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The Butterfly Theorem: Given a chord $AB$ of a known circle, let $M$ be the midpoint of $AB$. Chords $CD$ and $EF$ pass through point $M$ and intersect $AB$ at points $L$ and $N$ respectively. Prove that $LM = MN$. | LM=MN | 1/8 |
How many different lines pass through at least two points in this 4-by-4 grid of lattice points? | 20 | 1/8 |
Define $n!!$ to be $n(n-2)(n-4)\cdots 3\cdot 1$ for $n$ odd and $n(n-2)(n-4)\cdots 4\cdot 2$ for $n$ even. When $\sum_{i=1}^{2009} \frac{(2i-1)!!}{(2i)!!}$ is expressed as a fraction in lowest terms, its denominator is $2^ab$ with $b$ odd. Find $\dfrac{ab}{10}$. | 401 | 1/8 |
Farmer Yang has a \(2015 \times 2015\) square grid of corn plants. One day, the plant in the very center of the grid becomes diseased. Every day, every plant adjacent to a diseased plant becomes diseased. After how many days will all of Yang's corn plants be diseased? | 2014 | 2/8 |
In triangle $ABC$, point $P$ lies inside the triangle such that $\angle PAB = \angle PBC = \angle PCA = \varphi$. Show that if the angles of the triangle are $\alpha, \beta$, and $\gamma$, then
$$
\frac{1}{\sin ^{2} \varphi}=\frac{1}{\sin ^{2} \alpha}+\frac{1}{\sin ^{2} \beta}+\frac{1}{\sin ^{2} \gamma}
$$ | \frac{1}{\sin^2\varphi}=\frac{1}{\sin^2\alpha}+\frac{1}{\sin^2\beta}+\frac{1}{\sin^2\gamma} | 2/8 |
In the triangular prism $P-ABC$, the vertex $P$ projects orthogonally onto the circumcenter $O$ of the base $\triangle ABC$, and the midpoint of $PO$ is $M$. Construct a cross-section $\alpha$ through $AM$ parallel to $BC$, and denote $\angle PAM$ as $\theta_1$. The acute dihedral angle between $\alpha$ and the base plane $ABC$ is denoted as $\theta_2$. When $\theta_1$ is maximized, find the value of $\tan \theta_2$. | \frac{\sqrt{2}}{2} | 2/8 |
Compute the number of positive four-digit multiples of 11 whose sum of digits (in base ten) is divisible by 11. | 72 | 7/8 |
Given the function \( f(x) = x^3 + ax^2 + bx + c \) where \( a, b, \) and \( c \) are nonzero integers, if \( f(a) = a^3 \) and \( f(b) = b^3 \), what is the value of \( c \)? | 16 | 7/8 |
Let's call a number \( \mathrm{X} \) "50-supportive" if for any 50 real numbers \( a_{1}, \ldots, a_{50} \) whose sum is an integer, there is at least one number for which \( \left|a_{i} - \frac{1}{2}\right| \geq X \).
Indicate the greatest 50-supportive \( X \), rounded to the nearest hundredth based on standard mathematical rules. | 0.01 | 1/8 |
A group of adventurers is displaying their loot. It is known that exactly 13 adventurers have rubies; exactly 9 have emeralds; exactly 15 have sapphires; exactly 6 have diamonds. Additionally, the following conditions are known:
- If an adventurer has sapphires, then they have either emeralds or diamonds (but not both simultaneously);
- If an adventurer has emeralds, then they have either rubies or sapphires (but not both simultaneously).
What is the minimum number of adventurers that can be in such a group? | 22 | 2/8 |
Given a positive integer $n$ such that $n \geq 2$, find all positive integers $m$ for which the following condition holds for any set of positive real numbers $a_1, a_2, \cdots, a_n$ that satisfy:
$$
a_{1} \cdot a_{2} \cdot \cdots \cdot a_{n}=1
$$
It must follow that the inequality:
$$
a_{1}^{m}+a_{2}^{m}+\cdots+a_{n}^{m} \geq \frac{1}{a_{1}}+\frac{1}{a_{2}}+\cdots+\frac{1}{a_{n}}
$$ | \gen-1 | 1/8 |
The centers of two circles are $41$ inches apart. The smaller circle has a radius of $4$ inches and the larger one has a radius of $5$ inches.
