An inscribed circle is drawn inside isosceles trapezoid \(ABCD\) with \(AB = CD\). Let \(M\) be the point where the circle touches side \(CD\), \(K\) be the intersection point of the circle with segment \(AM\), and \(L\) be the intersection point of the circle with segment \(BM\). Calculate the value of \(\frac{AM}{AK} + \frac{BM}{BL}\).

Answer: 10


In quadrilateral \(ABCD\), points \(X, Y, Z\) are the midpoints of segments \(AB, AD, BC\), respectively. It is known that \(XY\) is perpendicular to \(AB\), \(YZ\) is perpendicular to \(BC\), and the measure of angle \(ABC\) is \(100^\circ\). Find the measure of angle \(ACD\).

Answer: 90


Find the value of $x$ such that it is the mean, median, and mode of the $9$ data values $-10, -5, x, x, 0, 15, 20, 25, 30$.
A) $\frac{70}{7}$
B) $\frac{75}{7}$
C) $\frac{80}{7}$
D) $10$
E) $15$

Answer: \frac{75}{7}


Given that a parabola \( P \) has the center of an ellipse \( E \) as its focus, \( P \) passes through both foci of \( E \), and \( P \) and \( E \) intersect at exactly three points. Find the eccentricity of \( E \).

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


Jamie King invested some money in real estate and mutual funds. The total amount he invested was $\$200,\!000$. If he invested 5.5 times as much in real estate as he did in mutual funds, what was his total investment in real estate?

Answer: 169,230.77


What is the area of the region \( \mathcal{R} \) formed by all points \( L(r, t) \) with \( 0 \leq r \leq 10 \) and \( 0 \leq t \leq 10 \) such that the area of triangle \( JKL \) is less than or equal to 10, where \( J(2,7) \) and \( K(5,3) \)?

Answer: \frac{313}{4}


Parallelogram  $ABCD$  is such that angle  $B < 90$  and  $AB<BC$ . Points E and F are on the circumference of  $\omega$  inscribing triangle ABC, such that tangents to  $\omega$  in those points pass through D. If  $\angle EDA= \angle{FDC}$ , find  $\angle{ABC}$ .

Answer: 60


Let \(\tan x\) and \(\tan \left(\frac{\pi}{4} - x\right)\) be the two roots of the equation
\[ a x^{2} + b x + c = 0. \]
Then the coordinates of the point through which the graph of the function \( y = a x^{2} + b x - c \) must pass are ____ .

Answer: (1,0)


Let $n (\geq 2)$ be a positive integer, and let $S$ be a set with $n$ elements. Find the smallest positive integer $k$, such that there exists a collection of subsets $A_{1}, A_{2}, \cdots, A_{k}$ of $S$ with the following property: for any two distinct elements $a, b$ of $S$, there exists some $j \in \{1, 2, \cdots, k\}$ such that $A_{j} \cap \{a, b\}$ is a singleton set.

Answer: \lceil\log_2n\rceil


Answer: -1-7z


A $3 \times 3$ square is partitioned into $9$ unit squares. Each unit square is painted either white or black with each color being equally likely, chosen independently and at random. The square is then rotated $90\,^{\circ}$ clockwise about its center, and every white square in a position formerly occupied by a black square is painted black. The colors of all other squares are left unchanged. What is the probability the grid is now entirely black?
$\textbf{(A)}\ \frac{49}{512}\qquad\textbf{(B)}\ \frac{7}{64}\qquad\textbf{(C)}\ \frac{121}{1024}\qquad\textbf{(D)}\ \frac{81}{512}\qquad\textbf{(E)}\ \frac{9}{32}$

Answer: \textbf{(A)}\\frac{49}{512}


Let \( M \) be a finite set of distinct numbers. It is known that among any three elements of \( M \), there will be two elements whose sum belongs to \( M \). What is the maximum number of elements that \( M \) can have?

Answer: 7


A regular $n$-gon $P_{1} P_{2} \ldots P_{n}$ satisfies $\angle P_{1} P_{7} P_{8}=178^{\circ}$. Compute $n$.

Answer: 630


For any real numbers  $x,y$  that satisfies the equation   $$ x+y-xy=155 $$   and   $$ x^2+y^2=325 $$  , Find  $|x^3-y^3|$

Answer: 4375


The vertices of a 50-sided polygon divide a circle into 50 arcs, the lengths of which are 1, 2, 3, ..., 50 in some order. It is known that each pair of "opposite" arcs (corresponding to the opposite sides of the 50-gon) differs in length by 25. Prove that the 50-gon has two parallel sides.

Answer: 2


If the sum of the binomial coefficients of the odd terms in the expansion of ${(x-\frac{2}{x})}^{n}$ is $16$, then the coefficient of $x^{3}$ in the expansion is ______.

Answer: -10


One of the three cards had the number 18, another had the number 75, and the third had some two-digit number. If you sum all the distinct six-digit numbers that can be obtained by arranging these cards in a row, you get the number 2606058. What number is written on the third card?

Answer: 36


In convex quadrilateral \(ABCD\), the line segments connecting the midpoints of opposite sides intersect at an angle of \(60^\circ\), and their lengths are in the ratio \(1:3\). What is the length of the shorter diagonal of quadrilateral \(ABCD\) if the longer diagonal is \(\sqrt{39}\)?

Answer: \sqrt{21}


A cube with a side length of 20 is divided into 8000 unit cubes, and a number is written in each small cube. It is known that in every column of 20 cubes parallel to the edge of the cube, the sum of the numbers is 1 (columns in all three directions are considered). In a certain unit cube, the number 10 is written. Through this cube, there are three $1 \times 20 \times 20$ layers parallel to the faces of the large cube. Find the sum of all numbers outside these layers.

Answer: 333


Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfy $|\overrightarrow{a}|=2$, $|\overrightarrow{b}|=1$ and $(\overrightarrow{a} + \overrightarrow{b}) \perp \overrightarrow{b}$, calculate the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$.

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


For how many integers $n$ with $1 \le n \le 2023$ is the product
\[
  \prod_{k=0}^{n-1} \left( \left( 1 + e^{2 \pi i k / n} \right)^n + 1 \right)^2
\]equal to zero?

Answer: 337


Given that the domains of functions f(x) and g(x) are both R, and f(x) + g(2-x) = 5, g(x) - f(x-4) = 7. If the graph of y=g(x) is symmetric about the line x=2, g(2) = 4, find the sum of the values of f(k) for k from 1 to 22.

Answer: -24


Let \( AB \) be the diameter of a circle, and \( C \) be a point on the plane. Lines \( AC \) and \( BC \) intersect the circle at points \( M \) and \( N \), respectively. Lines \( MB \) and \( NA \) intersect at point \( K \). Find the angle between lines \( CK \) and \( AB \).

Answer: 90


Let  $ABCD$  be a square and  $P$  be a point on the shorter arc  $AB$  of the circumcircle of the square. Which values can the expression  $\frac{AP+BP}{CP+DP}$  take?

Answer: \sqrt{2} - 1


Given the functions \( f(x) = x^2 + 4x + 3 \) and \( g(x) = x^2 + 2x - 1 \), find all integer solutions to the equation \( f(g(f(x))) = g(f(g(x))) \).

Answer: -2


A number of friends met. Each of them shook hands with everyone else, except for Anatoliy Ugryumov, who, being in a bad mood, shook hands with some but not others. A total of 197 handshakes were made. How many handshakes did Anatoliy make?

Answer: 7


Circles  $C_1$  and  $C_2$  intersect at points  $X$  and  $Y$  . Point  $A$  is a point on  $C_1$  such that the tangent line with respect to  $C_1$  passing through  $A$  intersects  $C_2$  at  $B$  and  $C$ , with  $A$  closer to  $B$  than  $C$ , such that  $2016 \cdot AB = BC$ . Line  $XY$  intersects line  $AC$  at  $D$ . If circles  $C_1$  and  $C_2$  have radii of  $20$  and  $16$ , respectively, find  $\sqrt{1+BC/BD}$ .

Answer: 2017


Petya had a large wooden cube with dimensions $30 \times 30 \times 30$ cm$^3$. Petya decided to paint the entire cube, which used 100 grams of paint. However, later Petya needed smaller cubes and cut the large cube into 27 smaller cubes with dimensions $10 \times 10 \times 10$ cm$^3$ by making 6 cuts parallel to the faces of the cube (2 cuts parallel to each pair of opposite faces). How much paint is needed to finish painting all the smaller cubes? Petya does not repaint already painted surfaces.

Answer: 200\,


Jo adds up all the positive integers from 1 to 100. Kate does a similar thing with the first 100 positive integers; however, she first rounds every integer to its nearest multiple of 10 (rounding 5s up) and then adds the 100 values. What is the positive difference between Jo's sum and Kate's sum?

Answer: 50


There are 1001 numbers arranged in a circle. For any two neighboring numbers \(x\) and \(y\), the inequalities \(|x-y| \geqslant 4\) and \(x+y \geqslant 6\) are satisfied. Find the minimum possible sum of the numbers written.

Answer: 3009


The total GDP of the capital city in 2022 is 41600 billion yuan, express this number in scientific notation.

Answer: 4.16 \times 10^{4}


Petya and Vasya took a math test. Petya answered $80\%$ of all the questions correctly, while Vasya answered exactly 35 questions correctly. The number of questions both answered correctly is exactly half the total number of questions. No one answered 7 questions. How many questions were on the test?

Answer: 60


Given complex number $z$, define sequence $z_{0}, z_{1}, z_{2}, \ldots$ as $z_{0}=z$ and $z_{n+1}=2 z_{n}^{2}+2 z_{n}$ for $n \geq 0$. Given that $z_{10}=2017$, find the minimum possible value of $|z|$.

Answer: \frac{\sqrt[1024]{4035}-1}{2}


Given a rectangle $ABCD$. A circle intersects side $AB$ at points $K, L$ and $CD$ at points $M, N$. Find the length of segment $MN$, given $AK=10$, $KL=17$, $DN=7$.

Answer: 23


In $\triangle ABC$, given that $AC - AB = \frac{\sqrt{2}}{2} BC$. Point $D$ is within the triangle $\triangle ABC$ such that $AD = AB$. A perpendicular line is drawn from $D$ to $BC$, intersecting at point $E$. Given that $3BE = CE$, find the measure of $\angle ADE$.

Answer: 135


An octagon $ABCDEFGH$ is divided into eight smaller equilateral triangles, such as $\triangle ABJ$ (where $J$ is the center of the octagon), shown in boldface in the diagram. By connecting every third vertex, we obtain a larger equilateral triangle $\triangle ADE$, which is also shown in boldface. Compute the ratio $[\triangle ABJ]/[\triangle ADE]$.
[asy]
size(150); defaultpen(linewidth(0.8)); dotfactor=5;
pair[] oct = new pair[8];
string[] octlabels = {"$H$", "$G$", "$F$", "$E$", "$D$", "$C$", "$B$", "$A$"};
octlabels.cyclic=true;
oct[0] = dir(0);
for(int i = 1; i <= 8; ++i){
    oct[i] = dir(45*i);
    draw(oct[i] -- oct[i-1]);
    dot(octlabels[i],oct[i],oct[i]);
}
draw(oct[0]--oct[3]--oct[6]--cycle, linewidth(1.3));
draw(oct[0]--oct[1]--(0,0)--cycle, linewidth(1.3));
dot("$J$",(0,0),2*S);
[/asy]

Answer: \frac{1}{4}


For certain real values of $a, b, c,$ and $d_{},$ the equation $x^4+ax^3+bx^2+cx+d=0$ has four non-real roots. The product of two of these roots is $13+i$ and the sum of the other two roots is $3+4i,$ where $i=\sqrt{-1}.$ Find $b.$

Answer: 51


Given a  $\triangle ABC$  with the edges  $a,b$  and  $c$  and the area  $S$ :

(a) Prove that there exists  $\triangle A_1B_1C_1$  with the sides  $\sqrt a,\sqrt b$  and  $\sqrt c$ .
(b) If  $S_1$  is the area of  $\triangle A_1B_1C_1$ , prove that  $S_1^2\ge\frac{S\sqrt3}4$ .

Answer: S_1^2\ge\frac{S\sqrt{3}}{4}


**Q**  Let  $n\geq 2$  be a fixed positive integer and let  $d_1,d_2,...,d_m$  be all positive divisors of  $n$ . Prove that:  $$ \frac{d_1+d_2+...+d_m}{m}\geq \sqrt{n+\frac{1}{4}} $$  Also find the value of  $n$  for which the equality holds.

*Proposed by dangerousliri*

Answer: 2


Given the curve \(C_{1}: \frac{x^{2}}{4} + y^{2} = 1\), the curve \(C_{2}: y = x^{2} - 1\), and the point \(M(0, -1)\). A line \(l\) passing through the origin \(O\) intersects \(C_{2}\) at two points \(A\) and \(B\). The lines \(MA\) and \(MB\) intersect \(C_{1}\) at points \(D\) and \(E\) respectively (with \(D, E \neq M\)).

(1) Prove that \(MD \perp ME\);
(2) Let the areas of \(\triangle MAB\) and \(\triangle MDE\) be \(S_{1}\) and \(S_{2}\) respectively. Determine if there exists a line \(l\) such that \(\frac{S_{1}}{S_{2}} = \frac{17}{32}\) holds. If such a line exists, find the equation of the line \(l\); if it does not exist, provide an explanation.


In the trapezoid \(ABCD\), the bisector of the obtuse angle \(B\) intersects the base \(AD\) at point \(K\), which is its midpoint. \(M\) is the midpoint of \(BC\) and \(AB = BC\).

Find the ratio \(KM: BD\).

