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Find all natural numbers \( n \) such that the sum of the digits of the number \( 2^n \) in decimal notation equals 5. | 5 | 3/8 |
We have an \(n\)-gon, and each of its vertices is labeled with a number from the set \(\{1, \ldots, 10\}\). We know that for any pair of distinct numbers from this set, there is at least one side of the polygon whose endpoints have these two numbers. Find the smallest possible value of \(n\). | 50 | 3/8 |
In how many ways can we paint 16 seats in a row, each red or green, in such a way that the number of consecutive seats painted in the same colour is always odd? | 1974 | 4/8 |
The points $(0,0),(1,2),(2,1),(2,2)$ in the plane are colored red while the points $(1,0),(2,0),(0,1),(0,2)$ are colored blue. Four segments are drawn such that each one connects a red point to a blue point and each colored point is the endpoint of some segment. The smallest possible sum of the lengths of the segments can be expressed as $a+\sqrt{b}$, where $a, b$ are positive integers. Compute $100a+b$. | 305 | 6/8 |
Given \(\frac{\sin \theta}{\sqrt{3} \cos \theta+1}>1\), find the range of values for \(\tan \theta\). | (-\infty,-\sqrt{2})\cup(\frac{\sqrt{3}}{3},\sqrt{2}) | 1/8 |
There are exactly 120 ways to color five cells in a $5 \times 5$ grid such that exactly one cell in each row and column is colored.
There are exactly 96 ways to color five cells in a $5 \times 5$ grid without the corner cell such that exactly one cell in each row and column is colored.
How many ways are there to color five cells in a $5 \times 5$ grid without two corner cells such that exactly one cell in each row and column is colored? | 78 | 5/8 |
In a quadrilateral, the lengths of all its sides and diagonals are less than 1 meter. Prove that it can be placed inside a circle with a radius of 0.9 meters. | 0.9 | 7/8 |
Given a complex number $z$ satisfying $z+ \bar{z}=6$ and $|z|=5$.
$(1)$ Find the imaginary part of the complex number $z$;
$(2)$ Find the real part of the complex number $\dfrac{z}{1-i}$. | \dfrac{7}{2} | 7/8 |
Two people, $A$ and $B$, need to get from point $M$ to point $N$, which is 15 km away from $M$, as quickly as possible. They can walk at a speed of 6 km/h, and they have a bicycle that allows travel at 15 km/h. $A$ starts the journey on foot, while $B$ rides the bicycle until meeting with pedestrian $C$, who is walking from $N$ to $M$. After meeting, $B$ walks, and $C$ rides the bicycle to meet with $A$, then gives the bicycle to $A$, who then rides it to $N$. When should pedestrian $C$ leave $N$ so that the travel time for both $A$ and $B$ to reach $N$ is minimized? (Note: $C$ walks at the same speed as $A$ and $B$; the travel time is calculated from when $A$ and $B$ leave $M$ to when the last of them arrives at $N$.) | \frac{3}{11} | 2/8 |
What is the smallest base-10 integer that can be represented as $CC_6$ and $DD_8$, where $C$ and $D$ are valid digits in their respective bases? | 63 | 2/8 |
a) \(\operatorname{ctg}(\alpha / 2)+\operatorname{ctg}(\beta / 2)+\operatorname{ctg}(\gamma / 2) \geq 3 \sqrt{3}\).
b) For an acute-angled triangle
\[
\operatorname{tg} \alpha+\operatorname{tg} \beta+\operatorname{tg} \gamma \geq 3 \sqrt{3}
\] | \operatorname{tg}\alpha+\operatorname{tg}\beta+\operatorname{tg}\gamma\ge3\sqrt{3} | 3/8 |
The lateral edges \( PA, PB \) and \( PC \) of the pyramid \( PABC \) are equal to 2, 2, and 3, respectively, and its base is an equilateral triangle. It is known that the areas of the lateral faces of the pyramid are equal to each other. Find the volume of the pyramid \( PABC \). | \frac{5\sqrt{2}}{16} | 4/8 |
Find the $1314^{\text{th}}$ digit past the decimal point in the decimal expansion of $\dfrac{5}{14}$. | 2 | 7/8 |
The product of two positive integers plus their sum equals 119. The integers are relatively prime, and each is less than 25. What is the sum of the two integers? | 20 | 1/8 |
Determine all non negative integers $k$ such that there is a function $f : \mathbb{N} \to \mathbb{N}$ that satisfies
\[ f^n(n) = n + k \]
