Given that \( a, b, \) and \( c \) are all positive integers, and the parabola \( y = ax^2 + bx + c \) intersects the x-axis at two distinct points \( A \) and \( B \). If both \( A \) and \( B \) are less than 1 unit away from the origin, what is the minimum value of \( a + b + c \)?

Answer: 11


Given the parametric equations of curve $C_1$ are $$\begin{cases} x=2\cos\theta \\ y=\sin\theta\end{cases}(\theta \text{ is the parameter}),$$ and the parametric equations of curve $C_2$ are $$\begin{cases} x=-3+t \\ y= \frac {3+3t}{4}\end{cases}(t \text{ is the parameter}).$$
(1) Convert the parametric equations of curves $C_1$ and $C_2$ into standard equations;
(2) Find the maximum and minimum distances from a point on curve $C_1$ to curve $C_2$.

Answer: \frac {12-2 \sqrt {13}}{5}


Neznaika does not know about multiplication and exponentiation operations. However, he is good at addition, subtraction, division, and square root extraction, and he knows how to use parentheses. While practicing, Neznaika chose three numbers 20, 2, and 2, and formed the expression:

$$
\sqrt{(2+20): 2} .
$$

Can he use the same three numbers 20, 2, and 2 to form an expression whose value is greater than 30?

Answer: 20 + 10\sqrt{2}


$\triangle ABC$ ,  $\angle A=23^{\circ},\angle B=46^{\circ}$ . Let  $\Gamma$  be a circle with center  $C$ , radius  $AC$ . Let the external angle bisector of  $\angle B$  intersects  $\Gamma$  at  $M,N$ . Find  $\angle MAN$ .

Answer: 60


Squares \(ABRS\), \(BCPQ\), and \(CAMN\) are constructed on the sides \(AB\), \(BC\), and \(CA\) of an equilateral triangle \(ABC\) with side length 1, respectively. Identical pyramids with equal edge lengths are then constructed on these squares. These pyramids are rotated around their base edges \(AB\), \(BC\), and \(CA\), such that their apexes coincide. What is the edge length of the pyramids if, in this new position, points \(M\), \(N\), \(P\), \(Q\), \(R\), and \(S\) form the vertices of a regular hexagon?

Answer: 1


Let \( x_{1}, x_{2}, x_{3} \) be the three roots of the equation \( x^{3}-17 x-18=0 \), with \( -4 < x_{1} < -3 \) and \( 4 < x_{3} < 5 \).
(1) Find the integer part of \( x_{2} \).
(2) Find the value of \( \arctan x_{1} + \arctan x_{2} + \arctan x_{3} \).

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


Let \( x_0 > x_1 > \cdots > x_n \) be real numbers. Show that \( x_0 + \frac{1}{x_0 - x_1} + \frac{1}{x_1 - x_2} + \cdots + \frac{1}{x_{n-1} - x_n} \geq x_n + 2n \).

Answer: x_{0}+\frac{1}{x_{0}-x_{1}}+\frac{1}{x_{1}-x_{2}}+\cdots+\frac{1}{x_{n-1}-x_{n}}\gex_{n}+2n


Determine all functions \( f: \mathbb{R} \rightarrow \mathbb{R} \) for which the following conditions apply:

(a) \( f(x-1-f(x)) = f(x) - 1 - x \) for all \( x \in \mathbb{R} \),

(b) The set \( \left\{ \frac{f(x)}{x} \mid x \in \mathbb{R}, x \neq 0 \right\} \) is finite.

Answer: f(x)=x


At most, how many interior angles greater than $180^\circ$ can a 2006-sided polygon have?

Answer: 2003


Given triangle \( \triangle ABC \) and an arbitrary point \( O \) within it, prove:
1. \( \frac{AC'}{A'B} \cdot \frac{BA'}{A'C} \cdot \frac{CB'}{B'A} = 1 \)
2. \( \frac{AO}{AA'} + \frac{BO}{BB'} + \frac{CO}{CC'} = 2 \)

Answer: 2


In the foggy city of London, there are exactly $10^{10}$ clubs, and each club has exactly 10 gentlemen. An enemy spy wants to kidnap some gentlemen in such a way that there is at least one member from each club among the kidnapped. It turns out that for any two clubs, there is a gentleman who is a member of both clubs. Prove that the spy only needs to kidnap 9 gentlemen.

Answer: 9


How many roots does $\arctan x=x^{2}-1.6$ have, where the arctan function is defined in the range $-\frac{p i}{2}<\arctan x<\frac{p i}{2}$ ?

Answer: 2


There are 158 children standing in a line. Starting from the first child on the left, every other child receives an apple (the first child receives an apple). Starting from the first child on the right, every third child receives a banana (the first child receives a banana). Find the number of children who do not receive any fruit.