The length of the common internal tangent is:
$\textbf{(A)}\ 41\text{ inches} \qquad \textbf{(B)}\ 39\text{ inches} \qquad \textbf{(C)}\ 39.8\text{ inches} \qquad \textbf{(D)}\ 40.1\text{ inches}\\ \textbf{(E)}\ 40\text{ inches}$ | \textbf{(E)}\40 | 1/8 |
Find the flux of the vector field
$$
\vec{a} = x \vec{i} + y \vec{j} + z \vec{k}
$$
through the part of the surface
$$
x^2 + y^2 = 1
$$
bounded by the planes \( z = 0 \) and \( z = 2 \). (The normal vector is outward to the closed surface formed by these surfaces). | 4\pi | 7/8 |
Given an equilateral triangle $\triangle ABC$. Points $D$ and $E$ are taken on side $BC$ such that $BC = 3DE$. Construct an equilateral triangle $\triangle DEF$ and connect point $A$ to point $F$. Draw $DG$ parallel to $AF$ intersecting side $AB$ at point $G$, and draw $EH$ parallel to $AF$ intersecting side $AC$ at point $H$. Draw perpendiculars $GI \perp AF$ and $HJ \perp AF$. Given that the area of $\triangle BDF$ is $45$ and the area of $\triangle DEF$ is $30$, find the ratio $GI \div HJ$. | 3 | 1/8 |
A lot of snow has fallen, and the kids decided to make snowmen. They rolled 99 snowballs with masses of 1 kg, 2 kg, 3 kg, ..., up to 99 kg. A snowman consists of three snowballs stacked on top of each other, and one snowball can be placed on another if and only if the mass of the first is at least half the mass of the second. What is the maximum number of snowmen that the children will be able to make? | 24 | 1/8 |
In rectangle \(ABCD\), where \(AB = a\) and \(BC = b\) with \(a < b\), let a line through the center \(O\) of the rectangle intersect segments \(BC\) and \(DA\) at points \(E\) and \(F\) respectively. The quadrilateral \(ECDF\) is folded along segment \(EF\) onto the plane of quadrilateral \(BEFA\) such that point \(C\) coincides with point \(A\), resulting in quadrilateral \(EFGA\).
(1) Prove that the area of pentagon \(ABEFG\) is \(\frac{a(3b^2 - a^2)}{4b}\).
(2) If \(a = 1\) and \(b\) is a positive integer, find the minimum area of pentagon \(ABEFG\). | \frac{11}{8} | 6/8 |
A digital clock shows the time 4:56. How many minutes will pass until the clock next shows a time in which all of the digits are consecutive and are in increasing order? | 458 | 6/8 |
In triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively. Given vectors $\overrightarrow{m} = (2\sin B, -\sqrt{3})$ and $\overrightarrow{n} = (\cos 2B, 2\cos^2 B - 1)$ and $\overrightarrow{m} \parallel \overrightarrow{n}$:
(1) Find the measure of acute angle $B$;
(2) If $b = 2$, find the maximum value of the area $S_{\triangle ABC}$ of triangle $ABC$. | \sqrt{3} | 1/8 |
Points $A$, $B$, and $C$ lie in that order along a straight path where the distance from $A$ to $C$ is $1800$ meters. Ina runs twice as fast as Eve, and Paul runs twice as fast as Ina. The three runners start running at the same time with Ina starting at $A$ and running toward $C$, Paul starting at $B$ and running toward $C$, and Eve starting at $C$ and running toward $A$. When Paul meets Eve, he turns around and runs toward $A$. Paul and Ina both arrive at $B$ at the same time. Find the number of meters from $A$ to $B$. | 800 | 7/8 |
Find all functions $f: \mathbb{R}^+ \to \mathbb{R}^+$ such that
$$(z + 1)f(x + y) = f(xf(z) + y) + f(yf(z) + x),$$
for all positive real numbers $x, y, z$. | f(x) = x | 1/8 |
A circle is inscribed in a square, and within this circle, a smaller square is inscribed such that one of its sides coincides with a side of the larger square and two vertices lie on the circle. Calculate the percentage of the area of the larger square that is covered by the smaller square. | 50\% | 1/8 |
Margie's car can go $32$ miles on a gallon of gas, and gas currently costs $$4$ per gallon. How many miles can Margie drive on $\textdollar 20$ worth of gas?
$\textbf{(A) }64\qquad\textbf{(B) }128\qquad\textbf{(C) }160\qquad\textbf{(D) }320\qquad \textbf{(E) }640$ | \textbf{(C)}~160 | 1/8 |
Calculate the sum of the series $\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^{5}}$ with an accuracy of $10^{-3}$. | 0.973 | 2/8 |
(Dick Gibbs) For a given positive integer $k$ find, in terms of $k$, the minimum value of $N$ for which there is a set of $2k+1$ distinct positive integers that has sum greater than $N$ but every subset of size $k$ has sum at most $N/2$. | 2k^3+3k^2+3k | 1/8 |
The non-intersecting diagonals of two adjacent faces of a rectangular parallelepiped are inclined to the base plane at angles $\alpha$ and $\beta$. Find the angle between these diagonals. | \arccos(\sin\alpha\sin\beta) | 4/8 |
Consider 1000 distinct points in the plane such that every set of 4 points contains 2 points that are at a distance of 1 or less. Show that there exists a disk of radius 1 containing 334 points. | 334 | 4/8 |
Square $ABCD$ has side length $30$. Point $P$ lies inside the square so that $AP = 12$ and $BP = 26$. The centroids of $\triangle{ABP}$, $\triangle{BCP}$, $\triangle{CDP}$, and $\triangle{DAP}$ are the vertices of a convex quadrilateral. What is the area of that quadrilateral?