for all $n \in \mathbb{N}$ | 0 | 3/8 |
Given a positive integer $n \ge 2$ , determine the largest positive integer $N$ for which there exist $N+1$ real numbers $a_0, a_1, \dots, a_N$ such that $(1) \ $ $a_0+a_1 = -\frac{1}{n},$ and $(2) \ $ $(a_k+a_{k-1})(a_k+a_{k+1})=a_{k-1}-a_{k+1}$ for $1 \le k \le N-1$ . | n | 2/8 |
In honor of a holiday, 1% of the soldiers in the regiment received new uniforms. The soldiers are arranged in a rectangle such that those in new uniforms are in at least 30% of the columns and at least 40% of the rows. What is the smallest possible number of soldiers in the regiment? | 1200 | 7/8 |
In the figure, polygons $A$, $E$, and $F$ are isosceles right triangles; $B$, $C$, and $D$ are squares with sides of length $1$; and $G$ is an equilateral triangle. The figure can be folded along its edges to form a polyhedron having the polygons as faces. The volume of this polyhedron is | 5/6 | 1/8 |
Real numbers \( x, y, z \) are chosen such that the following equations hold: \( x y + y z + z x = 4 \) and \( x y z = 6 \). Prove that, for any such choice, the value of the expression
$$
\left(x y - \frac{3}{2}(x + y)\right) \left(y z - \frac{3}{2}(y + z)\right) \left(z x - \frac{3}{2}(z + x)\right)
$$
is always the same, and find this value. | \frac{81}{4} | 3/8 |
How many ways are there to arrange in a row $n$ crosses and 14 zeros such that among any three consecutive symbols there is at least one zero, if
(a) $n=29$;
(b) $n=28$? | 120 | 2/8 |
How many ways are there to choose 2010 functions \( f_{1}, \ldots, f_{2010} \) from \(\{0,1\}\) to \(\{0,1\}\) such that \( f_{2010} \circ f_{2009} \circ \cdots \circ f_{1} \) is constant? Note: a function \( g \) is constant if \( g(a)=g(b) \) for all \( a, b \) in the domain of \( g \). | 4^{2010}-2^{2010} | 1/8 |
A monkey becomes happy when it eats three different fruits. What is the maximum number of monkeys that can be made happy if you have 20 pears, 30 bananas, 40 peaches, and 50 mandarins? | 45 | 2/8 |
Given that \( B(-6,0) \) and \( C(6,0) \) are two vertices of triangle \( \triangle ABC \), and the interior angles \( \angle A, \angle B, \angle C \) satisfy \( \sin B - \sin C = \frac{1}{2} \sin A \). Find the equation of the locus of vertex \( A \). | \frac{x^2}{9}-\frac{y^2}{27}=1 | 7/8 |
Let \( S(n) \) be the sum of the digits of an integer \( n \). For example, \( S(327) = 3 + 2 + 7 = 12 \). Find the value of
\[
A = S(1) - S(2) + S(3) - S(4) + \ldots - S(2016) + S(2017)
\] | 1009 | 1/8 |
Do there exist natural numbers \( m \) and \( n \) such that
\[ n^2 + 2018mn + 2019m + n - 2019m^2 \]
is a prime number? | No | 5/8 |
Given that $\textstyle\binom{2k}k$ results in a number that ends in two zeros, find the smallest positive integer $k$. | 13 | 7/8 |
Given that \( m, n, t \) (\(m < n\)) are all positive integers, and points \( A(-m,0) \), \( B(n,0) \), \( C(0,t) \), and \( O \) is the origin. It is given that \( \angle ACB = 90^\circ \), and:
\[
OA^2 + OB^2 + OC^2 = 13(OA + OB - OC).
\]
1. Find the value of \(m + n + t\).
2. If a quadratic function passes through points \( A, B, \) and \( C \), find the expression of this quadratic function. | -\frac{1}{3}x^2+\frac{8}{3}x+3 | 7/8 |
A line is drawn through a vertex of a triangle and cuts two of its middle lines (i.e. lines connecting the midpoints of two sides) in the same ratio. Determine this ratio. | \frac{1+\sqrt{5}}{2} | 4/8 |
Three different positive integers have a mean of 7. What is the largest positive integer that could be one of them?
A 15
B 16
C 17
D 18
E 19 | 18 | 1/8 |
Given that vertices \(A\) and \(C\) of triangle \(\triangle ABC\) are on the graph of the inverse proportional function \(y = \frac{\sqrt{3}}{x}\) (where \(x > 0\)), \(\angle ACB = 90^\circ\), \(\angle ABC = 30^\circ\), \(AB \perp x\)-axis, point \(B\) is above point \(A\), and \(AB = 6\). Determine the coordinates of point \(C\). | (\frac{\sqrt{3}}{2},2) | 6/8 |
Let $f(x) = a\cos x - \left(x - \frac{\pi}{2}\right)\sin x$, where $x \in \left[0, \frac{\pi}{2}\right]$.
$(1)$ When $a = -1$, find the range of the derivative ${f'}(x)$ of the function $f(x)$.
