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Consider the infinite series $1 - \frac{1}{3} - \frac{1}{9} + \frac{1}{27} - \frac{1}{81} - \frac{1}{243} + \frac{1}{729} - \cdots$. Let \( T \) be the sum of this series. Find \( T \). | \frac{15}{26} | 7/8 |
For nonnegative real numbers $x,y,z$ and $t$ we know that $|x-y|+|y-z|+|z-t|+|t-x|=4$ .
Find the minimum of $x^2+y^2+z^2+t^2$ .
*proposed by Mohammadmahdi Yazdi, Mohammad Ahmadi* | 2 | 2/8 |
Let $T$ be a positive integer whose only digits are 0s and 1s. If $X = T \div 18$ and $X$ is an integer, what is the smallest possible value of $X$? | 6172839500 | 3/8 |
In a trapezoid, the lengths of the diagonals are known to be 6 and 8, and the length of the midline is 5. Find the height of the trapezoid. | 4.8 | 2/8 |
Given that $f(x)$ is a function defined on $\mathbb{R}$, and for any $x \in \mathbb{R}$, it holds that $f(x+2) = f(2-x) + 4f(2)$, if the graph of the function $y=f(x+1)$ is symmetric about the point $(-1,0)$ and $f(1)=3$, then find $f(2015)$. | -3 | 7/8 |
One side of a rectangle is $1 \mathrm{~cm}$. It is divided into four smaller rectangles by two perpendicular lines, such that the areas of three of the smaller rectangles are not less than $1 \mathrm{~cm}^{2}$, and the area of the fourth rectangle is not less than $2 \mathrm{~cm}^{2}$. What is the minimum length of the other side of the original rectangle? | 3+2\sqrt{2} | 3/8 |
Find the last two digits of \( 7 \times 19 \times 31 \times \cdots \times 1999 \). (Here \( 7, 19, 31, \ldots, 1999 \) form an arithmetic sequence of common difference 12.) | 75 | 7/8 |
A palindrome is a string that does not change when its characters are written in reverse order. Let S be a 40-digit string consisting only of 0's and 1's, chosen uniformly at random out of all such strings. Let $E$ be the expected number of nonempty contiguous substrings of $S$ which are palindromes. Compute the value of $\lfloor E\rfloor$. | 113 | 4/8 |
Given that the integers \( x \) and \( y \) satisfy \( \frac{1}{x} - \frac{1}{y} = \frac{1}{2023} \), how many such integer pairs \( (x, y) \) exist? | 29 | 2/8 |
Let \( N \) be a positive integer such that \( N+1 \) is a prime number. Consider \( a_{i} \in \{0,1\} \) for \( i=0,1,2, \cdots, N \), and \( a_{i} \) are not all the same. A polynomial \( f(x) \) satisfies \( f(i)=a_{i} \) for \( i=0,1,2, \cdots, N \). Prove that the degree of \( f(x) \) is at least \( N \). | N | 6/8 |
Define a $\it{great\ word}$ as a sequence of letters that consists only of the letters $D$, $E$, $F$, and $G$ --- some of these letters may not appear in the sequence --- and in which $D$ is never immediately followed by $E$, $E$ is never immediately followed by $F$, $F$ is never immediately followed by $G$, and $G$ is never immediately followed by $D$. How many six-letter great words are there? | 972 | 7/8 |
Does there exist a six-digit number \( A \) such that among the numbers \( A, 2A, \ldots, 500000A \), there is not a single number ending in six identical digits? | Yes | 1/8 |
Consider the set of numbers $\{1, 10, 10^2, 10^3, \ldots, 10^{10}\}$. The ratio of the largest element of the set to the sum of the other ten elements of the set is closest to which integer?