Answer: 52


In a class of 120 students, the teacher recorded the following scores for an exam. Calculate the average score for the class.

\begin{tabular}{|c|c|}
\multicolumn{2}{c}{}\\\hline
\textbf{Score (\%)}&\textbf{Number of Students}\\\hline
95&12\\\hline
85&24\\\hline
75&30\\\hline
65&20\\\hline
55&18\\\hline
45&10\\\hline
35&6\\\hline
\end{tabular}

Answer: 69.83


Let $f : \mathbb{R} \to \mathbb{R}$ be a function such that
\[f(x^2 + yf(z)) = xf(x) + zf(y)\]for all real numbers $x,$ $y,$ and $z.$

Let $n$ be the number of possible values of $f(5),$ and let $s$ be the sum of all possible values of $f(5).$  Find $n \times s.$

Answer: 10


Let the function \( f^{\prime}(x) \) be the derivative of an even function \( f(x) \) with \( x \neq 0 \), and suppose \( f(-1)=0 \). Given that \( x f^{\prime}(x) - f(x) < 0 \) for \( x > 0 \), determine the range of \( x \) such that \( f(x) > 0 \).

Answer: (-1,0)\cup(0,1)


One day, Xiao Ming took 100 yuan to go shopping. In the first store, he bought several items of product A. In the second store, he bought several items of product B. In the third store, he bought several items of product C. In the fourth store, he bought several items of product D. In the fifth store, he bought several items of product E. In the sixth store, he bought several items of product F. The prices of the six products are all different integers, and Xiao Ming spent the same amount of money in all six stores. How much money does Xiao Ming have left?

Answer: 28


The mean of the numbers 3, 7, 10, and 15 is twice the mean of $x$, 20, and 6. What is the value of $x$?

Answer: -12.875


Given that the product of positive numbers \( a, b, c \) is \( a b c = 1 \), prove that \(\left(a-1+\frac{1}{b}\right)\left(b-1+\frac{1}{c}\right)\left(c-1+\frac{1}{a}\right) \leq 1\).

Answer: 1


A rectangle  $ABEF$  is drawn on the leg  $AB$  of a right triangle  $ABC$ , whose apex  $F$  is on the leg  $AC$ . Let  $X$  be the intersection of the diagonal of the rectangle  $AE$  and the hypotenuse  $BC$  of the triangle. In what ratio does point  $X$  divide the hypotenuse  $BC$  if it is known that  $| AC | = 3 | AB |$  and  $| AF | = 2 | AB |$ ?

Answer: 2:3


How many employees of the enterprise can be insured by the given insurance company if the enterprise is willing to pay an insurance premium of 5,000,000 rubles?

The cost of service for one person in the clinic (without hospitalization), with which the insurance company has a contract, averages 18,000 rubles. The average cost of treatment for one insured person in case of hospitalization is 60,000 rubles (the proportion of hospitalized individuals is 25% of the insured number). Overhead costs of the insurance company for insuring one person are on average 2,800 rubles. The profit of the insurance company is 15% of the cost price.

Answer: 121


Five soccer teams play a match where each team plays every other team exactly once. Each match awards 3 points to the winner, 0 points to the loser, and 1 point to each team in the event of a draw. After all matches have been played, the total points of the five teams are found to be five consecutive natural numbers. Let the teams ranked 1st, 2nd, 3rd, 4th, and 5th have drawn $A$, $B$, $C$, $D$, and $E$ matches respectively. Determine the five-digit number $\overline{\mathrm{ABCDE}}$.

Answer: 13213


Aunt Wang needs 3 minutes to cut a paper-cut for window decoration. After cutting each one, she rests for 1 minute. She starts cutting at 9:40. After cutting 10 paper-cuts, it is \_\_\_\_ hour \_\_\_\_ minute.

Answer: 10:19


Given the numbers 2, 3, 4, 3, 1, 6, 3, 7, determine the sum of the mean, median, and mode of these numbers.

Answer: 9.625


Find the least number of elements of a finite set  $A$  such that there exists a function  $f : \left\{1,2,3,\ldots \right\}\rightarrow A$  with the property: if  $i$  and  $j$  are positive integers and  $i-j$  is a prime number, then  $f(i)$  and  $f(j)$  are distinct elements of  $A$ .

Answer: 4


A line \( l \) passes through the focus \( F \) of the parabola \( y^{2}=4 p x \) (\( p>0 \)) and intersects the parabola at two points \( P \) and \( Q \). Perpendiculars \( PR \) and \( QS \) are drawn from points \( P \) and \( Q \) to the directrix, with feet \( R \) and \( S \), respectively. Let \( |PF|=a \) and \( |QF|=b \). \( M \) is the midpoint of \( RS \). Find the length of \( |MF| \).