$(2)$ If $f(x) \leq 0$ always holds, find the maximum value of the real number $a$. | -1 | 3/8 |
Simplify $\left(\frac{a^2}{a+1}-a+1\right) \div \frac{a^2-1}{a^2+2a+1}$, then choose a suitable integer from the inequality $-2 \lt a \lt 3$ to substitute and evaluate. | -1 | 2/8 |
In a regular quadrilateral truncated pyramid \(A B C D A_{1} B_{1} C_{1} D_{1}\), a cross-section \(A B C_{1} D_{1}\) is made. Find the volume ratio of the resulting polyhedra \(\frac{V_{A B C D D_{1} C_{1}}}{V_{A_{1} B_{1} C_{1} D_{1} A B}}\), given that \(A B : A_{1} B_{1} = 3\). Round your answer to two decimal places. | 4.2 | 1/8 |
There are exactly 120 ways to color five cells in a \(5 \times 5\) grid such that each row and each column has exactly one colored cell.
There are exactly 96 ways to color five cells in a \(5 \times 5\) grid, excluding a corner cell, such that each row and each column has exactly one colored cell.
How many ways are there to color five cells in a \(5 \times 5\) grid, excluding two corner cells, such that each row and each column has exactly one colored cell? | 78 | 4/8 |
(1) Calculate the value of $(\frac{2}{3})^{0}+3\times(\frac{9}{4})^{{-\frac{1}{2}}}+(\log 4+\log 25)$.
(2) Given $\alpha \in (0,\frac{\pi }{2})$, and $2\sin^{2}\alpha - \sin \alpha \cdot \cos \alpha - 3\cos^{2}\alpha = 0$, find the value of $\frac{\sin \left( \alpha + \frac{\pi }{4} \right)}{\sin 2\alpha + \cos 2\alpha + 1}$.
(3) There are three cards, marked with $1$ and $2$, $1$ and $3$, and $2$ and $3$, respectively. Three people, A, B, and C, each take a card. A looks at B's card and says, "The number that my card and B's card have in common is not $2$." B looks at C's card and says, "The number that my card and C's card have in common is not $1$." C says, "The sum of the numbers on my card is not $5$." What is the number on A's card?
(4) Given $f\left( x \right)=x-\frac{1}{x+1}$ and $g\left( x \right)={{x}^{2}}-2ax+4$, for any ${{x}_{1}}\in \left[ 0,1 \right]$, there exists ${{x}_{2}}\in \left[ 1,2 \right]$ such that $f\left( {{x}_{1}} \right)\geqslant g\left( {{x}_{2}} \right)$. Find the minimum value of the real number $a$. | \frac{9}{4} | 1/8 |
Let \( n = \overline{abc} \) be a three-digit number, where \( a \), \( b \), and \( c \) represent the lengths of the sides of an isosceles (including equilateral) triangle. Find the number of such three-digit numbers \( n \). | 165 | 3/8 |
How many sides can a convex polygon have if all its diagonals are equal? | 5 | 6/8 |
A geometric sequence of positive integers is formed for which the first term is 3 and the fifth term is 375. What is the sixth term of the sequence? | 9375 | 1/8 |
Let $ a,b$ be integers greater than $ 1$ . What is the largest $ n$ which cannot be written in the form $ n \equal{} 7a \plus{} 5b$ ? | 47 | 6/8 |
In a cube $A B C D-A_{1} B_{1} C_{1} D_{1}$ with a side length of 1, points $E$ and $F$ are located on $A A_{1}$ and $C C_{1}$ respectively, such that $A E = C_{1} F$. Determine the minimum area of the quadrilateral $E B F D_{1}$. | \frac{\sqrt{6}}{2} | 7/8 |
Given the function $f\left( x \right)=2\sin (\omega x+\varphi )\left( \omega \gt 0,\left| \varphi \right|\lt \frac{\pi }{2} \right)$, the graph passes through point $A(0,-1)$, and is monotonically increasing on $\left( \frac{\pi }{18},\frac{\pi }{3} \right)$. The graph of $f\left( x \right)$ is shifted to the left by $\pi$ units and coincides with the original graph. When ${x}_{1}$, ${x}_{2} \in \left( -\frac{17\pi }{12},-\frac{2\pi }{3} \right)$ and ${x}_{1} \ne {x}_{2}$, if $f\left( {x}_{1} \right)=f\left( {x}_{2} \right)$, find $f({x}_{1}+{x}_{2})$. | -1 | 7/8 |
What is the minimum number of kings that must be placed on a chessboard so that they attack all the cells not occupied by them? (A king attacks the cells adjacent to its cell by side or corner). | 9 | 3/8 |
Given \( n \in \mathbf{N}, n > 4 \), and the set \( A = \{1, 2, \cdots, n\} \). Suppose there exists a positive integer \( m \) and sets \( A_1, A_2, \cdots, A_m \) with the following properties:
1. \( \bigcup_{i=1}^{m} A_i = A \);
2. \( |A_i| = 4 \) for \( i=1, 2, \cdots, m \);
3. Let \( X_1, X_2, \cdots, X_{\mathrm{C}_n^2} \) be all the 2-element subsets of \( A \). For every \( X_k \) \((k=1, 2, \cdots, \mathrm{C}_n^2)\), there exists a unique \( j_k \in\{1, 2, \cdots, m\} \) such that \( X_k \subseteq A_{j_k} \).