$\textbf{(A)}\ 1 \qquad\textbf{(B)}\ 9 \qquad\textbf{(C)}\ 10 \qquad\textbf{(D)}\ 11 \qquad\textbf{(E)} 101$ | \textbf{(B)}~9 | 1/8 |
Given the positive numbers \(a_{1}, b_{1}, c_{1}, a_{2}, b_{2}, c_{2}\) such that \( b_{1}^{2} \leq a_{1} c_{1} \) and \( b_{2}^{2} \leq a_{2} c_{2} \), prove that \[ \left(a_{1}+a_{2}+5\right)\left(c_{1}+c_{2}+2\right) > \left(b_{1}+b_{2}+3\right)^{2}. \] | (a_1+a_2+5)(c_1+c_2+2)>(b_1+b_2+3)^2 | 4/8 |
A choir director must select a group of singers from among his $6$ tenors and $8$ basses. The only requirements are that the difference between the number of tenors and basses must be a multiple of $4$, and the group must have at least one singer. Let $N$ be the number of different groups that could be selected. What is the remainder when $N$ is divided by $100$? | 95 | 7/8 |
From point $A$ to point $B$, three cars depart at equal time intervals. They all arrive at $B$ simultaneously, and then they proceed to point $C$, which is 120 km away from $B$. The first car arrives at point $C$ one hour after the second car. After reaching point $C$, the third car immediately turns back and meets the first car 40 km from $C$. Determine the speed of the first car, assuming that the speed of each car remains constant throughout the entire route. | 30 | 3/8 |
Consider a terminal with fifteen gates arranged in a straight line with exactly $90$ feet between adjacent gates. A passenger's departure gate is assigned at random. Later, the gate is changed to another randomly chosen gate. Calculate the probability that the passenger walks $360$ feet or less to the new gate. Express the probability as a simplified fraction $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, and find $m+n$. | 31 | 7/8 |
How many positive integers divide $5n^{11}-2n^5-3n$ for all positive integers $n$. | 12 | 5/8 |
The angles in a particular triangle are in arithmetic progression, and the side lengths are $4,5,x$. The sum of the possible values of x equals $a+\sqrt{b}+\sqrt{c}$ where $a, b$, and $c$ are positive integers. What is $a+b+c$?
$\textbf{(A)}\ 36\qquad\textbf{(B)}\ 38\qquad\textbf{(C)}\ 40\qquad\textbf{(D)}\ 42\qquad\textbf{(E)}\ 44$ | \textbf{(A)}\36 | 1/8 |
Pauline Bunyan can shovel snow at the rate of $20$ cubic yards for the first hour, $19$ cubic yards for the second, $18$ for the third, etc., always shoveling one cubic yard less per hour than the previous hour. If her driveway is $4$ yards wide, $10$ yards long, and covered with snow $3$ yards deep, then the number of hours it will take her to shovel it clean is closest to | 7 | 7/8 |
Given the ellipse $C$: $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1 (a > b > 0)$ has an eccentricity of $\dfrac{\sqrt{3}}{2}$, and it passes through point $A(2,1)$.
(Ⅰ) Find the equation of ellipse $C$;
(Ⅱ) If $P$, $Q$ are two points on ellipse $C$, and the angle bisector of $\angle PAQ$ always perpendicular to the x-axis, determine whether the slope of line $PQ$ is a constant value? If yes, find the value; if no, explain why. | \dfrac{1}{2} | 1/8 |
Let \( a_{1} = 3 \), and for \( n > 1 \), let \( a_{n} \) be the largest real number such that:
\[ 4\left(a_{n-1}^{2} + a_{n}^{2}\right) = 10 a_{n-1} a_{n} - 9 \]
What is the largest positive integer less than \( a_{8} \)? | 335 | 1/8 |
Circles $\mathcal{C}_1, \mathcal{C}_2,$ and $\mathcal{C}_3$ have their centers at (0,0), (12,0), and (24,0), and have radii 1, 2, and 4, respectively. Line $t_1$ is a common internal tangent to $\mathcal{C}_1$ and $\mathcal{C}_2$ and has a positive slope, and line $t_2$ is a common internal tangent to $\mathcal{C}_2$ and $\mathcal{C}_3$ and has a negative slope. Given that lines $t_1$ and $t_2$ intersect at $(x,y),$ and that $x=p-q\sqrt{r},$ where $p, q,$ and $r$ are positive integers and $r$ is not divisible by the square of any prime, find $p+q+r.$ | 27 | 1/8 |
In triangle \(ABC\), two identical rectangles \(PQRS\) and \(P_1Q_1R_1S_1\) are inscribed (with points \(P\) and \(P_1\) lying on side \(AB\), points \(Q\) and \(Q_1\) lying on side \(BC\), and points \(R, S, R_1,\) and \(S_1\) lying on side \(AC\)). It is known that \(PS = 3\) and \(P_1S_1 = 9\). Find the area of triangle \(ABC\). | 72 | 1/8 |
A ball thrown vertically upwards has its height above the ground expressed as a quadratic function with respect to its time of motion. Xiaohong throws two balls vertically upwards one after the other, with a 1-second interval between them. Assume the initial height above the ground for both balls is the same, and each reaches the same maximum height 1.1 seconds after being thrown. If the first ball's height matches the second ball's height at $t$ seconds after the first ball is thrown, determine $t = \qquad$ . | 1.6 | 6/8 |
Two water particles fall freely in succession from a $300 \mathrm{~m}$ high cliff. The first one has already fallen $\frac{1}{1000} \mathrm{~mm}$ when the second one starts to fall.