Answer: \sqrt{}


Given a stalk of bamboo with nine sections, with three sections from the bottom holding 3.9 liters, and the four sections from the top holding three liters, determine the combined volume of the middle two sections.

Answer: 2.1


Given a function $f(x) = \cos x \sin \left( x + \frac{\pi}{3} \right) - \sqrt{3} \cos^2 x + \frac{\sqrt{3}}{4}$, where $x \in \mathbb{R}$,
(1) Find the smallest positive period of $f(x)$ and the interval where $f(x)$ is monotonically decreasing;
(2) Find the maximum and minimum values of $f(x)$ in the closed interval $\left[-\frac{\pi}{4}, \frac{\pi}{4}\right]$.

Answer: -\frac{1}{2}


In the triangle  $ABC$  on the sides  $AB$  and  $AC$  outward constructed equilateral triangles  $ABD$  and  $ACE$ . The segments  $CD$  and  $BE$  intersect at point  $F$ . It turns out that point  $A$  is the center of the circle inscribed in triangle  $ DEF$ . Find the angle  $BAC$ .

(Rozhkova Maria)

Answer: 60


Given the ellipse \(\frac{x^{2}}{5^{2}}+\frac{y^{2}}{4^{2}}=1\), a line passing through its left focus \( F_{1} \) intersects the ellipse at points \( A \) and \( B \). Point \( D(a, 0) \) is located to the right of \( F_{1} \). Lines \( A D \) and \( B D \) intersect the ellipse's left directrix at \( M \) and \( N \), respectively. If the circle with diameter \( M N \) passes exactly through \( F_{1} \), find the value of \( a \).

Answer: 5


If \( N \) is the smallest positive integer whose digits have a product of 1728, what is the sum of the digits of \( N \)?

Answer: 28


On the extension of the chord $KL$ of the circle with center $O$, a point $A$ is taken, and from it, tangents $AP$ and $AQ$ are drawn. Let $M$ be the midpoint of the segment $PQ$. Prove that $\angle M K O = \angle M L O$.

Answer: \angleMKO=\angleMLO


A light pulse starts at a corner of a reflective square. It bounces around inside the square, reflecting off of the square's perimeter $n$ times before ending in a different corner. The path of the light pulse, when traced, divides the square into exactly 2021 regions. Compute the smallest possible value of $n$.

Answer: 129


A younger brother leaves home and walks to the park at a speed of 4 kilometers per hour. Two hours later, the older brother leaves home and rides a bicycle at a speed of 20 kilometers per hour to catch up with the younger brother. How long will it take for the older brother to catch up with the younger brother?

Answer: 0.5


Given real numbers $b$ and $c$, and the function $f(x) = x^2 + bx + c$, the equation $f(f(x)) = 0$ has exactly three different real roots. Find the maximum value of the sum of the roots of $f(x)$.

Answer: 1/2


Find the fraction \(\frac{p}{q}\) with the smallest possible natural denominator for which \(\frac{1}{2014} < \frac{p}{q} < \frac{1}{2013}\). Enter the denominator of this fraction in the provided field.

Answer: 4027


How many integers $n$ satisfy the condition $100 < n < 200$ and the condition $n$ has the same remainder whether it is divided by $6$ or by $8$?

Answer: 25


We color some cells in  $10000 \times 10000$  square, such that every  $10 \times 10$  square and every  $1 \times 100$  line have at least one coloring cell. What minimum number of cells we should color ?

Answer: 10000


Is there a positive integer that can be represented both as the sum of 1990 consecutive integers and in exactly 1990 different ways as the sum of at least two consecutive positive integers?

Answer: Yes


In a finite sequence of real numbers, the sum of any 7 consecutive terms is negative and the sum of any 11 consecutive terms is positive. How many terms can such a sequence have at most?

Answer: 16


In a row with 120 seats, some of the seats are already occupied. If a new person arrives and must sit next to someone regardless of their choice of seat, what is the minimum number of people who were already seated?

Answer: 40


In each cell of a $4 \times 4$ grid, one of the two diagonals is drawn uniformly at random. Compute the probability that the resulting 32 triangular regions can be colored red and blue so that any two regions sharing an edge have different colors.

Answer: \frac{1}{512}


Given that Sia and Kira count sequentially, where Sia skips every fifth number, find the 45th number said in this modified counting game.

Answer: 54


Let \[S = 1 + \frac 18 + \frac{1\cdot 5}{8\cdot 16} + \frac{1\cdot 5\cdot 9}{8\cdot 16\cdot 24} + \cdots + \frac{1\cdot 5\cdot 9\cdots (4k+1)}{8\cdot 16\cdot 24\cdots(8k+8)} + \cdots.\] Find the positive integer  $n$  such that  $2^n < S^{2007} < 2^{n+1}$ .

