Three numbers, $a_1\,$, $a_2\,$, $a_3\,$, are drawn randomly and without replacement from the set $\{1, 2, 3, \dots, 1000\}\,$. Three other numbers, $b_1\,$, $b_2\,$, $b_3\,$, are then drawn randomly and without replacement from the remaining set of 997 numbers. Let $p\,$ be the probability that, after a suitable rotation, a brick of dimensions $a_1 \times a_2 \times a_3\,$ can be enclosed in a box of dimensions $b_1 \times b_2 \times b_3\,$, with the sides of the brick parallel to the sides of the box. If $p\,$ is written as a fraction in lowest terms, what is the sum of the numerator and denominator?

Answer: 5


Given \( a < -1 \), the variable \( x \) satisfies \( x^{2} + ax \leq -x \), and the minimum value of \( x^{2} + ax \) is \( -\frac{1}{2} \). Find the value of \( a \).

Answer: -\frac{3}{2}


In parallelogram \(A B C D\), point \(E\) is the midpoint of side \(A D\), and point \(F\) is the foot of the perpendicular dropped from vertex \(B\) to line \(C E\).

Find the area of triangle \(A B F\), given that \(A B = a\) and \(\angle B A F = \alpha\).

Answer: \frac{1}{2}^2\sin(\alpha)


Let  $M = \{1,2,\cdots , 10\}$ , and let  $T$  be a set of 2-element subsets of  $M$ . For any two different elements  $\{a,b\}, \{x,y\}$  in  $T$ , the integer  $(ax+by)(ay+bx)$  is not divisible by 11. Find the maximum size of  $T$ .

Answer: 25


For any  $n\in\mathbb N$ , denote by  $a_n$  the sum  $2+22+222+\cdots+22\ldots2$ , where the last summand consists of  $n$  digits of  $2$ . Determine the greatest  $n$  for which  $a_n$  contains exactly  $222$  digits of  $2$ .

Answer: 222


Find the derivative.

\[ y = \frac{1}{4} \cdot \ln \frac{x-1}{x+1} - \frac{1}{2} \cdot \operatorname{arctg} \, x \]

Answer: \frac{1}{x^4-1}


Given the functions
$$
f(x)=a \sin x(a \in \mathbf{R}), g(x)=\mathrm{e}^{x}.
$$
(1) If $0<a \leqslant 1$, determine the monotonicity of
$$
F(x)=f(1-x)+\ln x
$$
in the interval $(0,1)$;

(2) Prove that:
$$
\sin \frac{1}{2^{2}}+\sin \frac{1}{3^{2}}+\cdots+\sin \frac{1}{(n+1)^{2}}
<\ln 2, \quad n \in \mathbf{Z}_{+};
$$

(3) Given $G(x)=g(x)-m x^{2}-2(x+1)+k$ $(k \in \mathbf{Z})$, for any $x>0$ and $m<0$, $G(x)>0$ always holds. Find the minimum value of $k$.

Answer: 2


Given the function \( f(x) \) defined on the set of real numbers \( \mathbf{R} \) that satisfies \( f^{\prime}(x) - 2 f(x) - 4 > 0 \) and \( f(0) = -1 \), determine the solution to the inequality \( f(x) > e^{2x} - 2 \).

Answer: (0,+\infty)


a tournament is playing between n persons. Everybody plays with everybody one time. There is no draw here. A number  $k$  is called  $n$  good if there is any tournament such that in that tournament they have any player in the tournament that has lost all of  $k$ 's.
prove that
1.  $n$  is greater than or equal to  $2^{k+1}-1$ 2.Find all  $n$  such that  $2$  is a n-good

Answer: n\ge7


Four points are randomly chosen from the vertices of a regular 12-sided polygon. Find the probability that the four chosen points form a rectangle (including square).

Answer: 1/33


The phrase "COLORFUL TARTAN'' is spelled out with wooden blocks, where blocks of the same letter are indistinguishable. How many ways are there to distribute the blocks among two bags of different color such that neither bag contains more than one of the same letter?

Answer: 16


If $p$ and $q$ are primes and $x^2-px+q=0$ has distinct positive integral roots, then which of the following statements are true?
$I.\ \text{The difference of the roots is odd.} \\ II.\ \text{At least one root is prime.} \\ III.\ p^2-q\ \text{is prime}. \\ IV.\ p+q\ \text{is prime}$
$\\ \textbf{(A)}\ I\ \text{only} \qquad \textbf{(B)}\ II\ \text{only} \qquad \textbf{(C)}\ II\ \text{and}\ III\ \text{only} \\ \textbf{(D)}\ I, II, \text{and}\ IV\ \text{only}\ \qquad \textbf{(E)}\ \text{All are true.}$

Answer: \textbf{(E)}


Let  $ABC$  be a right triangle with  $m(\widehat{A})=90^\circ$ . Let  $APQR$  be a square with area  $9$  such that  $P\in [AC]$ ,  $Q\in [BC]$ ,  $R\in [AB]$ . Let  $KLMN$  be a square with area  $8$  such that  $N,K\in [BC]$ ,  $M\in [AB]$ , and  $L\in [AC]$ . What is  $|AB|+|AC|$ ?

Answer: 12


There are 2013 cards with the digit 1 written on them and 2013 cards with the digit 2 written on them. Vasya arranges these cards into a 4026-digit number. On each turn, Petya can swap any two cards and pay Vasya 1 ruble. The process ends when Petya forms a number divisible by 11. What is the maximum amount of money Vasya can earn if Petya aims to pay as little as possible?

Answer: 5


Points $A_{1}$ and $C_{1}$ are located on the sides $BC$ and $AB$ of triangle $ABC$. Segments $AA_{1}$ and $CC_{1}$ intersect at point $M$.

In what ratio does line $BM$ divide side $AC$, if $AC_{1}: C_{1}B = 2: 3$ and $BA_{1}: A_{1}C = 1: 2$?

Answer: 1:3


Answer: \frac{1}{2}


Let $S$ be a set of size 3. How many collections $T$ of subsets of $S$ have the property that for any two subsets $U \in T$ and $V \in T$, both $U \cap V$ and $U \cup V$ are in $T$ ?

Answer: 74


If two different properties are randomly selected from the five types of properties (metal, wood, water, fire, and earth) where metal overcomes wood, wood overcomes earth, earth overcomes water, water overcomes fire, and fire overcomes metal, determine the probability that the two selected properties do not overcome each other.

Answer: \dfrac{1}{2}


How many natural numbers \( n \) less than \( 10^{6} \) have a digit sum that is even, and for which the digit sum of \( (n+1) \) is also even?

Answer: 45454


There are  $63$  houses at the distance of  $1, 2, 3, . . . , 63 \text{ km}$  from the north pole, respectively. Santa Clause wants to distribute vaccine to each house. To do so, he will let his assistants,  $63$  elfs named  $E_1, E_2, . . . , E_{63}$  , deliever the vaccine to each house; each elf will deliever vaccine to exactly one house and never return. Suppose that the elf  $E_n$   takes  $n$  minutes to travel  $1 \text{ km}$  for each  $n = 1,2,...,63$  , and that all elfs leave the north pole simultaneously. What is the minimum amount of time to complete the delivery?

Answer: 1024


In a game called "set," all possible four-digit numbers consisting of the digits 1, 2, and 3 (each used exactly once) are considered. A triplet of numbers forms a set if for each digit position, either all three numbers have the same digit, or all three numbers have different digits.

For example, the numbers 1232, 2213, 3221 form a set because in the first position, all three digits (1, 2, and 3) are present; in the second position, only the digit 2 is present; in the third position, all three digits are present; and in the fourth position, all three digits are present. The numbers 1123, 2231, and 3311 do not form a set because in the last position there are two 1s and a 3.

How many total sets exist in the game?

(Note: Rearranging the numbers does not create a new set. For instance, 1232, 2213, and 3221 is considered the same set as 2213, 1232, and 3221.)

Answer: 1080


How many ways can the integers from -7 to 7 be arranged in a sequence such that the absolute value of the numbers in the sequence is nondecreasing?

Answer: 128


In the rectangle \(ABCD\), it is known that \(AB = 2\) and \(AD < \sqrt{2}\). An ellipse \(K\) is drawn with \(AB\) as the major axis and the length of the minor axis equal to \(\sqrt{2}|AD|\). A point \(P\) on ellipse \(K\) is chosen, different from the endpoints of the major axis. Connect \(PC\) and \(PD\) which intersect \(AB\) at points \(M\) and \(L\), respectively. Find the value of \(AM^2 + LB^2\).

Answer: 4


Péter took one stamp each of values 1, 2, 3, ..., 37 forints from his stamp collection. He wants to group them so that each group has the same total value of stamps. In how many ways can he do this?

Answer: 2


In convex quadrilateral \(EFGH\), \(\angle E = \angle G\), \(EF = GH = 150\), and \(EH \neq FG\). The perimeter of \(EFGH\) is 580. Find \(\cos E\).

Answer: \frac{14}{15}


There are six rock specimens with weights of 8.5 kg, 6 kg, 4 kg, 4 kg, 3 kg, and 2 kg. They need to be distributed into three backpacks such that the heaviest backpack is as light as possible. What is the weight of the rock specimens in the heaviest backpack?

Answer: 10


Answer: 82


In the country of East Westmore, statisticians estimate there is a baby born every $8$ hours and a death every day. To the nearest hundred, how many people are added to the population of East Westmore each year?
$\textbf{(A)}\hspace{.05in}600\qquad\textbf{(B)}\hspace{.05in}700\qquad\textbf{(C)}\hspace{.05in}800\qquad\textbf{(D)}\hspace{.05in}900\qquad\textbf{(E)}\hspace{.05in}1000$

Answer: \textbf{(B)}\700


A quadrilateral \(ABCD\), whose diagonals are perpendicular to each other, is inscribed in a circle with center \(O\).

Find the distance from point \(O\) to side \(AB\), given that \(CD = 8\).

Answer: 4


Let \( p \) and \( q \) be two positive integers that are relatively prime. Sequences \(\{a_n\}\) and \(\{b_n\}\) satisfy \( a_1 = p \) and \( b_1 = q \). For \( n \ge 2 \), \( a_n \) and \( b_n \) are defined such that
\[
\frac{a_n}{b_n} - \frac{a_{n-1}}{b_{n-1}} = \frac{1}{b_n b_{n-1}},
\]
and \( a_n > a_{n-1} \) so that \( a_n \) is the smallest positive integer. Prove that \(\{a_n - b_n\}\) forms an arithmetic sequence.

Answer: {a_{n}-b_{n}}


The fictional country of Isoland uses a 6-letter license plate system using the same 12-letter alphabet as the Rotokas of Papua New Guinea (A, E, G, I, K, O, P, R, T, U, V). Design a license plate that starts with a vowel (A, E, I, O, U), ends with a consonant (G, K, P, R, T, V), contains no repeated letters and does not include the letter S.

Answer: 151200


Petya bought a "Constructor" set that contains 100 sticks of different lengths. The instructions for the "Constructor" state that any three sticks from the set can form a triangle. Petya decided to test this claim by trying to form triangles from the sticks. The sticks are sorted in ascending order of length. What is the minimum number of checks (in the worst-case scenario) Petya needs to make to prove or disprove the instruction's claim?

Answer: 1


Mila and Zhenya each thought of a number and wrote down all the natural divisors of their numbers on the board. Mila wrote 10 numbers, Zhenya 9 numbers, and the number 6 appeared twice. How many distinct numbers are on the board in total?

Answer: 13


Let  $ABC$  be a triangle in which  $\angle ABC = 60^{\circ}$ . Let  $I$  and  $O$  be the incentre and circumcentre of  $ABC$ , respectively. Let  $M$  be the midpoint of the arc  $BC$  of the circumcircle of  $ABC$ , which does not contain the point  $A$ . Determine  $\angle BAC$  given that  $MB = OI$ .

Answer: 30


Given a trapezoid \(ABCD\), where \(BC = a\) and \(AD = b\). A line parallel to the bases \(BC\) and \(AD\) is drawn, intersecting side \(AB\) at point \(P\), diagonal \(AC\) at point \(L\), diagonal \(BD\) at point \(R\), and side \(CD\) at point \(Q\). It is known that \(PL = LR\). Find \(PQ\).

Answer: \frac{3ab}{2a+b}


The lateral edges of a triangular pyramid are pairwise perpendicular and equal to \(a\), \(b\), and \(c\). Find the volume of the pyramid.

Answer: \frac{1}{6}abc


A fair $6$-sided die is repeatedly rolled until an odd number appears. What is the probability that every even number appears at least once before the first occurrence of an odd number?
$\textbf{(A)} ~\frac{1}{120} \qquad\textbf{(B)} ~\frac{1}{32} \qquad\textbf{(C)} ~\frac{1}{20} \qquad\textbf{(D)} ~\frac{3}{20} \qquad\textbf{(E)} ~\frac{1}{6}$

Answer: \textbf{(C)}~\frac{1}{20}


If the area of the circle $ EFG $ is $\frac{(c+4 \sqrt{3})\pi}{3}$, find the value of $ c $.

Answer: 7


The polynomial $f(x)=x^{2007}+17x^{2006}+1$ has distinct zeroes $r_1,\ldots,r_{2007}$.  A polynomial $P$ of degree $2007$ has the property that
\[P\left(r_j+\dfrac{1}{r_j}\right)=0\]for $j=1,\ldots,2007$.  Determine the value of $\frac{P(1)}{P(-1)}$.

Answer: \frac{289}{259}


In a Cartesian coordinate system, the points where both the x-coordinate and y-coordinate are integers are called lattice points. How many lattice points (x, y) satisfy the inequality \((|x|-1)^{2}+(|y|-1)^{2}<2\)?

