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In triangle $ABC$, the angle bisectors $AA'$, $BB'$, and $CC'$ are drawn. Find the angle $B'A'C'$ if the angle $BAC$ is $120^\circ$. | 90 | 2/8 |
Sector $OAB$ is a quarter of a circle with a radius of 6 cm. A circle is inscribed within this sector, tangent to both the radius lines $OA$ and $OB$, and the arc $AB$. Determine the radius of the inscribed circle in centimeters. Express your answer in simplest radical form. | 6\sqrt{2} - 6 | 2/8 |
Reimu and Sanae play a game using 4 fair coins. Initially both sides of each coin are white. Starting with Reimu, they take turns to color one of the white sides either red or green. After all sides are colored, the 4 coins are tossed. If there are more red sides showing up, then Reimu wins, and if there are more green sides showing up, then Sanae wins. However, if there is an equal number of red sides and green sides, then neither of them wins. Given that both of them play optimally to maximize the probability of winning, what is the probability that Reimu wins? | \frac{5}{16} | 1/8 |
Three positive reals \( x \), \( y \), and \( z \) are such that
\[
\begin{array}{l}
x^{2}+2(y-1)(z-1)=85 \\
y^{2}+2(z-1)(x-1)=84 \\
z^{2}+2(x-1)(y-1)=89
\end{array}
\]
Compute \( x + y + z \). | 18 | 7/8 |
How many integers $n>1$ are there such that $n$ divides $x^{13}-x$ for every positive integer $x$? | 31 | 7/8 |
Given real numbers \( a \), \( b \), \( x \), \( y \) that satisfy the equations \( a x + b y = 3 \), \( a x^{2} + b y^{2} = 7 \), \( a x^{3} + b y^{3} = 16 \), and \( a x^{4} + b y^{4} = 42 \), find the value of \( a x^{5} + b y^{5} \). | 20 | 6/8 |
The standard enthalpy of formation (ΔH_f°) of a substance is equal to the heat effect of the formation reaction of 1 mole of the substance from simple substances in their standard states (at 1 atm pressure and a given temperature). Therefore, it is necessary to find the heat effect of the reaction:
$$
\underset{\text {graphite}}{6 \mathrm{C}(\kappa)} + 3 \mathrm{H}_{2}(g) = \mathrm{C}_{6} \mathrm{H}_{6} (l) + \mathrm{Q}_{\text{formation}}\left(\mathrm{C}_{6} \mathrm{H}_{6}(l)\right) (4)
$$
According to Hess's law, the heat effect of the reaction depends only on the types and states of the reactants and products and does not depend on the path of the transition.
Hess's law allows dealing with thermochemical equations like algebraic expressions, i.e., based on it, by combining equations of reactions with known heat effects, one can calculate the unknown heat effect of the overall reaction.
Thus, we obtain:
$$
\mathrm{C}_{2}\mathrm{H}_{2}(g) = \underset{\text {graphite}}{2 \mathrm{C}(\kappa)} + \mathrm{H}_{2}(g) + 226.7 \text { kJ; } \quad -3
$$
$$
\begin{array}{lll}
3 \mathrm{C}_{2} \mathrm{H}_{2}(g) & = \mathrm{C}_{6} \mathrm{H}_{6}(l) + 631.1 \text { kJ;} & 1 \\
\mathrm{C}_{6} \mathrm{H}_{6}(l) & = \mathrm{C}_{6} \mathrm{H}_{6}(l) - 33.9 \text { kJ;} & 1
\end{array}
$$
$$
\underset{\text {graphite}}{6 \mathrm{C}(\kappa)} + 3 \mathrm{H}_{2}(g) = \mathrm{C}_{6} \mathrm{H}_{6}(l) + \mathrm{Q}_{\text{formation}}\left(\mathrm{C}_{6} \mathrm{H}_{6}(l)\right);
$$
$$
\mathrm{Q}_{\text{formation}}\left(\mathrm{C}_{6} \mathrm{H}_{6}(l)\right) = 226.7 \cdot (-3) + 631.1 - 33.9 = -82.9 \text{ kJ/mol}.
$$ | -82.9 | 1/8 |
On the $x O y$ coordinate plane, there is a Chinese chess "knight" at the origin $(0,0)$. The "knight" needs to be moved to the point $P(1991,1991)$ using the movement rules of the chess piece. Calculate the minimum number of moves required. | 1328 | 2/8 |
Let $k > 1, n > 2018$ be positive integers, and let $n$ be odd. The nonzero rational numbers $x_1,x_2,\ldots,x_n$ are not all equal and satisfy $$ x_1+\frac{k}{x_2}=x_2+\frac{k}{x_3}=x_3+\frac{k}{x_4}=\ldots=x_{n-1}+\frac{k}{x_n}=x_n+\frac{k}{x_1} $$ Find:
a) the product $x_1 x_2 \ldots x_n$ as a function of $k$ and $n$ b) the least value of $k$ , such that there exist $n,x_1,x_2,\ldots,x_n$ satisfying the given conditions. | 4 | 1/8 |
Let's call the distance between numbers the absolute value of their difference. It is known that the sum of the distances from sixteen consecutive natural numbers to a certain number \(a\) is 276, and the sum of the distances from these same sixteen numbers to a certain number \(b\) is 748. Find all possible values of \(a\) if it is known that \(a + b = 62.5\). | -0.75 | 1/8 |
On the surface of a spherical planet, there are four continents separated from each other by an ocean.