Answer: 501


Find the angle at the apex of a regular quadrilateral pyramid if it is equal to the angle between a lateral edge and the base plane of the pyramid.

Answer: \arccos(\frac{\sqrt{5}-1}{2})


The bases of the trapezoid \(ABCD\) are related by the equation \(AD = 4 \cdot BC\), and the sum of the angles \(\angle A\) and \(\angle D\) is \(120^\circ\). Points \(M\) and \(N\) are chosen on the legs such that \(CN : ND = BM : MA = 1 : 2\). Perpendiculars erected at points \(M\) and \(N\) to the legs of the trapezoid intersect at point \(O\). Find \(AD\) if \(AO = 1\).

Answer: \sqrt{3}


For  $n\in\mathbb N$ , define  $$ a_n=\frac1{\binom n1}+\frac1{\binom n2}+\ldots+\frac1{\binom nn}. $$  (a) Prove that the sequence  $b_n=a_n^n$  is convergent and determine the limit.
(b) Show that  $\lim_{n\to\infty}b_n>\left(\frac32\right)^{\sqrt3+\sqrt2}$ .

Answer: e^2


Find all real values of $x$ that satisfy $\frac{1}{x+1} + \frac{3}{x+7} \ge \frac23.$ (Give your answer in interval notation.)

Answer: (-7, -4] \cup (-1, 2]


Find the inclination angle of the line $\sqrt {2}x+ \sqrt {6}y+1=0$.

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


The expansion of (ax- \frac {3}{4x}+ \frac {2}{3})(x- \frac {3}{x})^{6} is given, and the sum of its coefficients is 16. Determine the coefficient of the x^{3} term in this expansion.

Answer: \frac{117}{2}


The storage capacity of two reservoirs, A and B, changes over time. The relationship between the storage capacity of reservoir A (in hundred tons) and time $t$ (in hours) is: $f(t) = 2 + \sin t$, where $t \in [0, 12]$. The relationship between the storage capacity of reservoir B (in hundred tons) and time $t$ (in hours) is: $g(t) = 5 - |t - 6|$, where $t \in [0, 12]$. The question is: When do the combined storage capacities of reservoirs A and B reach their maximum value? And what is this maximum value?
(Reference data: $\sin 6 \approx -0.279$).

Answer: 6.721


At what angle to the x-axis is the tangent to the graph of the function \( g(x) = x^2 \ln x \) inclined at the point \( x_0 = 1 \)?

Answer: \frac{\pi}{4}


What is the greatest number of consecutive integers whose sum is $45?$

Answer: 90


Two equal cones with a common vertex touch each other and some plane $\alpha$. Let $l$ be the line along which the planes of the bases of the cones intersect. Find the angle between the line $l$ and the plane $\alpha$ if the height of each cone is 2 and the radius of the base is 1.

Answer: 60


Find all functions  $f:\mathbb{Z}_{>0}\mapsto\mathbb{Z}_{>0}$  such that   $$ xf(x)+(f(y))^2+2xf(y) $$
is perfect square for all positive integers  $x,y$ .

**This problem was proposed by me for the BMO 2017 and it was shortlisted. We then used it in our TST.

Answer: f(x)=x


Find the integer $n,$ $0 \le n \le 180,$ such that $\cos n^\circ = \cos 259^\circ.$

Answer: 101


The bisector of  $\angle BAC$  in  $\triangle ABC$  intersects  $BC$  in point  $L$ . The external bisector of  $\angle ACB$  intersects  $\overrightarrow{BA}$  in point  $K$ . If the length of  $AK$  is equal to the perimeter of  $\triangle ACL$ ,  $LB=1$ , and  $\angle ABC=36^\circ$ , find the length of  $AC$ .

Answer: 1


Given a polynomial \( p(x) \) of degree at most \( 2n \), and for every integer \( k \in [-n, n] \), \( |p(k)| \leq 1 \). Prove that for all \( x \in [-n, n] \), \( |p(x)| \leq 2^{2n} \).

Answer: 2^{2n}


Determine the maximum number  $ h$  satisfying the following condition: for every  $ a\in [0,h]$  and every polynomial  $ P(x)$  of degree 99 such that  $ P(0)\equal{}P(1)\equal{}0$ , there exist  $ x_1,x_2\in [0,1]$  such that  $ P(x_1)\equal{}P(x_2)$  and  $ x_2\minus{}x_1\equal{}a$ .