Answer: 1:2


In the sequence $\left\{a_{n}\right\}$, for $1 \leqslant n \leqslant 5$, we have $a_{n}=n^{2}$. Additionally, for all positive integers $n$, the following holds: $a_{n+5}+a_{n+1}=a_{n+4}+a_{n}$. Find the value of $a_{2023}$.

Answer: 17


The English alphabet, which has 26 letters, is randomly permuted. Let \(p_{1}\) be the probability that \(\mathrm{AB}, \mathrm{CD}\), and \(\mathrm{EF}\) all appear as contiguous substrings. Let \(p_{2}\) be the probability that \(\mathrm{ABC}\) and \(\mathrm{DEF}\) both appear as contiguous substrings. Compute \(\frac{p_{1}}{p_{2}}\).

Answer: 23


A line of recruits was standing facing the sergeant. Upon the command "left turn," some turned to the left, some to the right, and the rest turned around. Can the sergeant always position himself in the line so that there are an equal number of recruits facing him from both sides?

Answer: Yes


For how many ordered pairs $(b,c)$ of positive integers does neither $x^2+bx+c=0$ nor $x^2+cx+b=0$ have two distinct real solutions?
$\textbf{(A) } 4 \qquad \textbf{(B) } 6 \qquad \textbf{(C) } 8 \qquad \textbf{(D) } 12 \qquad \textbf{(E) } 16 \qquad$

Answer: \textbf{(B)}6


For $n$ a positive integer, let $f(n)$ be the quotient obtained when the sum of all positive divisors of $n$ is divided by $n.$ For example, $f(14)=(1+2+7+14)\div 14=\frac{12}{7}$.
What is $f(768)-f(384)?$

Answer: \frac{1}{192}


In triangle $ABC$, the sides opposite to angles $A$, $B$, $C$ are denoted as $a$, $b$, $c$, respectively. If the function $f(x)=\frac{1}{3} x^{3}+bx^{2}+(a^{2}+c^{2}-ac)x+1$ has no extreme points, then the maximum value of angle $B$ is \_\_\_\_\_

Answer: \frac{\pi}{3}


Find the largest positive integer  $n$  such that  $n\varphi(n)$  is a perfect square.  ( $\varphi(n)$  is the number of integers  $k$ ,  $1 \leq k \leq n$  that are relatively prime to  $n$ )

Answer: 1


In the triangular pyramid \( SABC \), the altitudes \( AA_1 \) and \( BB_1 \) are drawn. It is known that the segment \( A_1B_1 \) is parallel to the edge \( AB \). Prove that some two faces of the pyramid have equal areas.

Answer: 2


Find all positive integers  $n$  such that the inequality   $$ \left( \sum\limits_{i=1}^n a_i^2\right) \left(\sum\limits_{i=1}^n a_i \right) -\sum\limits_{i=1}^n a_i^3 \geq 6 \prod\limits_{i=1}^n a_i $$   holds for any  $n$  positive numbers  $a_1, \dots, a_n$ .

Answer: 3


Square \(ABCD\) has side length 2, and \(X\) is a point outside the square such that \(AX = XB = \sqrt{2}\). What is the length of the longest diagonal of pentagon \(AXB\)?

Answer: \sqrt{10}


In triangle \( \triangle ABC \), \( O \) is the circumcenter, and the three altitudes \( AD \), \( BE \), and \( CF \) intersect at the orthocenter \( H \). Line \( ED \) intersects \( AB \) at point \( M \), and line \( FD \) intersects \( AC \) at point \( N \).
(2) Prove that \( OH \perp MN \).

Answer: OH\perpMN


Given that \( m, n, k \) are positive integers, if there exists a pair of positive integers \( (a, b) \) such that
\[
(1+a) n^{2}-4(m+a) n+4 m^{2}+4 a+b(k-1)^{2}<3,
\]
then the number of possible values of \( m+n+k \) is \(\quad\) .

Answer: 4


You are given 10 numbers - one one and nine zeros. You are allowed to select two numbers and replace each of them with their arithmetic mean.

What is the smallest number that can end up in the place of the one?

Answer: \frac{1}{512}


Francesca uses $100$ grams of lemon juice, $100$ grams of sugar, and $400$ grams of water to make lemonade. There are $25$ calories in $100$ grams of lemon juice and $386$ calories in $100$ grams of sugar. Water contains no calories. How many calories are in $200$ grams of her lemonade?
$\textbf{(A) } 129\qquad \textbf{(B) } 137\qquad \textbf{(C) } 174\qquad \textbf{(D) } 233\qquad \textbf{(E) } 411$

Answer: \textbf{(B)}137


Jia and Yi are playing a guessing game with the following rules: It is known that there are five cards, each with the numbers $1-\left( \frac{1}{2} \right)^n$ ($n\in \mathbf{N}^*, 1\leqslant n\leqslant 5$) written on them. Now, Jia and Yi each randomly draw one card and then try to guess who has the larger number based on the number they drew. After looking at his number, Jia thinks for a moment and says: "I don't know who has the larger number"; after hearing Jia's judgment, Yi thinks for a moment and says: "I also don't know who has the larger number." Assuming that the reasoning made by Jia and Yi is correct, then the number Yi holds is $\boxed{\frac{7}{8}}$.

Answer: \frac{7}{8}


For a positive integer $n$, let $\theta(n)$ denote the number of integers $0 \leq x<2010$ such that $x^{2}-n$ is divisible by 2010. Determine the remainder when $\sum_{n=0}^{2009} n \cdot \theta(n)$ is divided by 2010.

Answer: 335


There were 6 boxes in a store, each with a mass of 15, 16, 18, 19, 20, and 31 kilograms. Two companies bought five boxes, with one company taking twice the mass of apples than the other. Which box remained in the store?

Answer: 20


The product of three positive numbers is 1, and their sum is greater than the sum of their inverses. Prove that just one of the numbers is greater than 1.

Answer: 1


Two positive integers are written on the blackboard, one being 2002 and the other being a number less than 2002. If the arithmetic mean of the two numbers is an integer $m$, then the following operation can be performed: one of the numbers is erased and replaced by $m$. What is the maximum number of times this operation can be performed?

Answer: 10


In triangle $ABC$, $AB = AC = 17$, $BC = 16$, and point $G$ is the centroid of the triangle. Points $A'$, $B'$, and $C'$ are the images of $A$, $B$, and $C$, respectively, after a $90^\circ$ clockwise rotation about $G$. Determine the area of the union of the two regions enclosed by triangles $ABC$ and $A'B'C'$.

Answer: 240


In triangle \( \triangle ABC \), \( \angle A > 90^\circ \) and \( \angle B = 20^\circ \). Draw \( AD \perp AB \) intersecting \( BC \) at \( D \). Given \( AB = 1 \) and \( CD = 4 \), let \( S \) be the area of \( \triangle ABC \). The sum of the numerator and denominator of the simplest form of \( S^2 \) is \(\qquad\).

Answer: 7


Let  $P(x)$  be a nonconstant polynomial of degree  $n$  with rational coefficients which can not be presented as a product of two nonconstant polynomials with rational coefficients. Prove that the number of polynomials  $Q(x)$  of degree less than  $n$  with rational coefficients such that  $P(x)$  divides  $P(Q(x))$
a) is finite
b) does not exceed  $n$ .

Answer: n


Stock investor Li Jin bought shares of a certain company last Saturday for $27 per share. The table below shows the price changes of the stock within the week.

| Day of the Week | Monday | Tuesday | Wednesday | Thursday | Friday | Saturday |
|-----------------|--------|---------|-----------|----------|--------|----------|
| Price Change per Share (Compared to Previous Day) | $-1.5$ | $-1$ | $+1.5$ | $+0.5$ | $+1$ | $-0.5$ |

At the close of trading on Wednesday, the price per share was ____ dollars; the highest price during the week was ____ dollars per share; and the lowest price was ____ dollars per share.

Answer: 24.5


The numbers \(1\) to \(10\) are written in a circle randomly. Find the expected number of numbers which are at least 2 larger than an adjacent number.

Answer: \frac{17}{3}


A tennis player computes her win ratio by dividing the number of matches she has won by the total number of matches she has played. At the start of a weekend, her win ratio is exactly $.500$. During the weekend, she plays four matches, winning three and losing one. At the end of the weekend, her win ratio is greater than $.503$. What's the largest number of matches she could've won before the weekend began?

Answer: 164


Let C be the number of ways to arrange the letters of the word CATALYSIS, T be the number of ways to arrange the letters of the word TRANSPORT, S be the number of ways to arrange the letters of the word STRUCTURE, and M be the number of ways to arrange the letters of the word MOTION. What is  $\frac{C - T + S}{M}$  ?

Answer: 126


Roma thought of a natural number for which the sum of its digits is divisible by 8. He then added 2 to the number and again obtained a number whose sum of digits is divisible by 8. Find the smallest number that Roma could have thought of.

Answer: 699


Michael has $280 in $20 bills. How many $20 bills does he have?

(A) 10
(B) 12
(C) 14
(D) 16
(E) 18

Answer: 14


Find the number of positive integers $n \le 1500$ that can be expressed in the form
\[\lfloor x \rfloor + \lfloor 2x \rfloor + \lfloor 4x \rfloor = n\] for some real number $x.$

Answer: 854


The numbers \(a, b, c, d\) belong to the interval \([-4.5, 4.5]\). Find the maximum value of the expression \(a + 2b + c + 2d - ab - bc - cd - da\).

Answer: 90


Find the length of the diagonal and the area of a rectangle whose one corner is at (1, 1) and the opposite corner is at (9, 7).

Answer: 48


Let  $n\in\mathbb{Z}^+$ . Arrange  $n$  students  $A_1,A_2,...,A_n$  on a circle such that the distances between them are.equal. They each receives a number of candies such that the total amount of candies is  $m\geq n$ . A configuration is called *balance* if for an arbitrary student  $A_i$ , there will always be a regular polygon taking  $A_i$  as one of its vertices, and every student standing at the vertices of this polygon has an equal number of candies.

a) Given  $n$ , find the least  $m$  such that we can create a balance configuration.

b) In a *move*, a student can give a candy to the student standing next to him (no matter left or right) on one condition that the receiver has less candies than the giver. Prove that if  $n$  is the product of at most  $2$  prime numbers and  $m$  satisfies the condition in a), then no matter how we distribute the candies at the beginning, one can always create a balance configuration after a finite number of moves.

Answer: n


A plane passes through the side of the base of a regular quadrilateral pyramid and bisects the dihedral angle at this side. Find the area of the base of the pyramid with the smallest volume, given that the mentioned plane intersects the height of the pyramid at a point that is at a distance \(d\) from the base plane.

Answer: 8d^2


Answer: 1000


The following diagram shows equilateral triangle  $\vartriangle ABC$  and three other triangles congruent to it. The other three triangles are obtained by sliding copies of  $\vartriangle  ABC$  a distance  $\frac18 AB$  along a side of  $\vartriangle ABC$  in the directions from  $A$  to  $B$ , from  $B$  to  $C$ , and from  $C$  to  $A$ . The shaded region inside all four of the triangles has area  $300$ . Find the area of  $\vartriangle ABC$ .
![Image](https://cdn.artofproblemsolving.com/attachments/3/a/8d724563c7411547d3161076015b247e882122.png)

Answer: 768


A Gareth sequence is a sequence of numbers where each number after the second is the non-negative difference between the two previous numbers. For example, if a Gareth sequence begins 15, 12, then:

- The third number in the sequence is \(15 - 12 = 3\),
- The fourth number is \(12 - 3 = 9\),
- The fifth number is \(9 - 3 = 6\),

resulting in the sequence \(15, 12, 3, 9, 6, \ldots\).

If a Gareth sequence begins 10, 8, what is the sum of the first 30 numbers in the sequence?

Answer: 64


For each positive integer $n$, let $f(n) = n^4 - 360n^2 + 400$. What is the sum of all values of $f(n)$ that are prime numbers?

Answer: 802


A pentagon $ABCDE$ is circumscribed about a circle. The angles at the vertices $A{}$, $C{}$ and $E{}$ of the pentagon are equal to $100^\circ$. Find the measure of the angle $\angle ACE$.

Answer: 40^\circ


In the spring round of the 2000 City Tournament, high school students in country $N$ were presented with six problems. Each problem was solved by exactly 1000 students, and no two students solved all six problems together. What is the smallest possible number of high school students in country $N$ who participated in the spring round?

Answer: 2000


Given a point $P$ on the inscribed circle of a square $ABCD$, considering the angles $\angle APC = \alpha$ and $\angle BPD = \beta$, find the value of $\tan^2 \alpha + $\tan^2 \beta$.

Answer: 8


A circle is constructed on the longer leg of a right triangle as its diameter. Determine the radius of this circle if the shorter leg of the triangle is 7.5 cm, and the length of the chord connecting the vertex of the right angle with the point where the hypotenuse intersects the circle is 6 cm.

Answer: 5


The sequence \( a_n \) is defined by \( a_1 = a_2 = 1 \), \( a_{n+2} = a_{n+1} + 2a_n \). The sequence \( b_n \) is defined by \( b_1 = 1 \), \( b_2 = 7 \), \( b_{n+2} = 2b_{n+1} + 3b_n \). Show that the only integer in both sequences is 1.

Answer: 1


Answer: 199


Find the number of all trees planted at a five-foot distance from each other on a rectangular plot of land, the sides of which are 120 feet and 70 feet.