[asy] unitsize(120); pair B = (0, 0), A = (0, 1), D = (1, 1), C = (1, 0), P = (1/4, 2/3); draw(A--B--C--D--cycle); dot(P); defaultpen(fontsize(10pt)); draw(A--P--B); draw(C--P--D); label("$A$", A, W); label("$B$", B, W); label("$C$", C, E); label("$D$", D, E); label("$P$", P, N*1.5+E*0.5); dot(A); dot(B); dot(C); dot(D); [/asy]
$\textbf{(A) }100\sqrt{2}\qquad\textbf{(B) }100\sqrt{3}\qquad\textbf{(C) }200\qquad\textbf{(D) }200\sqrt{2}\qquad\textbf{(E) }200\sqrt{3}$
| 200 | 1/8 |
Let $BD$ be the altitude of triangle $ABC$, and let $E$ be the midpoint of $BC$. Compute the radius of the circumcircle of triangle $BDE$ if $AB = 30 \text{ cm}$, $BC = 26 \text{ cm}$, and $AC = 28 \text{ cm}$. | 16.9\, | 1/8 |
What is the minimum width required for an infinite strip of paper from which any triangle with an area of 1 can be cut? | \sqrt[4]{3} | 4/8 |
A car travels due east at a speed of $\frac{5}{4}$ miles per minute on a straight road. Simultaneously, a circular storm with a 51-mile radius moves south at $\frac{1}{2}$ mile per minute. Initially, the center of the storm is 110 miles due north of the car. Calculate the average of the times, $t_1$ and $t_2$, when the car enters and leaves the storm respectively. | \frac{880}{29} | 1/8 |
Denote by \(\langle x\rangle\) the fractional part of the real number \(x\) (for instance, \(\langle 3.2\rangle = 0.2\)). A positive integer \(N\) is selected randomly from the set \(\{1, 2, 3, \ldots, M\}\), with each integer having the same probability of being picked, and \(\left\langle\frac{87}{303} N\right\rangle\) is calculated. This procedure is repeated \(M\) times and the average value \(A(M)\) is obtained. What is \(\lim_{M \rightarrow \infty} A(M)\)? | \frac{50}{101} | 7/8 |
Find the smallest number $k$ , such that $ \frac{l_a+l_b}{a+b}<k$ for all triangles with sides $a$ and $b$ and bisectors $l_a$ and $l_b$ to them, respectively.
*Proposed by Sava Grodzev, Svetlozar Doichev, Oleg Mushkarov and Nikolai Nikolov* | \frac{4}{3} | 3/8 |
For each integer from 1 through 2019, Tala calculated the product of its digits. Compute the sum of all 2019 of Tala's products. | 184320 | 6/8 |
Let $ (a_n)^{\infty}_{n\equal{}1}$ be a sequence of integers with $ a_{n} < a_{n\plus{}1}, \quad \forall n \geq 1.$ For all quadruple $ (i,j,k,l)$ of indices such that $ 1 \leq i < j \leq k < l$ and $ i \plus{} l \equal{} j \plus{} k$ we have the inequality $ a_{i} \plus{} a_{l} > a_{j} \plus{} a_{k}.$ Determine the least possible value of $ a_{2008}.$ | 2015029 | 4/8 |
Let $\overline{AB}$ be a diameter of circle $\omega$. Extend $\overline{AB}$ through $A$ to $C$. Point $T$ lies on $\omega$ so that line $CT$ is tangent to $\omega$. Point $P$ is the foot of the perpendicular from $A$ to line $CT$. Suppose $\overline{AB} = 18$, and let $m$ denote the maximum possible length of segment $BP$. Find $m^{2}$. | 432 | 5/8 |
A regular tetrahedron $SABC$ of volume $V$ is given. The midpoints $D$ and $E$ are taken on $SA$ and $SB$ respectively and the point $F$ is taken on the edge $SC$ such that $SF: FC = 1: 3$ . Find the volume of the pentahedron $FDEABC$ . | \frac{15}{16}V | 1/8 |
In trapezoid $ABCD$, angles $A$ and $D$ are right angles, $AB=1$, $CD=4$, $AD=5$. A point $M$ is taken on side $AD$ such that $\angle CMD = 2 \angle BMA$.
In what ratio does point $M$ divide side $AD$? | \frac{2}{3} | 2/8 |
Given an isosceles triangle \( \triangle ABC \) with base angles \( \angle ABC = \angle ACB = 50^\circ \), points \( D \) and \( E \) lie on \( BC \) and \( AC \) respectively. Lines \( AD \) and \( BE \) intersect at point \( P \). Given \( \angle ABE = 30^\circ \) and \( \angle BAD = 50^\circ \), find \( \angle BED \). | 40 | 2/8 |
The classrooms at MIT are each identified with a positive integer (with no leading zeroes). One day, as President Reif walks down the Infinite Corridor, he notices that a digit zero on a room sign has fallen off. Let \( N \) be the original number of the room, and let \( M \) be the room number as shown on the sign.