Find the smallest value of \( n \). | 13 | 5/8 |
Given an ellipse $G$: $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 (a > b > 0)$ with an eccentricity of $\frac{\sqrt{6}}{3}$, a right focus at $(2 \sqrt{2},0)$, and a line $l$ with a slope of $1$ intersecting the ellipse $G$ at points $A$ and $B$. An isosceles triangle is constructed with $AB$ as the base and $P(-3,2)$ as the vertex.
(1) Find the equation of the ellipse $G$;
(2) If $M(m,n)$ is any point on the ellipse $G$, find the maximum and minimum values of $\frac{n}{m-4}$;
(3) Find the area of $\triangle PAB$. | \frac{9}{2} | 7/8 |
How many sequences of ten binary digits are there in which neither two zeroes nor three ones ever appear in a row? | 28 | 4/8 |
An alarm clock gains 9 minutes each day. When going to bed at 22:00, the precise current time is set on the clock. At what time should the alarm be set so that it rings exactly at 6:00? Explain your answer. | 6:03 | 1/8 |
Given that $\{a_{n}\}$ is an arithmetic progression, $\{b_{n}\}$ is a geometric progression, and $a_{2}+a_{5}=a_{3}+9=8b_{1}=b_{4}=16$.
$(1)$ Find the general formulas for $\{a_{n}\}$ and $\{b_{n}\}$.
$(2)$ Arrange the terms of $\{a_{n}\}$ and $\{b_{n}\}$ in ascending order to form a new sequence $\{c_{n}\}$. Let the sum of the first $n$ terms of $\{c_{n}\}$ be denoted as $S_{n}$. If $c_{k}=101$, find the value of $k$ and determine $S_{k}$. | 2726 | 2/8 |
Find all positive integers $x$ such that the product of all digits of $x$ is given by $x^2 - 10 \cdot x - 22.$ | 12 | 5/8 |
Arithmetic sequences $\left(a_n\right)$ and $\left(b_n\right)$ have integer terms with $a_1=b_1=1<a_2 \le b_2$ and $a_n b_n = 2010$ for some $n$. What is the largest possible value of $n$? | 8 | 7/8 |
At a recent math contest, Evan was asked to find \( 2^{2016} (\bmod p) \) for a given prime number \( p \) where \( 100 < p < 500 \). Evan first tried taking 2016 modulo \( p-1 \), resulting in a value \( e \) larger than 100. Evan noted that \( e - \frac{1}{2}(p-1) = 21 \) and then realized the answer was \( -2^{21} (\bmod p) \). What was the prime \( p \)? | 211 | 4/8 |
From the set \( M = \{1, 2, \cdots, 2008\} \) of the first 2008 positive integers, a \( k \)-element subset \( A \) is chosen such that the sum of any two numbers in \( A \) cannot be divisible by the difference of those two numbers. What is the maximum value of \( k \)? | 670 | 1/8 |
A pyramid with a square base has a base edge of 20 cm and a height of 40 cm. Two smaller similar pyramids are cut away from the original pyramid: one has an altitude that is one-third the original, and another that is one-fifth the original altitude, stacked atop the first smaller pyramid. What is the volume of the remaining solid as a fraction of the volume of the original pyramid? | \frac{3223}{3375} | 5/8 |
A quadrilateral is drawn on a sheet of transparent paper. What is the minimum number of times the sheet must be folded to verify that it is a square? | 2 | 4/8 |
In triangle $ABC$, $BC = 23$, $CA = 27$, and $AB = 30$. Points $V$ and $W$ are on $\overline{AC}$ with $V$ on $\overline{AW}$, points $X$ and $Y$ are on $\overline{BC}$ with $X$ on $\overline{CY}$, and points $Z$ and $U$ are on $\overline{AB}$ with $Z$ on $\overline{BU}$. In addition, the points are positioned so that $\overline{UV}\parallel\overline{BC}$, $\overline{WX}\parallel\overline{AB}$, and $\overline{YZ}\parallel\overline{CA}$. Right angle folds are then made along $\overline{UV}$, $\overline{WX}$, and $\overline{YZ}$. The resulting figure is placed on a leveled floor to make a table with triangular legs. Let $h$ be the maximum possible height of a table constructed from triangle $ABC$ whose top is parallel to the floor. Then $h$ can be written in the form $\frac{k\sqrt{m}}{n}$, where $k$ and $n$ are relatively prime positive integers and $m$ is a positive integer that is not divisible by the square of any prime. Find $k+m+n$.