A point in the ocean is called special if there are at least three closest land points to it, and all of them are on different continents. What is the maximum number of such special points that can exist on this planet? | 4 | 3/8 |
In any permutation of the numbers \(1, 2, 3, \ldots, 18\), we can always find a set of 6 consecutive numbers whose sum is at least \(m\). Find the maximum value of the real number \(m\). | 57 | 3/8 |
For a real number $x,$ let $\lfloor x\rfloor$ denote the greatest integer less than or equal to $x,$ and let $\{x\} = x -\lfloor x\rfloor$ denote the fractional part of $x.$ The sum of all real numbers $\alpha$ that satisfy the equation $$ \alpha^2+\{\alpha\}=21 $$ can be expressed in the form $$ \frac{\sqrt{a}-\sqrt{b}}{c}-d $$ where $a, b, c,$ and $d$ are positive integers, and $a$ and $b$ are not divisible by the square of any prime. Compute $a + b + c + d.$ | 169 | 7/8 |
A positive integer $N$ is a palindrome if the integer obtained by reversing the sequence of digits of $N$ is equal to $N$. The year 1991 is the only year in the current century with the following 2 properties:
(a) It is a palindrome
(b) It factors as a product of a 2-digit prime palindrome and a 3-digit prime palindrome.
How many years in the millenium between 1000 and 2000 have properties (a) and (b)?
$\text{(A) } 1\quad \text{(B) } 2\quad \text{(C) } 3\quad \text{(D) } 4\quad \text{(E) } 5$ | \textbf{(D)}4 | 1/8 |
A number is called flippy if its digits alternate between two distinct digits. For example, $2020$ and $37373$ are flippy, but $3883$ and $123123$ are not. How many five-digit flippy numbers are divisible by $15?$ | 4 | 7/8 |
A computer network is formed by connecting $2004$ computers by cables. A set $S$ of these computers is said to be independent if no pair of computers of $S$ is connected by a cable. Suppose that the number of cables used is the minimum number possible such that the size of any independent set is at most $50$ . Let $c(L)$ be the number of cables connected to computer $L$ . Show that for any distinct computers $A$ and $B$ , $c(A)=c(B)$ if they are connected by a cable and $|c(A)-c(B)| \le 1$ otherwise. Also, find the number of cables used in the network. | 39160 | 5/8 |
Let $n$ be a positive integer. A child builds a wall along a line with $n$ identical cubes. He lays the first cube on the line and at each subsequent step, he lays the next cube either on the ground or on the top of another cube, so that it has a common face with the previous one. How many such distinct walls exist? | 2^{n-1} | 6/8 |
A thin diverging lens with an optical power of $D_{p} = -6$ diopters is illuminated by a beam of light with a diameter $d_{1} = 10$ cm. On a screen positioned parallel to the lens, a light spot with a diameter $d_{2} = 20$ cm is observed. After replacing the thin diverging lens with a thin converging lens, the size of the spot on the screen remains unchanged. Determine the optical power $D_{c}$ of the converging lens. | 18 | 1/8 |
$P$ is a point inside triangle $A B C$, and lines $A P, B P, C P$ intersect the opposite sides $B C, C A, A B$ in points $D, E, F$, respectively. It is given that $\angle A P B=90^{\circ}$, and that $A C=B C$ and $A B=B D$. We also know that $B F=1$, and that $B C=999$. Find $A F$. | 499 / 500 | 1/8 |
Given that $A$, $B$, and $C$ are the three interior angles of $\triangle ABC$, $a$, $b$, and $c$ are the three sides, $a=2$, and $\cos C=-\frac{1}{4}$.