Answer: 16


The students' written work has a binary grading system, i.e., a work will either be accepted if it is done well or not accepted if done poorly. Initially, the works are checked by a neural network which makes an error in 10% of the cases. All works identified as not accepted by the neural network are then rechecked manually by experts who do not make mistakes. The neural network can misclassify good work as not accepted and bad work as accepted. It is known that among all the submitted works, 20% are actually bad. What is the minimum percentage of bad works among those rechecked by the experts after the selection by the neural network? Indicate the integer part of the number in your answer.

Answer: 66


Given a convex quadrilateral \(ABCD\) inscribed in circle \(O\), let diagonals \(AC\) and \(BD\) intersect at point \(P\). The circumcircles of triangles \(\triangle ABP\) and \(\triangle CDP\) intersect at points \(P\) and \(Q\), where \(O\), \(P\), and \(Q\) are distinct from each other. Prove that \(\angle OQP = 90^\circ\).

Answer: \angleOQP=90


A median $BM$ is drawn in triangle $ABC$, a median $BN$ is drawn in triangle $MCB$, and a median $NK$ is drawn in triangle $BNA$. It turns out that $NK \perp BM$. Find the ratio $AC : BC$.

Answer: 2


Find the value of: \(\frac{\left(\sqrt{3} \cdot \tan 12^{\circ} - 3\right) \cdot \csc 12^{\circ}}{4 \cos ^{2} 12^{\circ} - 2}\).

Answer: -4 \sqrt{3}


Find all odd natural numbers greater than 500 but less than 1000, each of which has the property that the sum of the last digits of all its divisors (including 1 and the number itself) is equal to 33.

Answer: 729


In an isosceles triangle, the heights drawn to the base and to the lateral side are respectively 10 cm and 12 cm. Find the length of the base.

Answer: 15\,


In the ellipse \(\Gamma\), \(A\) is an endpoint of the major axis, \(B\) is an endpoint of the minor axis, and \(F_{1}, F_{2}\) are the two foci. If \(\overrightarrow{A F_{1}} \cdot \overrightarrow{A F_{2}}+\overrightarrow{B F_{1}} \cdot \overrightarrow{B F_{2}}=0\), find the value of \(\tan \angle A B F_{1} \cdot \tan \angle A B F_{2}\).

Answer: -\frac{1}{5}


Odell and Kershaw run for $30$ minutes on a circular track. Odell runs clockwise at $250 m/min$ and uses the inner lane with a radius of $50$ meters. Kershaw runs counterclockwise at $300 m/min$ and uses the outer lane with a radius of $60$ meters, starting on the same radial line as Odell. How many times after the start do they pass each other?

Answer: 47


Given that $\alpha, \beta, \gamma$ are all acute angles and $\cos^{2} \alpha + \cos^{2} \beta + \cos^{2} \gamma = 1$, find the minimum value of $\tan \alpha \cdot \tan \beta \cdot \tan \gamma$.

Answer: 2\sqrt{2}


Let \(ABCD\) be a convex cyclic quadrilateral with the circumcenter \(U\) where the diagonals are perpendicular to each other. Let \(g\) be the line obtained by reflecting the diagonal \(AC\) over the angle bisector of \(\angle BAD\).

Prove that the point \(U\) lies on the line \(g\).

Answer: 0


Let point \( P \) be a moving point on the circle \( C: (x+2)^2 + y^2 = 4 \), and it is not at the origin. The coordinates of the fixed point \( A \) are \( (2, 0) \). The perpendicular bisector of the line segment \( AP \) intersects the line \( CP \) at point \( Q \). Given that \( M \) has coordinates \( (-1,0) \) and \( N \) has coordinates \( (1,0) \), find the product of the slopes of the lines \( MQ \) and \( NQ \).

Answer: 3


Four friends — Alex, Betty, Clara, and Dave — participated in a relay race by running in pairs, with one pair sitting out each race. Dave ran in 8 races, which was more than any other friend, and Betty ran in 3 races, which was fewer than any other friend. Determine the total number of races those pairs completed.

Answer: 10


In a triangle without obtuse angles, two squares are inscribed such that the vertices of the squares lie on the sides of the triangle. Prove that:
1) If the squares are equal, then the triangle is either isosceles or right-angled.
2) If the squares are not equal, then two vertices of the larger square lie on a side that is smaller than the side of the triangle on which two vertices of the smaller square lie.

Answer: 2


A sequence of twelve \(0\)s and/or \(1\)s is randomly generated and must start with a '1'. If the probability that this sequence does not contain two consecutive \(1\)s can be written in the form \(\dfrac{m}{n}\), where \(m,n\) are relatively prime positive integers, find \(m+n\).

Answer: 2281


There are three pairs of real numbers $(x_1,y_1)$, $(x_2,y_2)$, and $(x_3,y_3)$ that satisfy both $x^3 - 3xy^2 = 2010$ and $y^3 - 3x^2y = 2000$. Compute $\left(1-\frac{x_1}{y_1}\right)\left(1-\frac{x_2}{y_2}\right)\left(1-\frac{x_3}{y_3}\right)$.

Answer: \frac{1}{100}


An aluminum cube with an edge length of \( l = 10 \) cm is heated to a temperature of \( t_{1} = 100^{\circ} \mathrm{C} \). After this, it is placed on ice, which has a temperature of \( t_{2} = 0^{\circ} \mathrm{C} \). Determine the maximum depth to which the cube can sink. The specific heat capacity of aluminum is \( c_{a} = 900 \) J/kg\(^\circ \mathrm{C} \), the specific latent heat of fusion of ice is \( \lambda = 3.3 \times 10^{5} \) J/kg, the density of aluminum is \( \rho_{a} = 2700 \) kg/m\(^3 \), and the density of ice is \( \rho_{n} = 900 \) kg/m\(^3 \).

Answer: 0.0818


A point $(x,y)$ in the plane is called a lattice point if both $x$ and $y$ are integers. The area of the largest square that contains exactly three lattice points in its interior is closest to

Answer: 5.0


Given that $f(x)$ has a derivative and satisfies $\lim_{\Delta x \to 0} \, \frac{f(1)-f(1-2\Delta x)}{2\Delta x}=-1$, find the slope of the tangent line to the curve $y=f(x)$ at the point $(1,f(1))$.

Answer: -1


Let \\(f(x)=3\sin (\omega x+ \frac {\pi}{6})\\), where \\(\omega > 0\\) and \\(x\in(-\infty,+\infty)\\), and the function has a minimum period of \\(\frac {\pi}{2}\\).

\\((1)\\) Find \\(f(0)\\).

\\((2)\\) Find the expression for \\(f(x)\\).

\\((3)\\) Given that \\(f( \frac {\alpha}{4}+ \frac {\pi}{12})= \frac {9}{5}\\), find the value of \\(\sin \alpha\\).

Answer: \frac {4}{5}


Each of the two students wrote down three consecutive natural numbers, the middle of which is the cube of a natural number. Then they multiplied all six numbers together. Prove that the resulting product is divisible by 5184.

Answer: 5184


Professor Gamble buys a lottery ticket, which requires that he pick six different integers from $1$ through $46$, inclusive. He chooses his numbers so that the sum of the base-ten logarithms of his six numbers is an integer. It so happens that the integers on the winning ticket have the same property— the sum of the base-ten logarithms is an integer. What is the probability that Professor Gamble holds the winning ticket?

Answer: \frac{1}{4}


Given $0 < \beta < \frac{\pi}{2} < \alpha < \pi$ and $\cos \left(\alpha- \frac{\beta}{2}\right)=- \frac{1}{9}, \sin \left( \frac{\alpha}{2}-\beta\right)= \frac{2}{3}$, calculate the value of $\cos (\alpha+\beta)$.

Answer: -\frac{239}{729}


In a Cartesian coordinate system, the parametric equation for curve $C_1$ is
$$
\left\{ \begin{aligned}
x &= 2\cos\alpha, \\
y &= \sqrt{2}\sin\alpha
\end{aligned} \right.
$$
with $\alpha$ as the parameter. Using the origin as the pole, the positive half of the $x$-axis as the polar axis, and the same unit length as the Cartesian coordinate system, establish a polar coordinate system. The polar equation for curve $C_2$ is $\rho = \cos\theta$.

(1) Find the general equation for curve $C_1$ and the Cartesian coordinate equation for curve $C_2$;

(2) If $P$ and $Q$ are any points on the curves $C_1$ and $C_2$, respectively, find the minimum value of $|PQ|$.

Answer: \frac{\sqrt{7} - 1}{2}


In how many ways can the digits of $45,\!502,\!2$ be arranged to form a 6-digit number? (Remember, numbers cannot begin with 0.)

Answer: 150


In a group photo, 4 boys and 3 girls are to stand in a row such that no two boys or two girls stand next to each other. How many different arrangements are possible?

Answer: 144


Let \(a, b, c\) be positive integers such that \(ab + bc - ca = 0\) and \(a - c = 101\). Find \(b\).

Answer: 2550


A number \( n \) has a sum of digits equal to 100, while \( 44n \) has a sum of digits equal to 800. Find the sum of the digits of \( 3n \).

Answer: 300


For each integer $n\geq3$, let $f(n)$ be the number of $3$-element subsets of the vertices of the regular $n$-gon that are the vertices of an isosceles triangle (including equilateral triangles). Find the sum of all values of $n$ such that $f(n+1)=f(n)+78$.

Answer: 245


Consider the function defined piecewise by
\[ f(x) = \left\{ \begin{aligned}
2x + 1 & \quad \text{if } x < 1 \\
x^2 & \quad \text{if } x \ge 1
\end{aligned} \right. \]
Determine the value of \( f^{-1}(-3) + f^{-1}(-1) + f^{-1}(1) + f^{-1}(3) + f^{-1}(9) \).

Answer: 1 + \sqrt{3}


The equation \(2008=1111+444+222+99+77+55\) is an example of decomposing the number 2008 as a sum of distinct numbers with more than one digit, where each number's representation (in the decimal system) uses only one digit.

i) Find a similar decomposition for the number 2009.

ii) Determine all possible such decompositions of the number 2009 that use the minimum number of terms (the order of terms does not matter).

Answer: 1111 + 777 + 66 + 55


Let \( l \) and \( m \) be two skew lines. On \( l \), there are three points \( A \), \( B \), and \( C \) such that \( AB = BC \). Perpendicular lines \( AD \), \( BE \), and \( CF \) are drawn from \( A \), \( B \), and \( C \) to \( m \), respectively, with \( D \), \( E \), and \( F \) as the feet of the perpendiculars. It is given that \( AD = \sqrt{15} \), \( BE = \frac{7}{2} \), and \( CF = \sqrt{10} \). Find the distance between \( l \) and \( m \).

Answer: \sqrt{6}


Two congruent right circular cones each with base radius  $3$  and height  $8$  have axes of symmetry that intersect at right angles at a point in the interior of the cones a distance  $3$  from the base of each cone. A sphere with radius  $r$  lies inside both cones. The maximum possible value for  $r^2$  is  $\frac mn$ , where  $m$  and  $n$  are relatively prime positive integers. Find  $m+n$ .

Answer: 298


Let $A$ be a set of ten distinct positive numbers (not necessarily integers). Determine the maximum possible number of arithmetic progressions consisting of three distinct numbers from the set $A$.

Answer: 20


Randomly fill the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9 into a $3 \times 3$ grid, with each small square filled with one number and all numbers being distinct. What is the probability that the sum of the numbers in each row and each column is odd?

Answer: \frac{1}{14}


On the coordinate line, the points with coordinates \(0, 1, 2, \ldots, 2n-1\) are marked, where \(n\) is a positive integer. A flea starts jumping from the point with coordinate 0 and after \(2n\) jumps returns there, having visited all the marked points. It is known that the total length of all jumps except the last one is \(n(2n-1)\). Find the length of the last jump.

Answer: n


Let $\mathbf{p}$ be the projection of vector $\mathbf{v}$ onto vector $\mathbf{u},$ and let $\mathbf{q}$ be the projection of $\mathbf{p}$ onto $\mathbf{u}.$ If $\frac{\|\mathbf{p}\|}{\|\mathbf{v}\|} = \frac{3}{4},$ then find $\frac{\|\mathbf{q}\|}{\|\mathbf{u}\|}.$

Answer: \frac{9}{16}


What is $1254_6 - 432_6 + 221_6$? Express your answer in base $6$.

Answer: 1043_6


In triangle \( \triangle ABC \) with \( AC > AB \), let \( P \) be the intersection point of the perpendicular bisector of \( BC \) and the internal angle bisector of \( \angle A \). Construct \( PX \perp AB \), intersecting the extension of \( AB \) at point \( X \). Also, construct \( PY \perp AC \), intersecting \( AC \) at point \( Y \). Let \( Z \) be the intersection of \( XY \) and \( BC \). Determine the value of \( \frac{BZ}{ZC} \).

Answer: 1


Let \( P(x) \) be a polynomial with leading coefficient 1, and let the sequence of integers \( a_1, a_2, a_3, \ldots \) be such that \( P(a_1) = 0 \), \( P(a_2) = a_1 \), \( P(a_3) = a_2 \), and so on. The numbers in the sequence do not repeat. What degree can \( P(x) \) have?

Answer: 1


Prove that for any real  $ x $  and  $ y $  the inequality  $x^2 \sqrt {1+2y^2} + y^2 \sqrt {1+2x^2} \geq xy (x+y+\sqrt{2})$  .

Answer: x^2\sqrt{1+2y^2}+y^2\sqrt{1+2x^2}\gexy(x+y+\sqrt{2})


There are two points  $A$  and  $B$  in the plane.
a) Determine the set  $M$  of all points  $C$  in the plane for which  $|AC|^2 +|BC|^2 = 2\cdot|AB|^2.$ b) Decide whether there is a point  $C\in M$  such that   $\angle ACB$  is maximal and if so, determine this angle.

Answer: 60


In country A with 2011 cities and country B with 2011 cities, there is a unique bidirectional flight route between any city in country A and any city in country B. There are no routes between cities within the same country, and at most 19 different airlines operate these routes. Find the largest integer \( k \) such that no matter how the routes are assigned, there will always exist \( k \) cities that are connected (where you can travel from one city to another, although not necessarily directly) and all the routes are operated by the same airline.