[asy] unitsize(1 cm); pair translate; pair[] A, B, C, U, V, W, X, Y, Z; A[0] = (1.5,2.8); B[0] = (3.2,0); C[0] = (0,0); U[0] = (0.69*A[0] + 0.31*B[0]); V[0] = (0.69*A[0] + 0.31*C[0]); W[0] = (0.69*C[0] + 0.31*A[0]); X[0] = (0.69*C[0] + 0.31*B[0]); Y[0] = (0.69*B[0] + 0.31*C[0]); Z[0] = (0.69*B[0] + 0.31*A[0]); translate = (7,0); A[1] = (1.3,1.1) + translate; B[1] = (2.4,-0.7) + translate; C[1] = (0.6,-0.7) + translate; U[1] = U[0] + translate; V[1] = V[0] + translate; W[1] = W[0] + translate; X[1] = X[0] + translate; Y[1] = Y[0] + translate; Z[1] = Z[0] + translate; draw (A[0]--B[0]--C[0]--cycle); draw (U[0]--V[0],dashed); draw (W[0]--X[0],dashed); draw (Y[0]--Z[0],dashed); draw (U[1]--V[1]--W[1]--X[1]--Y[1]--Z[1]--cycle); draw (U[1]--A[1]--V[1],dashed); draw (W[1]--C[1]--X[1]); draw (Y[1]--B[1]--Z[1]); dot("$A$",A[0],N); dot("$B$",B[0],SE); dot("$C$",C[0],SW); dot("$U$",U[0],NE); dot("$V$",V[0],NW); dot("$W$",W[0],NW); dot("$X$",X[0],S); dot("$Y$",Y[0],S); dot("$Z$",Z[0],NE); dot(A[1]); dot(B[1]); dot(C[1]); dot("$U$",U[1],NE); dot("$V$",V[1],NW); dot("$W$",W[1],NW); dot("$X$",X[1],dir(-70)); dot("$Y$",Y[1],dir(250)); dot("$Z$",Z[1],NE);[/asy] | 318 | 1/8 |
Suppose that $k \geq 2$ is a positive integer. An in-shuffle is performed on a list with $2 k$ items to produce a new list of $2 k$ items in the following way: - The first $k$ items from the original are placed in the odd positions of the new list in the same order as they appeared in the original list. - The remaining $k$ items from the original are placed in the even positions of the new list, in the same order as they appeared in the original list. For example, an in-shuffle performed on the list $P Q R S T U$ gives the new list $P S Q T R U$. A second in-shuffle now gives the list $P T S R Q U$. Ping has a list of the 66 integers from 1 to 66, arranged in increasing order. He performs 1000 in-shuffles on this list, recording the new list each time. In how many of these 1001 lists is the number 47 in the 24th position? | 83 | 7/8 |
On side \(AC\) of triangle \(ABC\), point \(E\) is chosen. The angle bisector \(AL\) intersects segment \(BE\) at point \(X\). It is found that \(AX = XE\) and \(AL = BX\). What is the ratio of angles \(A\) and \(B\) of the triangle? | 2 | 1/8 |
Let $ABC$ be a triangle with circumcenter $O$ such that $AC = 7$ . Suppose that the circumcircle of $AOC$ is tangent to $BC$ at $C$ and intersects the line $AB$ at $A$ and $F$ . Let $FO$ intersect $BC$ at $E$ . Compute $BE$ . | \frac{7}{2} | 1/8 |
There are 10 numbers:
$$
21, 22, 34, 39, 44, 45, 65, 76, 133, \text{ and } 153
$$
Group them into two groups of 5 numbers each, such that the product of the numbers in both groups is the same. What is this product? | 349188840 | 1/8 |
Points \( M, N, \) and \( K \) are located on the lateral edges \( A A_{1}, B B_{1}, \) and \( C C_{1} \) of the triangular prism \( A B C A_{1} B_{1} C_{1} \) such that \( \frac{A M}{A A_{1}} = \frac{5}{6}, \frac{B N}{B B_{1}} = \frac{6}{7}, \) and \( \frac{C K}{C C_{1}} = \frac{2}{3} \). Point \( P \) belongs to the prism. Find the maximum possible volume of the pyramid \( M N K P \), given that the volume of the prism is 35. | 10 | 3/8 |
8. Let \( A B C \) be a triangle with sides \( A B = 6 \), \( B C = 10 \), and \( C A = 8 \). Let \( M \) and \( N \) be the midpoints of \( B A \) and \( B C \), respectively. Choose the point \( Y \) on ray \( C M \) so that the circumcircle of triangle \( A M Y \) is tangent to \( A N \). Find the area of triangle \( N A Y \). | \frac{600}{73} | 5/8 |
Suppose that $A_{2}, A_{3}, \ldots, A_{n}$ are independent events with
$$
P\left(A_{i}\right)=\frac{1}{2 i^{2}}
$$
What is the probability that an odd number of the events $A_{2}, A_{3}, \ldots, A_{n}$ occur? | \frac{n-1}{4n} | 7/8 |
Find all the functions \( f: \mathbb{Q} \rightarrow \mathbb{Q} \) (where \( \mathbb{Q} \) is the set of rational numbers) that satisfy \( f(1)=2 \) and
\[ f(x y) \equiv f(x) f(y) - f(x + y) + 1, \quad \text{for all} \ x, y \in \mathbb{Q}. \] | f(x)=x+1 | 1/8 |
In triangle $ABC$, the angles $\angle B = 30^\circ$ and $\angle A = 90^\circ$ are known. Point $K$ is marked on side $AC$, and points $L$ and $M$ are marked on side $BC$ such that $KL = KM$ (point $L$ is on segment $BM$).