*Proposed by F. Petrov, D. Rostovsky, A. Khrabrov*

Answer: \frac{1}{50}


In the geometric sequence ${a_{n}}$, $a_{n}=9$, $a_{5}=243$, find the sum of the first 4 terms.

Answer: 120


Determine the smallest integer $B$ such that there exist several consecutive integers, including $B$, that add up to 2024.

Answer: -2023


Given an ellipse $$C: \frac {x^{2}}{a^{2}}+ \frac {y^{2}}{b^{2}}=1$$ and a hyperbola $$\frac {x^{2}}{4-v}+ \frac {y^{2}}{1-v}=1 (1<v<4)$$ share a common focus. A line $l$ passes through the right vertex B of the ellipse C and intersects the parabola $y^2=2x$ at points P and Q, with $OP \perpendicular OQ$.

(Ⅰ) Find the equation of the ellipse C;

(Ⅱ) On the ellipse C, is there a point R $(m, n)$ such that the line $l: mx+ny=1$ intersects the circle $O: x^2+y^2=1$ at two distinct points M and N, and the area of $\triangle OMN$ is maximized? If such a point exists, find the coordinates of point R and the corresponding area of $\triangle OMN$; if not, explain why.

Answer: \frac {1}{2}


A granite pedestal. When constructing a square foundation and a cubic pedestal for a monument, granite cubic blocks of size \(1 \times 1\) meter were used. The pedestal used exactly as many blocks as the square foundation upon which it stood. All the blocks were used whole and uncut.

Look at the picture and try to determine the total number of blocks used. The foundation has a thickness of one block.

Answer: 128


Given that real numbers \( x \) and \( y \) satisfy the equation \((x+2)^{2}+y^{2}=1\), what is the minimum value of \(\frac{y-1}{x-2}\)?

Answer: 0


Suppose a function  $f:\mathbb{R}\to\mathbb{R}$  satisfies  $|f(x+y)|\geqslant|f(x)+f(y)|$  for all real numbers  $x$  and  $y$ . Prove that equality always holds. Is the conclusion valid if the sign of the inequality is reversed?

Answer: |f(x+y)|=|f(x)+f(y)|


Find all functions $f:\mathbb{R}\to\mathbb{R}$ which satisfy the following equality for all $x,y\in\mathbb{R}$ \[f(x)f(y)-f(x-1)-f(y+1)=f(xy)+2x-2y-4.\][i]

Answer: f(x) = x^2 + 1


In a regular $n$-gon ($n \geqslant 3$), the midpoints of all sides and diagonals are marked. What is the maximum number of marked points that lie on the same circle?

Answer: n


Given the parametric equation of curve $C_{1}$ is $\begin{cases} x=-2+2\cos \theta \\ y=2\sin \theta \end{cases}$ (with $\theta$ as the parameter), and establishing a coordinate system with the origin as the pole and the positive half-axis of $x$ as the polar axis, the polar equation of curve $C_{2}$ is $\rho=4\sin \theta$.
(Ⅰ) Find the coordinates of the intersection points of curves $C_{1}$ and $C_{2}$.
(Ⅱ) Points $A$ and $B$ are on curves $C_{1}$ and $C_{2}$, respectively. When $|AB|$ is maximized, find the area of $\triangle OAB$ (where $O$ is the origin).

Answer: 2+2\sqrt{2}


Svetlana takes a triplet of numbers and transforms it by the rule: at each step, each number is replaced by the sum of the other two. What is the difference between the largest and smallest numbers in the triplet on the 1580th step of applying this rule, if the initial triplet of numbers was $\{80, 71, 20\}$? If the problem allows for multiple answers, list them all as a set.

Answer: 60


The sequence \(\left(x_{n}\right)\) is defined recursively by \(x_{0}=1, x_{1}=1\), and:

\[ x_{n+2}=\frac{1+x_{n+1}}{x_{n}} \]

for all \(n \geq 0\). Calculate \(x_{2007}\).

Answer: 2


Answer: 6


Answer: 1/2


Suppose that $x, y, z$ are three distinct prime numbers such that $x + y + z = 49$. Find the maximum possible value for the product $xyz$.

Answer: 3059


In front of the elevator doors, there are people with masses of 50, 51, 55, 57, 58, 59, 60, 63, 75, and 140 kg. The elevator's load capacity is 180 kg. What is the minimum number of trips needed to transport all the people?

Answer: 4


A cube with side length $2$ is sliced by a plane that passes through a vertex $A$, the midpoint $M$ of an adjacent edge, and the midpoint $P$ of the face diagonal of the top face, not containing vertex $A$. Find the area of the triangle $AMP$.

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


Let \( S \) be the set of positive integers such that for any \( a \) and \( b \) in the set, we have \( \operatorname{GCD}(a, b) > 1 \). Additionally, for any \( a, b, \) and \( c \) in the set, we have \( \operatorname{GCD}(a, b, c) = 1 \). Is it possible that \( S \) has 2012 elements?