How far apart will the two particles be at the moment when the first particle reaches the base of the cliff? (The result should be calculated to the nearest $\frac{1}{10} \mathrm{~mm}$. Air resistance, etc., are not to be considered.) | 34.6 | 1/8 |
The quadrilateral \(ABCD\) is circumscribed around a circle with a radius of \(1\). Find the greatest possible value of \(\left| \frac{1}{AC^2} + \frac{1}{BD^2} \right|\). | 1/4 | 1/8 |
Let $n$ be the answer to this problem. Box $B$ initially contains $n$ balls, and Box $A$ contains half as many balls as Box $B$. After 80 balls are moved from Box $A$ to Box $B$, the ratio of balls in Box $A$ to Box $B$ is now $\frac{p}{q}$, where $p, q$ are positive integers with $\operatorname{gcd}(p, q)=1$. Find $100p+q$. | 720 | 3/8 |
When simplified, the third term in the expansion of $(\frac{a}{\sqrt{x}}-\frac{\sqrt{x}}{a^2})^6$ is:
$\textbf{(A)}\ \frac{15}{x}\qquad \textbf{(B)}\ -\frac{15}{x}\qquad \textbf{(C)}\ -\frac{6x^2}{a^9} \qquad \textbf{(D)}\ \frac{20}{a^3}\qquad \textbf{(E)}\ -\frac{20}{a^3}$ | \textbf{(A)}\\frac{15}{x} | 1/8 |
Find the number of permutations $x_1, x_2, x_3, x_4, x_5$ of numbers $1, 2, 3, 4, 5$ such that the sum of five products $$ x_1x_2x_3 + x_2x_3x_4 + x_3x_4x_5 + x_4x_5x_1 + x_5x_1x_2 $$ is divisible by $3$ . | 80 | 2/8 |
Consider the cubes whose vertices lie on the surface of a given cube. Which one is the smallest among them? | \frac{1}{\sqrt{2}} | 1/8 |
Find all functions \( f: \mathbb{Z} \rightarrow \mathbb{Z} \) such that:
1. \( f(p) > 0 \) for any prime \( p \),
2. \( p \mid (f(x) + f(p))^{f(p)} - x \) for any prime \( p \) and for any \( x \in \mathbb{Z} \). | f(x)=x | 2/8 |
The radius of the circumcircle of the acute-angled triangle \(ABC\) is 1. It is known that on this circumcircle lies the center of another circle passing through the vertices \(A\), \(C\), and the orthocenter of triangle \(ABC\). Find \(AC\). | \sqrt{3} | 1/8 |
The diagram shows a triangle \(ABC\) and two lines \(AD\) and \(BE\), where \(D\) is the midpoint of \(BC\) and \(E\) lies on \(CA\). The lines \(AD\) and \(BE\) meet at \(Z\), the midpoint of \(AD\). What is the ratio of the length \(CE\) to the length \(EA\)? | 2:1 | 3/8 |
Calculate the sum of $5.46$, $2.793$, and $3.1$ as a decimal. | 11.353 | 6/8 |
Two vertical chords are drawn in a circle, dividing the circle into 3 distinct regions. Two horizontal chords are added in such a way that there are now 9 regions in the circle. A fifth chord is added that does not lie on top of one of the previous four chords. The maximum possible number of resulting regions is \( M \) and the minimum possible number of resulting regions is \( m \). What is \( M^{2} + m^{2} \)? | 296 | 4/8 |
An astronomer discovered that the intervals between the appearances of comet $2011 Y$ near the planet $12 I V 1961$ are successive terms of a decreasing geometric progression. The three latest intervals (in years) are the roots of the cubic equation \( t^{3}-c t^{2}+350 t-1000=0 \), where \( c \) is a constant. What will be the duration of the next interval before the comet's appearance? | 2.5 | 4/8 |
Jerry and Silvia wanted to go from the southwest corner of a square field to the northeast corner. Jerry walked due east and then due north to reach the goal, but Silvia headed northeast and reached the goal walking in a straight line. Which of the following is closest to how much shorter Silvia's trip was, compared to Jerry's trip?