Answer: 375


Let \( n \) be a positive integer and define \(\mathrm{S}_{\mathrm{n}} = \left\{\left(\mathrm{a}_{1}, \mathrm{a}_{2}, \ldots, \mathrm{a}_{2^{n}}\right) \mid \mathrm{a}_{\mathrm{i}} \in\{0,1\}, 1 \leq \mathrm{i} \leq 2^{\mathrm{n}}\right\}\). For \( \mathrm{a}, \mathrm{b} \in \mathrm{S}_{\mathrm{n}} \) where \( a = \left(a_{1}, a_{2}, \ldots, a_{2^{n}}\right) \) and \( b = \left(b_{1}, b_{2}, \ldots, b_{2^{n}}\right) \), define \( d(a, b) = \sum_{i=1}^{2^{n}} \left|a_{i} - b_{i}\right| \). If for any \(a, b \in A\), \( d(a, b) \geq 2^{n-1} \), then \( A \subseteq \mathrm{S}_{\mathrm{n}} \) is called a "good subset". Find the maximum possible value of \( |A| \).

Answer: 2^{n+1}


Martha writes down a random mathematical expression consisting of 3 single-digit positive integers with an addition sign " $+$ " or a multiplication sign " $\times$ " between each pair of adjacent digits.  (For example, her expression could be  $4 + 3\times 3$ , with value 13.)  Each positive digit is equally likely, each arithmetic sign (" $+$ " or " $\times$ ") is equally likely, and all choices are independent.  What is the expected value (average value) of her expression?

Answer: 50


Five friends sat in a movie theater in a row containing $5$ seats, numbered $1$ to $5$ from left to right. (The directions "left" and "right" are from the point of view of the people as they sit in the seats.) During the movie Ada went to the lobby to get some popcorn. When she returned, she found that Bea had moved two seats to the right, Ceci had moved one seat to the left, and Dee and Edie had switched seats, leaving an end seat for Ada. In which seat had Ada been sitting before she got up?
$\textbf{(A) }1 \qquad \textbf{(B) } 2 \qquad \textbf{(C) } 3 \qquad \textbf{(D) } 4\qquad \textbf{(E) } 5$

Answer: \textbf{(B)}2


Let  $d_1, d_2, \ldots , d_{k}$  be the distinct positive integer divisors of  $6^8$ . Find the number of ordered pairs  $(i, j)$  such that  $d_i - d_j$  is divisible by  $11$ .

Answer: 665


How many ways to fill the board  $ 4\times 4$  by nonnegative integers, such that sum of the numbers of each row and each column is 3?

Answer: 2008


Given that the golden ratio $m = \frac{{\sqrt{5}-1}}{2}$, calculate the value of $\frac{{\sin{42}°+m}}{{\cos{42}°}}$.

Answer: \sqrt{3}


Given the equation of a circle $x^2 + y^2 - 6x - 8y = 0$, if the longest chord AC and the shortest chord BD both pass through the point (-1, 4) on the circle, find the area of the quadrilateral ABCD.

Answer: 30


A shopper plans to purchase an item that has a listed price greater than $\textdollar 100$ and can use any one of the three coupons. Coupon A gives $15\%$ off the listed price, Coupon B gives $\textdollar 30$ off the listed price, and Coupon C gives $25\%$ off the amount by which the listed price exceeds
$\textdollar 100$.
Let $x$ and $y$ be the smallest and largest prices, respectively, for which Coupon A saves at least as many dollars as Coupon B or C. What is $y - x$?
$\textbf{(A)}\ 50 \qquad \textbf{(B)}\ 60 \qquad \textbf{(C)}\ 75 \qquad \textbf{(D)}\ 80  \qquad \textbf{(E)}\ 100$

Answer: \textbf{(A)}\50


In a particular sequence, the first term is $a_1=1007$ and the second term is $a_2=1008$. Furthermore, the values of the remaining terms are chosen so that $a_n + a_{n+1} + a_{n+2} = 2n$ for all $n \geq 1$. Determine $a_{500}$.

Answer: 1339


Consider the operation "minus the reciprocal of," defined by $a\diamond b=a-\frac{1}{b}$. What is $((1\diamond2)\diamond3)-(1\diamond(2\diamond3))$?
$\textbf{(A) } -\dfrac{7}{30} \qquad\textbf{(B) } -\dfrac{1}{6} \qquad\textbf{(C) } 0 \qquad\textbf{(D) } \dfrac{1}{6} \qquad\textbf{(E) } \dfrac{7}{30}$

Answer: \textbf{(A)}\-\frac{7}{30}


Let $A B C$ be a triangle with $A B=A C=\frac{25}{14} B C$. Let $M$ denote the midpoint of $\overline{B C}$ and let $X$ and $Y$ denote the projections of $M$ onto $\overline{A B}$ and $\overline{A C}$, respectively. If the areas of triangle $A B C$ and quadrilateral $A X M Y$ are both positive integers, find the minimum possible sum of these areas.

Answer: 1201


A set \( \mathcal{T} \) of distinct positive integers has the following property: for every integer \( y \) in \( \mathcal{T}, \) the arithmetic mean of the set of values obtained by deleting \( y \) from \( \mathcal{T} \) is an integer. Given that 2 belongs to \( \mathcal{T} \) and that 1024 is the largest element of \( \mathcal{T}, \) what is the greatest number of elements that \( \mathcal{T} \) can have?

Answer: 15


Given a parameterized curve  $ C: x\equal{}e^t\minus{}e^{\minus{}t},\ y\equal{}e^{3t}\plus{}e^{\minus{}3t}$ .

Find the area bounded by the curve  $ C$ , the  $ x$  axis and two lines  $ x\equal{}\pm 1$ .

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


What is the probability that two trainees were born on the same day (not necessarily the same year)? Note: There are 62 trainees.

Answer: 99.59095749


A sledding hill consists of a straight incline \( AB \) and a horizontal section \( BC \). Point \( A \) is located 5 meters from the nearest horizontal ground point \( H \). The distance \( HC \) is 3 meters, with point \( B \) lying on the segment \( HC \). Find the distance from point \( H \) to point \( B \) such that the time it takes for the sled to move from rest along the broken line \( ABC \) is minimized. Assume a uniform gravitational field, and neglect friction, air resistance, and changes in the sled's speed vector as it passes through point \( B \). Assume the acceleration due to gravity is \( 10 \ \text{m/s}^2 \).

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


Rthea, a distant planet, is home to creatures whose DNA consists of two (distinguishable) strands of bases with a fixed orientation. Each base is one of the letters H, M, N, T, and each strand consists of a sequence of five bases, thus forming five pairs. Due to the chemical properties of the bases, each pair must consist of distinct bases. Also, the bases H and M cannot appear next to each other on the same strand; the same is true for N and T. How many possible DNA sequences are there on Rthea?

Answer: 28812


Solve the inequality

$$
\sqrt{10x - 21} - \sqrt{5x^2 - 21x + 21} \geq 5x^2 - 31x + 42
$$

Provide the sum of all integer solutions for \(x\) that satisfy the inequality.

Answer: 7


A deck of 8056 cards has 2014 ranks numbered 1-2014. Each rank has four suits: hearts, diamonds, clubs, and spades. Each card has a rank and a suit, and no two cards have the same rank and the same suit. How many subsets of the set of cards in this deck have cards from an odd number of distinct ranks?

Answer: \frac{1}{2}(16^{2014}-14^{2014})


Given the sequence $\left\{a_{n}\right\}$ satisfying $a_{1}=1$ and $a_{n+1}=a_{n}+\frac{1}{2 a_{n}}$, find $\lim _{n \rightarrow \infty}\left(a_{n}-\sqrt{n}\right)$.

Answer: 0


Find all real numbers $k$ for which there exists a nonzero, 2-dimensional vector $\mathbf{v}$ such that
\[\begin{pmatrix} 3 & 4 \\ 6 & 3 \end{pmatrix} \mathbf{v} = k \mathbf{v}.\]

Answer: 3 - 2\sqrt{6}


On a circle, there are 2018 points. Each of these points is labeled with an integer. Each number is greater than the sum of the two numbers immediately preceding it in a clockwise direction.

Determine the maximum possible number of positive numbers among these 2018 numbers.

(Walther Janous)

Answer: 1009


Suppose that for a positive integer \( n \), \( 2^n + 1 \) is a prime number. What remainder can this prime have when divided by 240?

Answer: 17


Valya had five colorful ribbons in her box with lengths of 15, 20, 24, 26, and 30 cm. Kolya cut a piece from some (possibly all) of the ribbons. The average length of the ribbons decreased by 5 cm, but the median and range of the lengths did not change. What are the new lengths of the ribbons?

Answer: 9,9,24,24,24


Given that \(7,999,999,999\) has at most two prime factors, find its largest prime factor.

Answer: 4,002,001


Find the smallest natural number, multiple of 36, in which all 10 digits appear exactly once.

Answer: 1023457896


In the quadrilateral \(ABCD\), diagonals \(AC\) and \(BD\) intersect at point \(K\). Points \(L\) and \(M\) are the midpoints of sides \(BC\) and \(AD\), respectively. Segment \(LM\) contains point \(K\). The quadrilateral \(ABCD\) is such that a circle can be inscribed in it. Find the radius of this circle, given that \(AB = 3\), \(AC = \sqrt{13}\), and \(LK: KM = 1: 3\).

Answer: 3/2


Given that \( n \) is a positive integer, find all values of \( n \) such that for any real numbers \( a \) and \( b \), if \( a > b \), then \( a^{n} > b^{n} \).

Answer: n


Let  $a,b$  be positive reals such that  $\frac{1}{a}+\frac{1}{b}\leq2\sqrt2$  and  $(a-b)^2=4(ab)^3$ . Find  $\log_a b$ .

Answer: -1


Square $ABCD$ is inscribed in a circle. Square $EFGH$ has vertices $E$ and $F$ on $\overline{CD}$ and vertices $G$ and $H$ on the circle. If the area of square $ABCD$ is $1$, then the area of square $EFGH$ can be expressed as $\frac {m}{n}$ where $m$ and $n$ are relatively prime positive integers and $m < n$. Find $10n + m$.

Answer: 251


Let $\Omega$ be a circle of radius 8 centered at point $O$, and let $M$ be a point on $\Omega$. Let $S$ be the set of points $P$ such that $P$ is contained within $\Omega$, or such that there exists some rectangle $A B C D$ containing $P$ whose center is on $\Omega$ with $A B=4, B C=5$, and $B C \| O M$. Find the area of $S$.

Answer: 164+64 \pi


Given a right triangle $ABC$. On the extension of the hypotenuse $BC$, a point $D$ is chosen such that the line $AD$ is tangent to the circumcircle $\omega$ of triangle $ABC$. The line $AC$ intersects the circumcircle of triangle $ABD$ at point $E$. It turns out that the angle bisector of $\angle ADE$ is tangent to the circle $\omega$. In what ratio does point $C$ divide segment $AE$?

Answer: 1:2


Let $\mathbf{v}$ be a vector such that
\[\left\| \mathbf{v} + \begin{pmatrix} 4 \\ 2 \end{pmatrix} \right\| = 10.\]
Find the smallest possible value of $\|\mathbf{v}\|$.

Answer: 10 - 2\sqrt{5}


A grocer, the owner of a small shop, decided to set aside a small amount of money for a rainy day. This money was in the form of dollar bills and coins of half and quarter dollar each. He distributed all the money into 8 bags in such a way that each bag contained an equal number of dollar bills, and an equal number of coins of each denomination. One evening, the grocer decided to redistribute the money into 7 bags so that each bag still had an equal number of dollar bills and each type of coin. The next evening he redistributed the money into 6 bags in the same manner.

The unfortunate madman then tried to redistribute all the money into 5 bags, but after several hours of hard work, utterly exhausted and failing to achieve his goal, he passed away, mourned by his neighbors. What is the smallest amount of money the grocer could have set aside for a rainy day?

Answer: 294


Each vertex of a regular dodecagon ($12$-gon) is to be colored either red or blue, and thus there are $2^{12}$ possible colorings. Find the number of these colorings with the property that no four vertices colored the same color are the four vertices of a rectangle.

Answer: 928


Given function $f(x)$ defined on $\mathbb{R}$ satisfies: (1) The graph of $y = f(x - 1)$ is symmetric about the point $(1, 0)$; (2) For all $x \in \mathbb{R}$, $$f\left( \frac {3}{4}-x\right) = f\left( \frac {3}{4}+x\right)$$ holds; (3) When $x \in \left(- \frac {3}{2}, - \frac {3}{4}\right]$, $f(x) = \log_{2}(-3x + 1)$, then find $f(2011)$.

Answer: -2


In $\triangle ABC$, the sides opposite to the three internal angles are $a$, $b$, and $c$, respectively. Given that $\cos A= \frac{ \sqrt {10}}{10}$ and $a\sin A+b\sin B-c\sin C= \frac{ 2 \sqrt {5}}{5}a\sin B$.
1. Find the value of $B$;
2. If $b=10$, find the area $S$ of $\triangle ABC$.

