One of the factors of $x^4+2x^2+9$ is:
$\textbf{(A)}\ x^2+3\qquad\textbf{(B)}\ x+1\qquad\textbf{(C)}\ x^2-3\qquad\textbf{(D)}\ x^2-2x-3\qquad\textbf{(E)}\ \text{none of these}$

Answer: (\textbf{E})


Problems 8, 9 and 10 use the data found in the accompanying paragraph and figures
Four friends, Art, Roger, Paul and Trisha, bake cookies, and all cookies have the same thickness. The shapes of the cookies differ, as shown.
$\circ$ Art's cookies are trapezoids:

$\circ$ Roger's cookies are rectangles:

$\circ$ Paul's cookies are parallelograms:

$\circ$ Trisha's cookies are triangles:

Each friend uses the same amount of dough, and Art makes exactly $12$ cookies. Art's cookies sell for $60$ cents each. To earn the same amount from a single batch, how much should one of Roger's cookies cost in cents?

Answer: 40


Given that the sequence $\{a\_n\}$ satisfies $\frac{1}{a_{n+1}} - \frac{1}{a_n} = d (n \in \mathbb{N}^*, d$ is a constant), the sequence $\{\frac{1}{b_n}\}$ is a harmonic sequence, and $b_1 + b_2 + b_3 + ... + b_9 = 90$, find the value of $b_4 + b_6$.

Answer: 20


12 Smurfs are seated around a round table. Each Smurf dislikes the 2 Smurfs next to them, but does not dislike the other 9 Smurfs. Papa Smurf wants to form a team of 5 Smurfs to rescue Smurfette, who was captured by Gargamel. The team must not include any Smurfs who dislike each other. How many ways are there to form such a team?

Answer: 36


In the triangle \( \triangle ABC \), the lengths of the sides opposite to angles \( \angle A \), \( \angle B \), and \( \angle C \) are \( a \), \( b \), and \( c \) respectively. Given that:
$$
\begin{array}{l}
\angle C = \max \{\angle A, \angle B, \angle C\}, \\
\sin C = 1 + \cos C \cdot \cos (A-B), \\
\frac{2}{a} + \frac{1}{b} = 1 .
\end{array}
$$

Find the minimum perimeter of the triangle \( \triangle ABC \).

Answer: 10


Given point $P(2,2)$, and circle $C$: $x^{2}+y^{2}-8y=0$. A moving line $l$ passing through point $P$ intersects circle $C$ at points $A$ and $B$, with the midpoint of segment $AB$ being $M$, and $O$ being the origin.

$(1)$ Find the equation of the trajectory of point $M$;

$(2)$ When $|OP|=|OM|$, find the equation of line $l$ and the area of $\Delta POM$.

Answer: \frac{16}{5}


What is the number of ways in which one can choose  $60$  unit squares from a  $11 \times 11$  chessboard such that no two chosen squares have a side in common?

Answer: 62


In triangle $\triangle ABC$, the opposite sides of angles $A$, $B$, and $C$ are $a$, $b$, $c$, and the vectors $\overrightarrow{m}=({\cos C, \cos({\frac{\pi}{2}-B})})$, $\overrightarrow{n}=({\cos({-4\pi+B}), -\sin C})$, and $\overrightarrow{m} \cdot \overrightarrow{n}=-\frac{\sqrt{2}}{2}$.
$(1)$ Find the measure of angle $A$;
$(2)$ If the altitude on side $AC$ is $2$, $a=3$, find the perimeter of $\triangle ABC$.

Answer: 5 + 2\sqrt{2} + \sqrt{5}


Answer: 54.6428571


How many integers from 1 to 1997 have a digit sum that is divisible by 5?

Answer: 399


Given $(625^{\log_2 250})^{\frac{1}{4}}$, find the value of the expression.

Answer: 250


What is the smallest natural number \( k \) for which the expression \( 2019 \cdot 2020 \cdot 2021 \cdot 2022 + k \) is a square of a natural number?

Answer: 1


Given a sequence $\{a_{n}\}$ where $a_{1}=1$ and $a_{n+1}-a_{n}=\left(-1\right)^{n+1}\frac{1}{n(n+2)}$, calculate the sum of the first 40 terms of the sequence $\{\left(-1\right)^{n}a_{n}\}$.

Answer: \frac{20}{41}


In the right triangular prism \( ABC-A_1B_1C_1 \), \( AB=1 \), \( BC=C_1C=\sqrt{3} \), and \( \angle ABC=90^\circ \). Point \( P \) is a moving point on plane \( ABC \). Find the minimum value of \( A_1P + \frac{1}{2}PC \).

Answer: \frac{5}{2}


What is the smallest natural number $n$ for which there exist natural numbers $x$ and $y$ satisfying the equation a) $x \cdot (x+n) = y^{2}$; b) $x \cdot (x+n) = y^{3}$?

Answer: 2


Several businessmen decided to start a company and share the profits equally. One of the businessmen was appointed as the director. One day, this director transferred part of the profit from the company's account to their personal account. This amount was three times larger than what each of the others would receive if they had divided the remaining profit equally among themselves. After this, the director left the company. The next director, one of the remaining businessmen, did the same thing as the previous director, and so on. Eventually, the penultimate director transferred to their personal account an amount that was also three times larger than what was left for the last businessman. As a result of these profit distributions, the last businessman received 190 times less money than the first director. How many businessmen started this company?

Answer: 19


The sequence \(\left(a_{n}\right)\) is defined such that \(a_{n}=n^{2}\) for \(1 \leq n \leq 5\) and for all natural numbers \(n\) the following equality holds: \(a_{n+5} + a_{n+1} = a_{n+4} + a_{n}\). Find \(a_{2015}\).

Answer: 17


How many numbers from 1 to 100 are divisible by 3, but do not contain the digit 3?

Answer: 26


Given that the terminal side of angle \\(\alpha\\) passes through the point \\(P(m,2\sqrt{2})\\), \\(\sin \alpha= \frac{2\sqrt{2}}{3}\\) and \\(\alpha\\) is in the second quadrant.
\\((1)\\) Find the value of \\(m\\);
\\((2)\\) If \\(\tan \beta= \sqrt{2}\\), find the value of \\( \frac{\sin \alpha\cos \beta+3\sin \left( \frac{\pi}{2}+\alpha\right)\sin \beta}{\cos (\pi+\alpha)\cos (-\beta)-3\sin \alpha\sin \beta}\\).

Answer: \frac{\sqrt{2}}{11}


Find all triples  $(p, q, r)$  of prime numbers for which  $4q - 1$  is a prime number and   $$ \frac{p + q}{p + r} = r - p $$  holds.

*(Walther Janous)*

Answer: (2,3,3)


Among the digits 0, 1, ..., 9, calculate the number of three-digit numbers that can be formed using repeated digits.

Answer: 252


There are  $2017$  distinct points in the plane. For each pair of these points, construct the midpoint of the segment joining the pair of points. What is the minimum number of distinct midpoints among all possible ways of placing the points?

Answer: 2016


Let $ \left(a_{n}\right)$ be the sequence of reals defined by $ a_{1}=\frac{1}{4}$ and the recurrence $ a_{n}= \frac{1}{4}(1+a_{n-1})^{2}, n\geq 2$. Find the minimum real $ \lambda$ such that for any non-negative reals $ x_{1},x_{2},\dots,x_{2002}$, it holds
\[ \sum_{k=1}^{2002}A_{k}\leq \lambda a_{2002}, \]
where $  A_{k}= \frac{x_{k}-k}{(x_{k}+\cdots+x_{2002}+\frac{k(k-1)}{2}+1)^{2}}, k\geq 1$.

Answer: \frac{1}{2005004}


Answer: 14


Given the line $y=kx+1$ intersects the parabola $C: x^2=4y$ at points $A$ and $B$, and a line $l$ is parallel to $AB$ and is tangent to the parabola $C$ at point $P$, find the minimum value of the area of triangle $PAB$ minus the length of $AB$.

Answer: -\frac{64}{27}


For each positive integer $n$, define $s(n)$ to equal the sum of the digits of $n$. The number of integers $n$ with $100 \leq n \leq 999$ and $7 \leq s(n) \leq 11$ is $S$. What is the integer formed by the rightmost two digits of $S$?

Answer: 24


Answer: 1


On standard dice, the total number of pips on each pair of opposite faces is 7. Two standard dice are placed in a stack, so that the total number of pips on the two touching faces is 5. What is the total number of pips on the top and bottom faces of the stack?

A 5
B 6
C 7
D 8
E 9

Answer: 9


On a faded piece of paper it is possible to read the following:
\[(x^2 + x + a)(x^{15}- \cdots ) = x^{17} + x^{13} + x^5 - 90x^4 + x - 90.\]
Some parts have got lost, partly the constant term of the first factor of the left side, partly the majority of the summands of the second factor. It would be possible to restore the polynomial forming the other factor, but we restrict ourselves to asking the following question: What is the value of the constant term  $a$ ? We assume that all polynomials in the statement have only integer coefficients.

Answer: 2


Gavrila is in an elevator cabin which is moving downward with a deceleration of 5 m/s². Find the force with which Gavrila presses on the floor. Gavrila's mass is 70 kg, and the acceleration due to gravity is 10 m/s². Give the answer in newtons, rounding to the nearest whole number if necessary.

Answer: 350


On an $8 \times 8$ chessboard, there are 16 rooks, each placed in a different square. What is the minimum number of pairs of rooks that can attack each other (a pair of rooks can attack each other if they are in the same row or column and there are no other rooks between them)?

Answer: 16


A cube has exactly six faces and twelve edges. How many vertices does a cube have?
(A) 4
(B) 5
(C) 6
(D) 7
(E) 8

Answer: 8


Answer: 1


Let \( A \), \( B \), and \( C \) be three points on the edge of a circular chord such that \( B \) is due west of \( C \) and \( ABC \) is an equilateral triangle whose side is 86 meters long. A boy swam from \( A \) directly toward \( B \). After covering a distance of \( x \) meters, he turned and swam westward, reaching the shore after covering a distance of \( y \) meters. If \( x \) and \( y \) are both positive integers, determine \( y \).

Answer: 12


With all angles measured in degrees, the product $\prod_{k=1}^{45} \csc^2(2k-1)^\circ=m^n$, where $m$ and $n$ are integers greater than 1. Find $m+n$.

Answer: 91


The $52$ cards in a deck are numbered $1, 2, \cdots, 52$. Alex, Blair, Corey, and Dylan each picks a card from the deck without replacement and with each card being equally likely to be picked, The two persons with lower numbered cards from a team, and the two persons with higher numbered cards form another team. Let $p(a)$ be the probability that Alex and Dylan are on the same team, given that Alex picks one of the cards $a$ and $a+9$, and Dylan picks the other of these two cards. The minimum value of $p(a)$ for which $p(a)\ge\frac{1}{2}$ can be written as $\frac{m}{n}$. where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Answer: 263


How many triples \((x, y, z) \in \mathbb{R}^{3}\) satisfy the following system of equations:
\[
\begin{array}{l}
x = 2018 - 2019 \cdot \operatorname{sign}(y + z) \\
y = 2018 - 2019 \cdot \operatorname{sign}(z + x) \\
z = 2018 - 2019 \cdot \operatorname{sign}(x + y)
\end{array}
\]
where for any real number \(\alpha\), \(\operatorname{sign}(\alpha)\) is defined as:
\[
\operatorname{sign}(\alpha) =
\begin{cases}
+1 & \text{if } \alpha > 0 \\
0 & \text{if } \alpha = 0 \\
-1 & \text{if } \alpha < 0
\end{cases}
\]

Answer: 3


A biologist wishes to estimate the fish population in a protected area. Initially, on March 1st, she captures and tags 80 fish, then releases them back into the water. Four months later, on July 1st, she captures another 90 fish for a follow-up study, finding that 4 of these are tagged. For her estimation, she assumes that 30% of these fish have left the area by July 1st due to various environmental factors, and that an additional 50% of the fish in the July sample weren't in the area on March 1st due to new arrivals. How many fish does she estimate were in the area on March 1st?

Answer: 900


A permutation of \{1,2, \ldots, 7\} is chosen uniformly at random. A partition of the permutation into contiguous blocks is correct if, when each block is sorted independently, the entire permutation becomes sorted. For example, the permutation $(3,4,2,1,6,5,7)$ can be partitioned correctly into the blocks $[3,4,2,1]$ and $[6,5,7]$, since when these blocks are sorted, the permutation becomes $(1,2,3,4,5,6,7)$. Find the expected value of the maximum number of blocks into which the permutation can be partitioned correctly.

Answer: \frac{151}{105}


Find the smallest possible value of \(x+y\) where \(x, y \geq 1\) and \(x\) and \(y\) are integers that satisfy \(x^{2}-29y^{2}=1\).

Answer: 11621


Suppose  $n$  and  $r$  are nonnegative integers such that no number of the form  $n^2+r-k(k+1) \text{ }(k\in\mathbb{N})$  equals to  $-1$  or a positive composite number.  Show that  $4n^2+4r+1$  is  $1$ ,  $9$ , or prime.

Answer: 1,9,


From vertex  $A$  of an equilateral triangle  $ABC$ , a ray  $Ax$  intersects  $BC$  at point  $D$ . Let  $E$  be a point on  $Ax$  such that  $BA =BE$ . Calculate  $\angle AEC$ .