$\textbf{(A)}\ 30\%\qquad\textbf{(B)}\ 40\%\qquad\textbf{(C)}\ 50\%\qquad\textbf{(D)}\ 60\%\qquad\textbf{(E)}\ 70\%$ | \textbf{(A)}\30 | 1/8 |
The integer parts of two finite decimals are 7 and 10, respectively. How many possible values are there for the integer part of the product of these two finite decimals? | 18 | 6/8 |
Given that \( f(x) \) is a polynomial with degree greater than 3 and coefficients in the interval \([1,4]\), and \( f(x) \) can be factored into two polynomials \( f(x) = g(x) h(x) \). If \( g(x) \) is a monic polynomial, prove that \( |g(6)| > 3 \). | |(6)|>3 | 2/8 |
Consider rectangle $ABCD$ with $AB = 6$ and $BC = 8$ . Pick points $E, F, G, H$ such that the angles $\angle AEB, \angle BFC, \angle CGD, \angle AHD$ are all right. What is the largest possible area of quadrilateral $EFGH$ ?
*Proposed by Akshar Yeccherla (TopNotchMath)* | 98 | 1/8 |
Given a non-zero sequence \(\{a_n\}\) that satisfies \(a_1 = 1\), \(a_2 = 3\), and \(a_n (a_{n+2} - 1) = a_{n+1} (a_{n+1} - 1)\) for \(n \in \mathbf{N}^*\), find the value of \(\mathrm{C}_{2023}^{0} a_1 + \mathrm{C}_{2023}^{1} a_2 + \mathrm{C}_{2023}^{2} a_3 + \cdots + \mathrm{C}_{2023}^{2023} a_{2024}\) expressed as an integer exponentiation. | 2\cdot3^{2023}-2^{2023} | 2/8 |
Find the minimum positive integer $n\ge 3$ , such that there exist $n$ points $A_1,A_2,\cdots, A_n$ satisfying no three points are collinear and for any $1\le i\le n$ , there exist $1\le j \le n (j\neq i)$ , segment $A_jA_{j+1}$ pass through the midpoint of segment $A_iA_{i+1}$ , where $A_{n+1}=A_1$ | 6 | 1/8 |
Let \( f(x) = ax^2 + bx + c \), where \( a, b, c \in \mathbf{R} \) and \( a > 100 \). Determine the maximum number of integer values \( x \) can take such that \( |f(x)| \leqslant 50 \). | 2 | 1/8 |
What is the largest natural number \( k \) such that there are infinitely many sequences of \( k \) consecutive natural numbers where each number can be expressed as the sum of two squares? (Note: 0 is considered a square number.) | 3 | 1/8 |
In the Cartesian coordinate system $(xOy)$, the parametric equation of line $l$ is given by $\begin{cases}x=1+t\cos a \\ y= \sqrt{3}+t\sin a\end{cases} (t\text{ is a parameter})$, where $0\leqslant α < π$. In the polar coordinate system with $O$ as the pole and the positive half of the $x$-axis as the polar axis, the curve $C_{1}$ is defined by $ρ=4\cos θ$. The line $l$ is tangent to the curve $C_{1}$.
(1) Convert the polar coordinate equation of curve $C_{1}$ to Cartesian coordinates and determine the value of $α$;
(2) Given point $Q(2,0)$, line $l$ intersects with curve $C_{2}$: $x^{2}+\frac{{{y}^{2}}}{3}=1$ at points $A$ and $B$. Calculate the area of triangle $ABQ$. | \frac{6 \sqrt{2}}{5} | 2/8 |
How many 1000-digit positive integers have all digits odd, and are such that any two adjacent digits differ by 2? | 8\times3^{499} | 2/8 |
When the set of natural numbers is listed in ascending order, what is the smallest prime number that occurs after a sequence of six consecutive positive integers, all of which are nonprime? | 37 | 1/8 |
Given that point $P$ is a moving point on the parabola $y=\frac{1}{4}x^2$, determine the minimum value of the sum of the distance from point $P$ to the line $x+2y+4=0$ and the $x$-axis. | \frac{6\sqrt{5}}{5}-1 | 2/8 |
A triangle is partitioned into three triangles and a quadrilateral by drawing two lines from vertices to their opposite sides. The areas of the three triangles are 3, 7, and 7, as shown. What is the area of the shaded quadrilateral? | 18 | 1/8 |
Let $ABC$ be triangle such that $|AB| = 5$ , $|BC| = 9$ and $|AC| = 8$ . The angle bisector of $\widehat{BCA}$ meets $BA$ at $X$ and the angle bisector of $\widehat{CAB}$ meets $BC$ at $Y$ . Let $Z$ be the intersection of lines $XY$ and $AC$ . What is $|AZ|$ ? | 10 | 3/8 |