Answer: 60


Answer: 828


Samuel's birthday cake is in the form of a $4 \times 4 \times 4$ inch cube. The cake has icing on the top and the four side faces, and no icing on the bottom. Suppose the cake is cut into $64$ smaller cubes, each measuring $1 \times 1 \times 1$ inch, as shown below. How many of the small pieces will have icing on exactly two sides?
[asy] /* Created by SirCalcsALot and sonone Code modfied from https://artofproblemsolving.com/community/c3114h2152994_the_old__aops_logo_with_asymptote */ import three; currentprojection=orthographic(1.75,7,2); //++++ edit colors, names are self-explainatory ++++ //pen top=rgb(27/255, 135/255, 212/255); //pen right=rgb(254/255,245/255,182/255); //pen left=rgb(153/255,200/255,99/255); pen top = rgb(170/255, 170/255, 170/255); pen left = rgb(81/255, 81/255, 81/255); pen right = rgb(165/255, 165/255, 165/255); pen edges=black; int max_side = 4; //+++++++++++++++++++++++++++++++++++++++ path3 leftface=(1,0,0)--(1,1,0)--(1,1,1)--(1,0,1)--cycle; path3 rightface=(0,1,0)--(1,1,0)--(1,1,1)--(0,1,1)--cycle; path3 topface=(0,0,1)--(1,0,1)--(1,1,1)--(0,1,1)--cycle; for(int i=0; i<max_side; ++i){ for(int j=0; j<max_side; ++j){ draw(shift(i,j,-1)*surface(topface),top); draw(shift(i,j,-1)*topface,edges); draw(shift(i,-1,j)*surface(rightface),right); draw(shift(i,-1,j)*rightface,edges); draw(shift(-1,j,i)*surface(leftface),left); draw(shift(-1,j,i)*leftface,edges); } } picture CUBE; draw(CUBE,surface(leftface),left,nolight); draw(CUBE,surface(rightface),right,nolight); draw(CUBE,surface(topface),top,nolight); draw(CUBE,topface,edges); draw(CUBE,leftface,edges); draw(CUBE,rightface,edges); // begin made by SirCalcsALot int[][] heights = {{4,4,4,4},{4,4,4,4},{4,4,4,4},{4,4,4,4}}; for (int i = 0; i < max_side; ++i) { for (int j = 0; j < max_side; ++j) { for (int k = 0; k < min(heights[i][j], max_side); ++k) { add(shift(i,j,k)*CUBE); } } } [/asy]
$\textbf{(A) }12 \qquad \textbf{(B) }16 \qquad \textbf{(C) }18 \qquad \textbf{(D) }20 \qquad \textbf{(E) }24$

Answer: \textbf{(D)}20


Given integer  $m\geq2$ , $x_1,...,x_m$  are non-negative real numbers,prove that:  $$ (m-1)^{m-1}(x_1^m+...+x_m^m)\geq(x_1+...+x_m)^m-m^mx_1...x_m $$  and please find out when the equality holds.

Answer: (-1)^{-1}(x_1^+x_2^+\cdots+x_m^)\ge(x_1+x_2+\cdots+x_m)^-^x_1x_2\cdotsx_m


We call a number greater than  $25$ , *semi-prime* if it is the sum of some two different prime numbers. What is the greatest number of consecutive natural numbers that can be *semi-prime*?

Answer: 5


The diagonal \( AC \) of a convex quadrilateral \( ABCD \) serves as the diameter of the circumscribed circle around it. Find the ratio of the areas of triangles \( ABC \) and \( ACD \), given that the diagonal \( BD \) divides \( AC \) in the ratio 2:5 (counting from point \( A \)), and \( \angle BAC = 45^\circ \).

Answer: \frac{29}{20}


Given Orvin goes to a store with just enough money to buy 40 balloons, and the store has a special promotion: for every balloon bought at full price, a second one can be bought at 1/2 off. Find the maximum number of balloons Orvin can buy.

Answer: 52


Two circles with radii 5 and 3 touch each other internally. A chord of the larger circle touches the smaller circle and is divided by the point of tangency in the ratio \( 3:1 \). Find the length of this chord.

Answer: 8


Given real numbers \( a, b, c, d \) that satisfy \( 5a + 6b - 7c + 4d = 1 \), what is the minimum value of \( 3a^2 + 2b^2 + 5c^2 + d^2 \)?

Answer: \frac{15}{782}


Let  $f: [0,1]\rightarrow(0,+\infty)$  be a continuous function.

a) Show that for any integer  $n\geq 1$ , there is a unique division  $0=a_{0}<a_{1}<\ldots<a_{n}=1$  such that  $\int_{a_{k}}^{a_{k+1}}f(x)\, dx=\frac{1}{n}\int_{0}^{1}f(x)\, dx$  holds for all  $k=0,1,\ldots,n-1$ .

b) For each  $n$ , consider the  $a_{i}$  above (that depend on  $n$ ) and define  $b_{n}=\frac{a_{1}+a_{2}+\ldots+a_{n}}{n}$ . Show that the sequence  $(b_{n})$  is convergent and compute it's limit.

Answer: \frac{\int_{0}^{1}xf(x)\,dx}{\int_{0}^{1}f(x)\,dx}


Answer: 0.04858


A $16$-quart radiator is filled with water. Four quarts are removed and replaced with pure antifreeze liquid. Then four quarts of the mixture are removed and replaced with pure antifreeze. This is done a third and a fourth time. The fractional part of the final mixture that is water is:
$\textbf{(A)}\ \frac{1}{4}\qquad  \textbf{(B)}\ \frac{81}{256}\qquad  \textbf{(C)}\ \frac{27}{64}\qquad  \textbf{(D)}\ \frac{37}{64}\qquad  \textbf{(E)}\ \frac{175}{256}$

Answer: \textbf{(B)}\\frac{81}{256}


Carl is on a vertex of a regular pentagon. Every minute, he randomly selects an adjacent vertex (each with probability $\frac{1}{2}$ ) and walks along the edge to it. What is the probability that after 10 minutes, he ends up where he had started?

Answer: \frac{127}{512}


Given that $F\_1$ and $F\_2$ are the two foci of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 (a > b > 0)$, and $P$ is a point on the hyperbola such that $\overrightarrow{PF\_1} \cdot \overrightarrow{PF\_2} = 0$ and $|\overrightarrow{PF\_1}| \cdot |\overrightarrow{PF\_2}| = 2ac (c$ is the semi-focal distance$)$, find the eccentricity of the hyperbola.

Answer: \frac{\sqrt{5} + 1}{2}


There are 55 boxes in a row, numbered consecutively from 1 to 55. Each box contains no more than 10 balls, and the number of balls in any two neighboring boxes differs by exactly 1. It is known that the total number of balls in the boxes with numbers $1, 4, 7, 10, \ldots, 55$ is 181. What is the minimum total number of balls that can be in all 55 boxes?

Answer: 487


In a scalene triangle, an inscribed circle is drawn, and the points of tangency with the sides are taken as the vertices of a second triangle. In this second triangle, another inscribed circle is drawn, and the points of tangency are taken as the vertices of a third triangle; this process continues indefinitely. Prove that there are no two similar triangles in the resulting sequence.

Answer: 2


The interior of a quadrilateral is bounded by the graphs of $(x+ay)^2 = 4a^2$ and $(ax-y)^2 = a^2$, where $a$ is a positive real number. What is the area of this region in terms of $a$, valid for all $a > 0$?

Answer: \frac{8a^2}{a^2+1}


Answer: \frac{2}{3}n^{3/2}<\sum_{r=1}^n\sqrt{r}<\frac{2}{3}n^{3/2}+\frac{1}{2}\sqrt{n}


A circle touches the sides of an angle at points $A$ and $B$. The distance from a point $C$ on the circle to the line $AB$ is 6. Find the sum of the distances from point $C$ to the sides of the angle, given that one of these distances is 5 greater than the other.

Answer: 13


Answer: 15


Once upon a time in a company, the following conversation took place:

- "We need to immediately call Misha!" exclaimed Vanya.

However, no one remembered Misha's phone number.

- "I remember exactly that the last three digits of the phone number are consecutive natural numbers," said Nastya.
- "And I recall that the first five digits formed a palindrome," noted Anton.
- "No one remembers seven-digit numbers as they are; they are broken into three groups: first three digits, and then two groups of two digits each. I believe that the three-digit number produced by this division was divisible by 9," observed Nikita.
- "Exactly, exactly," agreed Mitya, "and there were three consecutive ones in the phone number."
- "Only one of the two-digit numbers resulting from Nikita's division was prime," added Sasha.

Help the friends reconstruct Misha's phone number.

Answer: 7111765


Let  $ ABCD$  be a unit square (that is, the labels  $ A, B, C, D$  appear in that order around the square). Let  $ X$  be a point outside of the square such that the distance from  $ X$  to  $ AC$  is equal to the distance from  $ X$  to  $ BD$ , and also that  $ AX \equal{} \frac {\sqrt {2}}{2}$ .  Determine the value of  $ CX^2$ .

Answer: 5/2


Define a non-empty finite set as a trivial set if the sum of the squares of all its elements is odd.

Let the set \( A = \{1, 2, \cdots, 2017\} \). Determine the number of trivial subsets of \( A \) among all its proper subsets. Express the answer using a power of a number.

Answer: 2^{2016}-1


Given a right triangle \(ABC\) with a right angle at \(C\), a circle is drawn with diameter \(BC\) of length 26. A tangent \(AP\) from point \(A\) to this circle (distinct from \(AC\)) is drawn. The perpendicular \(PH\) dropped onto segment \(BC\) intersects segment \(AB\) at point \(Q\). Find the area of triangle \(BPQ\) given \(BH:CH = 4:9\).

Answer: 24


Three cones are placed on a table on their bases, touching each other. The radii of their bases are 1, 12, and 12, and the apex angles of the cones are $-4 \operatorname{arctg} \frac{1}{3}$, $4 \operatorname{arctg} \frac{2}{3}$, and $4 \operatorname{arctg} \frac{2}{3}$ respectively (the apex angle of a cone is the angle between its generatrices in an axial section). A sphere is placed on the table, touching all the cones. Find the radius of the sphere.

Answer: 40/21


Given $f(x) = e^{-x}$, calculate the limit $$\lim_{\Delta x \to 0} \frac{f(1 + \Delta x) - f(1 - 2\Delta x)}{\Delta x}$$.

Answer: -\frac{3}{e}


Let $S$ be the set $\{1,2,3,...,19\}$. For $a,b \in S$, define $a \succ b$ to mean that either $0 < a - b \le 9$ or $b - a > 9$. How many ordered triples $(x,y,z)$ of elements of $S$ have the property that $x \succ y$, $y \succ z$, and $z \succ x$?

Answer: 855


Given \( n(n \geqslant 3) \) points \( A_{1}, A_{2}, \cdots, A_{n} \) on a plane such that no three points are collinear. Let \( \alpha \) denote the minimum value of all angles \( \angle A_{i} A_{j} A_{k} \) where \( A_{i}, A_{j}, A_{k} \) are three different given points. For each \( n \), find the maximum value of \( \alpha \) and determine how these points should be distributed to achieve this maximum value.

Answer: \frac{180}{n}


Let \( n \) be an integer between 1 and 1990, and let \( x^2 + x - 3n \) be factored into the product of two linear factors with integer coefficients. Determine the number of integers \( n \) that satisfy this condition.

Answer: 50


Calculate the integral

$$
\int_{L} |z| \bar{z} \, dz
$$

where \( L \) is the upper semicircle \( |z|=1, \operatorname{Re} z > 0 \) oriented counterclockwise.

Answer: i\pi


A square has a side length of 4 units. Points $R$ and $S$ are chosen such that $R$ is the midpoint of the bottom side of the square, and $S$ is the midpoint of the right side of the square. What fraction of the interior of the square is not shaded if the triangle formed by $R$, $S$, and the upper left corner of the square is left unshaded?

[asy]
filldraw((0,0)--(4,0)--(4,4)--(0,4)--(0,0)--cycle,gray,linewidth(1));
filldraw((0,4)--(2,0)--(4,2)--(0,4)--cycle,white,linewidth(1));
label("R",(2,0),S);
label("S",(4,2),E);
[/asy]

Answer: \frac{3}{8}


At Jefferson Summer Camp, $60\%$ of the children play soccer, $30\%$ of the children swim, and $40\%$ of the soccer players swim. To the nearest whole percent, what percent of the non-swimmers play soccer?
$\mathrm{(A)}\ 30\% \qquad \mathrm{(B)}\ 40\% \qquad \mathrm{(C)}\ 49\% \qquad \mathrm{(D)}\ 51\% \qquad \mathrm{(E)}\ 70\%$

Answer: D


Answer: 144.9


Given that \( x \) and \( y \) are non-zero real numbers and they satisfy \(\frac{x \sin \frac{\pi}{5} + y \cos \frac{\pi}{5}}{x \cos \frac{\pi}{5} - y \sin \frac{\pi}{5}} = \tan \frac{9 \pi}{20}\),
1. Find the value of \(\frac{y}{x}\).
2. In \(\triangle ABC\), if \(\tan C = \frac{y}{x}\), find the maximum value of \(\sin 2A + 2 \cos B\).

Answer: \frac{3}{2}


On a table, there are 20 cards numbered from 1 to 20. Xiaoming picks 2 cards each time, such that the number on one card is 2 times the number on the other card plus 2. What is the maximum number of cards Xiaoming can pick?

Answer: 12


Let \( F_{1}, F_{2}, F_{3}, \ldots \) be the Fibonacci sequence, the sequence of positive integers satisfying
\[ F_{1}=F_{2}=1 \quad \text { and } \quad F_{n+2}=F_{n+1}+F_{n} \text { for all } n \geq 1. \]

Does there exist an \( n \geq 1 \) for which \( F_{n} \) is divisible by 2014?

Answer: Yes


We define two sequences: the first has 3, the second has 1 as the first element. For \(n \geq 2\), the \(n\)-th element of the first sequence is the arithmetic mean of the \((n-1)\)-th elements, and the \(n\)-th element of the second sequence is the harmonic mean of the \((n-1)\)-th elements. Prove that these two sequences are convergent and tend to the same limit. What is this limit? (The harmonic mean of two numbers \(a\) and \(b\) is \(\frac{2ab}{a+b}\).)

Answer: \sqrt{3}


In the plane, six lines are drawn such that no three of them meet at one point. Can it happen that there are exactly (a) 12 (b) 16 intersection points where two lines meet?