Answer: 212


From the digits 0, 1, 2, 3, 4, 5, select three different digits to form a three-digit even number. How many such numbers can be formed? (Answer with a number)

Answer: 52


For a natural number  $n$  denote by Map  $( n )$  the set of all functions  $f  :  \{ 1 , 2 , 3 , \cdots , n \}  \rightarrow  \{ 1 , 2 , 3 , \cdots , n \} . $  For  $ f , g \in $  Map $( n ) , f \circ g $  denotes the function in Map  $( n )$  that sends  $x  \rightarrow  f ( g ( x ) ) . $   $(a)$  Let  $ f \in$  Map  $( n ) . $  If for all  $x \in \{ 1 , 2 , 3 , \cdots , n \}  f ( x ) \neq x , $  show that  $ f \circ f \neq f $   $(b)$  Count the number of functions  $ f \in$   Map  $( n )$   such that  $ f \circ f = f $

Answer: \sum_{k=1}^n\binom{n}{k}k^{n-k}


In the trapezoid \(ABCD\), if \(AB = 8\), \(DC = 10\), the area of \(\triangle AMD\) is 10, and the area of \(\triangle BCM\) is 15, then the area of trapezoid \(ABCD\) is \(\quad\).

Answer: 45


An arithmetic sequence consists of two-digit even terms, where the sum of all odd terms is 100. Starting from the first term, each odd term is combined with the following adjacent even term to form a four-digit number without changing the order. Determine the difference between the sum of the new sequence and the sum of the original sequence.

Answer: 9900


Alice knows that $3$ red cards and $3$ black cards will be revealed to her one at a time in random order. Before each card is revealed, Alice must guess its color. If Alice plays optimally, the expected number of cards she will guess correctly is $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$

Answer: 051


Find all odd prime numbers  $p$  for which there exists a natural number  $g$  for which the sets \[A=\left\{ \left( {{k}^{2}}+1 \right)\,\bmod p|\,k=1,2,\ldots ,\frac{p-1}{2} \right\}\] and \[B=\left\{ {{g}^{k}}\bmod \,p|\,k=1,2,...,\frac{p-1}{2} \right\}\] are equal.

Answer: 3


The three sides of a right triangle have integral lengths which form an arithmetic progression. One of the sides could have length
$\textbf{(A)}\ 22\qquad\textbf{(B)}\ 58\qquad\textbf{(C)}\ 81\qquad\textbf{(D)}\ 91\qquad\textbf{(E)}\ 361$

Answer: \textbf{(C)}\81


Consider equilateral triangle  $ABC$  with side length  $1$ . Suppose that a point  $P$  in the plane of the triangle satisfies \[2AP=3BP=3CP=\kappa\] for some constant  $\kappa$ . Compute the sum of all possible values of  $\kappa$ .

*2018 CCA Math Bonanza Lightning Round #3.4*

Answer: \frac{18\sqrt{3}}{5}


If the graph of the power function $y=f(x)$ passes through the point $\left( -2,-\frac{1}{8} \right)$, find the value(s) of $x$ that satisfy $f(x)=27$.

Answer: \frac{1}{3}


From an 8x8 chessboard, 10 squares were cut out. It is known that among the removed squares, there are both black and white squares. What is the maximum number of two-square rectangles (dominoes) that can still be guaranteed to be cut out from this board?

Answer: 23


Rationalize the denominator of $\displaystyle \frac{1}{\sqrt{2} + \sqrt{3} + \sqrt{7}}$, and write your answer in the form \[
\frac{A\sqrt{2} + B\sqrt{3} + C\sqrt{7} + D\sqrt{E}}{F},
\]where everything is in simplest radical form and the fraction is in lowest terms, and $F$ is positive. What is $A + B + C + D + E + F$?

Answer: 57


Triangle $ABC$ has $AB=2 \cdot AC$. Let $D$ and $E$ be on $\overline{AB}$ and $\overline{BC}$, respectively, such that $\angle BAE = \angle ACD$. Let $F$ be the intersection of segments $AE$ and $CD$, and suppose that $\triangle CFE$ is equilateral. What is $\angle ACB$?
$\textbf{(A)}\ 60^\circ \qquad \textbf{(B)}\ 75^\circ \qquad \textbf{(C)}\ 90^\circ \qquad \textbf{(D)}\ 105^\circ \qquad \textbf{(E)}\ 120^\circ$

Answer: 90\,\textbf{(C)}


What is the largest \( n \), for which it is possible to place \( n \) points on the plane such that any 3 of them form the vertices of a right triangle?

Answer: 4


Find the area of the shape enclosed by the curve $y=x^2$ (where $x>0$), the tangent line at point A(2, 4), and the x-axis.

Answer: \frac{2}{3}


The sum of $n$ terms of an arithmetic progression is $153$, and the common difference is $2$.
If the first term is an integer, and $n>1$, then the number of possible values for $n$ is:
$\textbf{(A)}\ 2\qquad \textbf{(B)}\ 3\qquad \textbf{(C)}\ 4\qquad \textbf{(D)}\ 5\qquad \textbf{(E)}\ 6$

Answer: \textbf{(D)}\5


Find the remainder when 53! is divided by 59.

Answer: 30


The poetry lecture lasted 2 hours and $m$ minutes. The positions of the hour and minute hands on the clock at the end of the lecture are exactly swapped from their positions at the beginning of the lecture. If $[x]$ denotes the integer part of the decimal number $x$, find $[m]=$ $\qquad$ .

Answer: 46


Let   $K, L, M$ , and  $N$  be the midpoints of  $CD,DA,AB$  and  $BC$  of a square  $ABCD$  respectively. Find the are of the triangles  $AKB, BLC, CMD$  and  $DNA$  if the square  $ABCD$  has area  $1$ .

Answer: \frac{1}{2}


Answer: \frac{6}{11}


The sequence $(a_n)$ is defined recursively by $a_0=1$, $a_1=\sqrt[19]{2}$, and $a_n=a_{n-1}a_{n-2}^2$ for $n\geq 2$. What is the smallest positive integer $k$ such that the product $a_1a_2\cdots a_k$ is an integer?

Answer: 17


Alyosha is shading cells within a $6 \times 6$ square drawn on graph paper. Then he marks the grid nodes (intersections of the graph paper lines) that are adjacent to an equal number of shaded and unshaded squares. What is the maximum number of nodes that can be marked?

Answer: 45


Fill the numbers 1, 2, 3 into a 3×3 grid such that each row and each column contains no repeated numbers. How many different ways can this be done?

Answer: 12


An ordered pair  $(n,p)$  is *juicy* if  $n^{2} \equiv 1 \pmod{p^{2}}$  and  $n \equiv -1 \pmod{p}$  for positive integer  $n$  and odd prime  $p$ . How many juicy pairs exist such that  $n,p \leq 200$ ?

Proposed by Harry Chen (Extile)

Answer: 36


$K$ takes $30$ minutes less time than $M$ to travel a distance of $30$ miles. $K$ travels $\frac {1}{3}$ mile per hour faster than $M$. If $x$ is $K$'s rate of speed in miles per hours, then $K$'s time for the distance is:

Answer: \frac{30}{x}


The product of a million whole numbers is equal to million. What can be the greatest possible value of the sum of these numbers?

Answer: 1999999


Let \( S = \{1, 2, 3, 4\} \) and let the sequence \( a_{1}, a_{2}, \cdots, a_{n} \) have the following property: For any non-empty subset \( B \) of \( S \) (with the number of elements in set \( B \) denoted as \( |B| \)), there exist \( |B| \) consecutive terms in the sequence that exactly form the set \( B \). Find the smallest possible value of \( n \).

Answer: 8


There are some books and some exercise books. If each student is given 1 book and 2 exercise books, there are 2 books remaining when all exercise books are distributed. If each student is given 3 books and 5 exercise books, there is 1 exercise book remaining when all books are distributed. How many books and exercise books are there?


Each day Maria must work $8$ hours.  This does not include the $45$ minutes she takes for lunch.  If she begins working at $\text{7:25 A.M.}$ and takes her lunch break at noon, then her working day will end at
$\text{(A)}\ \text{3:40 P.M.} \qquad \text{(B)}\ \text{3:55 P.M.} \qquad \text{(C)}\ \text{4:10 P.M.} \qquad \text{(D)}\ \text{4:25 P.M.} \qquad \text{(E)}\ \text{4:40 P.M.}$

Answer: (C)\4:10P.M.


The volume of a rectangular block is \(120 \, \mathrm{cm}^3\). If the area of its base is \(24 \, \mathrm{cm}^2\), what is its height?
(A) \(5 \, \mathrm{cm}\)
(B) \(15 \, \mathrm{cm}\)
(C) \(0.2 \, \mathrm{cm}\)
(D) \(0.6 \, \mathrm{cm}\)
(E) \(1 \, \mathrm{cm}\)

Answer: 5\,


Let  $x_1,x_2,\cdots,x_n$  be postive real numbers such that  $x_1x_2\cdots x_n=1$  , $S=x^3_1+x^3_2+\cdots+x^3_n$ .Find the maximum of  $\frac{x_1}{S-x^3_1+x^2_1}+\frac{x_2}{S-x^3_2+x^2_2}+\cdots+\frac{x_n}{S-x^3_n+x^2_n}$

Answer: 1


Find the number of five-digit numbers that contain at least one digit 8 in their decimal representation.

Answer: 37512


$a$ , $b$ , $c$  are real. What is the highest value of  $a+b+c$  if  $a^2+4b^2+9c^2-2a-12b+6c+2=0$

Answer: \frac{17}{3}


Let \( D \) be a point inside the acute triangle \( \triangle ABC \) such that \( \angle ADB = \angle ACB + 90^\circ \) and \( AC \cdot BD = AD \cdot BC \).
1. Find the value of \( \frac{AB \cdot CD}{AC \cdot BD} \).
2. Prove that the tangents to the circumcircles of \( \triangle ACD \) and \( \triangle BCD \) at point \( C \) are perpendicular to each other.

Answer: \sqrt{2}


In a corridor 100 meters long, 20 carpet strips with a total length of 1000 meters are laid down. What could be the maximum number of uncovered sections (the width of the carpet strip is equal to the width of the corridor)?

Answer: 10


Given the sequences $\left\{a_{n}\right\}$ and $\left\{b_{n}\right\}$, where $a_{1} = p$ and $b_{1} = q$, and it is known that $a_{n} = p a_{n-1}$ and $b_{n} = q a_{n-1} + r b_{n-1}$ for constants $p$, $q$, $r$ (with $q > 0$, $p > r > 0$, $n \geq 2$), find the general term formula for the sequence $\left\{b_{n}\right\}$, denoted as $b_{n}=$ $\qquad$ .

Answer: \frac{q(p^n-r^n)}{p-r}


Majka examined multi-digit numbers in which odd and even digits alternate regularly. Those that start with an odd digit, she called "funny," and those that start with an even digit, she called "cheerful" (for example, the number 32387 is funny, the number 4529 is cheerful).

Majka created one three-digit funny number and one three-digit cheerful number, using six different digits without including 0. The sum of these two numbers was 1617. The product of these two numbers ended with the digits 40.

Determine Majka's numbers and calculate their product.

Answer: 635040


Michael picks a random subset of the complex numbers $\left\{1, \omega, \omega^{2}, \ldots, \omega^{2017}\right\}$ where $\omega$ is a primitive $2018^{\text {th }}$ root of unity and all subsets are equally likely to be chosen. If the sum of the elements in his subset is $S$, what is the expected value of $|S|^{2}$? (The sum of the elements of the empty set is 0.)

Answer: \frac{1009}{2}


Find the sum of all distinct possible values of  $x^2-4x+100$ , where  $x$  is an integer between 1 and 100, inclusive.

*Proposed by Robin Park*

Answer: 328053


A standard die is rolled eight times. What is the probability that the product of all eight rolls is odd and consists only of prime numbers? Express your answer as a common fraction.

Answer: \frac{1}{6561}


When \( q(x) = Dx^4 + Ex^2 + Fx + 6 \) is divided by \( x - 2 \), the remainder is 14. Find the remainder when \( q(x) \) is divided by \( x + 2 \).

Answer: 14


Kermit the frog enjoys hopping around the infinite square grid in his backyard. It takes him 1 Joule of energy to hop one step north or one step south, and 1 Joule of energy to hop one step east or one step west. He wakes up one morning on the grid with 100 Joules of energy, and hops till he falls asleep with 0 energy. How many different places could he have gone to sleep?

Answer: 10201


Determine the number of angles \(\theta\) between 0 and \(2\pi\), other than integer multiples of \(\pi / 2\), such that the quantities \(\sin \theta\), \(\cos \theta\), and \(\tan \theta\) form a geometric sequence in some order.

Answer: 4


Ann and Drew have purchased a mysterious slot machine; each time it is spun, it chooses a random positive integer such that  $k$  is chosen with probability  $2^{-k}$  for every positive integer  $k$ , and then it outputs  $k$  tokens. Let  $N$  be a fixed integer. Ann and Drew alternate turns spinning the machine, with Ann going first. Ann wins if she receives at least  $N$  total tokens from the slot machine before Drew receives at least  $M=2^{2018}$  total tokens, and Drew wins if he receives  $M$  tokens before Ann receives  $N$  tokens. If each person has the same probability of winning, compute the remainder when  $N$  is divided by  $2018$ .

*Proposed by Brandon Wang*

Answer: 2


Given \( n > 2 \) natural numbers such that there are no three equal numbers among them, and the sum of any two of these numbers is a prime number. What is the largest possible value of \( n \)?

Answer: 3


Find all integers \( x, y \geq 1 \) such that \( x^3 - y^3 = xy + 61 \).