Answer: 30


As shown in the diagram, there is a sequence of curves \(P_{0}, P_{1}, P_{2}, \cdots\). It is given that \(P_{0}\) is an equilateral triangle with an area of 1. Each \(P_{k+1}\) is obtained from \(P_{k}\) by performing the following operations: each side of \(P_{k}\) is divided into three equal parts, an equilateral triangle is constructed outwards on the middle segment of each side, and the middle segments are then removed (\(k=0,1,2, \cdots\)). Let \(S_{n}\) denote the area enclosed by the curve \(P_{n}\).
1. Find a general formula for the sequence \(\{S_{n}\}\).
2. Evaluate \(\lim _{n \rightarrow \infty} S_{n}\).

Answer: \frac{8}{5}


Let \(S\) be the set of all nonzero real numbers. Let \(f : S \to S\) be a function such that
\[f(x) + f(y) = cf(xyf(x + y))\]
for all \(x, y \in S\) such that \(x + y \neq 0\) and for some nonzero constant \(c\). Determine all possible functions \(f\) that satisfy this equation and calculate \(f(5)\).

Answer: \frac{1}{5}


$(1)$ $f(n)$ is a function defined on the set of positive integers, satisfying:
① When $n$ is a positive integer, $f(f(n))=4n+9$;
② When $k$ is a non-negative integer, $f(2^{k})=2^{k+1}+3$. Find the value of $f(1789)$.
$(2)$ The function $f$ is defined on the set of ordered pairs of positive integers, and satisfies the following properties:
① $f(x,x)=x$;
② $f(x,y)=f(y,x)$;
③ $(x+y)f(x,y)=yf(x,x+y)$. Find $f(14,52)$.

Answer: 364


How many paths are there from the starting point $C$ to the end point $D$, if every step must be up or to the right in a grid of 8 columns and 7 rows?

Answer: 6435


Let \( a \) and \( b \) be two real numbers. We set \( s = a + b \) and \( p = ab \). Express \( a^3 + b^3 \) in terms of \( s \) and \( p \) only.

Answer: ^3-3sp


Let  $l$  be a tangent line at the point  $T\left(t,\ \frac{t^2}{2}\right)\ (t>0)$  on the curve  $C: y=\frac{x^2}{2}.$  A circle  $S$  touches the curve  $C$  and the  $x$  axis.
Denote by  $(x(t),\ y(t))$  the center of the circle  $S$ . Find  $\lim_{r\rightarrow +0}\int_r^1 x(t)y(t)\ dt.$

Answer: \frac{\sqrt{2}+1}{30}


A point $P$ is chosen uniformly at random inside a square of side length 2. If $P_{1}, P_{2}, P_{3}$, and $P_{4}$ are the reflections of $P$ over each of the four sides of the square, find the expected value of the area of quadrilateral $P_{1} P_{2} P_{3} P_{4}$.

Answer: 8


A quadrangular prism \(A B C D A_{1} B_{1} C_{1} D_{1}\) has a rhombus \(A B C D\) at its base, where \(A C = 4\) and \(\angle D B C = 30^{\circ}\). A sphere passes through the vertices \(D, A, B, B_{1}, C_{1}, D_{1}\).

a) Find the area of the circle obtained by intersecting the sphere with the plane passing through points \(B, C\), and \(D\).

b) Find the angle \(A_{1} C D\).

c) It is additionally given that the radius of the sphere is 5. Find the volume of the prism.

Answer: 48\sqrt{3}


The base of the pyramid is a rectangle with an area of $S$. Two of the lateral faces are perpendicular to the plane of the base, while the other two are inclined to it at angles of 30° and $60^{\circ}$. Find the volume of the pyramid.

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


Let $f(n) = \frac{x_1 + x_2 + \cdots + x_n}{n}$, where $n$ is a positive integer. If $x_k = (-1)^k, k = 1, 2, \cdots, n$, the set of possible values of $f(n)$ is:

Answer: $\{0, -\frac{1}{n}\}$


In a convex polygon with an odd number of vertices \(2n + 1\), two diagonals are chosen independently at random. Find the probability that these diagonals intersect inside the polygon.

Answer: \frac{n(2n-1)}{3(2n^2-n-2)}


Let line \( l: y = kx + m \) (where \( k \) and \( m \) are integers) intersect the ellipse \( \frac{x^2}{16} + \frac{y^2}{12} = 1 \) at two distinct points \( A \) and \( B \), and intersect the hyperbola \( \frac{x^2}{4} - \frac{y^2}{12} = 1 \) at two distinct points \( C \) and \( D \). The vector sum \( \overrightarrow{AC} + \overrightarrow{BD} = \overrightarrow{0} \). The total number of such lines is _____.

Answer: 9


Given the set $M=\{x|2x^{2}-3x-2=0\}$ and the set $N=\{x|ax=1\}$. If $N \subset M$, what is the value of $a$?

Answer: \frac{1}{2}


Two swimmers, at opposite ends of a $90$-foot pool, start to swim the length of the pool,
one at the rate of $3$ feet per second, the other at $2$ feet per second.
They swim back and forth for $12$ minutes. Allowing no loss of times at the turns, find the number of times they pass each other.
$\textbf{(A)}\ 24\qquad \textbf{(B)}\ 21\qquad \textbf{(C)}\ 20\qquad \textbf{(D)}\ 19\qquad \textbf{(E)}\ 18$

Answer: \textbf{(C)}\20


Find the greatest real  $k$  such that, for every tetrahedron  $ABCD$  of volume  $V$ , the product of areas of faces  $ABC,ABD$  and  $ACD$  is at least  $kV^2$ .

Answer: 9/2


Let $a$ and $b$ be positive integers such that $ab \neq 1$. Determine all possible integer values of $f(a, b)=\frac{a^{2}+b^{2}+ab}{ab-1}$.

Answer: 7


The vertices of a $3 \times 1 \times 1$ rectangular prism are $A, B, C, D, E, F, G$, and $H$ so that $A E, B F$, $C G$, and $D H$ are edges of length 3. Point $I$ and point $J$ are on $A E$ so that $A I=I J=J E=1$. Similarly, points $K$ and $L$ are on $B F$ so that $B K=K L=L F=1$, points $M$ and $N$ are on $C G$ so that $C M=M N=N G=1$, and points $O$ and $P$ are on $D H$ so that $D O=O P=P H=1$. For every pair of the 16 points $A$ through $P$, Maria computes the distance between them and lists the 120 distances. How many of these 120 distances are equal to $\sqrt{2}$?

Answer: 32


In a \(7 \times 7\) table, some cells are black while the remaining ones are white. In each white cell, the total number of black cells located with it in the same row or column is written; nothing is written in the black cells. What is the maximum possible sum of the numbers in the entire table?

Answer: 168


Given positive numbers $x$ and $y$ satisfying $x^2+y^2=1$, find the maximum value of $\frac {1}{x}+ \frac {1}{y}$.

Answer: 2\sqrt{2}


Let point \( A \) (not on the y-axis) be a moving point on the line \( x - 2y + 13 = 0 \). From point \( A \), draw two tangents to the parabola \( y^2 = 8x \), with \( M \) and \( N \) being the points of tangency. Lines \( AM \) and \( AN \) intersect the y-axis at points \( B \) and \( C \), respectively. Prove:
1. The line \( MN \) always passes through a fixed point.
2. The circumcircle of triangle \( ABC \) always passes through a fixed point, and find the minimum radius of this circle.

Answer: \frac{3\sqrt{5}}{2}


Points with coordinates $(1,1),(5,1)$ and $(1,7)$ are three vertices of a rectangle. What are the coordinates of the fourth vertex of the rectangle?

Answer: (5,7)


Given a chessboard consisting of $16 \times 16$ small squares, with alternating black and white colors, define an operation $A$ as follows: change the color of all squares in the $i$-th row and all squares in the $j$-th column to their opposite color $(1 \leqslant i, j \leqslant 16)$. We call $A$ an operation on the small square located at the $i$-th row and $j$-th column.

Can we change all the squares on the chessboard to the same color through a certain number of operations? Prove your conclusion.

Answer: Yes


The diagram below shows \( \triangle ABC \), which is isosceles with \( AB = AC \) and \( \angle A = 20^\circ \). The point \( D \) lies on \( AC \) such that \( AD = BC \). The segment \( BD \) is constructed as shown. Determine \( \angle ABD \) in degrees.

Answer: 10


Consider those functions $f$ that satisfy $f(x+5)+f(x-5) = f(x)$ for all real $x$. Find the least common positive period $p$ for all such functions.

Answer: 30


Let  $s_1, s_2, s_3$  be the three roots of  $x^3 + x^2 +\frac92x + 9$ .  $$ \prod_{i=1}^{3}(4s^4_i + 81) $$   can be written as  $2^a3^b5^c$ . Find  $a + b + c$ .

Answer: 16


One standard balloon can lift a basket with contents weighing not more than 80 kg. Two standard balloons can lift the same basket with contents weighing not more than 180 kg. What is the weight, in kg, of the basket?

Answer: 20


Answer: 1-\frac{1}{n}<a_n<1


Find the minimum value of $n (n > 0)$ such that the function \\(f(x)= \begin{vmatrix} \sqrt {3} & \sin x \\\\ 1 & \cos x\\end{vmatrix} \\) when shifted $n$ units to the left becomes an even function.

Answer: \frac{5\pi}{6}


Let $P$ be a convex polygon with area 1. Show that there exists a rectangle with area 2 that contains it. Can this result be improved?

Answer: 2


Let  $ x \equal{} \sqrt[3]{\frac{4}{25}}\,$ .  There is a unique value of  $ y$  such that  $ 0 < y < x$  and  $ x^x \equal{} y^y$ .  What is the value of  $ y$ ?  Express your answer in the form  $ \sqrt[c]{\frac{a}{b}}\,$ , where  $ a$  and  $ b$  are relatively prime positive integers and  $ c$  is a prime number.

Answer: \sqrt[3]{\frac{32}{3125}}


The price of Type A remote control car is 46.5 yuan, and the price of Type B remote control car is 54.5 yuan. Lele has 120 yuan. If he buys both types of remote control cars, will he have enough money? If so, how much money will he have left after the purchase?

Answer: 19


Define a power cycle to be a set $S$ consisting of the nonnegative integer powers of an integer $a$, i.e. $S=\left\{1, a, a^{2}, \ldots\right\}$ for some integer $a$. What is the minimum number of power cycles required such that given any odd integer $n$, there exists some integer $k$ in one of the power cycles such that $n \equiv k$ $(\bmod 1024) ?$

Answer: 10


For any finite sequence of positive integers \pi, let $S(\pi)$ be the number of strictly increasing subsequences in \pi with length 2 or more. For example, in the sequence $\pi=\{3,1,2,4\}$, there are five increasing sub-sequences: $\{3,4\},\{1,2\},\{1,4\},\{2,4\}$, and $\{1,2,4\}$, so $S(\pi)=5$. In an eight-player game of Fish, Joy is dealt six cards of distinct values, which she puts in a random order \pi from left to right in her hand. Determine $\sum_{\pi} S(\pi)$ where the sum is taken over all possible orders \pi of the card values.

Answer: 8287


A prime number $ q $ is called***'Kowai'***number if  $ q = p^2 + 10$  where  $q$ ,   $p$ ,  $p^2-2$ ,   $p^2-8$ ,   $p^3+6$  are prime numbers. WE know that, at least one ***'Kowai'*** number can be found. Find the summation of all ***'Kowai'*** numbers.

Answer: 59


The Wolf and Ivan Tsarevich are 20 versts away from a source of living water, and the Wolf is taking Ivan Tsarevich there at a speed of 3 versts per hour. To revive Ivan Tsarevich, one liter of water is needed, which flows from the source at a rate of half a liter per hour. At the source, there is a Raven with unlimited carrying capacity; it must gather the water, after which it will fly towards the Wolf and Ivan Tsarevich at a speed of 6 versts per hour, spilling a quarter liter of water every hour. After how many hours will it be possible to revive Ivan Tsarevich?

Answer: 4


Let \( a, b, c \) be positive real numbers such that \( a b c = 1 \). Prove that:
\[
\frac{1}{a^{3}(b+c)}+\frac{1}{b^{3}(c+a)}+\frac{1}{c^{3}(a+b)} \geq \frac{3}{2}.
\]

Answer: \frac{3}{2}


In the Cartesian coordinate system $(xOy)$, a pole is established at the origin $O$ with the non-negative semi-axis of the $x$-axis as the polar axis, forming a polar coordinate system. Given that the equation of line $l$ is $4ρ\cos θ-ρ\sin θ-25=0$, and the curve $W$ is defined by the parametric equations $x=2t, y=t^{2}-1$.

1. Find the Cartesian equation of line $l$ and the general equation of curve $W$.
2. If point $P$ is on line $l$, and point $Q$ is on curve $W$, find the minimum value of $|PQ|$.

Answer: \frac{8\sqrt{17}}{17}


A deck consists of six red cards and six green cards, each with labels $A$, $B$, $C$, $D$, $E$ corresponding to each color. Two cards are dealt from this deck. A winning pair consists of cards that either share the same color or the same label. Calculate the probability of drawing a winning pair.
A) $\frac{1}{2}$
B) $\frac{10}{33}$
C) $\frac{30}{66}$
D) $\frac{35}{66}$
E) $\frac{40}{66}$

Answer: \frac{35}{66}


What is the largest four-digit negative integer congruent to $1 \pmod{17}$?