Find the length of segment $LM$, given that $AK = 4$, $BL = 31$, and $MC = 3$. | 14 | 7/8 |
Let $I_{m}=\textstyle\int_{0}^{2 \pi} \cos (x) \cos (2 x) \cdots \cos (m x) d x .$ For which integers $m, 1 \leq m \leq 10$ is $I_{m} \neq 0 ?$ | 3,4,7,8 | 1/8 |
An integer, whose decimal representation reads the same left to right and right to left, is called symmetrical. For example, the number 513151315 is symmetrical, while 513152315 is not. How many nine-digit symmetrical numbers exist such that adding the number 11000 to them leaves them symmetrical? | 8100 | 1/8 |
Let $C$ be a cube. Let $P$ , $Q$ , and $R$ be random vertices of $C$ , chosen uniformly and independently from the set of vertices of $C$ . (Note that $P$ , $Q$ , and $R$ might be equal.) Compute the probability that some face of $C$ contains $P$ , $Q$ , and $R$ . | \frac{37}{64} | 7/8 |
Let $x_0=1$ , and let $\delta$ be some constant satisfying $0<\delta<1$ . Iteratively, for $n=0,1,2,\dots$ , a point $x_{n+1}$ is chosen uniformly form the interval $[0,x_n]$ . Let $Z$ be the smallest value of $n$ for which $x_n<\delta$ . Find the expected value of $Z$ , as a function of $\delta$ . | 1-\ln\delta | 3/8 |
Let $a,$ $b,$ $c,$ $x,$ $y,$ $z$ be nonzero complex numbers such that
\[a = \frac{b + c}{x - 2}, \quad b = \frac{a + c}{y - 2}, \quad c = \frac{a + b}{z - 2},\]and $xy + xz + yz = 5$ and $x + y + z = 3,$ find $xyz.$ | 5 | 7/8 |
A net of a cube is shown with one integer on each face. A larger cube is constructed using 27 copies of this cube. What is the minimum possible sum of all of the integers showing on the six faces of the larger cube? | 90 | 1/8 |
Find the second-order derivative \( y_{xx}'' \) of the function given parametrically:
\[
\begin{cases}
x = \ln t \\
y = \operatorname{arctg} t
\end{cases}
\] | \frac{(1-^2)}{(1+^2)^2} | 7/8 |
Given the hyperbola $\frac{x^{2}}{m} + \frac{y^{2}}{n} = 1 (m < 0 < n)$ with asymptote equations $y = \pm \sqrt{2}x$, calculate the hyperbola's eccentricity. | \sqrt{3} | 3/8 |
An integer $n>0$ is written in decimal system as $\overline{a_ma_{m-1}\ldots a_1}$ . Find all $n$ such that
\[n=(a_m+1)(a_{m-1}+1)\cdots (a_1+1)\] | 18 | 1/8 |
There are 900 three-digit numbers $(100, 101, \cdots, 999)$. Each number is printed on a card; some of these numbers, when seen upside-down, still appear as three-digit numbers. For example, when 198 is seen upside-down, it still appears as a valid three-digit number. Therefore, at most $\qquad$ cards can be omitted from printing. | 34 | 1/8 |
Each side of a triangle is greater than 100. Can its area be less than $0.01?$ | Yes | 4/8 |
A circle has a center \( O \) and diameter \( AB = 2\sqrt{19} \). Points \( C \) and \( D \) are on the upper half of the circle. A line is drawn through \( C \) and \( D \). Points \( P \) and \( Q \) are on the line such that \( AP \) and \( BQ \) are both perpendicular to \( PQ \). \( QB \) intersects the circle at \( R \). If \( CP = DQ = 1 \) and \( 2AP = BQ \), what is the length of \( AP \)? | \frac{\sqrt{15}}{2} | 1/8 |
An automobile travels $a/6$ feet in $r$ seconds. If this rate is maintained for $3$ minutes, how many yards does it travel in $3$ minutes?