$(1)$ If $\sin A=2\sin B$, find $b$ and $c$;
$(2)$ If $\cos (A-\frac{π}{4})=\frac{4}{5}$, find $c$. | \frac{5\sqrt{30}}{2} | 7/8 |
Find the number of triples of natural numbers \((a, b, c)\) that satisfy the system of equations
$$
\left\{\begin{array}{l}
\gcd(a, b, c)=15 \\
\text{lcm}(a, b, c)=3^{15} \cdot 5^{18}
\end{array}\right.
$$ | 8568 | 7/8 |
In a quiz at the SFU Museum of Entertaining Sciences, 10 schoolchildren are participating. In each round, the participants are divided into pairs. Each participant meets every other participant exactly once. A victory in a duel earns 1 point, a draw 0.5 points, and a defeat 0 points. What is the minimum number of rounds required for an early winner to emerge? | 7 | 1/8 |
Points $X$ and $Y$ are the midpoints of arcs $AB$ and $BC$ of the circumscribed circle of triangle $ABC$ . Point $T$ lies on side $AC$ . It turned out that the bisectors of the angles $ATB$ and $BTC$ pass through points $X$ and $Y$ respectively. What angle $B$ can be in triangle $ABC$ ? | 90 | 4/8 |
In a $12\times 12$ square table some stones are placed in the cells with at most one stone per cell. If the number of stones on each line, column, and diagonal is even, what is the maximum number of the stones?**Note**. Each diagonal is parallel to one of two main diagonals of the table and consists of $1,2\ldots,11$ or $12$ cells. | 120 | 2/8 |
Find the smallest positive real number $k$ such that the following inequality holds $$ \left|z_{1}+\ldots+z_{n}\right| \geqslant \frac{1}{k}\big(\left|z_{1}\right|+\ldots+\left|z_{n}\right|\big) . $$ for every positive integer $n \geqslant 2$ and every choice $z_{1}, \ldots, z_{n}$ of complex numbers with non-negative real and imaginary parts.
[Hint: First find $k$ that works for $n=2$ . Then show that the same $k$ works for any $n \geqslant 2$ .] | \sqrt{2} | 3/8 |
William is biking from his home to his school and back, using the same route. When he travels to school, there is an initial $20^\circ$ incline for $0.5$ kilometers, a flat area for $2$ kilometers, and a $20^\circ$ decline for $1$ kilometer. If William travels at $8$ kilometers per hour during uphill $20^\circ$ sections, $16$ kilometers per hours during flat sections, and $20$ kilometers per hour during downhill $20^\circ$ sections, find the closest integer to the number of minutes it take William to get to school and back.
*Proposed by William Yue* | 29 | 3/8 |
The bases of a trapezoid are 3 cm and 5 cm. One of the diagonals of the trapezoid is 8 cm, and the angle between the diagonals is $60^{\circ}$. Find the perimeter of the trapezoid. | 22 | 6/8 |
Given \( f(x)=a \sin ((x+1) \pi)+b \sqrt[3]{x-1}+2 \), where \( a \) and \( b \) are real numbers and \( f(\lg 5) = 5 \), find \( f(\lg 20) \). | -1 | 7/8 |
Given the ellipse $\Gamma: \frac{x^{2}}{2}+y^{2}=1$ with the left focal point $F$, a tangent to the ellipse $\Gamma$ is drawn through an external point $P$ intersecting the ellipse at points $M$ and $N$. If $\angle M F N=60^{\circ}$, find the equation of the locus of point $P$. | \frac{(x-1)^2}{6}+\frac{y^2}{2}=1 | 1/8 |
A math competition has 10 questions. Each correct answer earns 5 points, while each incorrect or unanswered question results in a deduction of 2 points. A and B both participated in the competition and their total combined score is 58 points. A scored 14 points more than B. How many questions did A answer correctly? | 8 | 6/8 |
Given a moving circle $M$ that passes through the fixed point $F(0,-1)$ and is tangent to the line $y=1$. The trajectory of the circle's center $M$ forms a curve $C$. Let $P$ be a point on the line $l$: $x-y+2=0$. Draw two tangent lines $PA$ and $PB$ from point $P$ to the curve $C$, where $A$ and $B$ are the tangent points.