Answer: -1002


Given two vectors $\overrightarrow {a}$ and $\overrightarrow {b}$ with an angle of $\frac {2\pi}{3}$ between them, $|\overrightarrow {a}|=2$, $|\overrightarrow {b}|=3$, let $\overrightarrow {m}=3\overrightarrow {a}-2\overrightarrow {b}$ and $\overrightarrow {n}=2\overrightarrow {a}+k\overrightarrow {b}$:
1. If $\overrightarrow {m} \perp \overrightarrow {n}$, find the value of the real number $k$;
2. Discuss whether there exists a real number $k$ such that $\overrightarrow {m} \| \overrightarrow {n}$, and explain the reasoning.

Answer: \frac{4}{3}


Let  $\Delta$  denote the set of all triangles in a plane. Consider the function  $f: \Delta\to(0,\infty)$  defined by  $f(ABC) = \min \left( \dfrac ba, \dfrac cb \right)$ , for any triangle  $ABC$  with  $BC=a\leq CA=b\leq AB = c$ . Find the set of values of  $f$ .

Answer: [1,\frac{\sqrt{5}+1}{2})


In the diagram, square $ABCD$ has sides of length $4,$ and $\triangle ABE$ is equilateral. Line segments $BE$ and $AC$ intersect at $P.$ Point $Q$ is on $BC$ so that $PQ$ is perpendicular  to $BC$ and $PQ=x.$ [asy]
pair A, B, C, D, E, P, Q;
A=(0,0);
B=(4,0);
C=(4,-4);
D=(0,-4);
E=(2,-3.464);
P=(2.535,-2.535);
Q=(4,-2.535);
draw(A--B--C--D--A--E--B);
draw(A--C);
draw(P--Q, dashed);
label("A", A, NW);
label("B", B, NE);
label("C", C, SE);
label("D", D, SW);
label("E", E, S);
label("P", P, W);
label("Q", Q, dir(0));
label("$x$", (P+Q)/2, N);
label("4", (A+B)/2, N);
[/asy] Determine the measure of angle $BPC.$

Answer: 105^\circ


Given that \( a \) is a positive real number and
\[ f(x) = \log_{2} \left( ax + \sqrt{2x^2 + 1} \right) \]
is an odd function, find the solution set for \( f(x) > \frac{3}{2} \).

Answer: (\frac{7}{8},+\infty)


In an obtuse triangle, the longest side is 4, and the shortest side is 2. Could the area of the triangle be greater than $2 \sqrt{3}$?

Answer: No


Let $ABCD$ be a convex quadrilateral with $AB = AD$ and $CB = CD$. The bisector of $\angle BDC$ intersects $BC$ at $L$, and $AL$ intersects $BD$ at $M$, and it is known that $BL = BM$. Determine the value of $2\angle BAD + 3\angle BCD$.

Answer: 540^\circ


The bases of two equilateral triangles with side lengths \(a\) and \(3a\) lie on the same line. The triangles are situated on opposite sides of the line and do not share any common points. The distance between the nearest ends of their bases is \(2a\). Find the distance between the vertices of the triangles that do not lie on the given line.

Answer: 2a\sqrt{7}


A circle is inscribed in a convex quadrilateral \(ABCD\) with its center at point \(O\), and \(AO=OC\). Additionally, \(BC=5\), \(CD=12\), and \(\angle DAB\) is a right angle.

Find the area of the quadrilateral \(ABCD\).

Answer: 60


The numbers \(a_{1}, a_{2}, \ldots, a_{n}\) are such that the sum of any seven consecutive numbers is negative, and the sum of any eleven consecutive numbers is positive. What is the largest possible \(n\) for which this is true?

Answer: 16


In isosceles trapezoid $ABCD$, parallel bases $\overline{AB}$ and $\overline{CD}$ have lengths $500$ and $650$, respectively, and $AD=BC=333$. The angle bisectors of $\angle{A}$ and $\angle{D}$ meet at $P$, and the angle bisectors of $\angle{B}$ and $\angle{C}$ meet at $Q$. Find $PQ$.
Diagram
[asy] /* Made by MRENTHUSIASM */ size(300); pair A, B, C, D, A1, B1, C1, D1, P, Q; A = (-250,6*sqrt(731)); B = (250,6*sqrt(731)); C = (325,-6*sqrt(731)); D = (-325,-6*sqrt(731)); A1 = bisectorpoint(B,A,D); B1 = bisectorpoint(A,B,C); C1 = bisectorpoint(B,C,D); D1 = bisectorpoint(A,D,C); P = intersectionpoint(A--300*(A1-A)+A,D--300*(D1-D)+D); Q = intersectionpoint(B--300*(B1-B)+B,C--300*(C1-C)+C); draw(anglemark(P,A,B,1000),red); draw(anglemark(D,A,P,1000),red); draw(anglemark(A,B,Q,1000),red); draw(anglemark(Q,B,C,1000),red); draw(anglemark(P,D,A,1000),red); draw(anglemark(C,D,P,1000),red); draw(anglemark(Q,C,D,1000),red); draw(anglemark(B,C,Q,1000),red); add(pathticks(anglemark(P,A,B,1000), n = 1, r = 0.15, s = 750, red)); add(pathticks(anglemark(D,A,P,1000), n = 1, r = 0.15, s = 750, red)); add(pathticks(anglemark(A,B,Q,1000), n = 1, r = 0.15, s = 750, red)); add(pathticks(anglemark(Q,B,C,1000), n = 1, r = 0.15, s = 750, red)); add(pathticks(anglemark(P,D,A,1000), n = 2, r = 0.12, spacing = 150, s = 750, red)); add(pathticks(anglemark(C,D,P,1000), n = 2, r = 0.12, spacing = 150, s = 750, red)); add(pathticks(anglemark(Q,C,D,1000), n = 2, r = 0.12, spacing = 150, s = 750, red)); add(pathticks(anglemark(B,C,Q,1000), n = 2, r = 0.12, spacing = 150, s = 750, red)); dot("$A$",A,1.5*dir(A),linewidth(4)); dot("$B$",B,1.5*dir(B),linewidth(4)); dot("$C$",C,1.5*dir(C),linewidth(4)); dot("$D$",D,1.5*dir(D),linewidth(4)); dot("$P$",P,1.5*NE,linewidth(4)); dot("$Q$",Q,1.5*NW,linewidth(4)); draw(A--B--C--D--cycle^^A--P--D^^B--Q--C^^P--Q); label("$500$",midpoint(A--B),1.25N); label("$650$",midpoint(C--D),1.25S); label("$333$",midpoint(A--D),1.25W); label("$333$",midpoint(B--C),1.25E); [/asy] ~MRENTHUSIASM ~ihatemath123

Answer: 242


Let  $I$  be the incenter of  $\triangle ABC$ , and  $O$  be the excenter corresponding to  $B$ . If  $|BI|=12$ ,  $|IO|=18$ , and  $|BC|=15$ , then what is  $|AB|$ ?

Answer: 24


In the parallelogram $\mathrm{ABCD}$, points $\mathrm{E}$ and $\mathrm{F}$ lie on $\mathrm{AD}$ and $\mathrm{AB}$ respectively. Given that the area of $S_{A F I E} = 49$, the area of $\triangle B G F = 13$, and the area of $\triangle D E H = 35$, find the area of $S_{G C H I}$.

Answer: 97


Define a domino to be an ordered pair of distinct positive integers. A proper sequence of dominos is a list of distinct dominos in which the first coordinate of each pair after the first equals the second coordinate of the immediately preceding pair, and in which $(i,j)$ and $(j,i)$ do not both appear for any $i$ and $j$. Let $D_{40}$ be the set of all dominos whose coordinates are no larger than 40. Find the length of the longest proper sequence of dominos that can be formed using the dominos of $D_{40}.$

Answer: 761


Given that \( x, y, z \) are positive real numbers and \( x + y + z = 1 \), if \( \frac{a}{xyz} = \frac{1}{x} + \frac{1}{y} + \frac{1}{z} - 2 \), find the range of the real number \( a \).

Answer: (0,\frac{7}{27}]


The line $y - x \sqrt{3} + 3 = 0$ intersects the parabola $2y^2 = 2x + 3$ at points $A$ and $B.$  Let $P = (\sqrt{3},0).$  Find $|AP - BP|.$

Answer: \frac{2}{3}


Inside an angle of $30^\circ$ with vertex $A$, a point $K$ is chosen such that its distances to the sides of the angle are 1 and 2. Through point $K$, all possible straight lines intersecting the sides of the angle are drawn. Find the minimum area of the triangle cut off from the angle by one of these lines.

Answer: 8


A point $Q$ is chosen in the interior of $\triangle DEF$ such that when lines are drawn through $Q$ parallel to the sides of $\triangle DEF$, the resulting smaller triangles $u_{1}$, $u_{2}$, and $u_{3}$ have areas $9$, $16$, and $36$, respectively. Find the area of $\triangle DEF$.

Answer: 169


How many roots does the equation \(8x(1 - 2x^2)(8x^4 - 8x^2 + 1) = 1\) have on the interval \([0, 1]\)?

Answer: 4


The numbers \(a\) and \(b\) are such that each of the two quadratic polynomials \(x^2 + ax + b\) and \(x^2 + bx + a\) has two distinct roots, and the product of these polynomials has exactly three distinct roots. Find all possible values for the sum of these three roots.

Answer: 0


For \(a, b, c > 0\), find the maximum value of the expression

\[
A=\frac{a^{2}(b+c)+b^{2}(c+a)+c^{2}(a+b)}{a^{3}+b^{3}+c^{3}-2 abc}
\]

Answer: 6


How many integers between 2 and 100 inclusive cannot be written as \( m \cdot n \), where \( m \) and \( n \) have no common factors and neither \( m \) nor \( n \) is equal to 1?

Answer: 35


For an integer $x \geq 1$, let $p(x)$ be the least prime that does not divide $x$, and define $q(x)$ to be the product of all primes less than $p(x)$. In particular, $p(1) = 2.$ For $x$ having $p(x) = 2$, define $q(x) = 1$. Consider the sequence $x_0, x_1, x_2, \ldots$ defined by $x_0 = 1$ and \[ x_{n+1} = \frac{x_n p(x_n)}{q(x_n)} \] for $n \geq 0$. Find all $n$ such that $x_n = 1995$.

Answer: 142


An urn contains one red ball and one blue ball. A box of extra red and blue balls lies nearby. George performs the following operation four times: he draws a ball from the urn at random and then takes a ball of the same color from the box and returns those two matching balls to the urn. After the four iterations the urn contains six balls. What is the probability that the urn contains three balls of each color?
$\textbf{(A) } \frac16 \qquad \textbf{(B) }\frac15 \qquad \textbf{(C) } \frac14 \qquad \textbf{(D) } \frac13 \qquad \textbf{(E) } \frac12$

Answer: \textbf{(B)}\frac{1}{5}


Find the length of the arc of the curve \(r = a \cos^3 \left(\frac{\varphi}{3}\right)\) where \(a > 0\).

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


Let \( x \in \left(-\frac{3\pi}{4}, \frac{\pi}{4}\right) \), and \( \cos \left(\frac{\pi}{4} - x\right) = -\frac{3}{5} \). Find the value of \( \cos 2x \).

Answer: -\frac{24}{25}


A square with area $4$ is inscribed in a square with area $5$, with each vertex of the smaller square on a side of the larger square. A vertex of the smaller square divides a side of the larger square into two segments, one of length $a$, and the other of length $b$. What is the value of $ab$?

Answer: \frac{1}{2}


A cat has found $432_{9}$ methods in which to extend each of her nine lives.  How many methods are there in base 10?

Answer: 353


For any positive integer \( a \), let \( \tau(a) \) be the number of positive divisors of \( a \). Find, with proof, the largest possible value of \( 4 \tau(n) - n \) over all positive integers \( n \).

Answer: 12


Calculate the value of the expression

$$
\frac{\left(3^{4}+4\right) \cdot\left(7^{4}+4\right) \cdot\left(11^{4}+4\right) \cdot \ldots \cdot\left(2015^{4}+4\right) \cdot\left(2019^{4}+4\right)}{\left(1^{4}+4\right) \cdot\left(5^{4}+4\right) \cdot\left(9^{4}+4\right) \cdot \ldots \cdot\left(2013^{4}+4\right) \cdot\left(2017^{4}+4\right)}
$$

Answer: 4080401


In the quadrilateral pyramid $S-ABCD$ with a right trapezoid as its base, where $\angle ABC = 90^\circ$, $SA \perp$ plane $ABCD$, $SA = AB = BC = 1$, and $AD = \frac{1}{2}$, find the tangent of the angle between plane $SCD$ and plane $SBA$.

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


Given a 3x3 matrix where each row and each column forms an arithmetic sequence, and the middle element $a_{22} = 5$, find the sum of all nine elements.

Answer: 45


Maruška received a magical pouch from her grandmother, which doubled the amount of gold coins it contained every midnight. On Monday at noon, Maruška placed some gold coins into the empty pouch. On Tuesday and Wednesday, she took out 40 gold coins from the pouch each day and did not add any coins back in. On Thursday, she again took out 40 gold coins, and the pouch was empty.

How many gold coins did Maruška place into the pouch on Monday?

How many gold coins should she have placed into the empty pouch so that she could withdraw 40 gold coins every day, without adding any coins back in, and such that the amount of gold coins in the pouch was the same before each withdrawal?