\(\operatorname{GCD}(x, y)\) and \(\operatorname{GCD}(x, y, z)\) stand for the greatest common divisor of the numbers \( x \) and \( y \) and the numbers \( x, y, \) and \( z \), respectively.

Answer: Yes


Percival's castle had a square shape. One day Percival decided to expand his estate and added a square annex to the castle. As a result, the perimeter of the castle increased by $10\%$. By what percentage did the area of the castle increase?

Answer: 4


Using the data presented in the chart, what was the average daily high temperature in Brixton from September 15th, 2008 to September 21st, 2008, inclusive? The daily high temperatures in degrees Fahrenheit were recorded as 51, 64, 61, 59, 48, 63, and 55.

Answer: 57.3


A) For a sample of size $n$ taken from a normal population with a known standard deviation $\sigma$, the sample mean $\bar{x}$ is found. At a significance level $\alpha$, it is required to find the power function of the test of the null hypothesis $H_{0}: a=a_{0}$ regarding the population mean $a$ with the hypothetical value $a_{0}$, under the competing hypothesis $H_{1}: a=a_{1} \neq a_{0}$.

B) For a sample of size $n=16$ taken from a normal population with a known standard deviation $\sigma=5$, at a significance level of 0.05, the null hypothesis $H_{0}: a=a_{0}=20$ regarding the population mean $a$ with the hypothetical value $a_{0}=20$ is tested against the competing hypothesis $H_{1}: a \neq 20$. Calculate the power of the two-sided test for the hypothesized value of the population mean $a_{1}=24$.

Answer: 0.8925


Given points $A(-1,1)$, $B(1,2)$, $C(-2,-1)$, $D(2,2)$, the projection of vector $\overrightarrow{AB}$ in the direction of $\overrightarrow{CD}$ is ______.

Answer: \dfrac{11}{5}


If \( x \in \left[-\frac{5}{12} \pi, -\frac{\pi}{3}\right] \), what is the maximum value of \( y = \tan \left(x + \frac{2}{3} \pi\right) - \tan \left(x + \frac{\pi}{6}\right) + \cos \left(x + \frac{\pi}{6}\right) \)?

Answer: \frac{11\sqrt{3}}{6}


Leah has $13$ coins, all of which are pennies and nickels. If she had one more nickel than she has now, then she would have the same number of pennies and nickels. In cents, how much are Leah's coins worth?
$\textbf {(A) } 33 \qquad \textbf {(B) } 35 \qquad \textbf {(C) } 37 \qquad \textbf {(D) } 39 \qquad \textbf {(E) } 41$

Answer: 37cents\,(\textbf{C})


Pete liked the puzzle; he decided to glue it together and hang it on the wall. In one minute, he glued together two pieces (initial or previously glued). As a result, the entire puzzle was assembled into one complete picture in 2 hours. How much time would it take to assemble the picture if Pete glued three pieces together each minute instead of two?

Answer: 60


In a convex quadrilateral \(ABCD\), \(\angle BAC = \angle CAD\) and \(\angle ABC = \angle ACD\). The extensions of \(AD\) and \(BC\) meet at \(E\), and the extensions of \(AB\) and \(DC\) meet at \(F\). Determine the value of
\[
\frac{AB \cdot DE}{BC \cdot CE}.
\]

Answer: 1


Circles of radius 4 and 5 are externally tangent and are circumscribed by a third circle. Find the area of the shaded region. Express your answer in terms of $\pi$.

Answer: 40\pi


Consider seven different straight lines in the plane. A point is called good if it lies on at least three of these lines. Determine the largest possible number of good points.

Answer: 6


The sum of the numbers 300, 2020, and 10001 is

Answer: 12321


Answer: 110001_2


Let $x$ be the number of students in Danny's high school. If Maria's high school has $4$ times as many students as Danny's high school, then Maria's high school has $4x$ students. The difference between the number of students in the two high schools is $1800$, so we have the equation $4x-x=1800$.

Answer: 2400


In the Cartesian coordinate plane \( xOy \), \( \angle MON = \frac{\pi}{3} \), and the side \( MN \) moves along the line \( l: x = 3 \). Find the equation of the locus of the circumcenter of \( \triangle MON \).

Answer: \frac{(x-4)^2}{4}-\frac{y^2}{12}=1


Which of these numbers is less than its reciprocal?
$\text{(A)}\ -2 \qquad \text{(B)}\ -1 \qquad \text{(C)}\ 0 \qquad \text{(D)}\ 1 \qquad \text{(E)}\ 2$

Answer: (A)\-2


Given a square \(ABCD\) with side length \(a\), vertex \(A\) lies in plane \(\beta\), and the other vertices are on the same side of plane \(\beta\). The distances from points \(B\) and \(D\) to plane \(\beta\) are 1 and 2, respectively. If the dihedral angle between plane \(ABCD\) and plane \(\beta\) is 30 degrees, then \(a =\)?