(I) Find the equation of the curve $C$;
(II) When point $P(x_{0},y_{0})$ is a fixed point on line $l$, find the equation of line $AB$;
(III) When point $P$ moves along line $l$, find the minimum value of $|AF|⋅|BF|$. | \frac{9}{2} | 4/8 |
A section of maximum area is made through the vertex of a right circular cone. It is known that the area of this section is twice the area of the axial (vertical) section. Find the angle at the vertex of the axial section of the cone. | \frac{5\pi}{6} | 1/8 |
In a convex quadrilateral \(ABCD\), it is known that \(\angle ACB = 25^\circ\), \(\angle ACD = 40^\circ\), and \(\angle BAD = 115^\circ\). Find the angle \(\angle ADB\). | 25 | 4/8 |
Find the total number of times the digit ‘ $2$ ’ appears in the set of integers $\{1,2,..,1000\}$ . For example, the digit ’ $2$ ’ appears twice in the integer $229$ . | 300 | 7/8 |
In the plane Cartesian coordinate system \(xOy\), the set of points
$$
\begin{aligned}
K= & \{(x, y) \mid(|x|+|3 y|-6) \cdot \\
& (|3 x|+|y|-6) \leqslant 0\}
\end{aligned}
$$
corresponds to an area in the plane with the measurement of ______. | 24 | 2/8 |
In how many different ways can 3 men and 4 women be placed into two groups of two people and one group of three people if there must be at least one man and one woman in each group? Note that identically sized groups are indistinguishable. | 36 | 2/8 |
Rudolph bikes at a [constant](https://artofproblemsolving.com/wiki/index.php/Constant) rate and stops for a five-minute break at the end of every mile. Jennifer bikes at a constant rate which is three-quarters the rate that Rudolph bikes, but Jennifer takes a five-minute break at the end of every two miles. Jennifer and Rudolph begin biking at the same time and arrive at the $50$-mile mark at exactly the same time. How many minutes has it taken them? | 620 | 7/8 |
Given a circle $C: (x-3)^2 + (y-4)^2 = 25$, the shortest distance from a point on circle $C$ to line $l: 3x + 4y + m = 0 (m < 0)$ is $1$. If point $N(a, b)$ is located on the part of line $l$ in the first quadrant, find the minimum value of $\frac{1}{a} + \frac{1}{b}$. | \frac{7 + 4\sqrt{3}}{55} | 7/8 |
Find all positive integer solutions to the equation \(p^{m} + p^{n} + 1 = k^{2}\), where \(p\) is an odd prime, and \(n \leq m \leq 3n\). | Nosolutions | 1/8 |
Find all values of the parameter \(a\), for each of which the set of solutions to the inequality \(\frac{x^{2}+(a+1) x+a}{x^{2}+5 x+4} \geq 0\) is the union of three disjoint intervals. In the answer, specify the sum of the three smallest integer values of \(a\) from the resulting interval. | 9 | 2/8 |
Given an equilateral triangle \( \triangle ABC \) with side length 4, points \( D \), \( E \), and \( F \) are on \( BC \), \( CA \), and \( AB \) respectively, and \( |AE| = |BF| = |CD| = 1 \). The lines \( AD \), \( BE \), and \( CF \) intersect pairwise forming \( \triangle RQS \). Point \( P \) moves inside \( \triangle PQR \) and along its boundary. Let \( x \), \( y \), and \( z \) be the distances from \( P \) to the three sides of \( \triangle ABC \).
(1) Prove that when \( P \) is at one of the vertices of \( \triangle RQS \), the product \( xyz \) reaches a minimum value.
(2) Determine the minimum value of \( xyz \). | \frac{648}{2197}\sqrt{3} | 1/8 |
Find the smallest positive number \( c \) with the following property: For any integer \( n \geqslant 4 \) and any set \( A \subseteq \{1, 2, \ldots, n\} \), if \( |A| > c n \), then there exists a function \( f: A \rightarrow \{1, -1\} \) such that \( \left|\sum_{a \in A} f(a) \cdot a\right| \leq 1 \). | 2/3 | 1/8 |
In a deck of 52 cards, each player makes one cut. A cut consists of taking the top $N$ cards and placing them at the bottom of the deck without changing their order.
- First, Andrey cut 28 cards,
- then Boris cut 31 cards,
- then Vanya cut 2 cards,
- then Gena cut an unknown number of cards,
- then Dima cut 21 cards.
The last cut restored the original order. How many cards did Gena cut? | 22 | 7/8 |
(1) Given that $\tan \alpha = -2$, calculate the value of $\dfrac {3\sin \alpha + 2\cos \alpha}{5\cos \alpha - \sin \alpha}$.
(2) Given that $\sin \alpha = \dfrac {2\sqrt{5}}{5}$, calculate the value of $\tan (\alpha + \pi) + \dfrac {\sin \left( \dfrac {5\pi}{2} + \alpha \right)}{\cos \left( \dfrac {5\pi}{2} - \alpha \right)}$. | -\dfrac{5}{2} | 2/8 |
Divide the set \( M = \{1, 2, \ldots, 12\} \) of the first 12 positive integers into four subsets each containing three elements, such that in each subset, one number is the sum of the other two. Find the number of different ways to do this. | 8 | 1/8 |
Let \( h \) be a semicircle with diameter \( AB \). A point \( P \) is chosen arbitrarily within the segment \( AB \). The line perpendicular to \( AB \) through \( P \) intersects \( h \) at point \( C \). The segment \( PC \) divides the semicircular area into two parts. In each part, a circle is inscribed that touches \( AB \), \( PC \), and \( h \). The points of tangency of the two circles with \( AB \) are denoted by \( D \) and \( E \), where \( D \) lies between \( A \) and \( P \).
Prove that the measure of the angle \( \angle DCE \) does not depend on the choice of \( P \).