Answer: No


Given the function f(x) = a^x (a > 0, a ≠ 1).
(I) If $f(1) + f(-1) = \frac{5}{2}$, find the value of f(2) + f(-2).
(II) If the difference between the maximum and minimum values of the function f(x) on [-1, 1] is $\frac{8}{3}$, find the value of the real number a.

Answer: \frac{1}{3}


Calculate the value of $V_3$ in Horner's method (also known as Qin Jiushao algorithm) for finding the value of the polynomial $f(x) = 4x^6 + 3x^5 + 4x^4 + 2x^3 + 5x^2 - 7x + 9$ when $x = 4$.

Answer: 80


A quarter of the students in a class are non-swimmers. Half of the non-swimmers signed up for a swimming course. Four non-swimmers did not sign up for the course. How many students in the class can swim, and how many students are there in total in the class?

Answer: 32


A grasshopper starts at the origin in the coordinate plane and makes a sequence of hops. Each hop has length  $5$ , and after each hop the grasshopper is at a point whose coordinates are both integers; thus, there are  $12$  possible locations for the grasshopper after the first hop. What is the smallest number of hops needed for the grasshopper to reach the point  $(2021,2021)$ ?

Answer: 578


A package of seeds was passed around a table. The first person took 1 seed, the second person took 2 seeds, the third took 3 seeds, and so forth, with each subsequent person taking one more seed than the previous one. It is known that during the second round a total of 100 more seeds were taken than during the first round. How many people were sitting at the table?

Answer: 10


The natural number \( a \) is divisible by 21 and has 105 distinct divisors, including 1 and \( a \). Find the smallest such \( a \).

Answer: 254016


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

$$
f\left(x^{333}+y\right)=f\left(x^{2018}+2 y\right)+f\left(x^{42}\right)
$$

Answer: f(x)=0


Find the point in the closed unit disc  $D=\{ (x,y) | x^2+y^2\le 1 \}$  at which the function  $f(x,y)=x+y$  attains its maximum .

Answer: (\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}})


64 people are in a single elimination rock-paper-scissors tournament, which consists of a 6-round knockout bracket. Each person has a different rock-paper-scissors skill level, and in any game, the person with the higher skill level will always win. For how many players $P$ is it possible that $P$ wins the first four rounds that he plays?

Answer: 49


The triangles $ABC$ and $BDE$ are congruent isosceles triangles with bases $AB$ and $BD$ lying on the same line, and $C$ and $E$ are on the same side of this line. Let $O_{1}$ and $O_{2}$ be the centers of the circumscribed circles of triangles $ABE$ and $CDE$, respectively. What type of quadrilateral is $B O_{1} C O_{2}$?

Answer: parallelogram


Given a $5\times 5$ chess board, how many ways can you place five distinct pawns on the board such that each column and each row contains exactly one pawn and no two pawns are positioned as if they were "attacking" each other in the manner of queens in chess?

Answer: 1200


A four-digit number $a b c d$ is called balanced if

$$
a+b=c+d
$$

Calculate the following quantities:

a) How many numbers $a b c d$ are there such that $a+b=c+d=8$?

b) How many numbers $a b c d$ are there such that $a+b=c+d=16$?

c) How many balanced numbers are there?

Answer: 615


Given positive integers \( x, y, z \) that satisfy the condition \( x y z = (14 - x)(14 - y)(14 - z) \), and \( x + y + z < 28 \), what is the maximum value of \( x^2 + y^2 + z^2 \)?

Answer: 219


A point  $P$  is given on the curve  $x^4+y^4=1$ . Find the maximum distance from the point  $P$  to the origin.

Answer: \sqrt{2}


Find the smallest positive integer \( n \) such that \( 1^2 + 2^2 + 3^2 + 4^2 + \cdots + n^2 \) is divisible by 100.

Answer: 24


Answer: \frac{111110}{111111}<\frac{333331}{333334}<\frac{222221}{222223}


Let  $N$  be the midpoint of arc  $ABC$  of the circumcircle of  $\Delta ABC$ , and  $NP$ ,  $NT$  be the tangents to the incircle of this triangle. The lines  $BP$  and  $BT$  meet the circumcircle for the second time at points  $P_1$  and  $T_1$  respectively. Prove that  $PP_1 = TT_1$ .

Answer: PP_1=TT_1


A five-digit number $\overline{ABCDE}$ is a multiple of 2014, and $\overline{CDE}$ has exactly 16 factors. What is the smallest value of $\overline{ABCDE}$?

Answer: 24168


For what values of the parameter \( a \) does the equation \(\frac{\log _{a} x}{\log _{a} 2}+\frac{\log _{x}(2 a-x)}{\log _{x} 2}=\frac{1}{\log _{\left(a^{2}-1\right)} 2}\)

(1) Have a solution? (2) Have a unique solution?

Answer: 2


Fresh apples contain \(90\%\) water, while dried apples contain \(12\%\) water. Lena believes that if the compote contains more than \(95\%\) water, it will not taste good. What is the maximum amount of tasty compote she can make using 4 kg of fresh apples and 1 kg of dried apples (with any amount of added water)? Assume the evaporation of water during cooking is negligible.

Answer: 25.6\,


Given the ellipse $C$: $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1(a > b > 0)$, its right focus $F$, and the line passing through $F$ with a slope of $1$ intersects the ellipse at points $M$ and $N$. The perpendicular bisector of $MN$ intersects the $x$-axis at point $P$. If $\frac{|MN|}{|PF|}=4$, find the eccentricity of the ellipse $C$.

Answer: \frac{1}{2}


Given that a sequence satisfies $x_0=0$ and $|x_k|=|x_{k-1}+3|$ for all integers $k\ge 1,$ find the minimum possible value of $|x_1+x_2+\cdots+x_{2006}|.$

Answer: 27


The function \( f(x) \) defined on the set \( \left\{ x \in \mathbf{Z}_{+} \mid 1 \leqslant x \leqslant 12 \right\} \) satisfies
\[ |f(x+1) - f(x)| = 1 \quad \text{for} \; x = 1, 2, \ldots, 11, \]
and \( f(1), f(6), f(12) \) form a geometric sequence. Given that \( f(1) = 1 \), find the number of different functions \( f(x) \) that satisfy these conditions.

Answer: 155


Petya and Vasya were collecting mushrooms. It turned out that Petya collected as many mushrooms as the percentage of mushrooms Vasya collected from the total number of mushrooms they collected together. Additionally, Vasya collected an odd number of mushrooms. How many mushrooms did Petya and Vasya collect together?

Answer: 25


Compute  $\left\lceil\displaystyle\sum_{k=2018}^{\infty}\frac{2019!-2018!}{k!}\right\rceil$ . (The notation  $\left\lceil x\right\rceil$  denotes the least integer  $n$  such that  $n\geq x$ .)

*Proposed by Tristan Shin*

Answer: 2019


There is a peculiar computer with a button. If the current number on the screen is a multiple of 3, pressing the button will divide it by 3. If the current number is not a multiple of 3, pressing the button will multiply it by 6. Xiaoming pressed the button 6 times without looking at the screen, and the final number displayed on the computer was 12. What is the smallest possible initial number on the computer?

Answer: 27


What is the greatest integer less than or equal to \[\frac{5^{80} + 3^{80}}{5^{75} + 3^{75}}?\]

Answer: 3124


Answer: ^5+\frac{1}{8}b^5+\frac{1}{27}^5\ge14


Let  $x_1=1$  and  $x_{n+1} =x_n+\left\lfloor \frac{x_n}{n}\right\rfloor +2$ , for  $n=1,2,3,\ldots $  where  $x$  denotes the largest integer not greater than  $x$ . Determine  $x_{1997}$ .

Answer: 23913


Find the maximum value of the following expression:
$$
|\cdots||| x_{1}-x_{2}\left|-x_{3}\right|-x_{4}\left|-\cdots-x_{1990}\right|,
$$

where \( x_{1}, x_{2}, \cdots, x_{1990} \) are distinct natural numbers from 1 to 1990.

Answer: 1989


Count the number of sequences \(a_{1}, a_{2}, a_{3}, a_{4}, a_{5}\) of integers such that \(a_{i} \leq 1\) for all \(i\) and all partial sums \( (a_{1}, a_{1} + a_{2}, \ldots)\) are non-negative.

Answer: 132


In triangle $ABC,$ it is given that angles $B$ and $C$ are congruent. Points $P$ and $Q$ lie on $\overline{AC}$ and $\overline{AB},$ respectively, so that $AP = PQ = QB = BC.$ Angle $ACB$ is $r$ times as large as angle $APQ,$ where $r$ is a positive real number. Find $\lfloor 1000r \rfloor$.

Answer: 571


Given the function \( f(x) = ax^2 + 8x + 3 \) where \( a < 0 \), find the positive number \( l(a) \) such that the inequality \( |f(x)| \leq 5 \) holds in the interval \([0, l(a)]\). What value of \( a \) maximizes \( l(a) \), and what is the maximum \( l(a) \)?

Answer: \frac{\sqrt{5}+1}{2}


From the set of integers $\{1,2,3,\dots,3009\}$, choose $k$ pairs $\{a_i,b_i\}$ with $a_i<b_i$ so that no two pairs share a common element. Each sum $a_i+b_i$ must be distinct and less than or equal to $3009$. Determine the maximum possible value of $k$.

Answer: 1504


A trapezoid $ABCD$ is divided by a line parallel to its bases $AD$ and $BC$ into two trapezoids of equal area.

Find the segment of this line, enclosed between the non-parallel sides, if the lengths of the bases of the trapezoid are $a$ and $b$.

Answer: \sqrt{\frac{^{2}+b^{2}}{2}}


Given the function $f(x)= \begin{cases} kx^{2}+2x-1, & x\in (0,1] \\ kx+1, & x\in (1,+\infty) \end{cases}$ has two distinct zeros $x_{1}$ and $x_{2}$, then the maximum value of $\dfrac {1}{x_{1}}+ \dfrac {1}{x_{2}}$ is ______.

Answer: \dfrac {9}{4}


Evin and Jerry are playing a game with a pile of marbles. On each players' turn, they can remove  $2$ ,  $3$ ,  $7$ , or  $8$ marbles. If they can’t make a move, because there's  $0$  or  $1$  marble left, they lose the game. Given that Evin goes first and both players play optimally, for how many values of  $n$  from  $1$  to  $1434$  does Evin lose the game?

*Proposed by Evin Liang*

Solution*Solution.*  $\boxed{573}$ Observe that no matter how many marbles a one of them removes, the next player can always remove marbles such
that the total number of marbles removed is  $10$ . Thus, when the number of marbles is a multiple of  $10$ , the first player loses the game. We analyse this game based on the number of marbles modulo  $10$ :

If the number of marbles is  $0$  modulo  $10$ , the first player loses the game

If the number of marbles is  $2$ ,  $3$ ,  $7$ , or  $8$  modulo  $10$ , the first player wins the game by moving to  $0$  modulo 10

If the number of marbles is  $5$  modulo  $10$ , the first player loses the game because every move leads to  $2$ ,  $3$ ,  $7$ , or  $8$  modulo  $10$ In summary, the first player loses if it is  $0$  mod 5, and wins if it is  $2$  or  $3$  mod  $5$ . Now we solve the remaining cases by induction. The first player loses when it is  $1$  modulo  $5$  and wins when it is  $4$  modulo  $5$ . The base case is when there is  $1$  marble, where the first player loses because there is no move. When it is  $4$  modulo  $5$ , then the first player can always remove  $3$  marbles and win by the inductive hypothesis. When it is  $1$  modulo  $5$ , every move results in  $3$  or  $4$  modulo  $5$ , which allows the other player to win by the inductive hypothesis.
Thus, Evin loses the game if n is  $0$  or  $1$  modulo  $5$ . There are  $\boxed{573}$  such values of  $n$  from  $1$  to  $1434$ .

Answer: 573


A robotic grasshopper jumps 1 cm to the east, then 2 cm to the north, then 3 cm to the west, then 4 cm to the south. After every fourth jump, the grasshopper restarts the sequence of jumps. After a total of $n$ jumps, the position of the grasshopper is 162 cm to the west and 158 cm to the south of its original position. What is the sum of the squares of the digits of $n$?

Answer: 22


It is given that \( m \) and \( n \) are two natural numbers and both are not greater than 10. If \( c \) is the number of pairs of \( m \) and \( n \) satisfying the equation \( mx = n \), where \( \frac{1}{4} < x < \frac{1}{3} \), find \( c \).

Answer: 2


Four circles of radius 1 are externally tangent to each other and internally tangent to a larger circle. What is the radius of the large circle? Express your answer as a common fraction in simplest radical form.

Answer: 1 + \sqrt{2}


Answer: -0.006137


Triangle $ABC$ has $AB=40,AC=31,$ and $\sin{A}=\frac{1}{5}$. This triangle is inscribed in rectangle $AQRS$ with $B$ on $\overline{QR}$ and $C$ on $\overline{RS}$. Find the maximum possible area of $AQRS$.

Answer: 744


Numbers of the form \(\frac{n(n+1)(n+2)(n+3)}{1 \cdot 2 \cdot 3 \cdot 4}\) are called pentagonal numbers.

Show that the sum of the first \(n\) quadrilateral numbers is equal to the \(n\)-th pentagonal number.

Answer: \frac{n(n+1)(n+2)(n+3)}{24}


Answer: 0.0042


Let $n(n \geqslant 2)$ be an integer. $S$ is a subset of the set $\{1,2, \cdots, n\}$ such that in $S$, no element divides another, and no two elements are coprime. Find the maximum number of elements in $S$.

Answer: \lfloor\frac{n+2}{4}\rfloor


Determine all functions  $ f: \mathbb{N} \rightarrow \mathbb{N}$  which satisfy:

 $ f(x\plus{}f(y))\equal{}f(x)\plus{}y$  for all  $ x,y \in \mathbb{N}$ .