Answer: 40


Let $T$ be the triangle in the coordinate plane with vertices $(0,0), (4,0),$ and $(0,3).$ Consider the following five isometries (rigid transformations) of the plane: rotations of $90^{\circ}, 180^{\circ},$ and $270^{\circ}$ counterclockwise around the origin, reflection across the $x$-axis, and reflection across the $y$-axis. How many of the $125$ sequences of three of these transformations (not necessarily distinct) will return $T$ to its original position? (For example, a $180^{\circ}$ rotation, followed by a reflection across the $x$-axis, followed by a reflection across the $y$-axis will return $T$ to its original position, but a $90^{\circ}$ rotation, followed by a reflection across the $x$-axis, followed by another reflection across the $x$-axis will not return $T$ to its original position.)

Answer: 12


On the sphere \(\left\{(x, y, z) \mid x^{2}+y^{2}+z^{2}=1\right\}\), given \(n (n \geq 2)\) points \(A_{1}, A_{2}, \ldots, A_{n}\), find the maximum possible value of \(\sum_{1 \leq i < j \leq n} \left| A_i A_j \right|^2\).

Answer: n^2


Let $f(x)$ have a domain of $R$, $f(x+1)$ be an odd function, and $f(x+2)$ be an even function. When $x\in [1,2]$, $f(x)=ax^{2}+b$. If $f(0)+f(3)=6$, then calculate the value of $f\left(\frac{9}{2}\right)$.

Answer: \frac{5}{2}


There is a set of points \( M \) on a plane and seven different circles \( C_{1}, C_{2}, \cdots, C_{7} \). Circle \( C_{7} \) passes through exactly 7 points in \( M \), circle \( C_{6} \) passes through exactly 6 points in \( M \), and so on, with circle \( C_{1} \) passing through exactly 1 point in \( M \). What is the minimum number of points in \( M \)?

Answer: 12


A plane flies between two cities. With the tailwind, it takes 5 hours and 30 minutes, and against the wind, it takes 6 hours. Given that the wind speed is 24 kilometers per hour, and assuming the plane's flying speed is $x$ kilometers per hour, then the speed of the plane with the tailwind is     kilometers per hour, and the speed of the plane against the wind is     kilometers per hour.

Answer: 528


Find the number of positive integer solutions \((x, y, z, w)\) to the equation \(x + y + z + w = 25\) that satisfy \(x < y\).

Answer: 946


Let  $C$  be the set of all 100-digit numbers consisting of only the digits  $1$  and  $2$ . Given a number in  $C$ , we may transform the number by considering any  $10$  consecutive digits  $x_0x_1x_2 \dots x_9$  and transform it into  $x_5x_6\dots x_9x_0x_1\dots x_4$ . We say that two numbers in  $C$  are similar if one of them can be reached from the other by performing finitely many such transformations. Let  $D$  be a subset of  $C$  such that any two numbers in  $D$  are not similar. Determine the maximum possible size of  $D$ .

Answer: 21^5


Let \( x \) and \( y \) be positive integers such that \( \frac{100}{151} < \frac{y}{x} < \frac{200}{251} \). What is the minimum value of \( x \) ?

Answer: 3


Define
\[A = \frac{1}{1^2} + \frac{1}{5^2} - \frac{1}{7^2} - \frac{1}{11^2} + \frac{1}{13^2} + \frac{1}{17^2} - \dotsb,\]which omits all terms of the form $\frac{1}{n^2}$ where $n$ is an odd multiple of 3, and
\[B = \frac{1}{3^2} - \frac{1}{9^2} + \frac{1}{15^2} - \frac{1}{21^2} + \frac{1}{27^2} - \frac{1}{33^2} + \dotsb,\]which includes only terms of the form $\frac{1}{n^2}$ where $n$ is an odd multiple of 3.

Determine $\frac{A}{B}.$

Answer: 10


In the quadrilateral  $ABCD$ , we have  $\measuredangle BAD = 100^{\circ}$ ,  $\measuredangle BCD = 130^{\circ}$ , and  $AB=AD=1$  centimeter. Find the length of diagonal  $AC$ .

Answer: 1


In triangle $XYZ$, $XY=XZ$ and $W$ is on $XZ$ such that $XW=WY=YZ$. What is the measure of $\angle XYW$?

Answer: 36^{\circ}


A positive number \( a \) is given. It is known that the equation \( x^3 + 1 = ax \) has exactly two positive roots, and the ratio of the larger to the smaller is 2018. Prove that the equation \( x^3 + 1 = ax^2 \), which also has exactly two positive roots, has the same ratio of the larger root to the smaller root, which is also 2018.

Answer: 2018


Quadrilateral  $ABCD$  is cyclic. Line through point  $D$  parallel with line  $BC$  intersects  $CA$  in point  $P$ , line  $AB$  in point  $Q$  and circumcircle of  $ABCD$  in point  $R$ . Line through point  $D$  parallel with line  $AB$  intersects  $AC$  in point  $S$ , line  $BC$  in point  $T$  and circumcircle of  $ABCD$  in point  $U$ . If  $PQ=QR$ , prove that  $ST=TU$

Answer: ST=TU


In a plane, there are 7 points, with no three points being collinear. If 18 line segments are connected between these 7 points, then at most how many triangles can these segments form?

Answer: 23


Let the quadratic function \( f(x) = ax^2 + bx + c \) (\(a > 0\)). The equation \( f(x) - x = 0 \) has two roots \( x_1 \) and \( x_2 \) such that \( 0 < x_1 < x_2 < \frac{1}{a} \).

(1) Prove that \( x < f(x) < x_1 \) when \( x \in (0, x_1) \).

(2) Suppose the graph of the function \( f(x) \) is symmetric with respect to the line \( x = x_0 \). Prove that \( x_0 < \frac{x_1}{2} \).

Answer: x_0<\frac{x_1}{2}


Consider a square arrangement of tiles comprising 12 black and 23 white square tiles. A border consisting of an alternating pattern of black and white tiles is added around the square. The border follows the sequence: black, white, black, white, and so on. What is the ratio of black tiles to white tiles in the newly extended pattern?
A) $\frac{25}{37}$
B) $\frac{26}{36}$
C) $\frac{26}{37}$
D) $\frac{27}{37}$

Answer: \frac{26}{37}


Given the plane vectors $\overrightarrow{a}=(1,0)$ and $\overrightarrow{b}=\left( -\frac{1}{2}, \frac{\sqrt{3}}{2} \right)$, find the angle between $\overrightarrow{a}$ and $\overrightarrow{a}+ \overrightarrow{b}$.

Answer: \frac{\pi}{3}


The area of the triangle formed by the tangent line at point $(1,1)$ on the curve $y=x^3$, the x-axis, and the line $x=2$ is $\frac{4}{3}$.

Answer: \frac{8}{3}


Let  $k$  be  a real  number. Define on the set of reals the operation  $x*y$ =  $\frac{xy}{x+y+k}$  whenever  $x+y$  does not equal  $-k$ . Let  $x_1<x_2<x_3<x_4$  be the roots of  $t^4=27(t^2+t+1)$ .suppose that  $[(x_1*x_2)*x_3]*x_4=1$ . Find all possible values of  $k$

Answer: 3


Reduce one side of a square by $2 \mathrm{cm}$ and increase the other side by $20\%$ to form a rectangle with an area equal to the original square. What is the length of a side of the original square in $\mathrm{cm}$?

Answer: 12\,


Through the right focus of the hyperbola \( x^{2}-\frac{y^{2}}{2}=1 \), draw a line \( l \) that intersects the hyperbola at two points \( A \) and \( B \). If the real number \( \lambda \) such that \( |AB|=\lambda \) corresponds to exactly 3 such lines \( l \), then \( \lambda = \quad \).

Answer: 4


If $5(\cos a + \cos b) + 4(\cos a \cos b + 1) = 0,$ then find all possible values of
\[\tan \frac{a}{2} \tan \frac{b}{2}.\]Enter all the possible values, separated by commas.

Answer: 3,-3


In a volleyball tournament for the Euro-African cup, there were nine more teams from Europe than from Africa. Each pair of teams played exactly once and the Europeans teams won precisely nine times as many matches as the African teams, overall. What is the maximum number of matches that a single African team might have won?

Answer: 11


There are four weights of different masses. Katya weighs the weights in pairs. The results are 1800, 1970, 2110, 2330, and 2500 grams. How many grams does the sixth weighing result in?

Answer: 2190


For each of the  $9$  positive integers  $n,2n,3n,\dots , 9n$  Alice take the first decimal digit (from the left) and writes it onto a blackboard. She selected  $n$  so that among the nine digits on the blackboard there is the least possible number of different digits. What is this number of different digits equals to?

Answer: 4


There are some stamps with denominations of 0.5 yuan, 0.8 yuan, and 1.2 yuan, with a total value of 60 yuan. The number of 0.8 yuan stamps is four times the number of 0.5 yuan stamps. How many 1.2 yuan stamps are there?

Answer: 13


Two teams, Team A and Team B, are playing in a basketball finals series that uses a "best of seven" format (the first team to win four games wins the series and the finals end). Based on previous game results, Team A's home and away schedule is arranged as "home, home, away, away, home, away, home". The probability of Team A winning at home is 0.6, and the probability of winning away is 0.5. The results of each game are independent of each other. Calculate the probability that Team A wins the series with a 4:1 score.

Answer: 0.18


A tower is $45 \mathrm{~m}$ away from the bank of a river. If the width of the river is seen at an angle of $20^{\circ}$ from a height of $18 \mathrm{~m}$ in the tower, how wide is the river?

Answer: 16.38


Is the number of ordered 10-tuples of positive integers \((a_1, a_2, \ldots, a_{10})\) such that \(\frac{1}{a_1} + \frac{1}{a_2} + \cdots + \frac{1}{a_{10}} = 1\) even or odd?

Answer: odd


Wendy is playing darts with a circular dartboard of radius 20. Whenever she throws a dart, it lands uniformly at random on the dartboard. At the start of her game, there are 2020 darts placed randomly on the board. Every turn, she takes the dart farthest from the center, and throws it at the board again. What is the expected number of darts she has to throw before all the darts are within 10 units of the center?

Answer: 6060


The 8 eyelets for the lace of a sneaker all lie on a rectangle, four equally spaced on each of the longer sides. The rectangle has a width of 50 mm and a length of 80 mm. There is one eyelet at each vertex of the rectangle. The lace itself must pass between the vertex eyelets along a width side of the rectangle and then crisscross between successive eyelets until it reaches the two eyelets at the other width side of the rectangle as shown. After passing through these final eyelets, each of the ends of the lace must extend at least 200 mm farther to allow a knot to be tied. Find the minimum length of the lace in millimeters.
[asy] size(200); defaultpen(linewidth(0.7)); path laceL=(-20,-30)..tension 0.75 ..(-90,-135)..(-102,-147)..(-152,-150)..tension 2 ..(-155,-140)..(-135,-40)..(-50,-4)..tension 0.8 ..origin; path laceR=reflect((75,0),(75,-240))*laceL; draw(origin--(0,-240)--(150,-240)--(150,0)--cycle,gray); for(int i=0;i<=3;i=i+1) { path circ1=circle((0,-80*i),5),circ2=circle((150,-80*i),5); unfill(circ1); draw(circ1); unfill(circ2); draw(circ2); } draw(laceL--(150,-80)--(0,-160)--(150,-240)--(0,-240)--(150,-160)--(0,-80)--(150,0)^^laceR,linewidth(1));[/asy]

Answer: 790


Given that the sum of the binomial coefficients of the first two terms of the expansion of \\({(2x+\frac{1}{\sqrt{x}})}^{n}\\) is \\(10\\).

\\((1)\\) Find the value of \\(y' = 2x\\).

\\((2)\\) Find the constant term in this expansion.

Answer: 672


If the Cesaro sum of a sequence with 99 terms is 1000, calculate the Cesaro sum of the sequence with 100 terms consisting of the numbers 1 and the first 99 terms of the original sequence.

Answer: 991


A function $f$ is defined by $f(z) = (4 + i) z^2 + \alpha z + \gamma$ for all complex numbers $z$, where $\alpha$ and $\gamma$ are complex numbers and $i^2 = - 1$. Suppose that $f(1)$ and $f(i)$ are both real. What is the smallest possible value of $| \alpha | + |\gamma |$?

Answer: \sqrt{2}


Given that $a > 2b$ ($a, b \in \mathbb{R}$), the range of the function $f(x) = ax^2 + x + 2b$ is $[0, +\infty)$. Determine the minimum value of $$\frac{a^2 + 4b^2}{a - 2b}$$.

Answer: \sqrt{2}


If the point $\left(m,n\right)$ in the first quadrant is symmetric with respect to the line $x+y-2=0$ and lies on the line $2x+y+3=0$, calculate the minimum value of $\frac{1}{m}+\frac{8}{n}$.

Answer: \frac{25}{9}


George, Jeff, Brian, and Travis decide to play a game of hot potato. They begin by arranging themselves clockwise in a circle in that order. George and Jeff both start with a hot potato. On his turn, a player gives a hot potato (if he has one) to a randomly chosen player among the other three. If a player has two hot potatoes on his turn, he only passes one. If George goes first, and play proceeds clockwise, what is the probability that Travis has a hot potato after each player takes one turn?

Answer: \frac{5}{27}


In a round robin chess tournament each player plays every other player exactly once. The winner of each game gets  $ 1$  point and the loser gets  $ 0$  points. If the game is tied, each player gets  $ 0.5$  points. Given a positive integer  $ m$ , a tournament is said to have property  $ P(m)$  if the following holds: among every set  $ S$  of  $ m$  players, there is one player who won all her games against the other  $ m\minus{}1$  players in  $ S$  and one player who lost all her games against the other  $ m \minus{} 1$  players in  $ S$ . For a given integer  $ m \ge 4$ , determine the minimum value of  $ n$  (as a function of  $ m$ ) such that the following holds: in every  $ n$ -player round robin chess tournament with property  $ P(m)$ , the final scores of the  $ n$  players are all distinct.