Answer: 2\sqrt{5}


 $n_{1},\ldots ,n_{k}, a$  are integers that satisfies the above conditions A)For every  $i\neq j$ ,   $(n_{i}, n_{j})=1$  B)For every  $i, a^{n_{i}}\equiv 1 (mod n_{i})$  C)For every  $i, X^{a-1}\equiv 0(mod n_{i})$ .
Prove that  $a^{x}\equiv 1(mod x)$  congruence has at least  $2^{k+1}-2$  solutions. ( $x>1$ )

Answer: 2^{k+1}-2


Given a positive integer $ n$, for all positive integers $ a_1, a_2, \cdots, a_n$ that satisfy $ a_1 \equal{} 1$, $ a_{i \plus{} 1} \leq a_i \plus{} 1$, find $ \displaystyle \sum_{i \equal{} 1}^{n} a_1a_2 \cdots a_i$.

Answer: (2n-1)!!


The first four terms of a sequence are 1, 4, 2, and 3. Beginning with the fifth term in the sequence, each term is the sum of the previous four terms. Therefore, the fifth term is 10. What is the eighth term?
(A) 66
(B) 65
(C) 69
(D) 134
(E) 129

Answer: 66


Calculate the double integral

$$
\iint_{D} x \, dx \, dy
$$

where the region \( D \) is bounded by the lines

$$
y^{2} - 4y + x^{2} = 0, \quad y^{2} - 8y + x^{2} = 0, \quad y = \frac{x}{\sqrt{3}}, \quad x = 0
$$

Answer: 35


Given $w$ and $z$ are complex numbers such that $|w+z|=2$ and $|w^2+z^2|=10,$ find the smallest possible value of $|w^3+z^3|$.

Answer: 26


The angle bisector of the acute angle formed at the origin by the graphs of the lines $y = x$ and $y=3x$ has equation $y=kx.$ What is $k?$
$\textbf{(A)} \ \frac{1+\sqrt{5}}{2} \qquad \textbf{(B)} \ \frac{1+\sqrt{7}}{2} \qquad \textbf{(C)} \ \frac{2+\sqrt{3}}{2} \qquad \textbf{(D)} \ 2\qquad \textbf{(E)} \ \frac{2+\sqrt{5}}{2}$

Answer: \textbf{(A)}\\frac{1+\sqrt{5}}{2}


Let \( ABCD \) be a rectangle with \( AB = 3 \) and \( BC = 7 \). Let \( W \) be a point on segment \( AB \) such that \( AW = 1 \). Let \( X, Y, Z \) be points on segments \( BC, CD, DA \), respectively, so that quadrilateral \( WXYZ \) is a rectangle, and \( BX < XC \). Determine the length of segment \( BX \).

Answer: \frac{7-\sqrt{41}}{2}


Giraldo wrote five distinct natural numbers on the vertices of a pentagon. And next he wrote on each side of the pentagon the least common multiple of the numbers written of the two vertices who were on that side and noticed that the five numbers written on the sides were equal. What is the smallest number Giraldo could have written on the sides?

Answer: 30


Let \( S = \{8^k : k \text{ is an integer}, 0 \le k \le 3000\} \). Given that \( 8^{3000} \) has 2712 digits and that its first (leftmost) digit is 8, how many elements of \( S \) have 8 as their leftmost digit?

Answer: 153


Given a segment \( AB \) of length \( a \). Two circles of radius \( a \) with centers at points \( A \) and \( B \) are drawn, intersecting at point \( C \). An inscribed circle is drawn inside the curvilinear triangle \( ABC \), touching segment \( AB \) at point \( M \) and arc \( AC \) at point \( N \). Prove that the radius of the circle inscribed in the curvilinear triangle \( AMN \) is equal to \(\frac{27-12 \sqrt{2}}{98} a\).

Answer: \frac{27-12\sqrt{2}}{98}


At the 2013 Winnebago County Fair a vendor is offering a "fair special" on sandals. If you buy one pair of sandals at the regular price of $50, you get a second pair at a 40% discount, and a third pair at half the regular price. Javier took advantage of the "fair special" to buy three pairs of sandals. What percentage of the $150 regular price did he save?
$\textbf{(A)}\ 25 \qquad \textbf{(B)}\ 30 \qquad \textbf{(C)}\ 33 \qquad \textbf{(D)}\ 40 \qquad \textbf{(E)}\ 45$

Answer: \textbf{(B)}\30


Some middle school students in a city participated in a mathematics invitational competition, which consisted of 6 problems. It is known that each problem was solved by exactly 500 students, but for any two students, there is at least one problem that neither of them solved. What is the minimum number of middle school students who participated in this mathematics invitational competition?