Answer: f(x)=x


Given that $a$, $b$, $c$ are the sides opposite to angles $A$, $B$, $C$ respectively in $\triangle ABC$, with $a=4$ and $(4+b)(\sin A-\sin B)=(c-b)\sin C$, find the maximum value of the area of $\triangle ABC$.

Answer: 4\sqrt{3}


Given the sample contains 5 individuals with values a, 0, 1, 2, 3, and the average value of the sample is 1, calculate the standard deviation of the sample.

Answer: \sqrt{2}


In triangle $ABC$, $AB = 18$ and $BC = 12$. Find the largest possible value of $\tan A$.

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


Two noncongruent integer-sided isosceles triangles have the same perimeter and the same area. The ratio of the lengths of the bases of the two triangles is $8: 7$. Find the minimum possible value of their common perimeter.

Answer: 676


Let  $\triangle ABC$  be a triangle with  $AB=85$ ,  $BC=125$ ,  $CA=140$ , and incircle  $\omega$ .  Let  $D$ ,  $E$ ,  $F$  be the points of tangency of  $\omega$  with  $\overline{BC}$ ,  $\overline{CA}$ ,  $\overline{AB}$  respectively, and furthermore denote by  $X$ ,  $Y$ ,  and  $Z$  the incenters of  $\triangle AEF$ ,  $\triangle BFD$ , and  $\triangle CDE$ , also respectively.  Find the circumradius of  $\triangle XYZ$ .

*Proposed by David Altizio*

Answer: 30


Given point A (-2, 1) and circle C: $(x-2)^2+(y-2)^2=1$, a ray of light is emitted from point A to the x-axis and then reflects in the direction of the tangent to the circle. The distance traveled by the ray of light from point A to the tangent point is ______.

Answer: 2\sqrt{6}


\( f \) is a function from the integers to \(\{1, 2, 3, \ldots , n\} \) such that \( f(A) \) and \( f(B) \) are unequal whenever \( A \) and \( B \) differ by 5, 7, or 12. What is the smallest possible \( n \)?

Answer: 4


In recent years, China's scientific and technological achievements have been remarkable. The Beidou-3 global satellite navigation system has been operational for many years. The Beidou-3 global satellite navigation system consists of 24 medium Earth orbit satellites, 3 geostationary Earth orbit satellites, and 3 inclined geosynchronous orbit satellites, totaling 30 satellites. The global positioning accuracy of the Beidou-3 global satellite navigation system is better than 10 meters, and the measured navigation positioning accuracy is 2 to 3 meters. The global service availability is 99%, with better performance in the Asia-Pacific region. Now, two satellites are selected for signal analysis from the geostationary Earth orbit satellites and inclined geosynchronous orbit satellites.
$(1)$ Find the probability of selecting exactly one geostationary Earth orbit satellite and one inclined geosynchronous orbit satellite;
$(2)$ Find the probability of selecting at least one inclined geosynchronous orbit satellite.

Answer: \frac{4}{5}


We call *word* a sequence of letters  $\overline {l_1l_2...l_n}, n\ge 1$  .
A *word*  $\overline {l_1l_2...l_n}, n\ge 1$  is called *palindrome* if   $l_k=l_{n-k+1}$  , for any  $k, 1 \le k \le n$ .
Consider a *word*  $X=\overline {l_1l_2...l_{2014}}$  in which  $ l_k\in\{A,B\}$  , for any  $k, 1\le k \le 2014$ .
Prove that there are at least  $806$  *palindrome* *words* to ''stick" together to  get word  $X$ .

Answer: 806


Answer: 9


The sequence $\{a_n\}$ satisfies $a_n=13-3n$, $b_n=a_n⋅a_{n+1}⋅a_{n+2}$, $S_n$ is the sum of the first $n$ terms of $\{b_n\}$. Find the maximum value of $S_n$.

Answer: 310


Answer: \frac{7}{16}


N numbers are marked in the set  $\{1,2,...,2000\}$  so that any pair of the numbers  $(1,2),(2,4),...,(1000,2000)$  contains at least one marked number. Find the least possible value of  $N$ .

I.Gorodnin

Answer: 666


A rubber tire has an outer diameter of 25 inches. Calculate the approximate percentage increase in the number of rotations in one mile when the radius of the tire decreases by \(\frac{1}{4}\) inch.

Answer: 2\%


In how many ways can 25 forints be paid using 1, 2, 5, 10, and 20 forint coins?

Answer: 68


On January 15 in the stormy town of Stormville, there is a  $50\%$  chance of rain. Every day, the probability of it raining has a  $50\%$  chance of being  $\frac{2017}{2016}$  times that of the previous day (or  $100\%$  if this new quantity is over  $100\%$ ) and a  $50\%$  chance of being  $\frac{1007}{2016}$  times that of the previous day. What is the probability that it rains on January 20?

*2018 CCA Math Bonanza Lightning Round #3.3*

Answer: 243/2048


Answer: |(n)-k(n)|\le\frac{1}{(n-1)!n!}


Let  $x \leq y \leq z \leq t$  be real numbers such that  $xy + xz + xt + yz + yt + zt = 1.$  **a)** Prove that  $xt < \frac 13,$ b) Find the least constant  $C$  for which inequality  $xt < C$  holds for all possible values  $x$  and  $t.$

Answer: \frac{1}{3}


A circle with its center on the side \( AC \) of an isosceles triangle \( ABC \) (\( AB = BC \)) is tangent to the sides \( AB \) and \( BC \). Find the radius of the circle if the area of triangle \( ABC \) is 25 and the ratio of the height \( BD \) to the side \( AC \) is \( 3:8 \).

Answer: 2\sqrt{3}


If $p$, $q$, $r$, $s$, $t$, and $u$ are integers for which  $343x^3+64 = (px^2 + qx + r)(sx^2 + tx + u)$ for all $x$, then what is $p^2+q^2+r^2+s^2+t^2+u^2$?

Answer: 3506


In trapezoid \(ABCD\), \(AD\) is parallel to \(BC\). If \(AD = 52\), \(BC = 65\), \(AB = 20\), and \(CD = 11\), find the area of the trapezoid.

Answer: 594


In the isosceles trapezoid \(ABCD\) with bases \(AD\) and \(BC\), perpendiculars \(BH\) and \(DK\) are drawn from vertices \(B\) and \(D\) to the diagonal \(AC\). It is known that the feet of the perpendiculars lie on the segment \(AC\) and that \(AC = 20\), \(AK = 19\), and \(AH = 3\). Find the area of the trapezoid \(ABCD\).

Answer: 120


In a sequence of positive integers starting from 1, some numbers are colored red according to the following rules: first color 1 red, then color the next 2 even numbers 2 and 4 red; next, color the 4 closest consecutive odd numbers after 4, which are 5, 7, 9, red; then color the 4 closest consecutive even numbers after 9, which are 10, 12, 14, 16, red; and so on, coloring the closest 5 consecutive odd numbers after 16, which are 17, 19, 21, 23, 25, red. Continue this pattern to get a red subsequence: 1, 2, 4, 5, 7, 9, 12, 14, 16, 17,…. The 57th number in this red subsequence, starting from 1, is.

Answer: 103


For a positive integer  $n$ , written in decimal base, we denote by  $p(n)$  the product of its digits.

a) Prove that  $p(n) \leq n$ ;
b) Find all positive integers  $n$  such that
\[ 10p(n) = n^2+ 4n - 2005. \]

*Eugen Păltănea*

Answer: 45


Gavrila found that the front tires of a car last for 21,000 km, and the rear tires last for 28,000 km. He decided to switch them at some point to maximize the possible distance the car can travel. Find this maximum distance (in km).

Answer: 24000


In a large library storage room, there are $1584$ boxes, each containing $45$ books. The library dean asks for these books to be repacked so that each new box contains $47$ books. How many books will be left over after repacking the books into as many full boxes of $47$ books each as possible?

Answer: 28


Given \( x, y \in [0,+\infty) \) and satisfying \( x^{3} + y^{3} + 3xy = 1 \).
Find the maximum value of \( x^{2} y \).

Answer: \frac{4}{27}


Vasya thought of three natural numbers with a sum of 1003. After calculating their product, Vasya noticed that it ends in $N$ zeros. What is the maximum possible value of $N$?

Answer: 7


Given that Carl has 24 fence posts and places one on each of the four corners, with 3 yards between neighboring posts, where the number of posts on the longer side is three times the number of posts on the shorter side, determine the area, in square yards, of Carl's lawn.

Answer: 243


Prove that if the forest in the conditions of the previous problem is convex, then there exists a path guaranteeing an exit from the forest, with a length not exceeding $\sqrt{2 \pi S}$.

Answer: \sqrt{2\piS}


Given that the volume of tetrahedron \(ABCD\) is \(V\), point \(E\) is the midpoint of edge \(AD\), point \(F\) lies on the extension of \(AB\) such that \(BF = AB\). The plane passing through points \(C\), \(E\), and \(F\) intersects \(BD\) at point \(G\). Determine the volume of tetrahedron \(CDGE\).

Answer: \frac{V}{3}


The circumference of the circle with center $O$ is divided into $12$ equal arcs, marked the letters $A$ through $L$ as seen below. What is the number of degrees in the sum of the angles $x$ and $y$?

Answer: 90


Car A is traveling at 20 miles per hour. Car B is 1 mile behind, following at 30 miles per hour. A fast fly can move at 40 miles per hour. The fly begins on the front bumper of car B, and flies back and forth between the two cars. How many miles will the fly travel before it is crushed in the collision?

Answer: 4


What is the area of the region enclosed by the graph of the equation $x^2+y^2=|x|+|y|?$
$\textbf{(A)}\ \pi+\sqrt{2}\qquad\textbf{(B)}\ \pi+2\qquad\textbf{(C)}\ \pi+2\sqrt{2}\qquad\textbf{(D)}\ 2\pi+\sqrt{2}\qquad\textbf{(E)}\ 2\pi+2\sqrt{2}$

Answer: \textbf{(B)}\\pi+2


In a sealed box, there are three red chips and two green chips. Chips are randomly drawn from the box without replacement until either all three red chips or both green chips are drawn. What is the probability of drawing all three red chips?

Answer: $\frac{2}{5}$


From a regular octagon, a triangle is formed by connecting three randomly chosen vertices of the octagon. What is the probability that at least one of the sides of the triangle is also a side of the octagon?
[asy] size(3cm); pair A[]; for (int i=0; i<9; ++i) {   A[i] = rotate(22.5+45*i)*(1,0); } filldraw(A[0]--A[1]--A[2]--A[3]--A[4]--A[5]--A[6]--A[7]--cycle,gray,black); for (int i=0; i<8; ++i) { dot(A[i]); } [/asy]

$\textbf{(A) } \frac{2}{7} \qquad \textbf{(B) } \frac{5}{42} \qquad \textbf{(C) } \frac{11}{14} \qquad \textbf{(D) } \frac{5}{7} \qquad \textbf{(E) } \frac{6}{7}$

Answer: \textbf{(D)}\frac{5}{7}


The density of a body is defined as the ratio of its mass to the volume it occupies. There is a homogeneous cube with a volume of $V=8 \, m^{3}$. As a result of heating, each of its edges increased by 4 mm. By what percentage did the density of this cube change?

Answer: 0.6


A decorative window is made up of a rectangle with semicircles at either end. The ratio of $AD$ to $AB$ is $3:2$. And $AB$ is 30 inches. What is the ratio of the area of the rectangle to the combined area of the semicircles?

$\textbf{(A)}\ 2:3\qquad\textbf{(B)}\ 3:2\qquad\textbf{(C)}\ 6:\pi\qquad\textbf{(D)}\ 9:\pi\qquad\textbf{(E)}\ 30 :\pi$

Answer: \textbf{(C)}\6:\pi


The area of an equilateral triangle inscribed in a circle is 81 cm². Find the radius of the circle.

Answer: 6 \sqrt[4]{3}


Let \( x, y, z \) be the side lengths of a right-angled triangle, with \( z \) being the hypotenuse. Determine the maximum value of the parameter \( k \) for which the inequality \( x^{2}(y+z) + y^{2}(z+x) + z^{2}(x+y) > k \cdot x \cdot y \cdot z \) always holds. Also, indicate when equality occurs.

Answer: 2+3\sqrt{2}


When \( x^{2} \) is added to the quadratic polynomial \( f(x) \), its maximum value increases by \( \frac{27}{2} \), and when \( 4x^{2} \) is subtracted from it, its maximum value decreases by 9. How will the maximum value of \( f(x) \) change if \( 2x^{2} \) is subtracted from it?

Answer: \frac{27}{4}


In a high school's "Campus Microfilm Festival" event, the school will evaluate the films from two aspects: "viewing numbers" and "expert ratings". If a film A is higher than film B in at least one of these aspects, then film A is considered not inferior to film B. It is known that there are 10 microfilms participating. If a film is not inferior to the other 9 films, it is considered an excellent film. Therefore, the maximum possible number of excellent films among these 10 microfilms is __________.

Answer: 10


A circle passing through the vertices $B, C,$ and $D$ of parallelogram $ABCD$ touches line $AD$ and intersects line $AB$ at points $B$ and $E$. Find the length of segment $AE$, if $AD = 4$ and $CE = 5$.

Answer: \frac{16}{5}


Let \( n \) be a fixed positive integer. Initially, \( n \) 1's are written on a blackboard. Every minute, David picks two numbers \( x \) and \( y \) written on the blackboard, erases them, and writes the number \( (x+y)^{4} \) on the blackboard. Show that after \( n-1 \) minutes, the number written on the blackboard is at least \( 2^{\frac{4 n^{2}-4}{3}} \).