Answer: 2m-3


If \( n \) is a natural number and the value of \( n^{3} \) ends exactly with "2016", find the smallest value of \( n \).

Answer: 856


Find the range of the function \( f(x) = \frac{1}{g\left( \frac{64g(g(\ln x))}{1025} \right)} \), where

\[ g(x) = x^5 + \frac{1}{x^5} \]

Answer: [-\frac{32}{1025},0)\cup(0,\frac{32}{1025}]


$A B C D$ is any quadrilateral, $P$ and $Q$ are the midpoints of the diagonals $B D$ and $A C$ respectively. The parallels through $P$ and $Q$ to the other diagonal intersect at $O$.

If we join $O$ with the four midpoints of the sides $X, Y$, $Z$, and $T$, four quadrilaterals $O X B Y, O Y C Z, O Z D T$, and $OTAX$ are formed.

Prove that the four quadrilaterals have the same area.

Answer: 4


Find all triples  $(a,b,c)$  of distinct positive integers so that there exists a subset  $S$  of the positive integers for which for all positive integers  $n$  exactly one element of the triple  $(an,bn,cn)$  is in  $S$ .

Proposed by Carl Schildkraut, MIT

Answer: (,b,)


Let \( n \) and \( k \) be positive integers with \( n > k \). Prove that the greatest common divisor of \( C_{n}^{k}, C_{n+1}^{k}, \cdots, C_{n+k}^{k} \) is 1.

Answer: 1


Let \(r_1, r_2, r_3\) be the roots of the real-coefficient equation \(x^3 - x^2 + ax - b = 0\), where \(0 < r_i < 1\) for \(i=1,2,3\). Find the maximum possible value of \(7a - 9b\).

Answer: 2


Let $A$ denote the set of all integers $n$ such that $1 \leq n \leq 10000$, and moreover the sum of the decimal digits of $n$ is 2. Find the sum of the squares of the elements of $A$.

Answer: 7294927


There are $5$ people participating in a lottery, each drawing a ticket from a box containing $5$ tickets ($3$ of which are winning tickets) without replacement until all $3$ winning tickets have been drawn, ending the activity. The probability that the activity ends exactly after the $4$th person draws is $\_\_\_\_\_\_$.

Answer: \frac{3}{10}


Given a parallelogram $ABCD$ where $\angle B = 111^{\circ}$ and $BC = BD$. Point $H$ is marked on segment $BC$ such that $\angle BHD = 90^{\circ}$. Point $M$ is the midpoint of side $AB$. Find the angle $AMH$. Provide the answer in degrees.

Answer: 132


Let \( S=\{A_{1}, A_{2}, \cdots, A_{n}\} \), where \( A_{1}, A_{2}, \cdots, A_{n} \) are \( n \) distinct finite sets (\( n \geqslant 2 \)), satisfying that for any \( A_{i}, A_{j} \in S \), \( A_{i} \cup A_{j} \in S \). If \( k=\min_{1 \leqslant i \leqslant n}|A_{i}| \geqslant 2 \), prove that there exists an \( x \in \bigcup_{i=1}^{n} A_{i} \) such that \( x \) belongs to at least \(\frac{n}{k} \) of the sets \( A_{1}, A_{2}, \cdots, A_{n} \) (where \( |X| \) denotes the number of elements in the finite set \( X \)).

Answer: \frac{n}{k}


Next year, 2022, has the property that it may be written using at most two different digits, namely 2 and 0. How many such years will there be between 1 and 9999 inclusive?

Answer: 927


A paperboy delivers newspapers to 10 houses along Main Street. Wishing to save effort, he doesn't always deliver to every house, but to avoid being fired he never misses three consecutive houses. Compute the number of ways the paperboy could deliver papers in this manner.

Answer: 504


The lengths of the sides of the pentagon \(ABCDE\) are equal to 1. Let points \(P, Q, R,\) and \(S\) be the midpoints of the sides \(AB, BC, CD,\) and \(DE\) respectively, and let points \(K\) and \(L\) be the midpoints of segments \(PR\) and \(QS\) respectively. Find the length of segment \(KL\).

Answer: \frac{1}{4}


Consider a bug starting at vertex $A$ of a cube, where each edge of the cube is 1 meter long. At each vertex, the bug can move along any of the three edges emanating from that vertex, with each edge equally likely to be chosen. Let $p = \frac{n}{6561}$ represent the probability that the bug returns to vertex $A$ after exactly 8 meters of travel. Find the value of $n$.

Answer: 1641


Using one, two, and three fours, determine which numbers can be expressed using various arithmetic operations. For example, how can 64 be represented using two fours?

Answer: \sqrt{(\sqrt{\sqrt{4}})^{4!}}=64


The roots of a monic cubic polynomial  $p$  are positive real numbers forming a geometric sequence. Suppose that the sum of the roots is equal to  $10$ . Under these conditions, the largest possible value of  $|p(-1)|$  can be written as  $\frac{m}{n}$ , where  $m$ ,  $n$  are relatively prime integers. Find  $m + n$ .

Answer: 2224


Let $M$ be the midpoint of side $AB$ of triangle $ABC$. Let $P$ be a point on $AB$ between $A$ and $M$, and let $MD$ be drawn parallel to $PC$ and intersecting $BC$ at $D$. If the ratio of the area of triangle $BPD$ to that of triangle $ABC$ is denoted by $r$, then

Answer: r=\frac{1}{2}


In the convex polygon \( A_{1}, A_{2}, \ldots, A_{2n} \), let the interior angle at vertex \( A_{i} \) be denoted as \( \alpha_{i} \). Show that

$$
\alpha_{1}+\alpha_{3}+\ldots+\alpha_{2n-1} = \alpha_{2}+\alpha_{4}+\ldots+\alpha_{2n}
$$

Answer: \alpha_{1}+\alpha_{3}+\ldots+\alpha_{2n-1}=\alpha_{2}+\alpha_{4}+\ldots+\alpha_{2n}


In the number $52674.1892$, calculate the ratio of the value of the place occupied by the digit 6 to the value of the place occupied by the digit 8.

Answer: 10,000


From the numbers 0, 1, 2, 3, 4, select three different digits to form a three-digit number. What is the sum of the units digit of all these three-digit numbers?

Answer: 90


Define \(m \otimes n = m \times m - n \times n\). What is the value of \(2 \otimes 4 - 4 \otimes 6 - 6 \otimes 8 - \cdots - 98 \otimes 100\)?

Answer: 9972


Given a circle with a unit radius centered at O. Point \( A \) is at a distance \( a \) from the center ( \( 0 < a < 1 \) ). Through point \( A \), all possible chords \( MN \) are drawn.
a) Find the length of the shortest chord \( MN \).
b) Find the maximum area of the triangle \( OMN \).

Answer: \sqrt{1-^{2}}


Answer: 21


30 students from 5 grades participated in answering 40 questions. Each student answered at least 1 question. Every two students from the same grade answered the same number of questions, and students from different grades answered a different number of questions. How many students answered only 1 question?

Answer: 26


Let \( N \) be the smallest positive integer whose digits have a product of 2000. The sum of the digits of \( N \) is

Answer: 25


On weekdays, a scatterbrained scientist takes the Moscow metro ring line from Taganskaya station to Kievskaya station to get to work, and makes the return trip in the evening. The scientist boards the first available train that arrives. It is known that trains run at approximately equal intervals in both directions. The northern route (via Belorusskaya) takes 17 minutes to travel from Kievskaya to Taganskaya or vice versa, while the southern route (via Paveletskaya) takes 11 minutes.

The scientist, as usual, counts everything. One day, he calculated that, based on years of observations:

- A train traveling counterclockwise arrives at Kievskaya on average 1 minute and 15 seconds after a train traveling clockwise. The same applies to Taganskaya;
- The scientist's average commute from home to work takes 1 minute less than the commute from work to home.

Find the expected value of the interval between trains traveling in one direction.

Answer: 3


In the sequence
\[..., a, b, c, d, 0, 1, 1, 2, 3, 5, 8,...\]
each term is the sum of the two terms to its left. Find $a$.
$\textbf{(A)}\ -3 \qquad\textbf{(B)}\ -1 \qquad\textbf{(C)}\ 0 \qquad\textbf{(D)}\ 1 \qquad\textbf{(E)}\ 3$

Answer: \textbf{(A)}\-3


On the hypotenuse $AB$ of the right triangle $ABC$, a square $ABDE$ is constructed outward. It is known that $AC = 1$ and $BC = 3$.

In what ratio does the angle bisector of angle $C$ divide the side $DE$?

Answer: 1:3


In a diagram, the grid is composed of 1x1 squares. What is the area of the shaded region if the overall width of the grid is 15 units and its height is 5 units? Some parts are shaded in the following manner: A horizontal stretch from the left edge (6 units wide) that expands 3 units upward from the bottom, and another stretch that begins 6 units from the left and lasts for 9 units horizontally, extending from the 3 units height to the top of the grid.

Answer: 36


There are 11 integers written on a board (not necessarily distinct). Could it be possible that the product of any five of these integers is greater than the product of the remaining six?

Answer: Yes


Given a large circle with a radius of 11 and small circles with a radius of 1, find the maximum number of small circles that can be tangentially inscribed in the large circle without overlapping.

Answer: 31


The Absent-Minded Scientist had a sore knee. The doctor prescribed him 10 pills for his knee: take one pill daily. The pills are effective in $90\%$ of cases, and in $2\%$ of cases, there is a side effect—absent-mindedness disappears, if present.

Another doctor prescribed the Scientist pills for absent-mindedness—also one per day for 10 consecutive days. These pills cure absent-mindedness in $80\%$ of cases, but in $5\%$ of cases, there is a side effect—the knee stops hurting.

The bottles with the pills look similar, and when the Scientist went on a ten-day business trip, he took one bottle with him but didn't pay attention to which one. For ten days, he took one pill per day and returned completely healthy. Both the absent-mindedness and the knee pain were gone. Find the probability that the Scientist took pills for absent-mindedness.

Answer: 0.69


Positive integers $a, b$, and $c$ have the property that $a^{b}, b^{c}$, and $c^{a}$ end in 4, 2, and 9, respectively. Compute the minimum possible value of $a+b+c$.

Answer: 17


The domain of the function $f(x)=\log_{\frac12}(\log_4(\log_{\frac14}(\log_{16}(\log_{\frac1{16}}x))))$ is an interval of length $\tfrac mn$, where $m$ and $n$ are relatively prime positive integers.  What is $m+n$?
$\textbf{(A) }19\qquad \textbf{(B) }31\qquad \textbf{(C) }271\qquad \textbf{(D) }319\qquad \textbf{(E) }511\qquad$

Answer: \textbf{(C)}271


Let \( \triangle ABC \) be a triangle with \( AB = 3 \), \( AC = 8 \), \( BC = 7 \). \( M \) and \( N \) are the midpoints of \( \overline{AB} \) and \( \overline{AC} \), respectively. Point \( T \) is selected on side \( BC \) such that \( AT = TC \). The circumcircles of triangles \( BAT \) and \( MAN \) intersect at \( D \). Compute \( DC \).

Answer: \frac{7\sqrt{3}}{3}


The hour and minute hands of a clock move continuously and at constant speeds. A moment of time $X$ is called interesting if there exists such a moment $Y$ (the moments $X$ and $Y$ do not necessarily have to be different), so that the hour hand at moment $Y$ will be where the minute hand is at moment $X$, and the minute hand at moment $Y$ will be where the hour hand is at moment $X$. How many interesting moments will there be from 00:01 to 12:01?

Answer: 143


$ (a_n)_{n \equal{} 1}^\infty$  is defined on real numbers with  $ a_n \not \equal{} 0$ ,  $ a_na_{n \plus{} 3} \equal{} a_{n \plus{} 2}a_{n \plus{} 5}$  and  $ a_1a_2 \plus{} a_3a_4 \plus{} a_5a_6 \equal{} 6$ . So   $ a_1a_2 \plus{} a_3a_4 \plus{} \cdots \plus{}a_{41}a_{42} \equal{} ?$

Answer: 42


A bundle of wire was used in the following sequence:
- The first time, more than half of the total length was used, plus an additional 3 meters.
- The second time, half of the remaining length was used, minus 10 meters.
- The third time, 15 meters were used.
- Finally, 7 meters were left.

How many meters of wire were there originally in the bundle?

Answer: 54


Find the value of $(8x - 5)^2$ given that the number $x$ satisfies the equation $7x^2 + 6 = 5x + 11$.

Answer: \frac{2865 - 120\sqrt{165}}{49}


Calculate Mr. $X$'s net gain or loss from the transactions, given that he sells his home valued at $12,000$ to Mr. $Y$ for a $20\%$ profit and then buys it back from Mr. $Y$ at a $15\%$ loss.

Answer: 2160


Given $F(x) = \int_{0}^{x} (t^{2} + 2t - 8) \, dt$, where $x > 0$.
1. Determine the intervals of monotonicity for $F(x)$.
2. Find the maximum and minimum values of the function $F(x)$ on the interval $[1, 3]$.

Answer: -\frac{28}{3}


Find all integers \( n > 1 \) such that any prime divisor of \( n^6 - 1 \) is a divisor of \((n^3 - 1)(n^2 - 1)\).