The smallest interval containing all possible values of \( \frac{M}{N} \) can be expressed as \( \left[\frac{a}{b}, \frac{c}{d}\right) \) where \( a, b, c, d \) are positive integers with \( \operatorname{gcd}(a, b) = \operatorname{gcd}(c, d) = 1 \). Compute \( 1000a + 100b + 10c + d \). | 2031 | 1/8 |
Find all real numbers \( x \) that satisfy the equation \(\lg (x+1)=\frac{1}{2} \log _{3} x\). | 9 | 7/8 |
Given that a 1-step requires 4 toothpicks, a 2-step requires 10 toothpicks, a 3-step requires 18 toothpicks, and a 4-step requires 28 toothpicks, determine the number of additional toothpicks needed to build a 6-step staircase. | 26 | 1/8 |
For a positive integer \( n \), let \( \tau(n) \) be the number of positive integer divisors of \( n \). How many integers \( 1 \leq n \leq 50 \) are there such that \( \tau(\tau(n)) \) is odd? | 17 | 6/8 |
One of the mascots for the 2012 Olympic Games is called 'Wenlock' because the town of Wenlock in Shropshire first held the Wenlock Olympian Games in 1850. How many years ago was that?
A) 62
B) 152
C) 158
D) 162
E) 172 | 162 | 1/8 |
Given a sequence $\{a_{n}\}$ where $a_{1}=1$, and ${a}_{n}+(-1)^{n}{a}_{n+1}=1-\frac{n}{2022}$, let $S_{n}$ denote the sum of the first $n$ terms of the sequence $\{a_{n}\}$. Find $S_{2023}$. | 506 | 1/8 |
A competition consists of \( n \) true/false questions. After analyzing the responses of 8 participants, it was found that for any two questions, exactly two participants answered "True, True"; exactly two participants answered "False, False"; exactly two participants answered "True, False"; and exactly two participants answered "False, True". Determine the maximum value of \( n \). | 7 | 1/8 |
Given a number \\(x\\) randomly selected from the interval \\(\left[-\frac{\pi}{4}, \frac{2\pi}{3}\right]\\), find the probability that the function \\(f(x)=3\sin\left(2x- \frac{\pi}{6}\right)\\) is not less than \\(0\\). | \frac{6}{11} | 7/8 |
In duck language, only letters $q$ , $a$ , and $k$ are used. There is no word with two consonants after each other, because the ducks cannot pronounce them. However, all other four-letter words are meaningful in duck language. How many such words are there?
In duck language, too, the letter $a$ is a vowel, while $q$ and $k$ are consonants. | 21 | 2/8 |
In the diagram, \( PQ \) is perpendicular to \( QR \), \( QR \) is perpendicular to \( RS \), and \( RS \) is perpendicular to \( ST \). If \( PQ=4 \), \( QR=8 \), \( RS=8 \), and \( ST=3 \), then the distance from \( P \) to \( T \) is | 13 | 3/8 |
Let $a,$ $b,$ $c$ be nonzero real numbers such that $a + b + c = 0,$ and $ab + ac + bc \neq 0.$ Find all possible values of
\[
\frac{a^7 + b^7 + c^7}{abc (ab + ac + bc)}.
\] | -7 | 3/8 |
In the language of the AU tribe, there are two letters - "a" and "u". Some sequences of these letters are words, and each word has at least one and at most 13 letters. It is known that if any two words are written consecutively, the resulting sequence of letters will not be a word. Find the maximum possible number of words in such a language. | 16256 | 2/8 |
There are 19 weights with values $1, 2, 3, \ldots, 19$ grams: nine iron, nine bronze, and one gold. It is known that the total weight of all the iron weights is 90 grams more than the total weight of the bronze weights. Find the weight of the gold weight. | 10 | 7/8 |
How many pairs \((m, n)\) of non-negative integers are there such that \(m \neq n\) and \(\frac{50688}{m+n}\) is an odd positive power of 2? | 33760 | 4/8 |
Let $A$ and $B$ be the endpoints of a semicircular arc of radius $4$. This arc is divided into nine congruent arcs by eight equally spaced points $C_1$, $C_2$, $\dots$, $C_8$. Draw all chords of the form $\overline{AC_i}$ or $\overline{BC_i}$. Find the product of the lengths of these sixteen chords. | 38654705664 | 6/8 |
A volunteer organizes a spring sports event and wants to form a vibrant and well-trained volunteer team. They plan to randomly select 3 people from 4 male volunteers and 3 female volunteers to serve as the team leader. The probability of having at least one female volunteer as the team leader is ____; given the condition that "at least one male volunteer is selected from the 3 people drawn," the probability of "all 3 people drawn are male volunteers" is ____. | \frac{2}{17} | 7/8 |
When a positive integer $N$ is fed into a machine, the output is a number calculated according to the rule shown below.