Answer: 2^{\frac{4n^2-4}{3}}


Evaluate the expression $\dfrac{\sqrt[6]{5}}{\sqrt[4]{5}}$. What power of 5 does this expression represent?

Answer: -\frac{1}{12}


In a $3 \times 4$ rectangle, there are 4 points distributed. Prove that there must be a pair of points whose distance is not greater than $\frac{25}{8}$.

Answer: \frac{25}{8}


Factor the expression $81x^4 - 256y^4$ and find the sum of all integers in its complete factorization $(ax^2 + bxy + cy^2)(dx^2 + exy + fy^2)$.

Answer: 31


Find the least positive integer   $n$  satisfying the following statement: for eash pair of positive integers  $a$  and  $b$  such that  $36$  divides   $a+b$  and  $n$  divides  $ab$  it follows that  $36$  divides both   $a$  and  $b$ .

Answer: 1296


Vasya replaced the same digits in two numbers with the same letters, and different digits with different letters. It turned out that the number ZARAZA is divisible by 4, and ALMAZ is divisible by 28. Find the last two digits of the sum ZARAZA + ALMAZ.

Answer: 32


Given the function \( f(x)=x^{2}-(k^{2}-5ak+3)x+7 \) where \( a, k \in \mathbf{R} \), it is known that for any \( k \in [0,2] \), \( x_{1} \in [k, k+a] \) and \( x_{2} \in [k+2a, k+4a] \), then \( f(x_{1}) \geq f(x_{2}) \). Find the maximum value of the positive real number \( a \).

Answer: \frac{2\sqrt{6}-4}{5}


Isabella uses one-foot cubical blocks to build a rectangular fort that is $12$ feet long, $10$ feet wide, and $5$ feet high. The floor and the four walls are all one foot thick. How many blocks does the fort contain?

$\textbf{(A)}\ 204 \qquad \textbf{(B)}\ 280 \qquad \textbf{(C)}\ 320 \qquad \textbf{(D)}\ 340 \qquad \textbf{(E)}\ 600$

Answer: \textbf{(B)}\280


In trapezoid \(ABCD\), the smaller base \(BC\) is equal to 3, and the legs \(AB\) and \(CD\) are each equal to 3. The diagonals of the trapezoid form an angle of \(60^\circ\) with each other. Find the base \(AD\).

Answer: 6


The target below is made up of concentric circles with diameters  $4$ ,  $8$ ,  $12$ ,  $16$ , and  $20$ . The area of the dark region is  $n\pi$ . Find  $n$ .

[asy]
size(150);
defaultpen(linewidth(0.8));
int i;
for(i=5;i>=1;i=i-1)
{
if (floor(i/2)==i/2)
{
filldraw(circle(origin,4*i),white);
}
else
{
filldraw(circle(origin,4*i),red);
}
}
[/asy]

Answer: 60


Two cars leave points A and B simultaneously and meet at 12 PM. If the speed of the first car is doubled while keeping the speed of the second car the same, the meeting will occur 56 minutes earlier. If the speed of the second car is doubled while keeping the speed of the first car the same, they will meet 65 minutes earlier. Determine the meeting time if the speeds of both cars were doubled.

Answer: 10:29


Xiao Ming forgot the last two digits of his WeChat login password. He only remembers that the last digit is one of the letters \\(A\\), \\(a\\), \\(B\\), or \\(b\\), and the other digit is one of the numbers \\(4\\), \\(5\\), or \\(6\\). The probability that Xiao Ming can successfully log in with one attempt is \_\_\_\_\_\_.

Answer: \dfrac{1}{12}


Define the operation: \(a \quad b = \frac{a \times b}{a + b}\). Calculate the result of the expression \(\frac{20102010 \cdot 2010 \cdots 2010 \cdot 201 \text{ C}}{\text{ 9 " "}}\).

Answer: 201


Divers extracted a certain number of pearls, not exceeding 1000. The distribution of the pearls happens as follows: each diver in turn approaches the heap of pearls and takes either exactly half or exactly one-third of the remaining pearls. After all divers have taken their share, the remainder of the pearls is offered to the sea god. What is the maximum number of divers that could have participated in the pearl extraction?

Answer: 12


The letters $A$, $B$, $C$ and $D$ represent digits. If  $\begin{tabular}{ccc}&A&B\\ +&C&A\\ \hline &D&A\end{tabular}$and $\begin{tabular}{ccc}&A&B\\ -&C&A\\ \hline &&A\end{tabular}$,what digit does $D$ represent?
$\textbf{(A)}\ 5\qquad\textbf{(B)}\ 6\qquad\textbf{(C)}\ 7\qquad\textbf{(D)}\ 8\qquad\textbf{(E)}\ 9$

Answer: \textbf{(E)}\9


The six-digit number $M=\overline{abc321}$, where $a, b, c$ are three different numbers, and all are greater than 3. If $M$ is a multiple of 7, what is the smallest value of $M$?

Answer: 468321


In triangle $PQR$, $PQ = 8$, $PR = 17$, and the length of median $PM$ is 12. Additionally, the angle $\angle QPR = 60^\circ$. Find the area of triangle $PQR$.

Answer: 34\sqrt{3}


In a frame of dimensions \(8 \times 8\) with a width of 2 cells, there are a total of 48 cells.

How many cells are in a frame of dimensions \(254 \times 254\) with a width of 2 cells?

Answer: 2016


The focus of a vertically oriented, rotational paraboloid-shaped tall vessel is at a distance of 0.05 meters above the vertex. If a small amount of water is poured into the vessel, what angular velocity $\omega$ is needed to rotate the vessel around its axis so that the water overflows from the top of the vessel?

Answer: 9.9


Point \( A_{1} \) is located on side \( BC \) of triangle \( ABC \), and point \( B_{1} \) is located on side \( AC \). Let \( K \) be the intersection point of line segments \( AA_{1} \) and \( BB_{1} \). Find the area of quadrilateral \( A_{1}CB_{1}K \) if it is known that the area of triangle \( A_{1}BK \) is 5, the area of triangle \( AB_{1}K \) is 8, and the area of triangle \( ABK \) is 10.

Answer: 22


Two adjacent faces of a tetrahedron, which are equilateral triangles with a side length of 3, form a dihedral angle of 30 degrees. The tetrahedron is rotated around the common edge of these faces. Find the maximum area of the projection of the rotating tetrahedron onto the plane containing the given edge. (12 points)

Answer: \frac{9\sqrt{3}}{4}


Given a triangle \(ABC\) with sides \(AB = 13\), \(BC = 14\), and \(AC = 15\). Point \(K\) is marked on side \(AB\), point \(L\) is marked on side \(BC\), and point \(N\) is marked on side \(AC\). It is known that \(BK = \frac{14}{13}\), \(AN = 10\), and \(BL = 1\). A line is drawn through point \(N\) parallel to \(KL\) which intersects side \(BC\) at point \(M\). Find the area of the quadrilateral \(KLMN\).

Answer: \(\frac{36503}{1183}=30 \frac{1013}{1183}\).

Answer: \frac{36503}{1183}


Given the function $f(x) = \left( \frac{1}{3}\right)^{ax^2-4x+3}$,

$(1)$ If $a=-1$, find the intervals of monotonicity for $f(x)$;

$(2)$ If $f(x)$ has a maximum value of $3$, find the value of $a$;

$(3)$ If the range of $f(x)$ is $(0,+\infty)$, find the range of values for $a$.

Answer: \{0\}


The distance from \(A\) to \(B\) is 999 km. Along the road, there are kilometer markers indicating the distances to \(A\) and \(B\): 0।999, 1।998, \(\ldots, 999।0. How many of these markers have only two different digits?

Answer: 40


In a large bag of decorative ribbons, $\frac{1}{4}$ are yellow, $\frac{1}{3}$ are purple, $\frac{1}{8}$ are orange, and the remaining 45 ribbons are silver. How many of the ribbons are orange?

Answer: 19


Consider the solid with 4 triangles and 4 regular hexagons as faces, where each triangle borders 3 hexagons, and all the sides are of length 1. Compute the *square* of the volume of the solid. Express your result in reduced fraction and concatenate the numerator with the denominator (e.g., if you think that the square is  $\frac{1734}{274}$ , then you would submit 1734274).

Answer: 52972


For which values of the parameter \( a \) does the equation

$$
\log _{2}\left(2 x^{2}+(2 a+1) x-2 a\right)-2 \log _{4}\left(x^{2}+3 a x+2 a^{2}\right)=0
$$

have two distinct roots, the sum of the squares of which is greater than 4?

Answer: (-\infty,-1)\cup(\frac{3}{5},1)


Brian starts at the point  $\left(1,0\right)$  in the plane. Every second, he performs one of two moves: he can move from  $\left(a,b\right)$  to  $\left(a-b,a+b\right)$  or from  $\left(a,b\right)$  to  $\left(2a-b,a+2b\right)$ . How many different paths can he take to end up at  $\left(28,-96\right)$ ?

*2018 CCA Math Bonanza Individual Round #14*

Answer: 70


André trains for the marathon by running laps around a circular track with a radius of $100 \text{ m}$. To cover $42 \text{ km}$, the number of laps that André needs to run is between:
(a) 1 and 10 ;
(b) 10 and 50 ;
(c) 50 and 100 ;
(d) 100 and 500 ;
(e) 500 and 1000 .


(1) Let the real number \( t > 0 \). Prove: \(\left(1+\frac{2}{t}\right) \ln (1+t)>2\);

(2) From 100 cards numbered from 1 to 100, a card is randomly drawn each time and then placed back. This process is repeated 20 times. Let \( p \) be the probability that all 20 drawn numbers are different. Prove: \( p < \frac{1}{\mathrm{e}^{2}} \).

Answer: p<\frac{1}{\mathrm{e}^2}


A triangle $X Y Z$ and a circle $\omega$ of radius 2 are given in a plane, such that $\omega$ intersects segment $\overline{X Y}$ at the points $A, B$, segment $\overline{Y Z}$ at the points $C, D$, and segment $\overline{Z X}$ at the points $E, F$. Suppose that $X B>X A, Y D>Y C$, and $Z F>Z E$. In addition, $X A=1, Y C=2, Z E=3$, and $A B=C D=E F$. Compute $A B$.

Answer: \sqrt{10}-1


The sequence  $\left(a_n \right)$  is defined by  $a_1=1, \ a_2=2$  and  $$ a_{n+2} = 2a_{n+1}-pa_n, \ \forall n \ge 1, $$   for some prime  $p.$  Find all  $p$  for which there exists  $m$  such that  $a_m=-3.$

Answer: 7


Students were assigned to write several different three-digit numbers that do not contain the digit 7. How many such numbers can be written in total?

Answer: 648


In triangle \( ABC \) with \( AB = 8 \) and \( AC = 10 \), the incenter \( I \) is reflected across side \( AB \) to point \( X \) and across side \( AC \) to point \( Y \). Given that segment \( XY \) bisects \( AI \), compute \( BC^2 \). (The incenter \( I \) is the center of the inscribed circle of triangle \( ABC \).)

Answer: 84


In triangle \(ABC\), segment \(AD\) is a median, \(AD = m\), \(AB = a\), and \(AC = b\). Find \(\angle BAC\).

Answer: \arccos(\frac{4m^2-^2-b^2}{2ab})


Two brothers had tickets to a stadium located 10 km from their home. Initially, they planned to walk to the stadium. However, they changed their plan and decided to use a bicycle. They agreed that one would start on the bicycle and the other would walk simultaneously. After covering part of the distance, the first brother would leave the bicycle, and the second brother would ride the bicycle after reaching it, continuing until he caught up with the first brother at the entrance of the stadium. How much time do the brothers save compared to their initial plan to walk the entire way, given that each brother covers each kilometer 12 minutes faster on the bicycle than on foot?

Answer: 60


What is the maximum number of rooks that can be placed on a $300 \times 300$ chessboard such that each rook attacks at most one other rook? (A rook attacks all the squares it can reach according to chess rules without passing through other pieces.)

Answer: 400


Find all positive integer solutions \( x, y, z \) of the equation \( 3^{x} + 4^{y} = 5^{z} \).

Answer: 2,2,2


Define a cubic sequence:
\[
a_{n}=n^{3}+b n^{2}+c n+d \text {, }
\]
where \( b \), \( c \), and \( d \) are integers and constants, and \( n \) takes all integer values.
(1) Prove that there exists a cubic sequence such that only \( a_{2015} \) and \( a_{2016} \) are perfect squares in this sequence.
(2) If a cubic sequence satisfies condition (1), find all possible values of \( a_{2015} \cdot a_{2016} \).

Answer: 0


The number 770 is written on a blackboard. Melody repeatedly performs moves, where a move consists of subtracting either 40 or 41 from the number on the board. She performs moves until the number is not positive, and then she stops. Let $N$ be the number of sequences of moves that Melody could perform. Suppose $N=a \cdot 2^{b}$ where $a$ is an odd positive integer and $b$ is a nonnegative integer. Compute $100 a+b$.

Answer: 318


All $20$ diagonals are drawn in a regular octagon. At how many distinct points in the interior of the octagon (not on the boundary) do two or more diagonals intersect?

Answer: 70


The complete graph of $y=f(x)$, which consists of five line segments, is shown in red below. (On this graph, the distance between grid lines is $1$.)

What is the sum of the $x$-coordinates of all points where $f(x) = 1.8$?