$\textbf{(A)}\ \frac{a}{1080r}\qquad \textbf{(B)}\ \frac{30r}{a}\qquad \textbf{(C)}\ \frac{30a}{r}\qquad \textbf{(D)}\ \frac{10r}{a}\qquad \textbf{(E)}\ \frac{10a}{r}$ | \textbf{(E)}\\frac{10a}{r} | 1/8 |
Let \( \triangle ABC \) be a triangle with \( AB=6 \), \( BC=5 \), and \( AC=7 \). The tangents to the circumcircle of \( \triangle ABC \) at \( B \) and \( C \) meet at \( X \). Let \( Z \) be a point on the circumcircle of \( \triangle ABC \). Let \( Y \) be the foot of the perpendicular from \( X \) to \( CZ \). Let \( K \) be the intersection of the circumcircle of \( BCY \) with the line \( AB \). Given that \( Y \) is on the interior of segment \( CZ \) and \( YZ = 3CY \), compute \( AK \). | \frac{147}{10} | 1/8 |
Let $a_0 = 5/2$ and $a_k = a_{k-1}^2 - 2$ for $k \geq 1$. Compute \[ \prod_{k=0}^\infty \left(1 - \frac{1}{a_k} \right) \] in closed form. | \frac{3}{7} | 1/8 |
In a certain country, there are 47 cities. Each city has a bus station from which buses travel to other cities in the country and possibly abroad. A traveler studied the schedule and determined the number of internal bus routes originating from each city. It turned out that if we do not consider the city of Ozerny, then for each of the remaining 46 cities, the number of internal routes originating from it differs from the number of routes originating from other cities. Find out how many cities in the country have direct bus connections with the city of Ozerny.
The number of internal bus routes for a given city is the number of cities in the country that can be reached from that city by a direct bus without transfers. Routes are symmetric: if you can travel by bus from city $A$ to city $B$, you can also travel by bus from city $B$ to city $A$. | 23 | 1/8 |
A certain high school is planning to hold a coming-of-age ceremony for senior students on the "May Fourth" Youth Day to motivate the seniors who are preparing for the college entrance examination. The Student Affairs Office has prepared five inspirational songs, a video speech by an outstanding former student, a speech by a teacher representative, and a speech by a current student. Based on different requirements, find the arrangements for this event.<br/>$(1)$ If the three speeches cannot be adjacent, how many ways are there to arrange them?<br/>$(2)$ If song A cannot be the first one and song B cannot be the last one, how many ways are there to arrange them?<br/>$(3)$ If the video speech by the outstanding former student must be before the speech by the current student, how many ways are there to arrange them? (Provide the answer as a number) | 20160 | 7/8 |
Let $[n]$ denote the set of integers $\left\{ 1, 2, \ldots, n \right\}$ . We randomly choose a function $f:[n] \to [n]$ , out of the $n^n$ possible functions. We also choose an integer $a$ uniformly at random from $[n]$ . Find the probability that there exist positive integers $b, c \geq 1$ such that $f^b(1) = a$ and $f^c(a) = 1$ . ( $f^k(x)$ denotes the result of applying $f$ to $x$ $k$ times.) | \frac{1}{n} | 1/8 |
Let $\triangle ABC$ have sides $a$, $b$, and $c$ opposite to angles $A$, $B$, and $C$ respectively. Given that $\frac{{a^2 + c^2 - b^2}}{{\cos B}} = 4$. Find:<br/>
$(1)$ $ac$;<br/>
$(2)$ If $\frac{{2b\cos C - 2c\cos B}}{{b\cos C + c\cos B}} - \frac{c}{a} = 2$, find the area of $\triangle ABC$. | \frac{\sqrt{15}}{4} | 7/8 |
Let $f: \mathbb{N} \rightarrow \mathbb{N}$ be a strictly increasing function such that $f(1)=1$ and $f(2n)f(2n+1)=9f(n)^{2}+3f(n)$ for all $n \in \mathbb{N}$. Compute $f(137)$. | 2215 | 3/8 |
In a triangle with sides \(a, b, c\) and angles \(\alpha, \beta, \gamma\), the equality \(3\alpha + 2\beta = 180^\circ\) holds. The sides \(a, b, c\) are opposite to angles \(\alpha, \beta, \gamma\) respectively. Find the length of side \(c\) given that \(a = 2\) and \(b = 3\). | 4 | 6/8 |
For how many natural numbers \( n \) not exceeding 600 are the triples of numbers
\[ \left\lfloor \frac{n}{2} \right\rfloor, \left\lfloor \frac{n}{3} \right\rfloor, \left\lfloor \frac{n}{5} \right\rfloor \quad\text{and}\quad \left\lfloor \frac{n+1}{2} \right\rfloor, \left\lfloor \frac{n+1}{3} \right\rfloor, \left\lfloor \frac{n+1}{5} \right\rfloor \]