For example, starting with an input of $N=7,$ the machine will output $3 \cdot 7 +1 = 22.$ Then if the output is repeatedly inserted into the machine five more times, the final output is $26.$ $7 \to 22 \to 11 \to 34 \to 17 \to 52 \to 26$ When the same $6$-step process is applied to a different starting value of $N,$ the final output is $1.$ What is the sum of all such integers $N?$ $N \to \rule{0.5cm}{0.15mm} \to \rule{0.5cm}{0.15mm} \to \rule{0.5cm}{0.15mm} \to \rule{0.5cm}{0.15mm} \to \rule{0.5cm}{0.15mm} \to 1$ | 83 | 6/8 |
Among the natural numbers from 1 to 1000, there are a total of number 7s. | 300 | 7/8 |
Given an ellipse $C$ with one focus at $F_{1}(2,0)$ and the corresponding directrix $x=8$, and eccentricity $e=\frac{1}{2}$.
$(1)$ Find the equation of the ellipse $C$;
$(2)$ Find the length of the chord cut from the ellipse $C$ by a line passing through the other focus and having a slope of $45^{\circ}$. | \frac{48}{7} | 7/8 |
Jim starts with a positive integer $n$ and creates a sequence of numbers. Each successive number is obtained by subtracting the largest possible integer square less than or equal to the current number until zero is reached. For example, if Jim starts with $n = 55$, then his sequence contains $5$ numbers:
$\begin{array}{ccccc} {}&{}&{}&{}&55\\ 55&-&7^2&=&6\\ 6&-&2^2&=&2\\ 2&-&1^2&=&1\\ 1&-&1^2&=&0\\ \end{array}$
Let $N$ be the smallest number for which Jim’s sequence has $8$ numbers. What is the units digit of $N$? | 3 | 2/8 |
Below is the graph of an ellipse. (Assume that tick marks are placed every $1$ unit along the axes.)
[asy]
size(8cm);
int x, y;
for (y = -4; y <= 2; ++y) {
draw((-1,y)--(7,y),gray(0.7));
}
for (x = -1; x <= 7; ++x) {
draw((x,-4)--(x,2),gray(0.7));
}
draw(shift((3,-1))*xscale(1.5)*shift((-3,1))*Circle((3,-1),2));
draw((-1,0)--(7,0),EndArrow);
draw((0,-4)--(0,2),EndArrow);
//for (int i=-3; i<=1; ++i)
//draw((-0.15,i)--(0.15,i));
//for (int i=0; i<=6; ++i)
//draw((i,0.15)--(i,-0.15));
[/asy]
Compute the coordinates of the focus of the ellipse with the greater $x$-coordinate. | (3+\sqrt{5},-1) | 3/8 |
Let $ABC$ be a right-angled triangle with $\angle ABC=90^\circ$ , and let $D$ be on $AB$ such that $AD=2DB$ . What is the maximum possible value of $\angle ACD$ ? | 30 | 6/8 |
Given the ellipse $\frac{{x}^{2}}{3{{m}^{2}}}+\frac{{{y}^{2}}}{5{{n}^{2}}}=1$ and the hyperbola $\frac{{{x}^{2}}}{2{{m}^{2}}}-\frac{{{y}^{2}}}{3{{n}^{2}}}=1$ share a common focus, find the eccentricity of the hyperbola ( ). | \frac{\sqrt{19}}{4} | 5/8 |
A shooter fires 5 shots in succession, hitting the target with scores of: $9.7$, $9.9$, $10.1$, $10.2$, $10.1$. The variance of this set of data is __________. | 0.032 | 5/8 |
Mrs. Kučerová was on a seven-day vacation, and Káta walked her dog and fed her rabbits during this time. Káta received a large cake and 700 CZK as compensation. After another vacation, this time lasting four days, Káta received the same cake and 340 CZK for the same tasks.
What was the cost of the cake? | 140 | 1/8 |
Given \\(a > 0\\), the function \\(f(x)= \frac {1}{3}x^{3}+ \frac {1-a}{2}x^{2}-ax-a\\).