Find the sum of digits of all the numbers in the sequence $1,2,3,4,\cdots ,10000$. | 180001 | 6/8 |
The equality $(x+m)^2-(x+n)^2=(m-n)^2$, where $m$ and $n$ are unequal non-zero constants, is satisfied by $x=am+bn$, where:
$\textbf{(A)}\ a = 0, b \text{ } \text{has a unique non-zero value}\qquad \\ \textbf{(B)}\ a = 0, b \text{ } \text{has two non-zero values}\qquad \\ \textbf{(C)}\ b = 0, a \text{ } \text{has a unique non-zero value}\qquad \\ \textbf{(D)}\ b = 0, a \text{ } \text{has two non-zero values}\qquad \\ \textbf{(E)}\ a \text{ } \text{and} \text{ } b \text{ } \text{each have a unique non-zero value}$ | \textbf{(A)} | 1/8 |
Consider the multiplication of the two numbers $1,002,000,000,000,000,000$ and $999,999,999,999,999,999$. Calculate the number of digits in the product of these two numbers. | 38 | 1/8 |
Given $\sqrt{20} \approx 4.472, \sqrt{2} \approx 1.414$, find $-\sqrt{0.2} \approx$____. | -0.4472 | 2/8 |
Consider an \(n \times n\) array:
\[
\begin{array}{ccccc}
1 & 2 & 3 & \cdots & n \\
n+1 & n+2 & n+3 & \cdots & 2n \\
2n+1 & 2n+2 & 2n+3 & \cdots & 3n \\
\cdots & \cdots & \cdots & \cdots & \cdots \\
(n-1)n+1 & (n-1)n+2 & (n-1)n+3 & \cdots & n^2
\end{array}
\]
Select \(x_{1}\) from this array and remove the row and column containing \(x_{1}\). Repeat this process to choose \(x_{2}\) from the remaining array, then remove the row and column containing \(x_{2}\). Continue this process until only one number \(x_{n}\) is left. Find the value of \(x_{1} + x_{2} + \cdots + x_{n}\). | \frac{n(n^2+1)}{2} | 4/8 |
The probability of an event occurring in each of 900 independent trials is 0.5. Find the probability that the relative frequency of the event will deviate from its probability by no more than 0.02. | 0.7698 | 1/8 |
In a competition there are \(a\) contestants and \(b\) judges, where \(b \geq 3\) is an odd integer. Each judge rates each contestant as either "pass" or "fail". Suppose \(k\) is a number such that for any two judges their ratings coincide for at most \(k\) contestants. Prove \(\frac{k}{a} \geq \frac{b-1}{2b}\). | \frac{k}{}\ge\frac{1}{2b} | 1/8 |
How many integers are there from 1 to 16500 that
a) are not divisible by 5;
b) are not divisible by either 5 or 3;
c) are not divisible by 5, 3, or 11? | 8000 | 4/8 |
Find the greatest value of the expression \[ \frac{1}{x^2-4x+9}+\frac{1}{y^2-4y+9}+\frac{1}{z^2-4z+9} \] where $x$ , $y$ , $z$ are nonnegative real numbers such that $x+y+z=1$ . | \frac{7}{18} | 6/8 |
How many convex 33-sided polygons can be formed by selecting their vertices from the vertices of a given 100-sided polygon, such that the two polygons do not share any common sides? | \binom{67}{33}+\binom{66}{32} | 1/8 |
Through the edge \( BC \) of the triangular pyramid \( PABC \) and point \( M \), the midpoint of the edge \( PA \), a section \( BCM \) is drawn. The apex of a cone coincides with the apex \( P \) of the pyramid, and the base circle is inscribed in triangle \( BCM \) such that it touches the side \( BC \) at its midpoint. The points of tangency of the circle with segments \( BM \) and \( CM \) are the intersection points of the medians of faces \( APB \) and \( APC \). The height of the cone is twice the radius of the base. Find the ratio of the lateral surface area of the pyramid to the area of the base of the pyramid. | 2 | 1/8 |
To make a rectangular frame with lengths of 3cm, 4cm, and 5cm, the total length of wire needed is cm. If paper is then glued around the outside (seams not considered), the total area of paper needed is cm<sup>2</sup>. | 94 | 2/8 |
Consider the ellipse given by the equation \(\frac{x^{2}}{3}+\frac{y^{2}}{2}=1\). From the right focus \( F \) of the ellipse, draw two perpendicular chords \( AB \) and \( CD \). Let \( M \) and \( N \) be the midpoints of \( AB \) and \( CD \), respectively.
