \mathrm{~cm} \). When the cone is removed, its circular base and the lower portion of its lateral surface are covered in paint. The fraction of the total surface area of the cone that is covered in paint can be written as \( \frac{p}{q} \) where \( p \) and \( q \) are positive integers with no common divisor larger than 1. What is the value of \( p+q \)?
(The lateral surface of a cone is its external surface not including the circular base. A cone with radius \( r \), height \( h \), and slant height \( s \) has lateral surface area equal to \( \pi r s \).) | 59 | 7/8 |
Scientists found a fragment of an ancient mechanics manuscript. It was a piece of a book where the first page was numbered 435, and the last page was numbered with the same digits, but in some different order. How many sheets did this fragment contain? | 50 | 7/8 |
Given a triangle $KLM$. A circle is drawn passing through point $M$, tangent to segment $LK$ at point $A$ (which is the midpoint of $LK$), and intersecting sides $ML$ and $MK$ at points $C$ and $B$, respectively, such that $CB=4$. Point $C$ is equidistant from points $A$ and $L$, and $\cos \angle K = \frac{\sqrt{10}}{4}$. Find the length of segment $BK$. | \sqrt{6} | 1/8 |
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