distinct? As always, \(\lfloor x \rfloor\) denotes the greatest integer less than or equal to \( x \). | 440 | 6/8 |
Let $a$ and $b$ be constants. Suppose that the equation \[\frac{(x+a)(x+b)(x+12)}{(x+3)^2} = 0\]has exactly $3$ distinct roots, while the equation \[\frac{(x+2a)(x+3)(x+6)}{(x+b)(x+12)} = 0\]has exactly $1$ distinct root. Compute $100a + b.$ | 156 | 5/8 |
Fresh mushrooms contain $90\%$ water. When they are dried, they become 15 kg lighter at a moisture content of $60\%$. How many fresh mushrooms were there originally? | 20\, | 1/8 |
Convert $BD4_{16}$ to base 4. | 233110_4 | 5/8 |
All positive integers whose digits add up to 14 are listed in increasing order. What is the eleventh number in that list? | 194 | 7/8 |
On a river with a current speed of $5 \text{ km/h}$, there are docks $A$, $B$, and $C$ positioned in the direction of the current, with $B$ located halfway between $A$ and $C$. A raft, which moves with the current towards dock $C$, and a motorboat, which heads towards dock $A$ with a speed of $v \text{ km/h}$ in still water, simultaneously depart from dock $B$. After reaching dock $A$, the motorboat turns around and moves towards dock $C$. Find all values of $v$ for which the motorboat arrives at $C$ later than the raft. | 5<v<15 | 5/8 |
The king distributed ducats to his sons. To the eldest son, he gave a certain number of ducats, to the younger son he gave one less ducat, and to the next younger son again one less ducat, and so he continued until the youngest. Then he returned to the eldest son and gave him one less ducat than the youngest received last time, and distributed in the same manner as in the first round. In this round, the youngest son received one ducat. The eldest son received a total of 21 ducats.
Determine how many sons the king had and the total number of ducats he distributed. | 105 | 7/8 |
Ten chairs are arranged in a circle. Find the number of subsets of this set of chairs that contain at least three adjacent chairs. | 581 | 1/8 |
Let $a_0>0$ be a real number, and let $$ a_n=\frac{a_{n-1}}{\sqrt{1+2020\cdot a_{n-1}^2}}, \quad \textrm{for } n=1,2,\ldots ,2020. $$ Show that $a_{2020}<\frac1{2020}$ . | a_{2020}<\frac{1}{2020} | 4/8 |
Three simplest proper fractions with the same numerator, when converted to mixed numbers, are respectively $\mathrm{a} \frac{2}{3}, b \frac{3}{4}, c \frac{3}{5}$, where $a, b$, and $c$ are natural numbers not exceeding 10. Calculate $(2a + b) \div c = \qquad$. | 4.75 | 1/8 |
Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfy $|\overrightarrow{a}|=2$, $\overrightarrow{b}=(4\cos \alpha,-4\sin \alpha)$, and $\overrightarrow{a}\perp (\overrightarrow{a}- \overrightarrow{b})$, let the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ be $\theta$, then $\theta$ equals \_\_\_\_\_\_. | \dfrac {\pi}{3} | 6/8 |
A workshop has fewer than $60$ employees. When these employees are grouped in teams of $8$, $5$ employees remain without a team. When arranged in teams of $6$, $3$ are left without a team. How many employees are there in the workshop? | 45 | 5/8 |
Consider triangle \(ABC\) with side lengths \(AB=4\), \(BC=7\), and \(AC=8\). Let \(M\) be the midpoint of segment \(AB\), and let \(N\) be the point on the interior of segment \(AC\) that also lies on the circumcircle of triangle \(MBC\). Compute \(BN\). | \frac{\sqrt{105}}{2\sqrt{2}}\quad | 1/8 |
Given the vertices of a regular 100-sided polygon \( A_{1}, A_{2}, A_{3}, \ldots, A_{100} \), in how many ways can three vertices be selected such that they form an obtuse triangle? | 117600 | 2/8 |
Find all primes \( p \) for which the numbers \( p+1 \) and \( p^2+1 \) are twice the squares of natural numbers. | 7 | 5/8 |
Three CDs are bought at an average cost of \$15 each. If a fourth CD is purchased, the average cost becomes \$16. What is the cost of the fourth CD?
(A) \$16
(B) \$17
(C) \$18
(D) \$19
(E) \$20 | 19 | 1/8 |
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