1. Prove that the line \( MN \) always passes through a fixed point and find this fixed point.
2. Given that the slopes of the chords exist, find the maximum area of the triangle \( \triangle FMN \). | \frac{4}{25} | 1/8 |
In a regular octagon, find the ratio of the length of the shortest diagonal to the longest diagonal. Express your answer as a common fraction in simplest radical form. | \frac{\sqrt{2}}{2} | 6/8 |
Six real numbers $x_1<x_2<x_3<x_4<x_5<x_6$ are given. For each triplet of distinct numbers of those six Vitya calculated their sum. It turned out that the $20$ sums are pairwise distinct; denote those sums by $$ s_1<s_2<s_3<\cdots<s_{19}<s_{20}. $$ It is known that $x_2+x_3+x_4=s_{11}$ , $x_2+x_3+x_6=s_{15}$ and $x_1+x_2+x_6=s_{m}$ . Find all possible values of $m$ . | 7 | 1/8 |
Two counterfeit coins of equal weight are mixed with $8$ identical genuine coins. The weight of each of the counterfeit coins is different from the weight of each of the genuine coins. A pair of coins is selected at random without replacement from the $10$ coins. A second pair is selected at random without replacement from the remaining $8$ coins. The combined weight of the first pair is equal to the combined weight of the second pair. What is the probability that all $4$ selected coins are genuine?
$\textbf{(A)}\ \frac{7}{11}\qquad\textbf{(B)}\ \frac{9}{13}\qquad\textbf{(C)}\ \frac{11}{15}\qquad\textbf{(D)}\ \frac{15}{19}\qquad\textbf{(E)}\ \frac{15}{16}$ | \frac{15}{19}\\textbf{(D)} | 1/8 |
Given vectors $a=(\cos α, \sin α)$ and $b=(\cos β, \sin β)$, with $|a-b|= \frac{2 \sqrt{5}}{5}$, find the value of $\cos (α-β)$.
(2) Suppose $α∈(0,\frac{π}{2})$, $β∈(-\frac{π}{2},0)$, and $\cos (\frac{5π}{2}-β) = -\frac{5}{13}$, find the value of $\sin α$. | \frac{33}{65} | 7/8 |
On a circle, \(4n\) points are marked and colored alternately in red and blue. The points of each color are then paired up and connected with segments of the same color as the points (no three segments intersect at a single point). Prove that there are at least \(n\) intersection points of red segments with blue segments. | n | 1/8 |
In square $ABCD$, the length of $AB$ is 4 cm, $AE = AF = 1$, quadrilateral $EFGH$ is a rectangle, and $FG = 2EF$. What is the total area of the shaded region "风笔园"? $\qquad$ square centimeters. | 4\, | 1/8 |
The distance from point P(1, -1) to the line $ax+3y+2a-6=0$ is maximized when the line passing through P is perpendicular to the given line. | 3\sqrt{2} | 6/8 |
If the fractional equation $\frac{3}{{x-4}}+\frac{{x+m}}{{4-x}}=1$ has a root, determine the value of $m$. | -1 | 2/8 |
Let \( A_{1}, A_{2}, A_{3} \) be three points in the plane, and for convenience, let \( A_{4} = A_{1}, A_{5} = A_{2} \). For \( n = 1, 2, \) and 3, suppose that \( B_{n} \) is the midpoint of \( A_{n} A_{n+1} \), and suppose that \( C_{n} \) is the midpoint of \( A_{n} B_{n} \). Suppose that \( A_{n} C_{n+1} \) and \( B_{n} A_{n+2} \) meet at \( D_{n} \), and that \( A_{n} B_{n+1} \) and \( C_{n} A_{n+2} \) meet at \( E_{n} \). Calculate the ratio of the area of triangle \( D_{1} D_{2} D_{3} \) to the area of triangle \( E_{1} E_{2} E_{3} \). | \frac{25}{49} | 6/8 |
The sum of all the zeros of the function \( y = 2(5-x) \sin \pi x - 1 \) in the interval \( 0 \leq x \leq 10 \) is equal to __? | 60 | 1/8 |
The vertices of $\triangle ABC$ are labeled in counter-clockwise order, and its sides have lengths $CA = 2022$ , $AB = 2023$ , and $BC = 2024$ . Rotate $B$ $90^\circ$ counter-clockwise about $A$ to get a point $B'$ . Let $D$ be the orthogonal projection of $B'$ unto line $AC$ , and let $M$ be the midpoint of line segment $BB'$ . Then ray $BM$ intersects the circumcircle of $\triangle CDM$ at a point $N \neq M$ . Compute $MN$ .