\\((1)\\) Discuss the monotonicity of \\(f(x)\\);
\\((2)\\) When \\(a=1\\), let the function \\(g(t)\\) represent the difference between the maximum and minimum values of \\(f(x)\\) on the interval \\([t,t+3]\\). Find the minimum value of \\(g(t)\\) on the interval \\([-3,-1]\\). | \frac {4}{3} | 2/8 |
A right square pyramid with base edges of length $8\sqrt{2}$ units each and slant edges of length 10 units each is cut by a plane that is parallel to its base and 3 units above its base. What is the volume, in cubic units, of the new pyramid that is cut off by this plane? [asy]
import three;
size(2.5inch);
currentprojection = orthographic(1/2,-1,1/4);
triple A = (0,0,6);
triple[] base = new triple[4];
base[0] = (-4, -4, 0);
base[1] = (4, -4, 0);
base[2] = (4, 4, 0);
base[3] = (-4, 4, 0);
triple[] mid = new triple[4];
for(int i=0; i < 4; ++i)
mid[i] = (.6*xpart(base[i]) + .4*xpart(A), .6*ypart(base[i]) + .4*ypart(A), .6*zpart(base[i]) + .4*zpart(A));
for(int i=0; i < 4; ++i)
{
draw(A--base[i]);
draw(base[i]--base[(i+1)%4]);
draw(mid[i]--mid[(i+1)%4], dashed);
}
label("$8\sqrt{2}$ units", base[0]--base[1]);
label("10 units", base[0]--A, 2*W);
[/asy] | 32 | 7/8 |
A 6x6x6 cube is formed by assembling 216 unit cubes. Two 1x6 stripes are painted on each of the six faces of the cube parallel to the edges, with one stripe along the top edge and one along the bottom edge of each face. How many of the 216 unit cubes have no paint on them? | 144 | 1/8 |
If $x$ and $y$ are positive integers such that $xy - 8x + 7y = 775$, what is the minimal possible value of $|x - y|$? | 703 | 5/8 |
Fran writes the numbers \(1, 2, 3, \ldots, 20\) on a chalkboard. Then she erases all the numbers by making a series of moves; in each move, she chooses a number \(n\) uniformly at random from the set of all numbers still on the chalkboard, and then erases all of the divisors of \(n\) that are still on the chalkboard (including \(n\) itself). What is the expected number of moves that Fran must make to erase all the numbers? | \frac{131}{10} | 2/8 |
Let $ m,n\in \mathbb{N}^*$ . Find the least $ n$ for which exists $ m$ , such that rectangle $ (3m \plus{} 2)\times(4m \plus{} 3)$ can be covered with $ \dfrac{n(n \plus{} 1)}{2}$ squares, among which exist $ n$ squares of length $ 1$ , $ n \minus{} 1$ of length $ 2$ , $ ...$ , $ 1$ square of length $ n$ . For the found value of $ n$ give the example of covering. | 8 | 4/8 |
The formula for converting a Fahrenheit temperature $F$ to the corresponding Celsius temperature $C$ is $C = \frac{5}{9}(F-32).$ An integer Fahrenheit temperature is converted to Celsius, rounded to the nearest integer, converted back to Fahrenheit, and again rounded to the nearest integer.
For how many integer Fahrenheit temperatures between 32 and 1000 inclusive does the original temperature equal the final temperature? | 539 | 1/8 |
A number $p$ is $perfect$ if the sum of its divisors, except $p$ is $p$. Let $f$ be a function such that:
$f(n)=0$, if n is perfect
$f(n)=0$, if the last digit of n is 4
$f(a.b)=f(a)+f(b)$
Find $f(1998)$ | 0 | 7/8 |
Let \(\mathbb{Q}_{>0}\) be the set of positive rational numbers. Let \(f: \mathbb{Q}_{>0} \rightarrow \mathbb{R}\) be a function satisfying the conditions
\[
\begin{array}{l}
f(x) f(y) \geqslant f(x y), \\
f(x+y) \geqslant f(x)+f(y)
\end{array}
\]
for all \(x, y \in \mathbb{Q}_{>0}\). Given that \(f(a)=a\) for some rational \(a>1\), prove that \(f(x)=x\) for all \(x \in \mathbb{Q}_{>0}\). | f(x)=x | 1/8 |
In the cells of a $100 \times 100$ square, the numbers $1, 2, \ldots, 10000$ were placed, each exactly once, such that numbers differing by 1 are recorded in adjacent cells along the side. After that, the distances between the centers of each two cells, where the numbers in those cells differ exactly by 5000, were calculated. Let $S$ be the minimum of these distances. What is the maximum value that $S$ can take? | 50\sqrt{2} | 1/8 |
Assume that $x_1,x_2,\ldots,x_7$ are real numbers such that \begin{align*} x_1 + 4x_2 + 9x_3 + 16x_4 + 25x_5 + 36x_6 + 49x_7 &= 1, \\ 4x_1 + 9x_2 + 16x_3 + 25x_4 + 36x_5 + 49x_6 + 64x_7 &= 12, \\ 9x_1 + 16x_2 + 25x_3 + 36x_4 + 49x_5 + 64x_6 + 81x_7 &= 123. \end{align*} Find the value of $16x_1+25x_2+36x_3+49x_4+64x_5+81x_6+100x_7$. | 334 | 7/8 |
Given \(0 < a < \sqrt{3} \cos \theta\), \(\theta \in \left[-\frac{\pi}{4}, \frac{\pi}{3}\right]\), find the minimum value of \(f(a, \theta) = \cos^3 \theta + \frac{4}{3a \cos^2 \theta - a^3}\). | \frac{17 \sqrt{2}}{4} | 4/8 |
The sum \( 1 + \frac{1}{2} + \frac{1}{3} + \ldots + \frac{1}{45} \) is expressed as a fraction with the denominator \( 45! = 1 \cdot 2 \cdot \ldots \cdot 45 \). How many zeros (in decimal notation) does the numerator of this fraction end with? | 8 | 3/8 |
The Red Sox play the Yankees in a best-of-seven series that ends as soon as one team wins four games. Suppose that the probability that the Red Sox win Game $n$ is $\frac{n-1}{6}$. What is the probability that the Red Sox will win the series? | 1/2 | 1/8 |
If \( z \in \mathbf{C} \) satisfies \( 3 z^{6} + 2 \mathrm{i} z^{5} - 2 z - 3 \mathrm{i} = 0 \), find \( |z| \). | |z|=1 | 1/8 |
A wooden block is 4 inches long, 4 inches wide, and 1 inch high. The block is painted red on all six sides and then cut into sixteen 1 inch cubes. How many of the cubes each have a total number of red faces that is an even number?