(Walther Janous) | 45 | 2/8 |
$\mathbb{N}$ is the set of positive integers and $a\in\mathbb{N}$ . We know that for every $n\in\mathbb{N}$ , $4(a^n+1)$ is a perfect cube. Prove that $a=1$ . | 1 | 5/8 |
Let \(A\) and \(B\) be plane point sets defined as follows:
$$
\begin{array}{l}
A=\left\{(x, y) \left\lvert\,(y-x)\left(y-\frac{18}{25 x}\right) \geqslant 0\right.\right\}, \\
B=\left\{(x, y) \mid(x-1)^{2}+(y-1)^{2} \leqslant 1\right\} .
\end{array}
$$
If \((x, y) \in A \cap B\), find the minimum value of \(2x - y\). | -1 | 1/8 |
Let the triangle $ABC$ be fixed with a side length of $a$, and other points are not fixed. $F'$ is a point different from $F$, such that triangle $DEF'$ is equilateral. Find the locus of points $F'$ and its length. | 2a | 1/8 |
Andrey placed chips of 100 different colors on a $10 \times 10$ board. Every minute, one chip changes its color, and the chip that changes color must have been unique (distinguishable by color from all others) in its row or column before the operation. After $N$ minutes, it turned out that no chip could change its color anymore. What is the minimum possible value of $N$? | 75 | 1/8 |
Except for the first two terms, each term of the sequence $2000, y, 2000 - y,\ldots$ is obtained by subtracting the preceding term from the one before that. The last term of the sequence is the first negative term encountered. What positive integer $y$ produces a sequence of maximum length? | 1333 | 2/8 |
Let $a_1,$ $a_2,$ $\dots$ be a sequence of positive real numbers such that
\[a_n = 11a_{n - 1} - n\]for all $n > 1.$ Find the smallest possible value of $a_1.$ | \frac{21}{100} | 6/8 |
Prove that the number 11...1 (1986 ones) has at least
a) 8; b) 32 different divisors. | 32 | 4/8 |
Let $a,$ $b,$ and $c$ be complex numbers such that $|a| = |b| = |c| = 1$ and
\[\frac{a^2}{bc} + \frac{b^2}{ac} + \frac{c^2}{ab} = -1.\]Find all possible values of $|a + b + c|.$
Enter all the possible values, separated by commas. | 1,2 | 1/8 |
On a line, there are magical sheep. There are 22 blue ones, 18 red ones, and 15 green ones. When two sheep of different colors meet, they both take on the last color. After a certain number of meetings, all the sheep are the same color. What color are they? | blue | 3/8 |
There are 8 young people, among whom 5 are capable of doing English translation work, and 4 are capable of doing computer software design work (including one person who is capable of doing both tasks). Now, 5 young people are to be selected to undertake a task, with 3 people doing English translation work and 2 people doing software design work. The number of different ways to select them is ____. | 42 | 7/8 |
For all \(a, b, c \in \mathbf{R}^{+}\), find the minimum value of \(\frac{a}{\sqrt{a^{2}+8bc}} + \frac{b}{\sqrt{b^{2}+8ac}} + \frac{c}{\sqrt{c^{2}+8ab}}\). | 1 | 4/8 |
In triangle \( ABC \), angle \( B \) is \( 80^\circ \). On side \( BC \), point \( D \) is marked such that \( AB = AD = CD \). On side \( AB \), point \( F \) is marked such that \( AF = BD \). On segment \( AC \), point \( E \) is marked such that \( AB = AE \). Find angle \( AEF \). | 20 | 1/8 |
In a plane, there are \( n+4 \) points, four of which are located at the vertices of a square, and the remaining \( n \) lie inside this square. You are allowed to connect any of these points with segments, provided that no segment contains any of the marked points other than its endpoints, and no two segments share any points other than their endpoints. Find the maximum number of segments that can be constructed in this way. | 3n+5 | 2/8 |
Xiao Ming must stand in the very center, and Xiao Li and Xiao Zhang must stand together in a graduation photo with seven students. Find the number of different arrangements. | 192 | 7/8 |
How many ways are there to arrange $n$ crosses and 13 zeros in a row so that among any three consecutive symbols there is at least one zero, if
(a) $n = 27$
(b) $n = 26$? | 105 | 2/8 |
Given the function \( f(x) = a x^{3} + b x^{2} + c x + d \) (where \( a \neq 0 \)), and knowing that \( \left| f^{\prime}(x) \right| \leq 1 \) when \( 0 \leq x \leq 1 \), find the maximum value of \( a \). | \frac{8}{3} | 5/8 |
Let \( k \in \mathbb{N}^* \). Suppose that all positive integers are colored using \( k \) different colors, and there exists a function \( f: \mathbb{Z}^+ \rightarrow \mathbb{Z}^+ \) satisfying:
1. For positive integers \( m \) and \( n \) of the same color (they can be the same), \( f(m+n) = f(m) + f(n) \);
2. There exist positive integers \( m \) and \( n \) (they can be the same) such that \( f(m+n) \neq f(m) + f(n) \).
Find the minimum value of \( k \). | 3 | 2/8 |
In triangle \( ABC \), \( AB = 2 \), \( AC = 1 + \sqrt{5} \), and \( \angle CAB = 54^\circ \). Suppose \( D \) lies on the extension of \( AC \) through \( C \) such that \( CD = \sqrt{5} - 1 \). If \( M \) is the midpoint of \( BD \), determine the measure of \( \angle ACM \), in degrees. | 63 | 7/8 |
Positive integers $a_1, a_2, ... , a_7, b_1, b_2, ... , b_7$ satisfy $2 \leq a_i \leq 166$ and $a_i^{b_i} \cong a_{i+1}^2$ (mod 167) for each $1 \leq i \leq 7$ (where $a_8=a_1$ ). Compute the minimum possible value of $b_1b_2 ... b_7(b_1 + b_2 + ...+ b_7)$ . | 675 | 1/8 |
Given a rectangle \(ABCD\). A line through point \(A\) intersects segment \(CD\) at point \(X\) such that the areas of the resulting shapes satisfy \(S_{AXD}: S_{ABCX} = 1:2\). A line through point \(X\) intersects segment \(AB\) at point \(Y\) such that \(S_{AXY}: S_{YBCX} = 1:2\). Finally, a line through point \(Y\) intersects segment \(XC\) at point \(Z\) such that \(S_{XYZ}: S_{YBCZ} = 1:2\).
Calculate the ratio of the areas \(S_{AXD}: S_{AXZY}\). | \frac{9}{10} | 6/8 |
The base of a quadrilateral pyramid is a square \(ABCD\) with each side equal to 2. The lateral edge \(SA\) is perpendicular to the base plane and also equals 2. A plane is passed through the lateral edge \(SC\) and a point on side \(AB\) such that the resulting cross-section of the pyramid has the smallest perimeter. Find the area of this cross-section. | \sqrt{6} | 7/8 |
Let $n>4$ be a positive integer, which is divisible by $4$ . We denote by $A_n$ the sum of the odd positive divisors of $n$ . We also denote $B_n$ the sum of the even positive divisors of $n$ , excluding the number $n$ itself. Find the least possible value of the expression $$ f(n)=B_n-2A_n, $$ for all possible values of $n$ , as well as for which positive integers $n$ this minimum value is attained. | 4 | 5/8 |
A right circular cone is sliced into five pieces by planes parallel to its base. Each slice has the same height. What is the ratio of the volume of the second-largest piece to the volume of the largest piece? | \frac{37}{61} | 7/8 |
Given the coefficient of determination R^2 for four different regression models, where the R^2 values are 0.98, 0.67, 0.85, and 0.36, determine which model has the best fitting effect. | 0.98 | 7/8 |
Given a point \( P \) in space and a congruence transformation that leaves point \( P \) fixed, prove that there exist at most 3 planes passing through point \( P \) such that the composition of reflections in these planes is the given transformation. | 3 | 7/8 |
Let \( a, b, c, d \) be real numbers whose sum is 0. Prove that
$$
5(ab + bc + cd) + 8(ac + ad + bd) \leq 0
$$ | 5(++cd)+8(ac+ad+bd)\le0 | 6/8 |
Draw the graph of the function $y = \frac{| x^3 - x^2 - 2x | }{3} - | x + 1 |$ . | \frac{|x^3-x^2-2x|}{3}-|x+1| | 4/8 |
A projectile is launched with an initial velocity of $u$ at an angle of $\phi$ from the horizontal. The trajectory of the projectile is given by the parametric equations:
\[
x = ut \cos \phi,
\]
\[
y = ut \sin \phi - \frac{1}{2} gt^2,
\]
where $t$ is time and $g$ is the acceleration due to gravity. Suppose $u$ is constant but $\phi$ varies from $0^\circ$ to $180^\circ$. As $\phi$ changes, the highest points of the trajectories trace a closed curve. The area enclosed by this curve can be expressed as $d \cdot \frac{u^4}{g^2}$. Find the value of $d$. | \frac{\pi}{8} | 7/8 |
Find the largest four-digit number that is divisible by the sum of its digits. | 9990 | 5/8 |
Given that $D$ is a point on the side $AB$ of $\triangle ABC$, and $\overrightarrow{CD} = \frac{1}{3}\overrightarrow{AC} + \lambda \cdot \overrightarrow{BC}$, determine the value of the real number $\lambda$. | -\frac{4}{3} | 3/8 |
Let $f(n)$ be the sum of all the divisors of a positive integer $n$. If $f(f(n)) = n+2$, then call $n$ superdeficient. How many superdeficient positive integers are there? | 1 | 6/8 |
The $y$-intercepts, $P$ and $Q$, of two perpendicular lines intersecting at the point $A(6,8)$ have a sum of zero. What is the area of $\triangle APQ$?