*Proposed by Thomas Lam* | 2\sqrt{2} | 1/8 |
Find the largest positive integer \( n \) such that the inequality \(\frac{8}{15} < \frac{n}{n+k} < \frac{7}{13}\) holds for exactly one integer \( k \). (5th American Mathematics Invitational, 1987) | 112 | 1/8 |
The nonzero numbers $x{}$ and $y{}$ satisfy the inequalities $x^{2n}-y^{2n}>x$ and $y^{2n}-x^{2n}>y$ for some natural number $n{}$ . Can the product $xy$ be a negative number?
*Proposed by N. Agakhanov* | No | 6/8 |
A scanning code consists of a $7 \times 7$ grid of squares, with some of its squares colored black and the rest colored white. There must be at least one square of each color in this grid of $49$ squares. A scanning code is called $\textit{symmetric}$ if its look does not change when the entire square is rotated by a multiple of $90 ^{\circ}$ counterclockwise around its center, nor when it is reflected across a line joining opposite corners or a line joining midpoints of opposite sides. What is the total number of possible symmetric scanning codes?
$\textbf{(A)} \text{ 510} \qquad \textbf{(B)} \text{ 1022} \qquad \textbf{(C)} \text{ 8190} \qquad \textbf{(D)} \text{ 8192} \qquad \textbf{(E)} \text{ 65,534}$
| 1022 | 1/8 |
Call a natural number $n$ *good* if for any natural divisor $a$ of $n$ , we have that $a+1$ is also divisor of $n+1$ . Find all good natural numbers.
*S. Berlov* | 1 | 2/8 |
A sphere with radius \( \frac{3}{2} \) has its center at point \( N \). From point \( K \), which is at a distance of \( \frac{3 \sqrt{5}}{2} \) from the center of the sphere, two lines \( K L \) and \( K M \) are drawn, tangent to the sphere at points \( L \) and \( M \) respectively. Find the volume of the pyramid \( K L M N \), given that \( M L = 2 \). | 1 | 3/8 |
Given that vertex $E$ of right $\triangle ABE$ (with $\angle ABE = 90^\circ$) is inside square $ABCD$, and $F$ is the point of intersection of diagonal $BD$ and hypotenuse $AE$, find the area of $\triangle ABF$ where the length of $AB$ is $\sqrt{2}$. | \frac{1}{2} | 1/8 |
At the "Lukomorye" station, tickets are sold for one, five, and twenty trips. All tickets cost an integer number of gold coins. Five single-trip tickets are more expensive than one five-trip ticket, and four five-trip tickets are more expensive than one twenty-trip ticket. It turned out that the cheapest way for 33 warriors to travel was to buy tickets for 35 trips, spending 33 gold coins. How much does a five-trip ticket cost? | 5 | 1/8 |
How many (algebraically) different expressions can we obtain by placing parentheses in a 1 / a 2 / ... / a n ? | 2^{n-2} | 1/8 |
Let $ABCD$ be a convex quadrilateral such that \[\begin{array}{rl} E,F \in [AB],& AE = EF = FB G,H \in [BC],& BG = GH = HC K,L \in [CD],& CK = KL = LD M,N \in [DA],& DM = MN = NA \end{array}\] Let \[[NG] \cap [LE] = \{P\}, [NG]\cap [KF] = \{Q\},\] \[{[}MH] \cap [KF] = \{R\}, [MH]\cap [LE]=\{S\}\]
Prove that [list=a][*] $Area(ABCD) = 9 \cdot Area(PQRS)$ [*] $NP=PQ=QG$ [/list] | NP=PQ=QG | 2/8 |
As shown in the diagram, a cube with a side length of 12 cm is cut once. The cut is made along \( IJ \) and exits through \( LK \), such that \( AI = DL = 4 \) cm, \( JF = KG = 3 \) cm, and the section \( IJKL \) is a rectangle. The total surface area of the two resulting parts of the cube after the cut is \( \quad \) square centimeters. | 1176 | 3/8 |
If Mukesh got 80% on a test which has a total of 50 marks, how many marks did he get?