Answer: 2


The base of a rectangular parallelepiped is a square with a side length of \(2 \sqrt{3}\). The diagonal of a lateral face forms an angle of \(30^\circ\) with the plane of an adjacent lateral face. Find the volume of the parallelepiped.

Answer: 72


Prove that the sum of an odd number of unit vectors passing through the same point \( O \) and lying in the same half-plane whose border passes through \( O \) has length greater than or equal to 1.

Answer: 1


Finitely many polygons are placed in the plane. If for any two polygons of them, there exists a line through origin  $O$  that cuts them both, then these polygons are called "properly placed". Find the least  $m \in \mathbb{N}$ , such that for any group of properly placed polygons,  $m$  lines can drawn through  $O$  and every polygon is cut by at least one of these  $m$  lines.

Answer: 2


Let \( n \in \mathbf{Z}_{+} \), and
$$
\begin{array}{l}
a, b, c \in \{x \mid x \in \mathbf{Z} \text{ and } x \in [1,9]\}, \\
A_{n} = \underbrace{\overline{a a \cdots a}}_{n \text{ digits}}, B_{n} = \underbrace{b b \cdots b}_{2n \text{ digits}}, C_{n} = \underbrace{c c \cdots c}_{2n \text{ digits}}.
\end{array}
$$

The maximum value of \(a + b + c\) is ( ), given that there exist at least two \(n\) satisfying \(C_{n} - B_{n} = A_{n}^{2}\).

Answer: 18


What is the minimum number of cells that need to be colored in a square with a side length of 65 cells (a $65 \times 65$ square, with a total of 4225 cells) so that from any uncolored cell, it is impossible to reach another uncolored cell with a knight's move in chess?

Answer: 2112


Samantha leaves her house at 7:15 a.m. to catch the school bus, starts her classes at 8:00 a.m., and has 8 classes that last 45 minutes each, a 40-minute lunch break, and spends an additional 90 minutes in extracurricular activities. If she takes the bus home and arrives back at 5:15 p.m., calculate the total time spent on the bus.

Answer: 110


There are five positive integers that are divisors of each number in the list $$48, 144, 24, 192, 216, 120.$$ Find the sum of these positive integers.

Answer: 16


Given the ellipse \( C \):
\[ \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 \quad (a>b>0), \]

which passes through the point \( P\left(3, \frac{16}{5}\right) \) and has an eccentricity of \( \frac{3}{5} \). Draw a line \( l \) with slope \( k \) passing through the right focus of the ellipse \( C \), intersecting the ellipse at points \( A \) and \( B \). Let the slopes of \( PA \) and \( PB \) be \( k_{1} \) and \( k_{2} \) respectively.
(1) Find the standard equation of the ellipse.
(2) If \( k_{1} + k_{2} = 0 \), find the real number \( k \).

Answer: \frac{3}{5}


Edward, the author of this test, had to escape from prison to work in the grading room today. He stopped to rest at a place  $1,875$  feet from the prison and was spotted by a guard with a crossbow.

The guard fired an arrow with an initial velocity of  $100 \dfrac{\text{ft}}{\text{s}}$ .  At the same time, Edward started running away with an acceleration of  $1 \dfrac{\text{ft}}{\text{s}^2}$ .  Assuming that air resistance causes the arrow to decelerate at  $1 \dfrac{\text{ft}}{\text{s}^2}$ , and that it does hit Edward, how fast was the arrow moving at the moment of impact (in  $\dfrac{\text{ft}}{\text{s}}$ )?

Answer: 75\,


Given that none of the terms in the sequence \(\left\{a_{n}\right\}\) are zero. Let the sum of the first \(n\) terms of the sequence \(\left\{a_{n}\right\}\) be \(S_{n}\), and the sum of the first \(n\) terms of the sequence \(\left\{a_{n}^{2}\right\}\) be \(T_{n}\). It is known that
\[ S_{n}^{2}+4 S_{n}-3 T_{n}=0\ \ (n \in \mathbf{Z}_{+}). \]
Find the sum of the first \(n\) terms of the sequence \(\left\{\frac{(n-1)}{n(n+1)} a_{n}\right\}\).

Answer: \frac{2^{n+1}}{n+1}-2


Given an ellipse with its center at the origin \( O \), foci on the \( x \)-axis, and eccentricity \( \frac{\sqrt{3}}{2} \) which passes through the point \(\left(\sqrt{2}, \frac{\sqrt{2}}{2}\right)\). Suppose a line \( l \) that does not pass through the origin \( O \) intersects the ellipse at points \( P \) and \( Q \), and the slopes of lines \( OP \), \( PQ \), and \( OQ \) form a geometric sequence. Find the possible range of values for the area of triangle \( \triangle OPQ \).

Answer: (0,1)


Let  $p$  be a prime number for which  $\frac{p-1}{2}$  is also prime, and let  $a,b,c$  be integers not divisible by  $p$ . Prove that there are at most  $1+\sqrt {2p}$  positive integers  $n$  such that  $n<p$  and  $p$  divides  $a^n+b^n+c^n$ .

Answer: 1+\sqrt{2p}


$A$ and $B$ travel around an elliptical track at uniform speeds in opposite directions, starting from the vertices of the major axis. They start simultaneously and meet first after $B$ has traveled $150$ yards. They meet a second time $90$ yards before $A$ completes one lap. Find the total distance around the track in yards.
A) 600
B) 720
C) 840
D) 960
E) 1080

Answer: 720


Let $a$ be an integer such that $x^2 - x + a$ divides $x^{13} + x + 90$. Find the value of $a$.

Answer: 2


Let set $M$ consist of an odd number of elements. For each element $x$ in set $M$, there is a uniquely determined subset $H_{x} \subseteq M$ corresponding to $x$, satisfying the conditions:
(1) For each $x \in M$, $x \in H_{x}$;
(2) For any $x, y \in M$, $y \in H_{x}$ if and only if $x \in H_{y}$.
Prove that there is at least one $H_{x}$ with an odd number of elements.

Answer: 1


Compute the number of two digit positive integers that are divisible by both of their digits. For example,  $36$  is one of these two digit positive integers because it is divisible by both  $3$  and  $6$ .

*2021 CCA Math Bonanza Lightning Round #2.4*

Answer: 14


In triangle $ABC$ with an acute angle at vertex $A$, an angle bisector $AE$ and an altitude $BH$ are drawn. It is known that $\angle AEB = 45^{\circ}$. Find the angle $EHC$.

Answer: 45


When $s$ and $t$ range over all real numbers, what is the minimum value of $(s+5-3|\cos t|)^{2}+(s-2|\sin t|)^{2}$?

Answer: 2


A dice is thrown twice. Let $a$ be the number of dots that appear on the first throw and $b$ be the number of dots that appear on the second throw. Given the system of equations $\begin{cases} ax+by=2 \\\\ 2x+y=3\\end{cases}$.

(I) Find the probability that the system of equations has only one solution.

(II) If each solution of the system of equations corresponds to a point $P(x,y)$ in the Cartesian plane, find the probability that point $P$ lies in the fourth quadrant.

Answer: \frac{7}{12}


Person A and person B independently attempt to decrypt a password. Their probabilities of successfully decrypting the password are $\dfrac{1}{3}$ and $\dfrac{1}{4}$, respectively. Calculate:

$(1)$ The probability that exactly one of them decrypts the password.

$(2)$ If the probability of decrypting the password needs to be $\dfrac{99}{100}$, what is the minimum number of people like B required?

Answer: 17


The cost $C$ of sending a parcel post package weighing $P$ pounds, $P$ an integer, is $10$ cents for the first pound and $3$ cents for each additional pound. The formula for the cost is:

Answer: C=10+3(P-1)


Assume  $f:\mathbb N_0\to\mathbb N_0$  is a function such that  $f(1)>0$  and, for any nonnegative integers  $m$  and  $n$ ,  $$ f\left(m^2+n^2\right)=f(m)^2+f(n)^2. $$  (a) Calculate  $f(k)$  for  $0\le k\le12$ .
(b) Calculate  $f(n)$  for any natural number  $n$ .

Answer: f(n)=n


Let $\mathcal{C}$ be the hyperbola $y^2 - x^2 = 1$.  Given a point $P_0$ on the $x$-axis, we construct a sequence of points $(P_n)$ on the $x$-axis in the following manner: let $\ell_n$ be the line with slope 1 passing passing through $P_n$, then $P_{n+1}$ is the orthogonal projection of the point of intersection of $\ell_n$ and $\mathcal C$ onto the $x$-axis. (If $P_n = 0$, then the sequence simply terminates.)

Find the number of starting positions $P_0$ on the $x$-axis such that $P_0 = P_{2008}$.  Your answer should use exponential notation, in simplest form.

Answer: 2^{2008}-2


Let  $ABC$  be a triangle with  $AB=13$ ,  $BC=14$ , and  $CA=15$ . The incircle of  $ABC$  meets  $BC$  at  $D$ . Line  $AD$  meets the circle through  $B$ ,  $D$ , and the reflection of  $C$  over  $AD$  at a point  $P\neq D$ . Compute  $AP$ .

*2020 CCA Math Bonanza Tiebreaker Round #4*

Answer: 2\sqrt{145}


Let $ABCD$ be a square. Let $E, F, G$ and $H$ be the centers, respectively, of equilateral triangles with bases $\overline{AB}, \overline{BC}, \overline{CD},$ and $\overline{DA},$ each exterior to the square. What is the ratio of the area of square $EFGH$ to the area of square $ABCD$?

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


Find the number of solutions to $\sin x = \lg x$.

Answer: 3


Let  $n$  be a positive integer.  For each partition of the set  $\{1,2,\dots,3n\}$  into arithmetic  progressions,  we  consider  the  sum  $S$  of  the  respective  common differences  of  these  arithmetic  progressions.   What  is  the  maximal  value that  $S$  can attain?

(An *arithmetic progression* is a set of the form  $\{a,a+d,\dots,a+kd\}$ , where  $a,d,k$  are positive integers, and  $k\geqslant 2$ ; thus an arithmetic progression has at least three elements, and successive elements have difference  $d$ , called the *common difference* of the arithmetic progression.)

Answer: n^2


Determine the number of pairs of positive integers  $x,y$  such that  $x\le y$ ,  $\gcd (x,y)=5!$  and  $\text{lcm}(x,y)=50!$ .

Answer: 16384


Prove that the circumcenter $O$, centroid $G$, and orthocenter $H$ of triangle $\triangle ABC$ are collinear and that $OG:GH = 1:2$.

Answer: OG:GH=1:2


Let $S$ be the set of points $(a, b)$ with $0 \leq a, b \leq 1$ such that the equation $x^{4}+a x^{3}-b x^{2}+a x+1=0$ has at least one real root. Determine the area of the graph of $S$.

Answer: \frac{1}{4}


The point $(1,1,1)$ is rotated $180^\circ$ about the $y$-axis, then reflected through the $yz$-plane, reflected through the $xz$-plane, rotated $180^\circ$ about the $y$-axis, and reflected through the $xz$-plane.  Find the coordinates of the point now.

Answer: (-1,1,1)


It is known that \(a^{2} + b = b^{2} + c = c^{2} + a\). What values can the expression \(a\left(a^{2} - b^{2}\right) + b\left(b^{2} - c^{2}\right) + c\left(c^{2} - a^{2}\right)\) take?

Answer: 0


Rhombus $ABCD$ has side length $2$ and $\angle B = 120^\circ$. Region $R$ consists of all points inside the rhombus that are closer to vertex $B$ than any of the other three vertices. What is the area of $R$?

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


What is the smallest positive value of $m$ so that the equation $10x^2 - mx + 630 = 0$ has integral solutions?

Answer: 160


What is the probability that Hannah gets fewer than 4 heads if she flips 12 coins?

Answer: \frac{299}{4096}


For a group of children, it holds that in every trio of children from the group, there is a boy named Adam, and in every quartet, there is a girl named Beata.

What is the maximum number of children that can be in such a group, and what are their names?

Answer: 5


Olave sold 108 apples at a constant rate over 6 hours. If she continues to sell apples at the same rate, how many apples will she sell in the next 1 hour and 30 minutes?
(A) 27
(B) 33
(C) 45
(D) 36
(E) 21

Answer: 27


Answer: 2011^{2011}


In a convex quadrilateral \(ABCD\), side \(AB\) is equal to diagonal \(BD\), \(\angle A=65^\circ\), \(\angle B=80^\circ\), and \(\angle C=75^\circ\). What is \(\angle CAD\) (in degrees)?

Answer: 15


Consider arrangements of the $9$ numbers $1, 2, 3, \dots, 9$ in a $3 \times 3$ array. For each such arrangement, let $a_1$, $a_2$, and $a_3$ be the medians of the numbers in rows $1$, $2$, and $3$ respectively, and let $m$ be the median of $\{a_1, a_2, a_3\}$. Let $Q$ be the number of arrangements for which $m = 5$. Find the remainder when $Q$ is divided by $1000$.

Answer: 360


Point \( O \) is the center of the circle circumscribed around triangle \( ABC \) with sides \( BC = 8 \) and \( AC = 4 \). Find the length of side \( AB \) if the length of the vector \( 4 \overrightarrow{OA} - \overrightarrow{OB} - 3 \overrightarrow{OC} \) is 10.