[asy]
size(4cm,4cm);
pair A,B,C,D,E,F,G,a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p,q,r;
A=(0.5,0.1);
B=(0.5,0);
C=(0,0.5);
D=(1,0.5);
E=C+(D-A);
F=C+(B-A);
G=D+(B-A);
draw(A--D--E--C--A--B--G--D);
draw(C--F--B);
a=(3/4)*F+(1/4)*B;
b=(1/2)*F+(1/2)*B;
c=(1/4)*F+(3/4)*B;
m=(3/4)*C+(1/4)*A;
n=(1/2)*C+(1/2)*A;
o=(1/4)*C+(3/4)*A;
j=(3/4)*E+(1/4)*D;
k=(1/2)*E+(1/2)*D;
l=(1/4)*E+(3/4)*D;
draw(a--m--j);
draw(b--n--k);
draw(c--o--l);
f=(3/4)*G+(1/4)*B;
e=(1/2)*G+(1/2)*B;
d=(1/4)*G+(3/4)*B;
r=(3/4)*D+(1/4)*A;
q=(1/2)*D+(1/2)*A;
p=(1/4)*D+(3/4)*A;
i=(3/4)*E+(1/4)*C;
h=(1/2)*E+(1/2)*C;
g=(1/4)*E+(3/4)*C;
draw(d--p--g);
draw(e--q--h);
draw(f--r--i);
[/asy] | 8 | 3/8 |
Find all functions \( f: \mathbb{R} \rightarrow \mathbb{R} \) such that for all real numbers \( x \) and \( y \), the following equation holds:
\[ f\left(f(y)+x^{2}+1\right)+2x = y + f^{2}(x+1). \] | f(x)=x | 3/8 |
The hypotenuse of a right triangle whose legs are consecutive even numbers is 34 units. What is the sum of the lengths of the two legs? | 46 | 1/8 |
Jessica is tasked with placing four identical, dotless dominoes on a 4 by 5 grid to form a continuous path from the upper left-hand corner \(C\) to the lower right-hand corner \(D\). The dominoes are shaded 1 by 2 rectangles that must touch each other at their sides, not just at the corners, and cannot be placed diagonally. Each domino covers exactly two of the unit squares on the grid. Determine how many distinct arrangements are possible for Jessica to achieve this, assuming the path only moves right or down. | 35 | 1/8 |
An ant lies on each corner of a $20 \times 23$ rectangle. Each second, each ant independently and randomly chooses to move one unit vertically or horizontally away from its corner. After $10$ seconds, find the expected area of the convex quadrilateral whose vertices are the positions of the ants. | 130 | 4/8 |
The vertices of a regular nonagon (9-sided polygon) are to be labeled with the digits 1 through 9 in such a way that the sum of the numbers on every three consecutive vertices is a multiple of 3. Two acceptable arrangements are considered to be indistinguishable if one can be obtained from the other by rotating the nonagon in the plane. Find the number of distinguishable acceptable arrangements. | 144 | 2/8 |
What minimum number of colours is sufficient to colour all positive real numbers so that every two numbers whose ratio is 4 or 8 have different colours? | 3 | 1/8 |
Find the number of triples $(x,y,z)$ of real numbers that satisfy
\begin{align*}
x &= 2018 - 2019 \operatorname{sign}(y + z), \\
y &= 2018 - 2019 \operatorname{sign}(x + z), \\
z &= 2018 - 2019 \operatorname{sign}(x + y).
\end{align*}Note: For a real number $a,$
\[\operatorname{sign} (a) = \left\{
\begin{array}{cl}
1 & \text{if $a > 0$}, \\
0 & \text{if $a = 0$}, \\
-1 & \text{if $a < 0$}.
\end{array}
\right.\] | 3 | 4/8 |
Given an equilateral triangle \( \triangle ABC \) with side length 1, there are \( n \) equally spaced points on side \( BC \) named \( P_1, P_2, \dots, P_{n-1} \) in the direction from \( B \) to \( C \). Let \( S_n = \overrightarrow{AB} \cdot \overrightarrow{AP_1} + \overrightarrow{AP_1} \cdot \overrightarrow{AP_2} + \cdots + \overrightarrow{AP_{n-1}} \cdot \overrightarrow{AC} \). Prove that \( S_n = \frac{5n^2 - 2}{6n} \). | S_{n}=\frac{5n^{2}-2}{6n} | 1/8 |
The diagonal of a quadrilateral is a bisector of only one of its angles. Prove that the square of this diagonal is equal to
$$
a b+\frac{a c^{2}-b d^{2}}{a-b}
$$
where \( a, b, c, d \) are the lengths of the sides of the quadrilateral, and the diagonal converges with the sides of lengths \( a \) and \( b \) at one vertex. | ^2=+\frac{ac^2-bd^2}{b} | 1/8 |
If the square roots of a positive number are $x+1$ and $4-2x$, then the positive number is ______. | 36 | 7/8 |
Roll a die twice. Let $X$ be the maximum of the two numbers rolled. Which of the following numbers is closest to the expected value $E(X)$? | 4.5 | 2/8 |
Define the sequence \(a_{1}, a_{2}, \ldots\) as follows: \(a_{1}=1\) and for every \(n \geq 2\),
\[
a_{n}=\left\{
\begin{array}{ll}
n-2 & \text{if } a_{n-1}=0 \\
a_{n-1}-1 & \text{if } a_{n-1} \neq 0
\end{array}
\right.