(A) 40
(B) 62.5
(C) 10
(D) 45
(E) 35 | 40 | 4/8 |
A new window design consists of a rectangle topped with a semi-circle at both ends. The ratio of the length AD of the rectangle to its width AB is 4:3. If AB is 36 inches, calculate the ratio of the area of the rectangle to the combined area of the semicircles. | \frac{16}{3\pi} | 7/8 |
Let $ABC$ be an isosceles right-angled triangle, having the right angle at vertex $C$ . Let us consider the line through $C$ which is parallel to $AB$ and let $D$ be a point on this line such that $AB = BD$ and $D$ is closer to $B$ than to $A$ . Find the angle $\angle CBD$ . | 105 | 5/8 |
In the tetrahedron \(ABCD\), given that \(|BC| = |CD| = |DA|\), \(|BD| = |AC|\), \(|BD| > |BC|\), and the dihedral angle at edge \(AB\) is \(\pi / 3\), find the sum of the other dihedral angles. | \frac{5\pi}{3} | 1/8 |
Given that \( p \) is a prime number, the decimal part of \( \sqrt{p} \) is \( x \). The decimal part of \( \frac{1}{x} \) is \( \frac{\sqrt{p} - 31}{75} \). Find all prime numbers \( p \) that satisfy these conditions. | 2011 | 1/8 |
In a box, there are 6 cards labeled with numbers 1, 2, ..., 6. Now, one card is randomly drawn from the box, and its number is denoted as $a$. After adjusting the cards in the box to keep only those with numbers greater than $a$, a second card is drawn from the box. The probability of drawing an odd-numbered card in the first draw and an even-numbered card in the second draw is __________. | \frac{17}{45} | 7/8 |
Circle \( k_{2} \) touches circle \( k_{1} \) from the inside at point X. Point P lies on neither of the two circles and not on the line through the two circle centers. Point \( N_{1} \) is the point on \( k_{1} \) closest to P, and \( F_{1} \) is the point on \( k_{1} \) farthest from P. Similarly, point \( N_{2} \) is the point on \( k_{2} \) closest to P, and \( F_{2} \) is the point on \( k_{2} \) farthest from P.
Prove that \(\angle N_{1} X N_{2} = \angle F_{1} X F_{2} \). | \angleN_1XN_2=\angleF_1XF_2 | 6/8 |
Calculate the definite integral:
$$
\int_{0}^{\operatorname{arctg}(2 / 3)} \frac{6+\operatorname{tg} x}{9 \sin ^{2} x+4 \cos ^{2} x} \, dx
$$ | \frac{\pi}{4}+\frac{\ln2}{18} | 5/8 |
Given the quadrilateral \(ABCD\), it is known that \(\angle BAC = \angle CAD = 60^\circ\) and \(AB + AD = AC\). It is also known that \(\angle ACD = 23^\circ\). What is the measure, in degrees, of \(\angle ABC\)? | 83 | 3/8 |
In the number $74982.1035$ the value of the place occupied by the digit 9 is how many times as great as the value of the place occupied by the digit 3? | 100,000 | 6/8 |
We selected the vertices, the centroid, and the trisection points of the sides of an equilateral triangle. How many points can we keep among them such that no three of them form an equilateral triangle? | 6 | 1/8 |
Let $\mathcal{R}$ be the region consisting of the set of points in the coordinate plane that satisfy both $|8 - x| + y \le 10$ and $3y - x \ge 15$. When $\mathcal{R}$ is revolved around the line whose equation is $3y - x = 15$, the volume of the resulting solid is $\frac {m\pi}{n\sqrt {p}}$, where $m$, $n$, and $p$ are positive integers, $m$ and $n$ are relatively prime, and $p$ is not divisible by the square of any prime. Find $m + n + p$. | 365 | 7/8 |
What is the minimum number of participants that could have been in the school drama club if fifth-graders constituted more than $25\%$, but less than $35\%$; sixth-graders more than $30\%$, but less than $40\%$; and seventh-graders more than $35\%$, but less than $45\%$ (there were no participants from other grades)? | 11 | 6/8 |
For the NEMO, Kevin needs to compute the product
\[
9 \times 99 \times 999 \times \cdots \times 999999999.
\]
Kevin takes exactly $ab$ seconds to multiply an $a$ -digit integer by a $b$ -digit integer. Compute the minimum number of seconds necessary for Kevin to evaluate the expression together by performing eight such multiplications.
*Proposed by Evan Chen* | 870 | 5/8 |
In a math competition, there are 5 problems, each with a different natural number score. The smaller the problem number, the lower its score (for example, the score for problem 1 is less than the score for problem 2). Xiao Ming solved all the problems correctly. The total score for the first 2 problems is 10 points, and the total score for the last 2 problems is 18 points. How many points did Xiao Ming score in total? | 35 | 6/8 |
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