Answer: 5


Find the smallest \( a \in \mathbf{N}^{*} \) such that the following equation has real roots:
$$
\cos ^{2} \pi(a-x)-2 \cos \pi(a-x)+\cos \frac{3 \pi x}{2 a} \cdot \cos \left(\frac{\pi x}{2 a}+\frac{\pi}{3}\right)+2=0 .
$$

Answer: 6


In an acute-angled triangle \(ABC\), median \(BM\) and altitude \(CH\) are drawn. It is given that \(BM = CH = \sqrt{3}\), and \(\angle MBC = \angle ACH\). Find the perimeter of triangle \(ABC\).

Answer: 6


The number of passengers on the bus remained constant from departure to the second stop. Then, it was observed that at each stop, as many people boarded as had boarded at the two previous stops, and as many people disembarked as had boarded at the previous stop. The tenth stop from departure is the final one, and at this stop, 55 passengers disembarked.

How many passengers were on the bus between the seventh and eighth stops?

Answer: 21


Let  $S$  be a subset of  $\{1, 2, . . . , 500\}$  such that no two distinct elements of S have a
product that is a perfect square. Find, with proof, the maximum possible number of elements
in  $S$ .

Answer: 306


You can arrange 15 balls in the shape of a triangle, but you cannot arrange 96 balls in the shape of a square (missing one ball). Out of how many balls, not exceeding 50, can you arrange them both in the shape of a triangle and a square?

Answer: 36


The altitude drawn from the vertex of the right angle to the hypotenuse divides the triangle into two triangles, each of which is inscribed with a circle. Find the angles and the area of the triangle formed by the legs of the original triangle and the line passing through the centers of these circles, given that the height of the original triangle is \( h \).

Answer: \frac{^2}{2}


Let  $ ABCD$  be a convex quadrilateral. We have that  $ \angle BAC\equal{}3\angle CAD$ ,  $ AB\equal{}CD$ ,  $ \angle ACD\equal{}\angle CBD$ . Find angle  $ \angle ACD$

Answer: 30


On another island of knights, liars, and normal people, the king held opposite views and gave his daughter different paternal advice: "Dear, I don't want you to marry any knight or liar. I would like you to marry a solid, normal person with a good reputation. You should not marry a knight, because all knights are hypocrites. You should not marry a liar either, because all liars are treacherous. No, no matter what, a decent normal person would be just right for you!"

Assume you are an inhabitant of this island and a normal person. Your task is to convince the king that you are a normal person.
a) How many true statements will you need for this?

b) How many false statements will you need for the same purpose?

(In both cases, the task is to find the minimum number of statements.).

Answer: 1


A regular dodecagon \( Q_1 Q_2 \dotsb Q_{12} \) is drawn in the coordinate plane with \( Q_1 \) at \( (4,0) \) and \( Q_7 \) at \( (2,0) \). If \( Q_n \) is the point \( (x_n,y_n) \), compute the numerical value of the product
\[
(x_1 + y_1 i)(x_2 + y_2 i)(x_3 + y_3 i) \dotsm (x_{12} + y_{12} i).
\]

Answer: 531440


Inside a convex $n$-gon there are 100 points positioned in such a way that no three of these $n+100$ points are collinear. The polygon is divided into triangles, each having vertices among any 3 of the $n+100$ points. For what maximum value of $n$ can no more than 300 triangles be formed?

Answer: 102


Find all functions \( f: \mathbb{R} \rightarrow \mathbb{R} \) such that for all \( x, y \in \mathbb{R} \):

\[ f(f(f(x) + f(y))) = f(x) + y. \]

Answer: f(x)=x


A convex quadrilateral is determined by the points of intersection of the curves \(x^{4} + y^{4} = 100\) and \(xy = 4\); determine its area.

Answer: 4\sqrt{17}


Given that there is a gathering attended by 1982 people, and among any group of 4 people, at least 1 person knows the other 3. How many people, at minimum, must know all the attendees at this gathering?

Answer: 1979


For any positive integer \( n \), let \( f(n) \) denote the number of 1's in the base-2 representation of \( n \). For how many values of \( n \) with \( 1 \leq n \leq 2002 \) do we have \( f(n) = f(n+1) \)?

Answer: 501


Find the smallest positive integer \( n \) such that any simple graph with 10 vertices and \( n \) edges, with edges colored in two colors, always contains a monochromatic triangle or a monochromatic quadrilateral.

Answer: 31


Given the function \( f(x) = \left|\log_{2} x\right| \), if the real numbers \( a \) and \( b \) (where \( a < b \)) satisfy \( f(a) = f(b) \), then find the range of values for \( a + 2014b \).

Answer: (2015,+\infty)


Answer: 8


A rectangular table of dimensions \( x \) cm \(\times 80\) cm is covered with identical sheets of paper of size \( 5 \) cm \(\times 8 \) cm. The first sheet is placed in the bottom-left corner, and each subsequent sheet is placed one centimeter higher and one centimeter to the right of the previous one. The last sheet is placed in the top-right corner. What is the length \( x \) in centimeters?

Answer: 77


Find a positive integer that is divisible by 21 and has a square root between 30 and 30.5.

Answer: 903


A mouse is sitting in a toy car on a negligibly small turntable. The car cannot turn on its own, but the mouse can control when the car is launched and when the car stops (the car has brakes). When the mouse chooses to launch, the car will immediately leave the turntable on a straight trajectory at 1 meter per second. Suddenly someone turns on the turntable; it spins at $30 \mathrm{rpm}$. Consider the set $S$ of points the mouse can reach in his car within 1 second after the turntable is set in motion. What is the area of $S$, in square meters?

Answer: \frac{\pi}{6}


Humanity has discovered 15 habitable planets, where 7 are "Earth-like" and 8 are "Mars-like". Colonizing an Earth-like planet requires 3 units of colonization, while a Mars-like planet requires 1 unit. If humanity has 21 units available for colonization, determine how many different combinations of planets can be occupied given that all planets are distinct.

Answer: 981


Given that in quadrilateral ABCD, $\angle A : \angle B : \angle C : \angle D = 1 : 3 : 5 : 6$, express the degrees of $\angle A$ and $\angle D$ in terms of a common variable.

Answer: 144


Xiaohua wrote a computer program. After one minute of running the program, some bubbles first appear on the computer screen. Thereafter, new bubbles will appear at every whole minute, in the same quantity as the first minute. Half a minute after the 11th appearance of the bubbles, one bubble bursts. Subsequently, every minute, more bubbles will burst, with the quantity increasing by 1 each minute (i.e., half a minute after the 12th appearance of bubbles, 2 bubbles will burst, etc.). At a certain point, the total number of burst bubbles exactly equals the total number of bubbles that have appeared on the screen, meaning that all bubbles have disappeared. What is the maximum number of bubbles that can appear on the screen simultaneously during the entire process?

Answer: 1026


Let \( n \) be a strictly positive integer. Let \( m_{n} \) be the largest real number satisfying the following property: "Let \( x_{0}, x_{1}, \ldots, x_{n} \) be mutually distinct integers. For any monic polynomial \( F(x) \) of degree \( n \) with integer coefficients, at least one of the numbers \( \left|F\left(x_{0}\right)\right|, \left|F\left(x_{1}\right)\right|, \ldots,\left|F\left(x_{n}\right)\right| \) is greater than or equal to \( m_{n} \)."

Find the problem corresponding to the integer part of \( m_{9} \).

Answer: 708


Several players try out for the USAMTS basketball team, and they all have integer heights and weights when measured in centimeters and pounds, respectively. In addition, they all weigh less in pounds than they are tall in centimeters. All of the players weigh at least  $190$  pounds and are at most  $197$  centimeters tall, and there is exactly one player with
every possible height-weight combination.

The USAMTS wants to field a competitive team, so there are some strict requirements.

- If person  $P$  is on the team, then anyone who is at least as tall and at most as heavy as  $P$  must also be on the team.
- If person  $P$  is on the team, then no one whose weight is the same as  $P$ ’s height can also be on the team.

Assuming the USAMTS team can have any number of members (including zero), how many different basketball teams can be constructed?

Answer: 128


The set $\{[x] + [2x] + [3x] \mid x \in \mathbb{R}\} \mid \{x \mid 1 \leq x \leq 100, x \in \mathbb{Z}\}$ has how many elements, where $[x]$ denotes the greatest integer less than or equal to $x$.

Answer: 67


A paint brush is swept along both diagonals of a square to produce the symmetric painted area, as shown. Half the area of the square is painted. What is the ratio of the side length of the square to the brush width?

Answer: 2\sqrt{2}+2


The sum of an infinite geometric series is \( 16 \) times the series that results if the first two terms of the original series are removed. What is the value of the series' common ratio?

Answer: -\frac{1}{4}


A and B agreed to meet at a restaurant for a meal. Due to the restaurant being very busy, A arrived first and took a waiting number and waited for B. After a while, B also arrived but did not see A, so he took another waiting number. While waiting, B saw A, and they took out their waiting numbers and discovered that the digits of these two numbers are reversed two-digit numbers, and the sum of the digits of both numbers (for example, the sum of the digits of 23 is $2+3=5$) is 8, and B's number is 18 greater than A's. What is A's number?

Answer: 35


Answer: \frac{3\sqrt{3}}{8}


Define the sequence \(x_{i}\) by \(x_{1}=a\) and \(x_{i+1}=2x_{i}+1\). Define the sequence \(y_{i}=2^{x_{i}}-1\). Determine the largest integer \(k\) such that \(y_{1}, \ldots, y_{k}\) are all prime numbers.

Answer: 2


Let $S_{n}$ be the sum of the first $n$ terms of the sequence $\{a_{n}\}$, $a_{2}=5$, $S_{n+1}=S_{n}+a_{n}+4$; $\{b_{n}\}$ is a geometric sequence, $b_{2}=9$, $b_{1}+b_{3}=30$, with a common ratio $q \gt 1$.
$(1)$ Find the general formulas for sequences $\{a_{n}\}$ and $\{b_{n}\}$;
$(2)$ Let all terms of sequences $\{a_{n}\}$ and $\{b_{n}\}$ form sets $A$ and $B$ respectively. Arrange the elements of $A\cup B$ in ascending order to form a new sequence $\{c_{n}\}$. Find $T_{20}=c_{1}+c_{2}+c_{3}+\cdots +c_{20}$.

Answer: 660


Let \( Q_1 \) be a regular \( t \)-gon and \( Q_2 \) be a regular \( u \)-gon \((t \geq u \geq 3)\) such that each interior angle of \( Q_1 \) is \( \frac{60}{59} \) as large as each interior angle of \( Q_2 \). What is the largest possible value of \( u \)?

Answer: 119


Given 4 points \( A, B, C, D \) on a plane, with the distances \( AB = 1, BC = 9, CA = 9, AD = 7 \). Prove that at least one of the lengths \( BD \) or \( CD \) is not an integer.

Answer: 1


Given an obtuse triangle \( \triangle ABC \) with the following conditions:
1. The lengths of \( AB \), \( BC \), and \( CA \) are positive integers.
2. The lengths of \( AB \), \( BC \), and \( CA \) do not exceed 50.
3. The lengths of \( AB \), \( BC \), and \( CA \) form an arithmetic sequence with a positive common difference.

Determine the number of obtuse triangles that satisfy the above conditions, and identify the side lengths of the obtuse triangle with the largest perimeter.

Answer: 157


Suppose we need to divide 15 dogs into three groups, one with 4 dogs, one with 7 dogs, and one with 4 dogs. We want to form the groups such that Fluffy is in the 4-dog group, Nipper is in the 7-dog group, and Daisy is in the other 4-dog group. How many ways can we arrange the remaining dogs into these groups?

Answer: 18480


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

Answer: \frac{289}{259}


The inscribed circle of the right triangle \(ABC\) (with the right angle at \(C\)) touches the sides \(AB\), \(BC\), and \(CA\) at the points \(C_1\), \(A_1\), and \(B_1\) respectively. The altitudes of the triangle \(A_1B_1C_1\) intersect at the point \(D\). Find the distance between the points \(C\) and \(D\), given that the lengths of the legs of the triangle \(ABC\) are 3 and 4.

Answer: 1


The expression $\frac{k^{2}}{1.001^{k}}$ reaches its maximum value with which natural number $k$?

Answer: 2001


Let the probability of germination for each seed be 0.9, and 1000 seeds have been planted. For each seed that does not germinate, 2 more seeds need to be replanted. Let X be the number of replanted seeds. Calculate the expected value of X.

Answer: 200


For testing a certain product, there are 6 different genuine items and 4 different defective items. The test continues until all defective items are identified. If all defective items are exactly identified by the 5th test, how many possible testing methods are there?

Answer: 576


A plastic snap-together cube has a protruding snap on one side and receptacle holes on the other five sides as shown.  What is the smallest number of these cubes that can be snapped together so that only receptacle holes are showing?
[asy] draw((0,0)--(4,0)--(4,4)--(0,4)--cycle); draw(circle((2,2),1)); draw((4,0)--(6,1)--(6,5)--(4,4)); draw((6,5)--(2,5)--(0,4)); draw(ellipse((5,2.5),0.5,1)); fill(ellipse((3,4.5),1,0.25),black); fill((2,4.5)--(2,5.25)--(4,5.25)--(4,4.5)--cycle,black); fill(ellipse((3,5.25),1,0.25),black); [/asy]
$\text{(A)}\ 3 \qquad \text{(B)}\ 4 \qquad \text{(C)}\ 5 \qquad \text{(D)}\ 6 \qquad \text{(E)}\ 8$

Answer: (B)\4


The sides of a triangle with area \( T \) are \( a, b, c \). Prove that

$$
T^{2} \leq \frac{a b c(a+b+c)}{16}
$$

Answer: T^2\le\frac{(+b+)}{16}


The nonzero numbers \( a \), \( b \), and \( c \) satisfy the equations \( a^{2}(b+c-a) = b^{2}(a+c-b) = c^{2}(b+a-c) \). What is the maximum value that the expression \(\frac{2b + 3c}{a}\) can take?