$\textbf{(A)}\ 45\qquad\textbf{(B)}\ 48\qquad\textbf{(C)}\ 54\qquad\textbf{(D)}\ 60\qquad\textbf{(E)}\ 72$ | \textbf{(D)}\:60 | 1/8 |
A quartic (4th degree) polynomial \( p(x) \) satisfies:
\[ p(n) = \frac{1}{n^2} \] for \( n = 1, 2, 3, 4, \) and \( 5 \). Find \( p(6) \). | \frac{1}{18} | 3/8 |
What is the area, in square units, of a trapezoid bounded by the lines $y = x$, $y = 15$, $y = 5$ and the line $x = 5$? | 50 | 4/8 |
A frustum of a right circular cone is formed by cutting a small cone off of the top of a larger cone. If this frustum has a lower base radius of 8 inches, an upper base radius of 5 inches, and a height of 6 inches, what is its lateral surface area? Additionally, there is a cylindrical section of height 2 inches and radius equal to the upper base of the frustum attached to the top of the frustum. Calculate the total surface area excluding the bases. | 39\pi\sqrt{5} + 20\pi | 5/8 |
The altitudes of a scalene acute triangle \(ABC\) intersect at point \(H\). \(O\) is the circumcenter of triangle \(BHC\). The incenter \(I\) of triangle \(ABC\) lies on the segment \(OA\). Find the angle \(BAC\). | 60 | 3/8 |
In the diagram, $\triangle QRS$ is an isosceles right-angled triangle with $QR=SR$ and $\angle QRS=90^{\circ}$. Line segment $PT$ intersects $SQ$ at $U$ and $SR$ at $V$. If $\angle PUQ=\angle RVT=y^{\circ}$, the value of $y$ is | 67.5 | 3/8 |
The sequence \(\left\{a_{n}\right\}\) is defined as follows: \(a_{1}=2, a_{n+1}=a_{n}^{2}-a_{n}+1\) for \(n=1,2, \cdots\). Prove:
\[1-\frac{1}{2003^{2003}}<\frac{1}{a_{1}}+\frac{1}{a_{2}}+\cdots+\frac{1}{a_{2003}}<1.\]
(2003 Chinese Girls' Mathematical Olympiad problem) | 1-\frac{1}{2003^{2003}}<\frac{1}{a_1}+\frac{1}{a_2}+\cdots+\frac{1}{a_{2003}}<1 | 7/8 |
Prove that \(\left(1+\frac{1}{3}\right)\left(1+\frac{1}{5}\right) \cdots\left(1+\frac{1}{2 n-1}\right)>\frac{\sqrt{2 n+1}}{2}\), where \(n \in \mathbf{N}\), and \(n \geq 2\). | (1+\frac{1}{3})(1+\frac{1}{5})\cdots(1+\frac{1}{2n-1})>\frac{\sqrt{2n+1}}{2} | 3/8 |
Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ that satisfy $|\overrightarrow{a}| = 1$, $|\overrightarrow{b}| = 2$, and $\overrightarrow{a} \cdot \overrightarrow{b} = -\sqrt{3}$, find the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$. | \frac{5\pi}{6} | 5/8 |
Allie and Betty play a game where they take turns rolling a standard die. If a player rolls $n$, she is awarded $g(n)$ points, where \[g(n) = \left\{
\begin{array}{cl}
8 & \text{ if } n \text{ is a multiple of 3 and 4}, \\
3 & \text{ if } n \text{ is only a multiple of 3}, \\
1 & \text{ if } n \text{ is only a multiple of 4}, \\
0 & \text{ if } n \text{ is neither a multiple of 3 nor 4}.