Answer: (6,5)


In a math test at Yangcun Primary School for the fourth grade, there were 10 questions. If Xiaoxi, Xiaofei, Xianguan, and Xialan each answered exactly 8 questions correctly, then the minimum number of questions that all four of them answered correctly is $\qquad$.

Answer: 2


Given that the first term of a geometric sequence $\{{a_n}\}$ is $\frac{3}{2}$ and the sum of its first $n$ terms is $S_n$ $(n \in \mathbb{N^*})$, and $-2S_2, S_3, 4S_4$ form an arithmetic sequence.

(I) Find the general term formula of the sequence $\{{a_n}\}$.

(II) Find the maximum and minimum values of $S_n$ $(n \in \mathbb{N^*})$.

Answer: \frac{3}{4}


What is the maximal number of solutions can the equation have   $$ \max  \{a_1x+b_1, a_2x+b_2, \ldots, a_{10}x+b_{10}\}=0 $$  where  $a_1,b_1, a_2, b_2, \ldots , a_{10},b_{10}$  are real numbers, all  $a_i$  not equal to  $0$ .

Answer: 2


Given the function
$$
f_{n}(x)=\frac{n x^{2}-x}{x^{2}+1} \quad (n \in \mathbf{Z}_{+}),
$$

let \( x_{1}, x_{2}, \cdots, x_{n} \) be positive real numbers such that \( \sum_{i=1}^{n} x_{i}=1 \). Prove that \( \sum_{i=1}^{n} f_{n}\left(x_{i}\right) \geqslant 0 \).

Answer: 0


A company sells detergent in three different sized boxes: small (S), medium (M) and large (L). The medium size costs 50% more than the small size and contains 20% less detergent than the large size. The large size contains twice as much detergent as the small size and costs 30% more than the medium size. Rank the three sizes from best to worst buy.
$\text{(A)}\, SML \qquad \text{(B)}\, LMS \qquad \text{(C)}\, MSL \qquad \text{(D)}\, LSM \qquad \text{(E)}\, MLS$

Answer: \textbf{(E)}\


Find the sum of all the digits in the decimal representations of all the positive integers less than  $1000.$

Answer: 13500


There are 5 female students and 2 male students in a class. Find the number of different distribution schemes in which they can be divided into two groups, with each group having both female and male students.

Answer: 60


A regular [octahedron](https://artofproblemsolving.com/wiki/index.php/Octahedron) has side length $1$. A [plane](https://artofproblemsolving.com/wiki/index.php/Plane) [parallel](https://artofproblemsolving.com/wiki/index.php/Parallel) to two of its opposite faces cuts the octahedron into the two congruent solids. The [polygon](https://artofproblemsolving.com/wiki/index.php/Polygon) formed by the intersection of the plane and the octahedron has area $\frac {a\sqrt {b}}{c}$, where $a$, $b$, and $c$ are positive integers, $a$ and $c$ are relatively prime, and $b$ is not divisible by the square of any prime. What is $a + b + c$?
$\textbf{(A)}\ 10\qquad \textbf{(B)}\ 11\qquad \textbf{(C)}\ 12\qquad \textbf{(D)}\ 13\qquad \textbf{(E)}\ 14$

Answer: \textbf{(E)}\14


Find the largest real number \( p \) such that all three roots of the equation below are positive integers:
\[
5x^{3} - 5(p+1)x^{2} + (71p-1)x + 1 = 66p .
\]

Answer: 76


It is known that the number of birch trees in a certain mixed forest plot ranges from $13\%$ to $14\%$ of the total number of trees. Find the minimum possible total number of trees in this plot.

Answer: 15


In the table, 8 is located in the 3rd row and 2nd column, 2017 is located in the $a$th row and $b$th column. Find $a - b = \quad$
\begin{tabular}{|c|c|c|}
\hline 1 & 4 & 5 \\
\hline 2 & 3 & 6 \\
\hline 9 & 8 & 7 \\
\hline 10 & 13 & 14 \\
\hline 11 & 12 & 15 \\
\hline 18 & 17 & 16 \\
\hline 19 & 22 & 23 \\
\hline 20 & 21 & $\cdots$ \\
\hline
\end{tabular}

Answer: 672


Find all positive integer bases  $b \ge 9$  so that the number
\[ \frac{{\overbrace{11 \cdots 1}^{n-1 \ 1's}0\overbrace{77 \cdots 7}^{n-1\ 7's}8\overbrace{11 \cdots 1}^{n \ 1's}}_b}{3} \]
is a perfect cube in base 10 for all sufficiently large positive integers  $n$ .

*Proposed by Yang Liu*

Answer: 10


Tom is chasing Jerry on the coordinate plane. Tom starts at  $(x, y)$  and Jerry starts at  $(0, 0)$ . Jerry moves to the right at  $1$  unit per second. At each positive integer time  $t$ , if Tom is within  $1$  unit of Jerry, he hops to Jerry’s location and catches him. Otherwise, Tom hops to the midpoint of his and Jerry’s location.

*Example. If Tom starts at  $(3, 2)$ , then at time  $t = 1$  Tom will be at  $(2, 1)$  and Jerry will be at  $(1, 0)$ . At  $t = 2$  Tom will catch Jerry.*

Assume that Tom catches Jerry at some integer time  $n$ .
(a) Show that  $x \geq 0$ .
(b) Find the maximum possible value of  $\frac{y}{x+1}$ .

Answer: \frac{\sqrt{3}}{3}


The points on this graph show the end-of-year sales amounts for each year. During what year after 1994 did sales increase the most number of dollars?  Express your answer as a 4-digit year.

[asy]unitsize(101/10);
real textsize = 10, largetext = 12;
draw((10,0)--(0,0)--(0,10),Arrows(4,20));
path verticalmark = (0,-.3)--(0,.3), horizontalmark = (-.3,0)--(.3,0);
draw(verticalmark);
for(int i = 1; i<10;++i)

draw(shift(i*right)*verticalmark);
for(int i = 1; i<7; ++i)

draw(shift(9*i/6*up)*horizontalmark);
for(int i = 0; i<3; ++i)
label(string(5(2i+1)),(-.25,9/6 + 9/3*i),left,fontsize(textsize*pt));
for(int i = 0; i<3; ++i)
label("'"+string(94+2i),(2i,-1),fontsize(textsize*pt));
label("'00",(6,-1),fontsize(textsize*pt));
label("'02",(8,-1),fontsize(textsize*pt));
label("\textbf{Year}",(5,-2.5),fontsize(largetext*pt));
label(rotate(90)*"\textbf{Millions of Dollars}",(-2.5,5),fontsize(largetext*pt));
pair A = (0,9/6*2), B = (1,9/6*2.4), C = (2,9/6*3), D = (3,9/6*3.25), E = (4,9/6*4.2), F = (5,9/6*4.5),G = (6,9/6*4.8),H = (7,9/6*5.2), I = (8,9/6*4.75),J = (9,9/6*3.25);
draw(A--B--C--D--E--F--G--H--I--J);
dot(A);dot(B);dot(C);dot(D);dot(E);dot(F);dot(G);dot(H);dot(I);dot(J);
[/asy]

Answer: 1998


Compute the number of ordered pairs of integers $(x,y)$ with $1\le x<y\le 50$ such that $i^x+i^y$ is a real number, and additionally, $x+y$ is divisible by $4$.

Answer: 288


Given the hyperbola $mx^{2}+y^{2}=1$ and one of its asymptotes has a slope of $2$, find the value of $m$.

Answer: -4


A group consisting of 7 young men and 7 young women is randomly paired up. Find the probability that at least one pair consists of two young women. Round your answer to the nearest hundredth.

Answer: 0.96


Circle $C$ with radius 2 has diameter $\overline{AB}$. Circle D is internally tangent to circle $C$ at $A$. Circle $E$ is internally tangent to circle $C$, externally tangent to circle $D$, and tangent to $\overline{AB}$. The radius of circle $D$ is three times the radius of circle $E$, and can be written in the form $\sqrt{m}-n$, where $m$ and $n$ are positive integers. Find $m+n$.

Answer: 254


How many integers  $n$  with  $10 \le n \le 500$  have the property that the hundreds digit of  $17n$  and  $17n+17$  are different?

*Proposed by Evan Chen*

Answer: 84


Let $S$ be the set of all positive factors of 6000. What is the probability of a random quadruple $(a, b, c, d) \in S^{4}$ satisfies $$\operatorname{lcm}(\operatorname{gcd}(a, b), \operatorname{gcd}(c, d))=\operatorname{gcd}(\operatorname{lcm}(a, b), \operatorname{lcm}(c, d)) ?$$

Answer: \frac{41}{512}


A sphere is inscribed in a cone. What maximum fraction of the cone's volume does the sphere occupy?

Answer: \frac{1}{2}


The faces of a hexahedron and an octahedron are equilateral triangles with side length \(a\). The ratio of the radii of the inscribed spheres of these two polyhedra is a simplified fraction \(\frac{m}{n}\). What is the product \(m \cdot n\)?

Answer: 6


Real numbers $x, y$, and $z$ are chosen from the interval $[-1,1]$ independently and uniformly at random. What is the probability that $|x|+|y|+|z|+|x+y+z|=|x+y|+|y+z|+|z+x|$?

Answer: \frac{3}{8}


A rectangle has dimensions $8 \times 12$, and a circle centered at one of its corners has a radius of 10. Calculate the area of the union of the regions enclosed by the rectangle and the circle.

Answer: 96 + 75\pi


For which natural numbers \( n \) is the inequality

$$
\sin n \alpha + \sin n \beta + \sin n \gamma < 0
$$

true for any angles \( \alpha, \beta, \gamma \) that are the angles of an acute triangle?

Answer: 4


The sequence $\left\{x_{n}\right\}$ satisfies $x_{1}=1$, and for any $n \in \mathbb{Z}^{+}$, it holds that $x_{n+1}=x_{n}+3 \sqrt{x_{n}}+\frac{n}{\sqrt{x_{n}}}$. Find the value of $\lim _{n \rightarrow+\infty} \frac{n^{2}}{x_{n}}$.

Answer: \frac{4}{9}


If the legs of a right triangle are in the ratio $3:4$, find the ratio of the areas of the two triangles created by dropping an altitude from the right-angle vertex to the hypotenuse.

Answer: \frac{9}{16}


For how many positive integers  $n \le 500$  is  $n!$  divisible by  $2^{n-2}$ ?

*Proposed by Eugene Chen*

Answer: 44


Point \( O \), lying inside a convex quadrilateral with area \( S \), is reflected symmetrically with respect to the midpoints of its sides.

Find the area of the quadrilateral with vertices at the resulting points.

Answer: 2S


Simplify and write the result as a common fraction: $$\sqrt[4]{\sqrt[3]{\sqrt{\frac{1}{65536}}}}$$

Answer: \frac{1}{\sqrt[3]{4}}


Let $ABC$ be an acute-angled triangle and $D, E,$ and $F$ be the feet of the altitudes from $A, B,$ and $C,$ respectively. Let $P$ (respectively $Q$, respectively $R$) be the foot of the perpendicular from $A$ (respectively $B$, respectively $C$) to $(EF)$ (respectively $(FD)$, respectively $(DE)$). Show that the lines $(AP),(BQ)$, and $(CR)$ are concurrent.

Answer: O


2008 persons take part in a programming contest. In one round, the 2008 programmers are divided into two groups. Find the minimum number of groups such that every two programmers ever be in the same group.

Answer: 11


For any \( x, y \in \mathbb{R} \), find the minimum value of the algebraic expression \( M = \sqrt{2x^2 - 6x + 5} + \sqrt{y^2 - 4y + 5} + \sqrt{2x^2 - 2xy + y^2} \).

Answer: \sqrt{10}


Two circles are constructed on a plane in such a way that each passes through the center of the other. Points $P$ and $Q$ are the points of their intersection. A line passing through point $P$ intersects the first circle at point $A$ and the second circle at point $B$ such that $P$ is between $A$ and $B$. This line makes an angle of $15^{\circ}$ with the line connecting the centers. Find the area of triangle $ABQ$ if it is known that $PQ = 2\sqrt{3}$.

Answer: 3+2\sqrt{3}


The pensioners on one of the planets of Alpha Centauri enjoy spending their free time solving numeric puzzles: they choose natural numbers from a given range $[A, B]$ such that the sum of any two chosen numbers is not divisible by a certain number $N$. Last week, the newspaper "Alpha Centaurian Panorama" offered its readers a puzzle with the values $A=1353$, $B=2134$, and $N=11$. What is the maximum number of numbers that can be the solution to such a puzzle?

Answer: 356


The natural number \( A \) is represented in decimal as \( \overline{a_{n} a_{n-1} \cdots a_{0}} \). Define \( f(A)=2^{n} a_{0}+2^{n-1} a_{1}+\cdots+2 a_{n-1}+a_{n} \). Let \( A_{1}=f(A) \) and \( A_{i+1}=f(A_{i}) \) for \( i=1,2, \cdots, n \).

Prove:
1. There exists a natural number \( k \) such that \( A_{k+1}=A_{k} \).
2. If \( A=19^{86} \), what is the value of \( A_{k} \)? Provide an explanation.

Answer: 19


There are 2 teachers, 3 male students, and 4 female students taking a photo together. How many different standing arrangements are there under the following conditions? (Show the process, and represent the final result with numbers)
(1) The male students must stand together;
(2) The female students cannot stand next to each other;
(3) If the 4 female students have different heights, they must stand from left to right in order from tallest to shortest;
(4) The teachers cannot stand at the ends, and the male students must stand in the middle.