Answer: 5


Prove that if the $k$-th, $n$-th, and $p$-th terms of an arithmetic progression form three consecutive terms of a geometric progression, then its common ratio is $\frac{n-p}{k-n}$.

Answer: \frac{n-p}{k-n}


Given the sets $M={x|m\leqslant x\leqslant m+ \frac {7}{10}}$ and $N={x|n- \frac {2}{5}\leqslant x\leqslant n}$, both of which are subsets of ${x|0\leqslant x\leqslant 1}$, find the minimum value of the "length" of the set $M\cap N$. (Note: The "length" of a set ${x|a\leqslant x\leqslant b}$ is defined as $b-a$.)

Answer: \frac{1}{10}


In triangle \( \triangle ABC \), \(AB = AC\) and \(\angle BAC = 100^\circ\). Point \(D\) is on the extension of side \(AB\) such that \(AD = BC\). Find the measure of \(\angle BCD\).

Answer: 10


At a meeting of $ 12k$ people, each person exchanges greetings with exactly $ 3k\plus{}6$ others. For any two people, the number who exchange greetings with both is the same. How many people are at the meeting?

Answer: 36


Answer: 32


To prevent the spread of the novel coronavirus, individuals need to maintain a safe distance of at least one meter between each other. In a certain meeting room with four rows and four columns of seats, the distance between adjacent seats is more than one meter. During the epidemic, for added safety, it is stipulated that when holding a meeting in this room, there should not be three people seated consecutively in any row or column. For example, the situation shown in the first column of the table below does not meet the condition (where "$\surd $" indicates a seated person). According to this rule, the maximum number of participants that can be accommodated in this meeting room is ____.

|       |       |       |       |
|-------|-------|-------|-------|
| $\surd $ |       |       |       |
| $\surd $ |       |       |       |
| $\surd $ |       |       |       |

Answer: 11


In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $B= \frac {\pi}{3}$ and $(a-b+c)(a+b-c)= \frac {3}{7}bc$.
(Ⅰ) Find the value of $\cos C$;
(Ⅱ) If $a=5$, find the area of $\triangle ABC$.

Answer: 10 \sqrt {3}


Let  $F$  be a finite field having an odd number  $m$  of elements. Let  $p(x)$  be an irreducible (i.e. nonfactorable) polynomial over  $F$  of the form   $$ x^2+bx+c, ~~~~~~ b,c \in F. $$   For how many elements  $k$  in  $F$  is  $p(x)+k$  irreducible over  $F$ ?

Answer: \frac{-1}{2}


$A_1, A_2, ..., A_n$  are the subsets of  $|S|=2019$  such that union of any three of them gives  $S$  but if we combine two of subsets it doesn't give us  $S$ . Find the maximum value of  $n$ .

Answer: 64


Answer: a_{100}>2^{99}


Determine the sum of all distinct real values of $x$ such that $|||\cdots||x|+x|\cdots|+x|+x|=1$ where there are 2017 $x$ 's in the equation.

Answer: -\frac{2016}{2017}


Points  $A_1, A_2, \ldots, A_{2022}$  are chosen on a plane so that no three of them are collinear. Consider all angles  $A_iA_jA_k$  for distinct points  $A_i, A_j, A_k$ . What largest possible number of these angles can be equal to  $90^\circ$ ?

*Proposed by Anton Trygub*

Answer: 2,042,220


The extensions of a telephone exchange have only 2 digits, from 00 to 99. Not all extensions are in use. By swapping the order of two digits of an extension in use, you either get the same number or the number of an extension not in use. What is the highest possible number of extensions in use?
(a) Less than 45
(b) 45
(c) Between 45 and 55
(d) More than 55
(e) 55

Answer: 55


Each side of triangle \(ABC\) is divided into 8 equal segments. How many different triangles with vertices at the division points (points \(A, B, C\) cannot be the vertices of these triangles) exist such that none of their sides is parallel to any side of triangle \(ABC\)?

Answer: 216


Given the product sequence $\frac{5}{3} \cdot \frac{6}{5} \cdot \frac{7}{6} \cdot \ldots \cdot \frac{a}{b} = 12$, determine the sum of $a$ and $b$.

Answer: 71


Amy, Beth, and Claire each have some sweets. Amy gives one third of her sweets to Beth. Beth gives one third of all the sweets she now has to Claire. Then Claire gives one third of all the sweets she now has to Amy. All the girls end up having the same number of sweets.

Claire begins with 40 sweets. How many sweets does Beth have originally?

Answer: 50


The line $c$ is defined by the equation $y = 2x$. Points $A$ and $B$ have coordinates $A(2, 2)$ and $B(6, 2)$. On line $c$, find the point $C$ from which the segment $AB$ is seen at the largest angle.

Answer: (2,4)


In the production of a steel cable, it was found that the cable has the same length as the curve defined by the system of equations:

$$
\left\{\begin{array}{l}
x + y + z = 8 \\
xy + yz + xz = -18
\end{array}\right.
$$

Find the length of the cable.

Answer: 4\pi \sqrt{\frac{59}{3}}


The base of a pyramid is a triangle with sides 6, 5, and 5. The lateral faces of the pyramid form $45^{\circ}$ angles with the plane of the base. Find the volume of the pyramid.

Answer: 6


A five-digit natural number $\overline{a_1a_2a_3a_4a_5}$ is considered a "concave number" if and only if $a_1 > a_2 > a_3$ and $a_3 < a_4 < a_5$, with each $a_i \in \{0,1,2,3,4,5\}$ for $i=1,2,3,4,5$. Calculate the number of possible "concave numbers".

Answer: 146


We have an empty equilateral triangle with length of a side  $l$ . We put the triangle, horizontally, over a sphere of radius  $r$ . Clearly, if the triangle is small enough, the triangle is held by the sphere. Which is the distance between any vertex of the triangle and the centre of the sphere (as a function of  $l$  and  $r$ )?

Answer: \sqrt{r^2+\frac{^2}{4}}


Determine the range of $w(w + x)(w + y)(w + z)$, where $x, y, z$, and $w$ are real numbers such that
\[x + y + z + w = x^7 + y^7 + z^7 + w^7 = 0.\]

Answer: 0


The union of sets \( A \) and \( B \), \( A \cup B = \{a_1, a_2, a_3\} \), and \( A \neq B \). When \( (A, B) \) and \( (B, A) \) are considered as different pairs, how many such pairs \( (A, B) \) are there?

Answer: 27


Three squirrels typically eat porridge for breakfast: semolina (M), millet (P), oatmeal (O), buckwheat (G). No porridge is liked by all three squirrels, but for every pair of squirrels, there is at least one porridge that they both like. How many different tables can be made where each cell contains either a plus (if liked) or a minus (if not liked)?

|  | M | P | O | G |
| :--- | :--- | :--- | :--- | :--- |
| Squirrel 1 |  |  |  |  |
| Squirrel 2 |  |  |  |  |
| Squirrel 3 |  |  |  |  |

Answer: 132


The radius of a sphere is \( r = 10 \text{ cm} \). Determine the volume of the spherical segment whose surface area is in the ratio 10:7 compared to the area of its base.

Answer: 288 \pi


A shooter, in a shooting training session, has the probabilities of hitting the 10, 9, 8, and 7 rings as follows:

0.21, 0.23, 0.25, 0.28, respectively. Calculate the probability that the shooter in a single shot:

(1) Hits either the 10 or 9 ring; (2) Scores less than 7 rings.

Answer: 0.03


In a store, there are 21 white shirts and 21 purple shirts hanging in a row. Find the minimum number \( k \) such that, regardless of the initial order of the shirts, it is possible to take down \( k \) white shirts and \( k \) purple shirts so that the remaining white shirts are all hanging consecutively and the remaining purple shirts are also hanging consecutively.

Answer: 10


28 apples weigh 3 kilograms. If they are evenly divided into 7 portions, each portion accounts for $\boxed{\frac{1}{7}}$ of all the apples, and each portion weighs $\boxed{\frac{3}{7}}$ kilograms.

Answer: \frac{3}{7}


The values of $a$ in the equation:  $\log_{10}(a^2 - 15a) = 2$ are:

Answer: 20, -5


From the consecutive natural numbers \(1, 2, 3, \ldots, 2014\), select \( n \) numbers such that any two selected numbers do not have the property that one number is five times the other. Find the maximum value of \( n \) and provide a justification.

Answer: 1679


There are 10 cities in a state, and some pairs of cities are connected by 40 roads altogether. A city is called a "hub" if it is directly connected to every other city. What is the largest possible number of hubs?

Answer: 6


Given that $x$ is a multiple of $2520$, what is the greatest common divisor of $g(x) = (4x+5)(5x+2)(11x+8)(3x+7)$ and $x$?

Answer: 280


The side lengths of a triangle are integers. It is known that the altitude drawn to one of the sides divides it into integer segments, the difference of which is 7. For what minimum value of the length of this side, the altitude drawn to it is also an integer?

Answer: 25


Find all possible three-digit numbers that can be obtained by removing three digits from the number 112277. Sum them and write the result as the answer.

Answer: 1159


There are very many symmetrical dice. They are thrown simultaneously. With a certain probability \( p > 0 \), it is possible to get a sum of 2022 points. What is the smallest sum of points that can fall with the same probability \( p \)?

Answer: 337


Solve for the range of real numbers \( k \) such that the equation \( k \cdot 9^{x} - k \cdot 3^{x+1} + 6(k-5) = 0 \) has solutions for \( x \) in the interval \([0, 2]\).

Answer: [\frac{1}{2},8]


Let $[x]$ denote the greatest integer not exceeding the real number $x$, and $\{x\} = x - [x]$. Find the solutions to the equation
$$
[x]^{4}+\{x\}^{4}+x^{4}=2048
$$

Answer: -3-\sqrt{5}


If \( z_{1}, z_{2}, \cdots, z_{n} \in \mathbf{C}, \left|z_{1}\right|=\left|z_{2}\right|=\cdots=\left|z_{n}\right|=r \neq 0, \) and \( T_{s} \) represents the sum of all possible products of \( s \) out of these \( n \) numbers, find \( \left|\frac{T_{s}}{T_{n-s}}\right| \) in terms of \( r \) given that \( T_{n-s} \neq 0 \).

Answer: r^{2s-n}


Egor, Nikita, and Innokentiy took turns playing chess with each other (two play, one watches). After each game, the loser gave up their spot to the spectator (there were no draws). It turned out that Egor participated in 13 games, and Nikita in 27 games. How many games did Innokentiy play?

Answer: 14


Given a regular quadrilateral pyramid. The side of the base is 12 and the length of the lateral edge is 10. Sphere \( Q_{1} \) is inscribed in the pyramid. Sphere \( Q_{2} \) touches \( Q_{1} \) and all lateral faces of the pyramid. Find the radius of sphere \( Q_{2} \).

Answer: \frac{6\sqrt{7}}{49}


The teacher gave the children a task to multiply the number written on the board by three and add seven to the result. Kamča solved the task correctly. Her friend Růža also calculated correctly, but a different example: she multiplied the given number by seven and added three to the result. Růža's result was 84 more than Kamča's result.

What number was written on the board?

Answer: 22


Horizontal parallel segments $A B=10$ and $C D=15$ are the bases of trapezoid $A B C D$. Circle $\gamma$ of radius 6 has center within the trapezoid and is tangent to sides $A B, B C$, and $D A$. If side $C D$ cuts out an arc of $\gamma$ measuring $120^{\circ}$, find the area of $A B C D$.

Answer: \frac{225}{2}


Sixteen 6-inch wide square posts are evenly spaced with 6 feet between them to enclose a square field. What is the outer perimeter, in feet, of the fence?

Answer: 106


A circle with radius \( u_a \) is inscribed in angle \( A \) of triangle \( ABC \), and a circle with radius \( u_b \) is inscribed in angle \( B \); these circles touch each other externally. Prove that the radius of the circumscribed circle of the triangle with sides \( a_1 = \sqrt{u_a \cot \frac{\alpha}{2}}, b_1 = \sqrt{u_b \cot \frac{\beta}{2}}, \quad c_1=\sqrt{c} \) is equal to \( \frac{\sqrt{p}}{2} \), where \( p \) is the semiperimeter of triangle \( ABC \).

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


A school is hosting a Mathematics Culture Festival, and it was recorded that on that day, there were more than 980 (at least 980 and less than 990) students visiting. Each student visits the school for a period of time and then leaves, and once they leave, they do not return. Regardless of how these students schedule their visit, we can always find \( k \) students such that either all \( k \) students are present in the school at the same time, or at any time, no two of them are present in the school simultaneously. Find the maximum value of \( k \).

Answer: 32


In the triangular pyramid \(ABCD\) with base \(ABC\), the lateral edges are pairwise perpendicular, \(DA = DB = 5, DC = 1\). A ray of light is emitted from a point on the base. After reflecting exactly once from each lateral face (the ray does not reflect from the edges), the ray hits a point on the pyramid's base. What is the minimum distance the ray could travel?

Answer: \frac{10 \sqrt{3}}{9}