\end{array}
\right.\]
Allie rolls the die four times and gets a 6, 3, 4, and 1. Betty rolls and gets 12, 9, 4, and 2. Compute the product of Allie's total points and Betty's total points. | 84 | 7/8 |
In the convex quadrilateral \(ABCD\), diagonals \(AC\) and \(BD\) intersect at point \(P\). Given that \(\angle DBC = 60^{\circ}\), \(\angle ACB = 50^{\circ}\), \(\angle ABD = 20^{\circ}\), and \(\angle ACD = 30^{\circ}\), find the measure of \(\angle ADB\). | 30 | 2/8 |
Joe has a rectangular lawn measuring 120 feet by 180 feet. His lawn mower has a cutting swath of 30 inches, and he overlaps each cut by 6 inches to ensure no grass is missed. Joe mows at a rate of 4000 feet per hour. Calculate the time it will take Joe to mow his entire lawn. | 2.7 | 7/8 |
Cities $A$, $B$, $C$, $D$, and $E$ are connected by roads $\widetilde{AB}$, $\widetilde{AD}$, $\widetilde{AE}$, $\widetilde{BC}$, $\widetilde{BD}$, $\widetilde{CD}$, and $\widetilde{DE}$. How many different routes are there from $A$ to $B$ that use each road exactly once? (Such a route will necessarily visit some cities more than once.) [asy]
size(5cm);
pair A=(1,0), B=(4.24,0), C=(5.24,3.08), D=(2.62,4.98), E=(0,3.08);
dot (A);
dot (B);
dot (C);
dot (D);
dot (E);
label("$A$",A,S);
label("$B$",B,SE);
label("$C$",C,E);
label("$D$",D,N);
label("$E$",E,W);
guide squiggly(path g, real stepsize, real slope=45)
{
real len = arclength(g);
real step = len / round(len / stepsize);
guide squig;
for (real u = 0; u < len; u += step){
real a = arctime(g, u);
real b = arctime(g, u + step / 2);
pair p = point(g, a);
pair q = point(g, b);
pair np = unit( rotate(slope) * dir(g,a));
pair nq = unit( rotate(0 - slope) * dir(g,b));
squig = squig .. p{np} .. q{nq};
}
squig = squig .. point(g, length(g)){unit(rotate(slope)*dir(g,length(g)))};
return squig;
}
pen pp = defaultpen + 2.718;
draw(squiggly(A--B, 4.04, 30), pp);
draw(squiggly(A--D, 7.777, 20), pp);
draw(squiggly(A--E, 5.050, 15), pp);
draw(squiggly(B--C, 5.050, 15), pp);
draw(squiggly(B--D, 4.04, 20), pp);
draw(squiggly(C--D, 2.718, 20), pp);
draw(squiggly(D--E, 2.718, -60), pp);[/asy] | 16 | 1/8 |
Evaluate the following product of sequences: $\frac{1}{3} \cdot \frac{9}{1} \cdot \frac{1}{27} \cdot \frac{81}{1} \dotsm \frac{1}{2187} \cdot \frac{6561}{1}$. | 81 | 5/8 |
In the quadrilateral \(ABCD\), it is known that \(AB = BD\), \(\angle ABD = \angle DBC\), and \(\angle BCD = 90^\circ\). On the segment \(BC\), there is a point \(E\) such that \(AD = DE\). What is the length of segment \(BD\) if it is known that \(BE = 7\) and \(EC = 5\)? | 17 | 4/8 |
A six digit number (base 10) is squarish if it satisfies the following conditions:
(i) none of its digits are zero;
(ii) it is a perfect square; and
(iii) the first of two digits, the middle two digits and the last two digits of the number are all perfect squares when considered as two digit numbers.
How many squarish numbers are there? | 2 | 3/8 |
Given that in the expansion of \\((1+2x)^{n}\\), only the coefficient of the fourth term is the maximum, then the constant term in the expansion of \\((1+ \dfrac {1}{x^{2}})(1+2x)^{n}\\) is \_\_\_\_\_\_. | 61 | 1/8 |
Lee can make 24 cookies with four cups of flour. If the ratio of flour to sugar needed is 2:1 and he has 3 cups of sugar available, how many cookies can he make? | 36 | 2/8 |
From the numbers $-1, 0, 1, 2, 3$, select three (without repetition) to form a quadratic function $y = ax^2 + bx + c$ as coefficients. Determine the number of different parabolas that have at least one intersection point with the negative direction of the x-axis. | 26 | 1/8 |
In the acute triangle \( \triangle ABC \), if
\[ \sin A = 2 \sin B \cdot \sin C, \]
find the minimum value of \( \tan A + 2 \tan B \cdot \tan C + \tan A \cdot \tan B \cdot \tan C \). | 16 | 7/8 |
Find all $a,$ $0^\circ < a < 360^\circ,$ such that $\cos a,$ $\cos 2a,$ and $\cos 3a$ form an arithmetic sequence, in that order. Enter the solutions, separated by commas, in degrees. | 45^\circ, 135^\circ, 225^\circ, 315^\circ | 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? | 9 | 7/8 |
Find the number of functions \( f \) from the set \( S = \{ 0, 1, 2, \ldots, 2020 \} \) to itself such that, for all \( a, b, c \in S \), all three of the following conditions are satisfied:
(i) If \( f(a) = a \), then \( a = 0 \);
(ii) If \( f(a) = f(b) \), then \( a = b \); and
(iii) If \( c \equiv a + b \pmod{2021} \), then \( f(c) \equiv f(a) + f(b) \pmod{2021} \). | 1845 | 3/8 |
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