Answer: 1728


After the appearance of purple sand flowerpots in the Ming and Qing dynasties, their development momentum was soaring, gradually becoming the collection target of collectors. With the development of pot-making technology, purple sand flowerpots have been integrated into the daily life of ordinary people. A certain purple sand product factory is ready to mass-produce a batch of purple sand flowerpots. The initial cost of purchasing equipment for the factory is $10,000. In addition, $27 is needed to produce one purple sand flowerpot. When producing and selling a thousand purple sand flowerpots, the total sales of the factory is given by $P(x)=\left\{\begin{array}{l}5.7x+19,0＜x⩽10,\\ 108-\frac{1000}{3x},x＞10.\end{array}\right.($unit: ten thousand dollars).
$(1)$ Find the function relationship of total profit $r\left(x\right)$ (unit: ten thousand dollars) with respect to the output $x$ (unit: thousand pieces). (Total profit $=$ total sales $-$ cost)
$(2)$ At what output $x$ is the total profit maximized? And find the maximum value of the total profit.

Answer: 39


On the leg \(AC\) of the right triangle \(ABC\) with hypotenuse \(AB\), a point \(P\) is marked. Point \(D\) is the foot of the perpendicular dropped from the vertex \(A\) to the line \(BP\), and point \(E\) is the foot of the perpendicular dropped from point \(P\) to the side \(AB\). On the plane, a point \(T\) is chosen such that the lines \(TA\) and \(TP\) are tangent to the circumcircle of triangle \(PAB\), and point \(O\) is the center of this circumcircle. The perpendicular dropped from point \(T\) to the line \(DE\) intersects the line \(BC\) at point \(Q\), and the line passing through point \(C\) parallel to \(OQ\) intersects the segment \(BO\) at point \(K\). Find the angle \(\angle OKE\).

Answer: 90


Let $a$ be a positive integer such that there are exactly 10 perfect squares greater than $a$ and less than $2a$.

a) Find the smallest possible value of $a$.

b) Find the largest possible value of $a$.

Answer: 684


The letter T is formed by placing two $2\:\text{inch}\!\times\!4\:\text{inch}$ rectangles next to each other, as shown.  What is the perimeter of the T, in inches? [asy]
draw((1,0)--(3,0)--(3,4)--(4,4)--(4,6)--(0,6)--(0,4)--(1,4)--cycle);
[/asy]

Answer: 20


What is the highest power of 3 that divides the number whose decimal representation consists of $3^n$ ones?

Answer: 3^n


Let $p$ be a prime number, and let $k = 1 + 2 + \cdots + (p-1)$. Prove that:
$$
k \mid (p-1)! - (p-1).
$$

Answer: k\mid(p-1)!-(p-1)


Find all triples  $(a, b, c)$  of positive integers for which   $$ \begin{cases} a + bc=2010  b + ca = 250\end{cases} $$

Answer: (3,223,9)


Determine the greatest positive integer  $ n$  such that in three-dimensional space, there exist n points  $ P_{1},P_{2},\cdots,P_{n},$  among  $ n$  points no three points are collinear, and for arbitary  $ 1\leq i < j < k\leq n$ ,  $ P_{i}P_{j}P_{k}$  isn't obtuse triangle.

Answer: 8


For a real number \( x \), find the maximum value of
\[
\frac{x^6}{x^{12} + 3x^8 - 6x^6 + 12x^4 + 36}
\]

Answer: \frac{1}{18}


How many rearrangements of the letters of " $HMMTHMMT$ " do not contain the substring " $HMMT$ "? (For instance, one such arrangement is  $HMMHMTMT$ .)

Answer: 361


The number \( n \) has exactly six divisors (including 1 and itself). These divisors are arranged in ascending order. It turns out that the third divisor is seven times larger than the second one, and the fourth divisor is 10 larger than the third one. What is \( n \)?

Answer: 2891


Given \( a, b > 0 \). Find the maximum values of \( \sqrt{\frac{a}{2a+b}} + \sqrt{\frac{b}{2b+a}} \) and \( \sqrt{\frac{a}{a+2b}} + \sqrt{\frac{b}{b+2a}} \).

Answer: \frac{2\sqrt{3}}{3}


If person A is said to be not inferior to person B if at least one of A's height or weight is greater than that of B. Among 100 boys, a boy is called an outstanding boy if he is not inferior to the other 99 boys. What is the maximum number of outstanding boys possible among 100 boys?
(A) 1
(B) 2
(C) 50
(D) 100

Answer: 100


Given the real numbers \(a\) and \(b\) satisfying \(\left(a - \frac{b}{2}\right)^2 = 1 - \frac{7}{4} b^2\), let \(t_{\max}\) and \(t_{\min}\) denote the maximum and minimum values of \(t = a^2 + 2b^2\), respectively. Find the value of \(t_{\text{max}} + t_{\text{min}}\).

Answer: \frac{16}{7}


What three-digit integer is equal to the sum of the factorials of its digits?

Answer: 145


Calculate the volume of the solid of revolution obtained by rotating a right triangle with sides 3, 4, and 5 around one of its legs that form the right angle.

Answer: 12 \pi


Given that $\overrightarrow {e_{1}}$ and $\overrightarrow {e_{2}}$ are unit vectors with an angle of $\frac {2π}{3}$ between them, and $\overrightarrow {a}$ = 3 $\overrightarrow {e_{1}}$ + 2 $\overrightarrow {e_{2}}$, $\overrightarrow {b}$ = 3 $\overrightarrow {e_{2}}$, find the projection of $\overrightarrow {a}$ onto $\overrightarrow {b}$.

Answer: \frac {1}{2}


Given a triangle with  $a,b,c$  sides and with the area  $1$  ( $a \ge  b \ge  c$ ). Prove that  $b^2 \ge 2$ .

Answer: b^2\ge2


For a nonnegative integer $n$ and a strictly increasing sequence of real numbers $t_0,t_1,\dots,t_n$, let $f(t)$ be the corresponding real-valued function defined for $t \geq t_0$ by the following properties: \begin{enumerate} \item[(a)] $f(t)$ is continuous for $t \geq t_0$, and is twice differentiable for all $t>t_0$ other than $t_1,\dots,t_n$; \item[(b)] $f(t_0) = 1/2$; \item[(c)] $\lim_{t \to t_k^+} f'(t) = 0$ for $0 \leq k \leq n$; \item[(d)] For $0 \leq k \leq n-1$, we have $f''(t) = k+1$ when $t_k < t< t_{k+1}$, and $f''(t) = n+1$ when $t>t_n$. \end{enumerate} Considering all choices of $n$ and $t_0,t_1,\dots,t_n$ such that $t_k \geq t_{k-1}+1$ for $1 \leq k \leq n$, what is the least possible value of $T$ for which $f(t_0+T) = 2023$?

Answer: 29


Given natural numbers \( M \) and \( N \), both greater than ten, consisting of the same number of digits, and such that \( M = 3N \). To obtain the number \( M \), one of the digits in \( N \) is increased by 2, and each of the remaining digits is increased by an odd digit. What digit could the number \( N \) end with? Find all possible answers.

(N. Agahanov)

Answer: 6


There are three pairs of real numbers $(x_1,y_1)$, $(x_2,y_2)$, and $(x_3,y_3)$ that satisfy both $x^3-3xy^2=2005$ and $y^3-3x^2y=2004$. Compute $\left(1-\frac{x_1}{y_1}\right)\left(1-\frac{x_2}{y_2}\right)\left(1-\frac{x_3}{y_3}\right)$.

Answer: \frac{1}{1002}


In the diagram below, $ABCD$ is a square. Point $E$ is the midpoint of $\overline{AD}$. Points $F$ and $G$ lie on $\overline{CE}$, and $H$ and $J$ lie on $\overline{AB}$ and $\overline{BC}$, respectively, so that $FGHJ$ is a square. Points $K$ and $L$ lie on $\overline{GH}$, and $M$ and $N$ lie on $\overline{AD}$ and $\overline{AB}$, respectively, so that $KLMN$ is a square. The area of $KLMN$ is 99. Find the area of $FGHJ$.
[asy] pair A,B,C,D,E,F,G,H,J,K,L,M,N; B=(0,0); real m=7*sqrt(55)/5; J=(m,0); C=(7*m/2,0); A=(0,7*m/2); D=(7*m/2,7*m/2); E=(A+D)/2; H=(0,2m); N=(0,2m+3*sqrt(55)/2); G=foot(H,E,C); F=foot(J,E,C); draw(A--B--C--D--cycle); draw(C--E); draw(G--H--J--F); pair X=foot(N,E,C); M=extension(N,X,A,D); K=foot(N,H,G); L=foot(M,H,G); draw(K--N--M--L); label("$A$",A,NW); label("$B$",B,SW); label("$C$",C,SE); label("$D$",D,NE); label("$E$",E,dir(90)); label("$F$",F,NE); label("$G$",G,NE); label("$H$",H,W); label("$J$",J,S); label("$K$",K,SE); label("$L$",L,SE); label("$M$",M,dir(90)); label("$N$",N,dir(180)); [/asy]

Answer: 539


Find the relationship between \(\arcsin \cos \arcsin x\) and \(\arccos \sin \arccos x\).

Answer: \frac{\pi}{2}


Find the real solution(s) to the equation \((x+y)^{2} = (x+1)(y-1)\).

Answer: (-1,1)


Let  $a_0=1,a_1=2,$  and  $a_n=4a_{n-1}-a_{n-2}$  for  $n\ge 2.$ Find an odd prime factor of  $a_{2015}.$

Answer: 181


What is the largest real number $\theta$ less than $\pi$ (i.e. $\theta<\pi$ ) such that $\prod_{k=0}^{10} \cos \left(2^{k} \theta\right) \neq 0$ and $\prod_{k=0}^{10}\left(1+\frac{1}{\cos \left(2^{k} \theta\right)}\right)=1 ?

Answer: \frac{2046 \pi}{2047}


Let  $n$  be a positive integer. All numbers  $m$  which are coprime to  $n$  all satisfy  $m^6\equiv 1\pmod n$ . Find the maximum possible value of  $n$ .

Answer: 504


What are the prime numbers \( p \) such that \( p \) divides \( 29^{p} + 1 \)?

Answer: 2,3,5


A regular $n$-gon has $n$ diagonals, its perimeter is $p$, and the sum of the lengths of all the diagonals is $q$. What is $\frac{p}{q} + \frac{q}{p}$?

Answer: \sqrt{5}


Fill in the blanks with appropriate numbers to make the equation true: $x^2+5x+\_\_=(x+\_\_)^2.$

Answer: \frac{5}{2}


There are 4 spheres in space with radii 2, 2, 3, and 3, respectively. Each sphere is externally tangent to the other 3 spheres. Additionally, there is a small sphere that is externally tangent to all 4 of these spheres. Find the radius of the small sphere.

Answer: 6/11


Write $x^{10} + x^5 + 1$ as the product of two polynomials with integer coefficients.

Answer: (x^2 + x + 1)(x^8 - x^7 + x^5 - x^4 + x^3 - x + 1)


From point \(O\), \(n\) vectors of unit length emanate, such that in any half-plane bounded by a line passing through point \(O\), there are at least \(k\) vectors (the boundary line is assumed to be included in the half-plane). Prove that the length of the sum of these vectors does not exceed \(n - 2k\).

Answer: n-2k


A finite set  $S$  of points in the coordinate plane is called *overdetermined* if  $|S|\ge 2$  and there exists a nonzero polynomial  $P(t)$ , with real coefficients and of degree at most  $|S|-2$ , satisfying  $P(x)=y$  for every point  $(x,y)\in S$ .

For each integer  $n\ge 2$ , find the largest integer  $k$  (in terms of  $n$ ) such that there exists a set of  $n$  distinct points that is *not* overdetermined, but has  $k$  overdetermined subsets.

*Proposed by Carl Schildkraut*

Answer: 2^{n-1}-n


When $0.000315$ is multiplied by $7,928,564$ the product is closest to which of the following?
$\textbf{(A) }210\qquad\textbf{(B) }240\qquad\textbf{(C) }2100\qquad\textbf{(D) }2400\qquad\textbf{(E) }24000$

Answer: \textbf{(D)}\2400


Given a convex quadrilateral \(ABCD\) in which \(\angle BAC=20^\circ\), \(\angle CAD=60^\circ\), \(\angle ADB=50^\circ\), and \(\angle BDC=10^\circ\). Find \(\angle ACB\).

Answer: 80


Let $ABCD$ be a square with side length $1$. How many points $P$ inside the square (not on its sides) have the property that the square can be cut into $10$ triangles of equal area such that all of them have $P$ as a vertex?

Answer: 16


$ABCD$  is a cyclic quadrilateral inscribed in a circle of radius  $5$ , with  $AB=6$ ,  $BC=7$ ,  $CD=8$ . Find  $AD$ .

Answer: \sqrt{51}


The ten-digit natural numbers \(a, b, c\) are such that \(a + b = c\). What is the maximum number of odd digits that can appear among the 30 digits in total?

Answer: 29


Calculate the arc lengths of the curves given by the equations in the rectangular coordinate system.

$$
y=2-e^{x}, \ln \sqrt{3} \leq x \leq \ln \sqrt{8}
$$

Answer: 1 + \frac{1}{2} \ln \frac{3}{2}


Let the fourth-degree polynomial \( f(x) \) with integer coefficients satisfy:
\[ f(1 + \sqrt[3]{3}) = 1 + \sqrt[3]{3} \]
\[ f(1 + \sqrt{3}) = 7 + \sqrt{3} \]
Find \( f(x) = \qquad \)

Answer: x^4-3x^3+3x^2-3x


A circle with its center at point $M$ on the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ ($a > 0, b > 0$) is tangent to the $x$-axis exactly at one of the foci $F$ of the hyperbola, and intersects the $y$-axis at points $P$ and $Q$. If $\triangle MPQ$ is an equilateral triangle, calculate the eccentricity of the hyperbola.

Answer: \sqrt{3}


Answer: \frac{p_n}{q_n}=a_0+\frac{1}{q_0q_1}-\frac{1}{q_1q_2}+\frac{1}{q_2q_3}-\ldots+\frac{(-1)^{n-1}}{q_{n-1}q_n}


Find the smallest positive integer $n$ that satisfies:
(1) $n$ has exactly 144 distinct positive divisors;
(2) Among the positive divisors of $n$, there are 10 consecutive integers.