\]
A non-negative integer \(d\) is said to be jet-lagged if there are non-negative integers \(r, s\) and a positive integer \(n\) such that \(d=r+s\) and \(a_{n+r}=a_{n}+s\). How many integers in \(\{1,2, \ldots, 2016\}\) are jet-lagged? | 51 | 1/8 |
Determine the number of three-element subsets of the set \(\{1, 2, 3, 4, \ldots, 120\}\) for which the sum of the three elements is a multiple of 3. | 93640 | 3/8 |
Let $p(x)$ be the product of all digits of the decimal integer $x$. Find all positive integers $x$ such that $p(x) = x^2 - 10x - 22$. | 12 | 5/8 |
Let \( A_{1} A_{2} A_{3} A_{4} \), \( B_{1} B_{2} B_{3} B_{4} \), and \( C_{1} C_{2} C_{3} C_{4} \) be three regular tetrahedra in 3-dimensional space, no two of which are congruent. Suppose that, for each \( i \in \{1, 2, 3, 4\} \), \( C_{i} \) is the midpoint of the line segment \( A_{i} B_{i} \). Determine whether the four lines \( A_{1} B_{1} \), \( A_{2} B_{2} \), \( A_{3} B_{3} \), and \( A_{4} B_{4} \) must concur. | Yes | 2/8 |
Given that Chelsea is ahead by 60 points halfway through a 120-shot archery contest, with each shot scoring 10, 8, 5, 3, or 0 points and Chelsea scoring at least 5 points on every shot, determine the smallest number of bullseyes (10 points) Chelsea needs to shoot in her next n attempts to ensure victory, assuming her opponent can score a maximum of 10 points on each remaining shot. | 49 | 3/8 |
In triangle \( \triangle ABC \), the sides opposite to angles \( \angle A \), \( \angle B \), and \( \angle C \) are \( a \), \( b \), and \( c \) respectively. Given that:
\[ a^2 + 2(b^2 + c^2) = 2\sqrt{2} \]
find the maximum value of the area of triangle \( \triangle ABC \). | \frac{1}{4} | 5/8 |
In triangle \(ABC\), the angle bisector \(AL\) (where \(L \in BC\)) is drawn. Points \(M\) and \(N\) lie on the other two angle bisectors (or their extensions) such that \(MA = ML\) and \(NA = NL\). Given that \(\angle BAC = 50^\circ\).
Find the measure of \(\angle MAN\) in degrees. | 65 | 1/8 |
If \( n \) is a positive integer, we denote by \( n! \) the product of all integers from 1 to \( n \). For example, \( 5! = 1 \times 2 \times 3 \times 4 \times 5 = 120 \) and \( 13! = 1 \times 2 \times 3 \times 4 \times 5 \times \cdots \times 12 \times 13 \). By convention, we write \( 0! = 1! = 1 \). Find three distinct integers \( a \), \( b \), and \( c \) between 0 and 9 such that the three-digit number \( abc \) is equal to \( a! + b! + c! \). | 1,4,5 | 3/8 |
Given that point \\(A\\) on the terminal side of angle \\(\alpha\\) has coordinates \\(\left( \sqrt{3}, -1\right)\\),
\\((1)\\) Find the set of angle \\(\alpha\\)
\\((2)\\) Simplify the following expression and find its value: \\( \dfrac{\sin (2\pi-\alpha)\tan (\pi+\alpha)\cot (-\alpha-\pi)}{\csc (-\alpha)\cos (\pi-\alpha)\tan (3\pi-\alpha)} \\) | \dfrac{1}{2} | 7/8 |
Let \( A B C \) be a right triangle with \(\angle BAC = 90^{\circ}\) and \(I\) the intersection point of its angle bisectors. A line through \(I\) intersects the sides \(AB\) and \(AC\) at \(P\) and \(Q\) respectively. The distance from \(I\) to the side \(BC\) is \(1 \, \text{cm}\).
a) Find the value of \( PM \cdot NQ \).
b) Determine the minimum possible value for the area of triangle \(APQ\).
Hint: If \(x\) and \(y\) are two non-negative real numbers, then \(x + y \geq 2 \sqrt{xy}\). | 2 | 6/8 |
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