[asy]

size(150);

real ticklen=3;

real tickspace=2;

real ticklength=0.1cm;

real axisarrowsize=0.14cm;

pen axispen=black+1.3bp;

real vectorarrowsize=0.2cm;

real tickdown=-0.5;

real tickdownlength=-0.15inch;

real tickdownbase=0.3;

real wholetickdown=tickdown;

void rr_cartesian_axes(real xleft, real xright, real ybottom, real ytop, real xstep=1, real ystep=1, bool useticks=false, bool complexplane=false, bool usegrid=true) {

import graph;

real i;

if(complexplane) {

label("$\textnormal{Re}$",(xright,0),SE);

label("$\textnormal{Im}$",(0,ytop),NW);

} else {

label("$x$",(xright+0.4,-0.5));

label("$y$",(-0.5,ytop+0.2));

}

ylimits(ybottom,ytop);

xlimits( xleft, xright);

real[] TicksArrx,TicksArry;

for(i=xleft+xstep; i<xright; i+=xstep) {

if(abs(i) >0.1) {

TicksArrx.push(i);

}

}

for(i=ybottom+ystep; i<ytop; i+=ystep) {

if(abs(i) >0.1) {

TicksArry.push(i);

}

}

if(usegrid) {

xaxis(BottomTop(extend=false), Ticks("%", TicksArrx ,pTick=gray(0.22),extend=true),p=invisible);//,above=true);

yaxis(LeftRight(extend=false),Ticks("%", TicksArry ,pTick=gray(0.22),extend=true), p=invisible);//,Arrows);

}

if(useticks) {

xequals(0, ymin=ybottom, ymax=ytop, p=axispen, Ticks("%",TicksArry , pTick=black+0.8bp,Size=ticklength), above=true, Arrows(size=axisarrowsize));

yequals(0, xmin=xleft, xmax=xright, p=axispen, Ticks("%",TicksArrx , pTick=black+0.8bp,Size=ticklength), above=true, Arrows(size=axisarrowsize));

} else {

xequals(0, ymin=ybottom, ymax=ytop, p=axispen, above=true, Arrows(size=axisarrowsize));

yequals(0, xmin=xleft, xmax=xright, p=axispen, above=true, Arrows(size=axisarrowsize));

}

};

rr_cartesian_axes(-5,5,-5,5);

draw((-4,-5)--(-2,-1)--(-1,-2)--(1,2)--(2,1)--(4,5),red);

[/asy]

Answer: 4.5


If $\triangle ABC$ satisfies that $\cot A, \cot B, \cot C$ form an arithmetic sequence, then the maximum value of $\angle B$ is ______.

Answer: \frac{\pi}{3}


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

Tanya performed such an operation five times, and the number 100 appeared on the screen. What is the largest possible initial number for which this could happen?

Answer: 89


In a box, there are \( n \) white and \( n \) black balls. Pairs of balls are drawn randomly from the box two at a time.

a) Find the expected number of mixed-color pairs that are drawn from the box by the time the box is empty.

b) Suppose that if a pair is mixed-color, it is set aside, and if it is same-color, it is returned to the box. We call this operation an "attempt." Find the expected number of attempts needed such that all balls are set aside and the box is empty.

Answer: 2n-H_n


Given a prime number \( p \). All natural numbers from 1 to \( p \) are written in a sequence in ascending order. Find all \( p \) for which this sequence can be divided into several consecutive blocks such that the sums of the numbers in all blocks are identical.

Answer: 3


Some people know each other in a group of people, where "knowing" is a symmetric relation. For a person, we say that it is  $social$  if it knows at least  $20$  other persons and at least  $2$  of those  $20$  know each other. For a person, we say that it is  $shy$  if it doesn't know at least  $20$  other persons and at least  $2$  of those  $20$  don't know each other. Find the maximal number of people in that group, if we know that group doesn't have any  $social$  nor  $shy$  persons.

Answer: 40


Five runners, $P$, $Q$, $R$, $S$, $T$, have a race, and $P$ beats $Q$, $P$ beats $R$, $Q$ beats $S$, and $T$ finishes after $P$ and before $Q$.  Who could NOT have finished third in the race?
$\text{(A)}\ P\text{ and }Q \qquad \text{(B)}\ P\text{ and }R \qquad \text{(C)}\ P\text{ and }S \qquad \text{(D)}\ P\text{ and }T \qquad \text{(E)}\ P,S\text{ and }T$

Answer: (C)\P\\S


Misha has accepted a job in the mines and will produce one ore each day. At the market, he is able to buy or sell one ore for \ $3, buy or sell bundles of three wheat for \$ 12 each, or  $\textit{sell}$  one wheat for one ore. His ultimate goal is to build a city, which requires three ore and two wheat. How many dollars must Misha begin with in order to build a city after three days of working?

Answer: 9


What is the smallest positive integer representable as the sum of the cubes of three positive integers in two different ways?

Answer: 251


There are 130 trees planted in a circle: birches and linden trees (both types are present). Each tree has a sign saying: "Two different trees are growing next to this tree." It is known that this statement is false for all linden trees and for exactly one birch tree. How many birches could have been planted? Indicate all possible options.

Answer: 87


Four distinct points are arranged on a plane such that they have segments connecting them with lengths $a$, $a$, $a$, $b$, $b$, and $2a$. Determine the ratio $\frac{b}{a}$ assuming the formation of a non-degenerate triangle with one of the side lengths being $2a$.

Answer: \sqrt{2}


Given the obtuse angle $\alpha$ that satisfies the equation $$\frac {sin\alpha-3cos\alpha}{cos\alpha -sin\alpha }=tan2\alpha$$, find the value of $tan\alpha$.

Answer: 2 - \sqrt{7}


Find the sum of the values of $x$ such that $\cos^3 3x+ \cos^3 5x = 8 \cos^3 4x \cos^3 x$, where $x$ is measured in degrees and $100< x< 200.$

Answer: 906


Given that the sequence $\left\{a_{n}\right\}$ has a period of 7 and the sequence $\left\{b_{n}\right\}$ has a period of 13, determine the maximum value of $k$ such that there exist $k$ consecutive terms satisfying
\[ a_{1} = b_{1}, \; a_{2} = b_{2}, \; \cdots , \; a_{k} = b_{k} \]

Answer: 91


A rectangle with a length of 6 cm and a width of 4 cm is rotated around one of its sides. Find the volume of the resulting geometric solid (express the answer in terms of $\pi$).

Answer: 144\pi


Let the sequence $(a_n)$ be defined as $a_1=\frac{1}{3}$ and $a_{n+1}=\frac{a_n}{\sqrt{1+13a_n^2}}$ for every $n\geq 1$. If $a_k$ is the largest term of the sequence satisfying $a_k < \frac{1}{50}$, find the value of $k$.

Answer: 193


Given a set $I=\{1,2,3,4,5\}$, select two non-empty subsets $A$ and $B$ such that the largest number in set $A$ is less than the smallest number in set $B$. The total number of different selection methods is $\_\_\_\_\_\_$.

Answer: 49


Let  $n$  be a positive integer. For each  $4n$ -tuple of nonnegative real numbers  $a_1,\ldots,a_{2n}$ ,  $b_1,\ldots,b_{2n}$  that satisfy  $\sum_{i=1}^{2n}a_i=\sum_{j=1}^{2n}b_j=n$ , define the sets
\[A:=\left\{\sum_{j=1}^{2n}\frac{a_ib_j}{a_ib_j+1}:i\in\{1,\ldots,2n\} \textup{ s.t. }\sum_{j=1}^{2n}\frac{a_ib_j}{a_ib_j+1}\neq 0\right\},\]
\[B:=\left\{\sum_{i=1}^{2n}\frac{a_ib_j}{a_ib_j+1}:j\in\{1,\ldots,2n\} \textup{ s.t. }\sum_{i=1}^{2n}\frac{a_ib_j}{a_ib_j+1}\neq 0\right\}.\]
Let  $m$  be the minimum element of  $A\cup B$ . Determine the maximum value of  $m$  among those derived from all such  $4n$ -tuples  $a_1,\ldots,a_{2n},b_1,\ldots,b_{2n}$ .

*Proposed by usjl.*

Answer: \frac{n}{2}


Find all the solutions to
\[\frac{1}{x^2 + 11x - 8} + \frac{1}{x^2 + 2x - 8} + \frac{1}{x^2 - 13x - 8} = 0.\]Enter all the solutions, separated by commas.

Answer: 8,1,-1,-8


Two people, A and B, are playing table tennis. The probability that A wins a game is \(\frac{2}{3}\). The match ends when one of the players is ahead by two games. The probability that the total number of games is \(x_i \left(i \in \mathbf{N}_{+}\right)\) is \(p_i\), where \(x_i < x_{i+1} \left(i \in \mathbf{N}_{+}\right)\). Find \(S=\sum_{i=1}^{+\infty} x_{i} p_{i}\).

Answer: \frac{18}{5}


Derek fills a square  $10$  by  $10$  grid with  $50$   $1$ s and  $50$   $2$ s. He takes the product of the numbers in each of the  $10$  rows. He takes the product of the numbers in each of the  $10$  columns. He then sums these  $20$  products up to get an integer  $N.$  Find the minimum possible value of  $N.$

Answer: 640


A straight one-way city street has 8 consecutive traffic lights. Every light remains green for 1.5 minutes, yellow for 3 seconds, and red for 1.5 minutes. The lights are synchronized so that each light turns red 10 seconds after the preceding one turns red. What is the longest interval of time, in seconds, during which all 8 lights are green?

Answer: 20


Determine all real numbers  $a$  for which the system of equations
\begin{align*}
3x^2+2y^2+2z^2&=a
4x^2+4y^2+5z^2&=1-a
\end{align*}
has at least one solution  $(x,y,z)$  in the real numbers.

Answer: [\frac{2}{7},\frac{3}{7}]


Compute the sum of all positive integers \( n \) for which
$$
9 \sqrt{n} + 4 \sqrt{n+2} - 3 \sqrt{n+16}
$$
is an integer.

Answer: 18


On a coordinate plane, mark all the points $(x, y)$ such that $x$ and $y$ are integers satisfying the inequalities $0 \leqslant x \leqslant 2$ and $0 \leqslant y \leqslant 26$. How many lines exist that pass through exactly 3 of the marked points?

Answer: 365


A stationery store sells a certain type of pen bag for $18$ yuan each. Xiao Hua went to buy this pen bag. When checking out, the clerk said, "If you buy one more, you can get a 10% discount, which is $36 cheaper than now." Xiao Hua said, "Then I'll buy one more, thank you." According to the conversation between the two, Xiao Hua actually paid ____ yuan at checkout.

Answer: 486


Form a four-digit number using the digits 0, 1, 2, 3, 4, 5 without repetition.
(I) How many different four-digit numbers can be formed?
(II) How many of these four-digit numbers have a tens digit that is larger than both the units digit and the hundreds digit?
(III) Arrange the four-digit numbers from part (I) in ascending order. What is the 85th number in this sequence?

Answer: 2301


Suppose three boba drinks and four burgers cost  $28$  dollars, while two boba drinks and six burgers cost  $\$  37.70$. If you paid for one boba drink using only pennies, nickels, dimes, and quarters, determine the least number of coins you could use.

Answer: 10


Let $p(x) = x^{2008} + x^{2007} + x^{2006} + \cdots + x + 1,$
and let $r(x)$ be the polynomial remainder when $p(x)$ is divided by $x^4+x^3+2x^2+x+1$. Find the remainder when $|r(2008)|$ is divided by $1000$.

Answer: 64


Given the set \( A = \{1, 2, \cdots, 6\} \), the mapping \( f: A \rightarrow A \) is a bijection,
$$
\begin{array}{l}
f^{(2)}(x) = f(f(x)), \\
f^{(n+1)}(x) = f\left(f^{(n)}(x)\right) \quad (n \geqslant 1).
\end{array}
$$

The number of mappings \( f \) that satisfy \( f^{(12)}(x) = x \) for any \( x \in A \) is .

Answer: 576


Consider the assertion that for each positive integer $n \ge 2$, the remainder upon dividing $2^{2^n}$ by $2^n-1$ is a power of 4. Either prove the assertion or find (with proof) a counter-example.

Answer: 25


**U**ndecillion years ago in a galaxy far, far away, there were four space stations in the three-dimensional space, each pair spaced 1 light year away from each other. Admiral Ackbar wanted to establish a base somewhere in space such that the sum of squares of the distances from the base to each of the stations does not exceed 15 square light years. (The sizes of the space stations and the base are negligible.) Determine the volume, in cubic light years, of the set of all possible locations for the Admiral’s base.

Answer: \frac{27\sqrt{6}\pi}{8}


In a plane, equilateral triangle \(ABC\), square \(BCDE\), and regular dodecagon \(DEFGHIJKLMNOP\) each have side length 1 and do not overlap. Find the area of the circumcircle of \(\triangle AFN\).

Answer: (2+\sqrt{3})\pi


Find the derivative.

\[ y = x \cdot \cos \alpha + \sin \alpha \cdot \ln \sin (x - \alpha) \]

Answer: \frac{\sinx}{\sin(x-\alpha)}


Let  $ABCDE$  be a convex pentagon such that  $AB = BC = CD$  and  $\angle BDE =
\angle EAC = 30 ^{\circ}$ . Find the possible values of  $\angle BEC$ .

*Proposed by Josef Tkadlec (Czech Republic)*

Answer: 60


Let \(\omega\) and \(\Gamma\) be circles such that \(\omega\) is internally tangent to \(\Gamma\) at a point \(P\). Let \(AB\) be a chord of \(\Gamma\) tangent to \(\omega\) at a point \(Q\). Let \(R \neq P\) be the second intersection of line \(PQ\) with \(\Gamma\). If the radius of \(\Gamma\) is 17, the radius of \(\omega\) is 7, and \(\frac{AQ}{BQ} = 3\), find the circumradius of triangle \(AQR\).

Answer: \sqrt{170}