Answer: 110880


Prove that when \( n = 9002 \), the sum of binomial coefficients with a step of 6, i.e., \( \binom{n}{1} + \binom{n}{7} + \ldots + \binom{n}{n-1} \), gives a remainder of 1 when divided by 3.

Where \( \binom{n}{k} \) is the number of ways to choose \( k \) items from \( n \), which is \( \frac{n!}{k!(n-k)!} \) if \( 0 \leqslant k \leqslant n \) and 0 otherwise.

Answer: 1


<u>Set 1</u>**p1.** Assume the speed of sound is  $343$  m/s. Anastasia and Bananastasia are standing in a field in front of you. When they both yell at the same time, you hear Anastasia’s yell  $5$  seconds before Bananastasia’s yell. If Bananastasia yells first, and then Anastasia yells when she hears Bananastasia yell, you hear Anastasia’s yell  $5$  seconds after Bananastasia’s yell. What is the distance between Anastasia and Bananastasia in meters?**p2.** Michelle picks a five digit number with distinct digits. She then reverses the digits of her number and adds that to her original number. What is the largest possible sum she can get?**p3.** Twain is trying to crack a  $4$ -digit number combination lock. They know that the second digit must be even, the third must be odd, and the fourth must be different from the previous three. If it takes Twain  $10$  seconds to enter a combination, how many hours would it take them to try every possible combination that satisfies these rules?

PS. You should use hide for answers.

Answer: 1715


Given an ellipse $E: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \left(a > b > 0\right)$ that passes through the point $(0,1)$, and its eccentricity is $\frac{\sqrt{3}}{2}$.

$(1)$ Find the standard equation of the ellipse $E$;

$(2)$ Suppose a line $l: y = \frac{1}{2}x + m$ intersects the ellipse $E$ at points $A$ and $C$. A square $ABCD$ is constructed with $AC$ as its diagonal. Let the intersection of line $l$ and the $x$-axis be $N$. Is the distance between points $B$ and $N$ a constant value? If yes, find this constant value; if no, explain why.

Answer: \frac{\sqrt{10}}{2}


(My problem. :D)
Call the number of times that the digits of a number change from increasing to decreasing, or vice versa, from the left to right while ignoring consecutive digits that are equal the *flux* of the number. For example, the flux of 123 is 0 (since the digits are always increasing from left to right) and the flux of 12333332 is 1, while the flux of 9182736450 is 8. What is the average value of the flux of the positive integers from 1 to 999, inclusive?

Answer: \frac{175}{333}


Let \( P \) be a point inside the isosceles trapezoid \( ABCD \) where \( AD \) is one of the bases, and let \( PA, PB, PC, \) and \( PD \) bisect angles \( A, B, C, \) and \( D \) respectively. If \( PA = 3 \) and \( \angle APD = 120^\circ \), find the area of trapezoid \( ABCD \).

Answer: 6\sqrt{3}


Find all positive integers $n$ such that $n \cdot 2^{n-1} + 1$ is a perfect square.

Answer: 5


Reimu has 2019 coins \( C_{0}, C_{1}, \ldots, C_{2018} \), one of which is fake. These coins look identical, making each coin equally likely to be fake. She uses a machine that takes any two coins and picks one that is not fake. If both coins are not fake, the machine selects one uniformly at random. For \( i = 1, 2, \ldots, 1009 \), she puts \( C_{0} \) and \( C_{i} \) into the machine once, and the machine picks \( C_{i} \). What is the probability that \( C_{0} \) is fake?

Answer: \frac{2^{1009}}{2^{1009}+1009}


Find the smallest positive real number \( k \) such that the inequality
\[ ab + bc + ca + k\left(\frac{1}{a} + \frac{1}{b} + \frac{1}{c}\right) \geq 9 \]
holds for all positive real numbers \( a, b, c \).

Answer: 2


When selecting the first trial point using the 0.618 method during the process, if the experimental interval is $[2000, 3000]$, the first trial point $x_1$ should be chosen at ______.

Answer: 2618


If for all positive real numbers \(a, b, c, d\), the inequality \[\left(\frac{a^{3}}{a^{3}+15 b c d}\right)^{\frac{1}{2}} \geqslant \frac{a^{x}}{a^{x}+b^{x}+c^{x}+d^{x}}\] always holds, find all real numbers \(x\) that satisfy the condition.

Answer: \frac{15}{8}


Find the maximum constant \( k \) such that for \( x, y, z \in \mathbb{R}_+ \), the following inequality holds:
$$
\sum \frac{x}{\sqrt{y+z}} \geqslant k \sqrt{\sum x},
$$

where " \( \sum \) " denotes a cyclic sum.

Answer: \sqrt{\frac{3}{2}}


Given that point \( E(m, n) \) is a certain point on the parabola \( y^{2} = 2px \) (where \( p > 0 \)), lines through \( E \) with slopes \( k_{1} \) and \( k_{2} \) intersect the parabola at points \( A \) and \( B \), and \( C \) and \( D \) respectively. Let \( M \) and \( N \) be the midpoints of segments \( AB \) and \( CD \).
(1) When \( n = 0 \) and \( k_{1} \cdot k_{2} = -1 \), find the minimum area of \( \triangle EMN \).
(2) If \( k_{1} + k_{2} = \lambda \) (where \( \lambda \neq 0 \) and \( \lambda \) is a constant), prove that line \( MN \) passes through a fixed point.

Answer: p^2


A natural number \( N \) ends with the digit 5. A ninth-grader Dima found all its divisors and discovered that the sum of the two largest proper divisors is not divisible by the sum of the two smallest proper divisors. Find the smallest possible value of the number \( N \). A divisor of a natural number is called proper if it is different from 1 and the number itself.

Answer: 725


Given ellipse $C_1$ and parabola $C_2$ whose foci are both on the x-axis, the center of $C_1$ and the vertex of $C_2$ are both at the origin $O$. Two points are taken from each curve, and their coordinates are recorded in the table. Calculate the distance between the left focus of $C_1$ and the directrix of $C_2$.

Answer: \sqrt {3}-1


Four lines form four triangles. Prove that the orthocenters of these triangles lie on a single line.

Answer: 8


Let $m$ be the product of all positive integers less than $4!$ which are invertible modulo $4!$. Find the remainder when $m$ is divided by $4!$.

(Here $n!$ denotes $1\times\cdots\times n$ for each positive integer $n$.)

Answer: 1


Sarah's six assignment scores are 87, 90, 86, 93, 89, and 92. What is the arithmetic mean of these six scores?

Answer: 89.5


Answer: a_{2019}<\frac{1}{2}<a_{2018}


Determine all positive integers  $n$  with at least  $4$  factors such that  $n$  is the sum the squares of its  $4$  smallest factors.

Answer: 130


The novel takes 630 minutes to read aloud. The disc can hold 80 minutes of reading with at most 4 minutes of unused space. Calculate the number of minutes of reading each disc will contain.

Answer: 70


Given \( f(x) = \lg (x+1) - \frac{1}{2} \log_{3} x \).
(1) Solve the equation \( f(x) = 0 \);
(2) Determine the number of subsets of the set \( M = \{ n \mid f(n^2 - 214n - 1998) \geq 0, n \in \mathbf{Z} \} \).

Answer: 4


If $x$ is such that $\frac{1}{x}<2$ and $\frac{1}{x}>-3$, then:

Answer: x>\frac{1}{2} \text{ or } x<-\frac{1}{3}


In the right triangle \(ABC\), segment \(BH\) is the altitude dropped onto the hypotenuse, and \(BL\) is the median in triangle \(BHC\). Find the angle \(LBC\), given that \(BL=4\) and \(AH=\frac{9}{2\sqrt{7}}\).

Answer: \arccos(\frac{23}{4\sqrt{37}})


Using three rectangular pieces of paper (A, C, D) and one square piece of paper (B), an area of 480 square centimeters can be assembled into a large rectangle. It is known that the areas of B, C, and D are all 3 times the area of A. Find the total perimeter of the four pieces of paper A, B, C, and D in centimeters.

Answer: 184


The base of the pyramid \( S A B C D \) is the trapezoid \( A B C D \) with bases \( B C \) and \( A D \). Points \( P_{1}, P_{2}, P_{3} \) lie on the side \( B C \), with the distances satisfying \( B P_{1} < B P_{2} < B P_{3} < B C \). Points \( Q_{1}, Q_{2}, Q_{3} \) lie on the side \( A D \), with the distances satisfying \( A Q_{1} < A Q_{2} < A Q_{3} < A D \). Let the intersection points of \( B Q_{1} \) with \( A P_{1} \), \( P_{2} Q_{1} \) with \( P_{1} Q_{2} \), \( P_{3} Q_{2} \) with \( P_{2} Q_{3} \), and \( C Q_{3} \) with \( P_{3} D \) be denoted as \( R_{1}, R_{2}, R_{3}, \) and \( R_{4} \) respectively. It is known that the sum of the volumes of the pyramids \( S R_{1} P_{1} R_{2} Q_{1} \) and \( S R_{3} P_{3} R_{4} Q_{3} \) is 78. Find the minimum value of

\[
V_{S A B R_{1}}^{2} + V_{S R_{2} P_{2} R_{3} Q_{2}}^{2} + V_{S C D R_{4}}^{2}
\]

Provide the integer nearest to the found value.

Answer: 2028


A cinema has its seats arranged in \( n \) rows and \( m \) columns. It sold \( mn \) tickets but sold some seats more than once. The usher managed to allocate seats so that every ticket holder was in the correct row or column. Show that he could have allocated seats so that every ticket holder was in the correct row or column and at least one person was in the correct seat. What is the maximum \( k \) such that he could have always put every ticket holder in the correct row or column and at least \( k \) people in the correct seat?

Answer: 1


In the diagram, the circle has radius \(\sqrt{5}\). Rectangle \(ABCD\) has \(C\) and \(D\) on the circle, \(A\) and \(B\) outside the circle, and \(AB\) tangent to the circle. What is the area of \(ABCD\) if \(AB = 4AD\)?

Answer: 16/5


Experts and Viewers play "What? Where? When?" until one side wins six rounds. The probability of Experts winning a single round is 0.6, and there are no ties. Currently, the Experts are losing with a score of 3 to 4. Find the probability that the Experts will eventually win.

Answer: 0.4752


The area of this figure is $100\text{ cm}^2$.  Its perimeter is
[asy] draw((0,2)--(2,2)--(2,1)--(3,1)--(3,0)--(1,0)--(1,1)--(0,1)--cycle,linewidth(1)); draw((1,2)--(1,1)--(2,1)--(2,0),dashed); [/asy]

[figure consists of four identical squares]

Answer: 50 cm


From the 20 natural numbers 1, 2, 3, ..., 20, if three numbers are randomly selected and their sum is an even number greater than 10, then there are $\boxed{\text{answer}}$ such sets of numbers.

Answer: 563


How many  $4-$ digit numbers  $\overline{abcd}$  are there such that  $a<b<c<d$  and  $b-a<c-b<d-c$  ?

Answer: 7


Real numbers  $X_1, X_2, \dots, X_{10}$  are chosen uniformly at random from the interval  $[0,1]$ . If the expected value of  $\min(X_1,X_2,\dots, X_{10})^4$  can be expressed as a rational number  $\frac{m}{n}$  for relatively prime positive integers  $m$  and  $n$ , what is  $m+n$ ?

*2016 CCA Math Bonanza Lightning #4.4*

Answer: 1002


Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ that satisfy: $|\overrightarrow{a}| = 1$, $| \overrightarrow{b}| = 6$, and $\overrightarrow{a} \cdot (\overrightarrow{b} - \overrightarrow{a}) = 2$, find the angle between vectors $\overrightarrow{a}$ and $\overrightarrow{b}$.

Answer: \dfrac{\pi}{3}


The following bar graph represents the length (in letters) of the names of 19 people. What is the median length of these names?

Answer: 4


Given the line \(6 x - 5 y - 28 = 0\) and the ellipse \(\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1\) with \(a > b > 0\), and \(a^{2}, b \in \mathbf{Z}\), intersects the ellipse at points \(A\) and \(C\). Let \(B(0, b)\) be the top vertex of the ellipse, and the centroid of \(\triangle ABC\) be the right focus \(F_{2}\) of the ellipse. Find the equation of the ellipse.

Answer: \frac{x^2}{20}+\frac{y^2}{16}=1


Consider the expression \(1 \ast 2 \ast 3 \ast 4 \ast 5 \ast 6\).
Each star in the expression is to be replaced with either ' + ' or ' \times '.
\(N\) is the largest possible value of the expression. What is the largest prime factor of \(N\)?

Answer: 103


Let \( \triangle ABC \) be a triangle with \( AB = 8 \), \( BC = 15 \), and \( AC = 17 \). Point \( X \) is chosen at random on line segment \( AB \). Point \( Y \) is chosen at random on line segment \( BC \). Point \( Z \) is chosen at random on line segment \( CA \). What is the expected area of triangle \( XYZ \)?

Answer: 15


A given rectangle  $ R$  is divided into  $mn$  small rectangles by straight lines parallel to its sides. (The distances between the parallel lines may not be equal.) What is the minimum number of appropriately selected rectangles’ areas that should be known in order to determine the area of  $ R$ ?

Answer: +n-1


The faces of a 12-sided die are numbered 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, and 12 such that the sum of the numbers on opposite faces is 13. The die is meticulously carved so that it is biased: the probability of obtaining a particular face \( F \) is greater than \( \frac{1}{12} \), the probability of obtaining the face opposite \( F \) is less than \( \frac{1}{12} \) while the probability of obtaining any one of the other ten faces is \( \frac{1}{12} \).

When two such dice are rolled, the probability of obtaining a sum of 13 is \( \frac{29}{384} \).

What is the probability of obtaining face \( F \)?

Answer: \frac{7}{48}


On a certain island, there are 10 people. Some of them always tell the truth, while others always lie. Each person has thought of a real number (not necessarily an integer). The first person says, "My number is greater than 1." The second person says, "My number is greater than 2," and so on up to the tenth person who says, "My number is greater than 10." After that, these ten people stand in a line in some order, and each one then says, "My number is less than 1," "My number is less than 2," and so on up to "My number is less than 10" (each person speaks exactly once). The question is: What is the maximum number of truth-tellers among these people?

Answer: 9


There is a circle drawn on a sheet of paper. Is it possible to draw 5 segments on the same sheet such that each of them intersects with each other and for each segment, half of its intersection points with other segments lie on the circle?

Answer: Yes


The centers of three circles, which are pairwise tangent to each other externally, are located at the vertices of a right triangle. These circles are internally tangent to a fourth circle. Find the radius of the fourth circle if the perimeter of the right triangle is \( 2p \).

Answer: p


A train passenger knows that the speed of their train is 40 km/h. As soon as a passing train started to go by the window, the passenger started a stopwatch and noted that the passing train took 3 seconds to pass completely. Determine the speed of the passing train, given that its length is 75 meters.

Answer: 50


Find the value of \( \cos \frac{\pi}{11} + \cos \frac{3\pi}{11} + \cos \frac{5\pi}{11} + \cos \frac{7\pi}{11} + \cos \frac{9\pi}{11} \).

Answer: \frac{1}{2}


Find the $y$-intercepts of the following system of equations:
1. $2x - 3y = 6$
2. $x + 4y = -8$

Answer: -2


Determine the number of 0-1 binary sequences of ten 0's and ten 1's which do not contain three 0's together.

Answer: 24068


Let's call a number "wonderful" if it has exactly 3 different odd natural divisors (and any number of even ones). How many "wonderful" two-digit numbers exist?

Answer: 7


For any number $y$, define the operations $\&y = 2(7-y)$ and $\&y = 2(y-7)$. What is the value of $\&(-13\&)$?

Answer: 66


Let  $a>1$  be a positive integer.  The sequence of natural numbers  $\{a_n\}_{n\geq 1}$  is defined such that  $a_1 = a$  and for all  $n\geq 1$ ,  $a_{n+1}$  is the largest prime factor of  $a_n^2 - 1$ .  Determine the smallest possible value of  $a$  such that the numbers  $a_1$ ,  $a_2$ , $\ldots$ ,  $a_7$  are all distinct.

Answer: 46


Using the result of problem $\underline{61403}$, prove the inequalities:

a) $\sqrt[n]{a_{1} \cdots a_{n}} \leq \frac{a_{1} + \cdots + a_{n}}{n}$ (Cauchy's inequality).

b) $\left(\frac{b_{1} + \cdots + b_{n}}{n}\right)^{b_{1} + \cdots + b_{n}} \leq b_{1}^{b_{1}} \cdots b_{n}^{b_{n}}$.

Answer: (\frac{b_{1}+\cdots+b_{n}}{n})^{b_{1}+\cdots+b_{n}}\leb_{1}^{b_{1}}\cdotsb_{n}^{b_{n}}


Within a cube with edge length 6, there is a regular tetrahedron with edge length \( x \) that can rotate freely inside the cube. What is the maximum value of \( x \)?

Answer: 2\sqrt{6}


Maria buys computer disks at a price of 5 for $6 and sells them at a price of 4 for $7. Find how many computer disks Maria must sell to make a profit of $120.

Answer: 219


Given the function \( f(x) = x^2 \cos \frac{\pi x}{2} \), and the sequence \(\left\{a_n\right\}\) in which \( a_n = f(n) + f(n+1) \) where \( n \in \mathbf{Z}_{+} \). Find the sum of the first 100 terms of the sequence \(\left\{a_n\right\}\), denoted as \( S_{100} \).

Answer: 10200


$ABCDE$ is a regular pentagon.  $AP$, $AQ$ and $AR$ are the perpendiculars dropped from $A$ onto $CD$, $CB$ extended and $DE$ extended, respectively.  Let $O$ be the center of the pentagon.  If $OP = 1$, then find $AO + AQ + AR$.

[asy]

unitsize(2 cm);

pair A, B, C, D, E, O, P, Q, R;

A = dir(90);

B = dir(90 - 360/5);

C = dir(90 - 2*360/5);

D = dir(90 - 3*360/5);

E = dir(90 - 4*360/5);

O = (0,0);

P = (C + D)/2;

Q = (A + reflect(B,C)*(A))/2;

R = (A + reflect(D,E)*(A))/2;

draw((2*R - E)--D--C--(2*Q - B));

draw(A--P);

draw(A--Q);

draw(A--R);

draw(B--A--E);

label("$A$", A, N);

label("$B$", B, dir(0));

label("$C$", C, SE);

label("$D$", D, SW);

label("$E$", E, W);

dot("$O$", O, dir(0));

label("$P$", P, S);

label("$Q$", Q, dir(0));

label("$R$", R, W);

label("$1$", (O + P)/2, dir(0));

[/asy]

Answer: 4


Ms. $D$ places 13 cards with numbers $1, 2, \dots, 13$ in ascending order clockwise around a circle, all initially face up. Each turn, Ms. $D$ can turn over a card to its face-down side (once a card is face down, it cannot be flipped again). Ms. $D$'s flipping rule is: if a face-up card has the number $A$, and the card two positions away from it is also face up, then Ms. $D$ can turn over the card with number $A$. For example, if the cards with numbers 9 or 2 are face up, Ms. $D$ can flip the card numbered 12 (assuming the card numbered 12 is still face up). Ms. $D$ succeeds if she turns over 12 cards, leaving only one card face up. How many different sequences of flips can Ms. $D$ perform to succeed?

Answer: 26624


Find the maximum real number \( \theta \) (\(\theta < \pi\)) that satisfies
\[
\prod_{k=0}^{10} \cos 2^{k} \theta \neq 0
\]
and
\[
\prod_{k=0}^{10}\left(1+\frac{1}{\cos 2^{k} \theta}\right)=1.
\]

Answer: \frac{2046\pi}{2047}


How many Fridays? What is the maximum number of Fridays that can be in a year?

Answer: 53


Answer: 16


Given the numbers \(\log _{\sqrt{2 x-8}}(x-4), \log _{(x-4)^{2}}(5 x-26), \log _{\sqrt{5 x-26}}(2 x-8)\). For which values of \(x\) are two of these numbers equal, and the third one is greater by 1?

Answer: 6


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 \( \frac{A M}{A A_{1}} = \frac{3}{7} \), \( \frac{B N}{B B_{1}} = \frac{2}{5} \), and \( \frac{C K}{C C_{1}} = \frac{4}{9} \). Point \( P \) lies within the prism. Find the maximum possible volume of the pyramid \( M N K P \) if the volume of the prism is 40.

Answer: 8


A car and a truck start traveling towards each other simultaneously from points $A$ and $B$, respectively. It is known that the car's speed is twice the speed of the truck. The car arrives at point $C$ at 8:30, and the truck arrives at point $C$ at 15:00 on the same day. Both vehicles continue moving without stopping at point $C$. Determine the time at which the car and the truck meet.

Answer: 10:40


Among the first 1000 positive integers, how many can be expressed in the form \([2x] + [4x] + [6x] + [8x]\)?

Answer: 600


Given two skew lines \( l \) and \( m \), there are three points \( A, B, \) and \( C \) on \( l \), such that \( AB = BC \). Perpendicular lines \( AD, BE, \) and \( CF \) are drawn from \( A, B, \) and \( C \) respectively to the line \( m \), with foot points being \( D, E, \) and \( F \) accordingly. Given that \( AD = \sqrt{15} \), \( BE = \frac{7}{2} \), and \( CF = \sqrt{10} \), find the distance between \( l \) and \( m \).

Answer: \sqrt{6}


The speed of a motorboat is \( a \) km/hour, and the speed of a steamboat is \( b \) km/hour. Both vehicles travel the path \( AB \), but the motorboat starts its journey only after the steamboat has already traveled \( d \) km. Nevertheless, the steamboat reaches point \( B \) \( n \) hours later than the motorboat. What is the distance \( AB \)?

Answer: \frac{(bn)}{b}


Let  $ABC$  be a triangle with incenter  $I$  and  $AB = 1400$ ,  $AC = 1800$ ,  $BC = 2014$ .  The circle centered at  $I$  passing through  $A$  intersects line  $BC$  at two points  $X$  and  $Y$ . Compute the length  $XY$ .

*Proposed by Evan Chen*

Answer: 1186


Mark has a cursed six-sided die that never rolls the same number twice in a row, and all other outcomes are equally likely. Compute the expected number of rolls it takes for Mark to roll every number at least once.

Answer: \frac{149}{12}


Let the real number \( r \) satisfy \( r^{3} - r - 1 = 0 \), and let \( r + \sqrt{2} \) be a root of the polynomial \( P(x) = 0 \). If the leading coefficient of \( p(x) \) is 1, and \( p(x) \) is the polynomial of least degree with integer coefficients that satisfies the conditions, then \( p(x) = \) _____.

Answer: x^6-8x^4-2x^3+13x^2-10x-1


Initially, a natural number was displayed on the calculator screen. Each time, Olya added to the current number $n$ on the calculator screen a natural number by which $n$ was not divisible. For example, if the number 10 was displayed on the screen, Olya could add 7 and get 17.

Olya repeated this operation five times, and the number 200 appeared on the screen. What is the largest possible initial number for which this could have happened?

Answer: 189


Answer: 804095


How many solutions in integers \(x, y\) does the equation \( |3x + 2y| + |2x + y| = 100 \) have?

Answer: 400


Let \( A, B, C, D \) be four points on a line in that order. The circles with diameters \( AC \) and \( BD \) intersect at points \( X \) and \( Y \). The line \( XY \) intersects \( BC \) at \( Z \). Let \( P \) be a point on \( XY \) other than \( Z \). The line \( CP \) intersects the circle with diameter \( AC \) at \( C \) and \( M \), and the line \( BP \) intersects the circle with diameter \( BD \) at \( B \) and \( N \). Prove that the lines \( AM \), \( DN \), and \( XY \) are concurrent.

Answer: 0


Given positive numbers \(a_{i} (i=1,2, \cdots, n)\) such that \(\sum_{i=1}^{n} a_{i} = 1\), prove: \(\prod_{i=1}^{n}\left(a_{i}+\frac{1}{a_{i}}\right) \geq \left(n+\frac{1}{n}\right)^{n}\).

Answer: \prod_{i=1}^{n}(a_i+\frac{1}{a_i})\ge(n+\frac{1}{n})^n


Given 6 different thin rods with lengths $a, b, c, d, e, f$, any three of which can form a triangle. How many distinct tetrahedral edge frameworks can be assembled from the rods, where the frameworks are not equivalent by rotation or reflection?

Answer: 30


Answer: a_{50}+b_{50}>20


On side \( AB \) of parallelogram \( ABCD \), point \( F \) is selected, and on the extension of side \( BC \) beyond vertex \( B \), point \( H \) is chosen such that \( \frac{AB}{BF} = \frac{BC}{BH} = 5 \). Point \( G \) is selected so that \( BFGH \) is a parallelogram. Line \( GD \) intersects \( AC \) at point \( X \). Find \( AX \), if \( AC = 100 \).

Answer: 40


The five books "Poetry," "Documents," "Rites," "Changes," and "Spring and Autumn" all have different numbers of pages. The differences in the number of pages between the books are as follows:
1. "Poetry" and "Documents" differ by 24 pages.
2. "Documents" and "Rites" differ by 17 pages.
3. "Rites" and "Changes" differ by 27 pages.
4. "Changes" and "Spring and Autumn" differ by 19 pages.
5. "Spring and Autumn" and "Poetry" differ by 15 pages.

Determine the difference in the number of pages between the book with the most pages and the book with the least pages.

Answer: 34


Determinants of nine digits. Nine positive digits can be arranged in the form of a 3rd-order determinant in 9! ways. Find the sum of all such determinants.

Answer: 0


Given a right triangle \(ABC\) with a right angle at \(A\). On the leg \(AC\), a point \(D\) is marked such that \(AD:DC = 1:3\). Circles \(\Gamma_1\) and \(\Gamma_2\) are then drawn with centers at \(A\) and \(C\) respectively, both passing through point \(D\). \(\Gamma_2\) intersects the hypotenuse at point \(E\). Another circle \(\Gamma_3\) with center at \(B\) and radius \(BE\) intersects \(\Gamma_1\) inside the triangle at a point \(F\) such that \(\angle AFB\) is a right angle. Find \(BC\), given that \(AB = 5\).

Answer: 13


Let $S$ be the set of $3^{4}$ points in four-dimensional space where each coordinate is in $\{-1,0,1\}$. Let $N$ be the number of sequences of points $P_{1}, P_{2}, \ldots, P_{2020}$ in $S$ such that $P_{i} P_{i+1}=2$ for all $1 \leq i \leq 2020$ and $P_{1}=(0,0,0,0)$. (Here $P_{2021}=P_{1}$.) Find the largest integer $n$ such that $2^{n}$ divides $N$.

Answer: 4041


There are  $2^{2n+1}$  towns with  $2n+1$  companies and each two towns are connected with airlines from one of the companies. What’s the greatest number  $k$  with the following property:
We can close  $k$  of the companies and their airlines in such way that we can still reach each town from any other (connected graph).

Answer: n


The diameter $AB$ of a circle of radius $2$ is extended to a point $D$ outside the circle so that $BD=3$. Point $E$ is chosen so that $ED=5$ and line $ED$ is perpendicular to line $AD$. Segment $AE$ intersects the circle at a point $C$ between $A$ and $E$. What is the area of $\triangle  ABC$?
$\textbf{(A)}\ \frac{120}{37}\qquad\textbf{(B)}\ \frac{140}{39}\qquad\textbf{(C)}\ \frac{145}{39}\qquad\textbf{(D)}\ \frac{140}{37}\qquad\textbf{(E)}\ \frac{120}{31}$

Answer: \frac{140}{37}


Yu Semo and Yu Sejmo have created sequences of symbols  $\mathcal{U} = (\text{U}_1, \ldots, \text{U}_6)$  and  $\mathcal{J} = (\text{J}_1, \ldots, \text{J}_6)$ . These sequences satisfy the following properties.

- Each of the twelve symbols must be  $\Sigma$ ,  $\#$ ,  $\triangle$ , or  $\mathbb{Z}$ .
- In each of the sets  $\{\text{U}_1, \text{U}_2, \text{U}_4, \text{U}_5\}$ ,  $\{\text{J}_1, \text{J}_2, \text{J}_4, \text{J}_5\}$ ,  $\{\text{U}_1, \text{U}_2, \text{U}_3\}$ ,  $\{\text{U}_4, \text{U}_5, \text{U}_6\}$ ,  $\{\text{J}_1, \text{J}_2, \text{J}_3\}$ ,  $\{\text{J}_4, \text{J}_5, \text{J}_6\}$ , no two symbols may be the same.
- If integers  $d \in \{0, 1\}$  and  $i, j \in \{1, 2, 3\}$  satisfy  $\text{U}_{i + 3d} = \text{J}_{j + 3d}$ , then  $i < j$ .

How many possible values are there for the pair  $(\mathcal{U}, \mathcal{J})$ ?

Answer: 24


Let $p_{k}$ be the number of permutations of the set $\{1, \ldots, n\}$ having $k$ fixed points. Compute:

$$
\sum_{k=0}^{n} k p_{k}
$$

Answer: n!


A plane \( P \) slices through a cube of volume 1 with a cross-section in the shape of a regular hexagon. This cube also has an inscribed sphere, whose intersection with \( P \) is a circle. What is the area of the region inside the regular hexagon but outside the circle?

Answer: \frac{3\sqrt{3}-\pi}{4}


On an island, there are only 50 knights, who always tell the truth, and 15 civilians, who can either tell the truth or lie. A forgetful professor, who came to the island to give a lecture, forgot what color hat he is wearing. What is the minimum number of local inhabitants that the professor needs to ask about the color of his hat in order to be certain about its color?

Answer: 31


5 people are standing in a row for a photo, among them one person must stand in the middle. There are     ways to arrange them.

Answer: 24


Let $\mathcal{T}$ be the set of real numbers that can be represented as repeating decimals of the form $0.\overline{abcd}$ where $a, b, c, d$ are distinct digits. Find the sum of the elements of $\mathcal{T}.$

Answer: 2520


Ivan Semenovich leaves for work at the same time every day, travels at the same speed, and arrives exactly at 9:00 AM. One day, he overslept and left 40 minutes later than usual. To avoid being late, Ivan Semenovich increased his speed by 60% and arrived at 8:35 AM. By what percentage should he have increased his usual speed to arrive exactly at 9:00 AM?

Answer: 30


If  $ABCDE$  is a regular pentagon and  $X$  is a point in its interior such that  $CDX$  is equilateral, compute  $\angle{AXE}$  in degrees.

*2020 CCA Math Bonanza Lightning Round #1.3*

Answer: 84


Let  $n{}$  be a positive integer. What is the smallest sum of digits that  $5^n + 6^n + 2022^n$  can take?

Answer: 8


A fair coin is tossed. If the first toss shows heads, it is tossed again: if the second toss shows heads, record 0; if the second toss shows tails, record 1. If the first toss shows tails, record any random real number in the closed interval $[0, 1]$. This process is repeated two independent times, obtaining two real numbers $x$ and $y$. What is the probability that $|x-y| > \frac{1}{2}$?

Answer: $\frac{7}{16}$


Using the distinct digits \( a, b, \) and \( c \), Araceli wrote the number \( abc \), and Luana wrote the numbers \( ab, bc, \) and \( ca \). Find the digits \( a, b, \) and \( c \), knowing that the sum of the numbers written by Luana is equal to the number written by Araceli.

Answer: 198


Given an isosceles triangle $ABC$ ($AB = BC$). Point $E$ is marked on ray $BA$ beyond point $A$, and point $D$ is marked on side $BC$. It is known that

$$
\angle ADC = \angle AEC = 60^\circ, \quad AD = CE = 13.
$$

Find the length of segment $AE$ if $DC = 9$.

Answer: 4


A positive integer $a$ is input into a machine. If $a$ is odd, the output is $a+3$. If $a$ is even, the output is $a+5$. This process can be repeated using each successive output as the next input. If the input is $a=15$ and the machine is used 51 times, what is the final output?

Answer: 218


In triangle  $ABC$ ,  $\angle ACB=50^{\circ}$  and  $\angle CBA=70^{\circ}$ . Let  $D$  be a foot of perpendicular from point  $A$  to side  $BC$ ,  $O$  circumcenter of  $ABC$  and  $E$  antipode of  $A$  in circumcircle  $ABC$ . Find  $\angle DAE$

Answer: 30


The area of rectangle $ABCD$ is 2011 square centimeters. The vertex $F$ of trapezoid $AFGE$ is on $BC$, and $D$ is the midpoint of leg $EG$. Find the area of trapezoid $AFGE$.

Answer: 2011


Line $l_1$ has equation $4x - 3y = 2$ and passes through point $D = (-2, -3)$. Line $l_2$ has equation $y = 2$ and intersects line $l_1$ at point $E$. Line $l_3$ has a positive slope, passes through point $D$, and meets $l_2$ at point $F$. The area of $\triangle DEF$ is $6$. What is the slope of $l_3$?

Answer: \frac{25}{32}


At what angle does the hypotenuse of a right triangle with a perimeter of 2 units appear from a point on the internal angle bisector of the right angle that is at a distance of \(\sqrt{2}\) from the vertex?

Answer: 45


Given a triangle $ABC$ with sides opposite angles $A$, $B$, $C$ denoted by $a$, $b$, $c$ respectively, and with $b = 3$, $c = 2$, and the area $S_{\triangle ABC} = \frac{3\sqrt{3}}{2}$:
1. Determine the value of angle $A$;
2. When angle $A$ is obtuse, find the height from vertex $B$ to side $BC$.

Answer: \frac{3\sqrt{57}}{19}


(a) Determine the maximum possible value $M$ that $x + y + z$ can take, given that $x$, $y$, and $z$ are positive real numbers that satisfy
\[ 16xyz = (x + y)^2 (x + z)^2. \]

(b) Show that there are infinitely many triples $(x, y, z)$ of positive rational numbers for which
\[ 16xyz = (x + y)^2 (x + z)^2 \quad \text{and} \quad x + y + z = M. \]

Answer: 4


Consider the problem where \( n \geq 4, \alpha_{1}, \alpha_{2}, \cdots, \alpha_{n}, \beta_{1}, \beta_{2}, \cdots, \beta_{n} \) are two sets of real numbers, satisfying \( \sum_{i=1}^{n} \alpha_{i}^{2}<1 \) and \( \sum_{i=1}^{n} \beta_{i}^{2}<1 \). Define \( A^{2}=1-\sum_{i=1}^{n} \alpha_{i}^{2} \), \( B^{2}=1-\sum_{i=1}^{n} \beta_{i}^{2} \), and \( W=\frac{1}{2}\left(1-\sum_{i=1}^{n} \alpha_{i} \beta_{i}\right)^{2} \). Determine all real numbers \( \lambda \) such that the equation

\[ x^{n}+\lambda \left(x^{n-1}+x^{n-2}+\cdots+x^{3}+W x^{2}+A B x+1\right)=0 \]

has only real roots.

Answer: 0


Let \( n \) be a positive integer. For each of the integers \( n+1, \ldots, 2n \), consider its largest odd divisor. Prove that the sum of all these divisors is equal to \( n^2 \).

Answer: n^2


In a country there are  $15$  cities, some pairs of which are connected by a single two-way airline of a company. There are  $3$  companies and if any of them cancels all its flights, then it would still be possible to reach every city from every other city using the other two companies. At least how many two-way airlines are there?

Answer: 21


The addition below is incorrect. The display can be made correct by changing one digit $d$, wherever it occurs, to another digit $e$. Find the sum of $d$ and $e$.
$\begin{tabular}{ccccccc} & 7 & 4 & 2 & 5 & 8 & 6 \\ + & 8 & 2 & 9 & 4 & 3 & 0 \\ \hline 1 & 2 & 1 & 2 & 0 & 1 & 6 \end{tabular}$

Answer: 8


In a circle,  let  $AB$  and  $BC$  be chords , with   $AB =\sqrt3,  BC =3\sqrt3, \angle ABC =60^o$ . Find the length of the circle chord that divides angle  $ \angle ABC$  in half.

Answer: 4


Let $P$ be a moving point on curve $C_1$, and $Q$ be a moving point on curve $C_2$. The minimum value of $|PQ|$ is called the distance between curves $C_1$ and $C_2$, denoted as $d(C_1,C_2)$. If $C_1: x^{2}+y^{2}=2$, $C_2: (x-3)^{2}+(y-3)^{2}=2$, then $d(C_1,C_2)=$ \_\_\_\_\_\_ ; if $C_3: e^{x}-2y=0$, $C_4: \ln x+\ln 2=y$, then $d(C_3,C_4)=$ \_\_\_\_\_\_ .

Answer: \sqrt{2}


Let be a circumcircle of radius  $ 1 $  of a triangle whose centered representation in the complex plane is given by the affixes of  $ a,b,c, $  and for which the equation  $ a+b\cos x +c\sin x=0 $  has a real root in  $ \left( 0,\frac{\pi }{2} \right) . $  prove that the area of the triangle is a real number from the interval  $ \left( 1,\frac{1+\sqrt 2}{2} \right] . $ *Gheorghe Iurea*

Answer: (1,\frac{1+\sqrt{2}}{2}]


Let $A$ and $B$ be points in space for which $A B=1$. Let $\mathcal{R}$ be the region of points $P$ for which $A P \leq 1$ and $B P \leq 1$. Compute the largest possible side length of a cube contained within $\mathcal{R}$.

Answer: \frac{\sqrt{10}-1}{3}


There are 10,001 students at a university. Some students join together to form several clubs (a student may belong to different clubs). Some clubs join together to form several societies (a club may belong to different societies). There are a total of \( k \) societies. Suppose that the following conditions hold:
1. Each pair of students are in exactly one club.
2. For each student and each society, the student is in exactly one club of the society.
3. Each club has an odd number of students. In addition, a club with \( 2m + 1 \) students (where \( m \) is a positive integer) is in exactly \( m \) societies.

Find all possible values of \( k \).

Answer: 5000


There are $N$ prime numbers written on a board (not necessarily distinct). It turns out that the sum of any three numbers on the board is also a prime number. What is the largest possible value of $N$?

Answer: 4


Using three colors $\mathbf{R}$, $\mathbf{G}$, and $\mathbf{B}$ to color a $2 \times 5$ table in a way that two squares sharing a common edge must be colored differently. How many different coloring methods are there?

Answer: 486


Among all such numbers \( n \) such that any convex 100-gon can be represented as the intersection (i.e., the common part) of \( n \) triangles, find the smallest.

Answer: 50


Find the equation of the circle passing through the intersection points of the circles \( x^{2}+y^{2}+4x-4y=0 \) and \( x^{2}+y^{2}-x=0 \), and also passing through the point \( (1,-1) \).

Answer: 9x^{2}+9y^{2}-14x+4y=0


Oleg drew an empty $50 \times 50$ table and wrote a number above each column and to the left of each row. It turned out that all 100 written numbers are different, with 50 of them being rational and the remaining 50 irrational. Then, in each cell of the table, he wrote the sum of the numbers written next to its row and its column ("addition table"). What is the maximum number of sums in this table that could end up being rational numbers?

Answer: 1250


In the acute-angled triangle $ABC$, the altitude $AD$, median $BE$, and angle bisector $CF$ intersect at point $O$. Find $\angle C$ if $OE = 2OC$.

Answer: \arccos(\frac{1}{7})


The equation of a line is given by $Ax+By=0$. If we choose two different numbers from the set $\{1, 2, 3, 4, 5\}$ to be the values of $A$ and $B$ each time, then the number of different lines that can be obtained is     ．

Answer: 18


Determine the volume of the released gas:

\[ \omega\left(\mathrm{SO}_{2}\right) = n\left(\mathrm{SO}_{2}\right) \cdot V_{m} = 0.1122 \cdot 22.4 = 2.52 \text{ liters} \]

Answer: 2.52


In a trapezoid, the midline is equal to 7, the height is $\frac{15 \sqrt{3}}{7}$, and the angle between the diagonals opposite the base is $120^{\circ}$.

Find the diagonals of the trapezoid.

Answer: 10


How many ordered triples  $(a, b, c)$  of positive integers are there such that none of  $a, b, c$  exceeds  $2010$  and each of  $a, b, c$  divides  $a + b + c$ ?

Answer: 9045


There is a board of size 7×12 cells and a cube, the side of which is equal to a cell. One face of the cube is painted with non-drying paint. The cube can be placed in a certain cell of the board and rolled over an edge to an adjacent face. The cube cannot be placed twice in the same cell. What is the maximum number of cells that the cube can visit without soiling the board with paint?

Answer: 84


An integer between $1000$ and $9999$, inclusive, is called balanced if the sum of its two leftmost digits equals the sum of its two rightmost digits. How many balanced integers are there?

Answer: 615


Let $A B C$ be a triangle, and let $D, E$, and $F$ be the midpoints of sides $B C, C A$, and $A B$, respectively. Let the angle bisectors of $\angle F D E$ and $\angle F B D$ meet at $P$. Given that $\angle B A C=37^{\circ}$ and $\angle C B A=85^{\circ}$, determine the degree measure of $\angle B P D$.

Answer: 61^{\circ}