Answer: 1000


Among $n$ new students, any group of 3 students has 2 students who know each other, and any group of 4 students has 2 students who do not know each other. Find the maximum value of $n$.

Answer: 8


In chess tournaments, each victory is worth 1 point, each draw is worth 0.5 points, and each loss is worth zero points. In the "Magistral Championship" of chess, only International Masters (IMs) and Grand Masters (GMs) participated. The number of GMs was ten times the number of IMs. Each player played only once against all other opponents, and thus, if \( n \) is the number of players, then there were \( \frac{n(n-1)}{2} \) games. The sum of the points of all GMs was 4.5 times the sum of all IMs' points. Therefore, the questions are:

a) How many International Masters participated in this competition?

b) How many Grand Masters participated in this championship?

c) How many games were played in the tournament?

Answer: 55


Given a linear function \( f(x) \), it is known that the distance between the intersection points of the graphs \( y=x^{2}+2 \) and \( y=f(x) \) is \(\sqrt{10}\), and the distance between the intersection points of the graphs \( y=x^{2}-1 \) and \( y=f(x)+1 \) is \(\sqrt{42}\). Find the distance between the intersection points of the graphs \( y=x^{2} \) and \( y=f(x)+1\).

Answer: \sqrt{34}


We call the polynomial  $P (x)$  simple if the coefficient of each of its members belongs to the set  $\{-1, 0, 1\}$ .
Let  $n$  be a positive integer,  $n> 1$ . Find the smallest possible number of terms with a non-zero coefficient in a simple  $n$ -th degree polynomial with all values at integer places are divisible by  $n$ .

Answer: 2


In a magical country, there are only two types of people, Type A people who always tell the truth, and Type B people who always lie. One day, 2014 citizens of this country lined up in a row, and each person said: "There are more Type B people behind me than there are Type A people in front of me." How many Type A people are there among these 2014 citizens?

Answer: 1007


On the sides $BC, CA, AB$ of an equilateral triangle $ABC$ with a side length of 7, points $A_1, B_1, C_1$ are chosen respectively. It is known that $AC_1 = BA_1 = CB_1 = 3$. Find the ratio of the area of triangle $ABC$ to the area of the triangle formed by the lines $AA_1, BB_1, CC_1$.

Answer: 37


\( p(x) \) is a polynomial of degree \( n \) with leading coefficient \( c \), and \( q(x) \) is a polynomial of degree \( m \) with leading coefficient \( c \), such that \( p(x)^2 = (x^2 - 1)q(x)^2 + 1 \). Show that \( p'(x) = nq(x) \).

Answer: p'(x)=nq(x)


Semicircles of diameter 4 inches are aligned in a linear pattern, with a second row staggered under the first such that the flat edges of the semicircles in the second row touch the midpoints of the arcs in the first row. What is the area, in square inches, of the shaded region in an 18-inch length of this pattern? Express your answer in terms of $\pi$.

Answer: 16\pi


Calculate the number of terms in the simplified expression of \[(x+y+z)^{2020} + (x-y-z)^{2020},\] by expanding it and combining like terms.

Answer: 1,022,121


Each of the three cutlets needs to be fried on a pan on both sides for five minutes each side. The pan can hold only two cutlets at a time. Is it possible to fry all three cutlets in less than 20 minutes (ignoring the time for flipping and transferring the cutlets)?

Answer: 15


Solve the following system of equations. It has a solution if and only if each term equals zero:

$$
\left\{\begin{array}{c}
3 x^{2}+8 x-3=0 \\
3 x^{4}+2 x^{3}-10 x^{2}+30 x-9=0
\end{array}\right.
$$

Answer: -3


Given sets \( A, B, C \) (not necessarily distinct) such that \( A \cup B \cup C = \{1, 2, 3, \ldots, 11\} \), find the number of ordered pairs \((A, B, C)\) that satisfy this condition.

Answer: 7^{11}


Each of five, standard, six-sided dice is rolled once. What is the probability that there is at least one pair but not a three-of-a-kind (that is, there are two dice showing the same value, but no three dice show the same value)?

Answer: \frac{25}{36}


The sequence \(\left\{B_{n}\right\}\) is defined as
\[ B_{n} = \cot \left(\operatorname{arccot} A_{n} + \operatorname{arccot} A_{n+1} + \operatorname{arccot} A_{n+2}\right), \]
where the sequence \(\left\{A_{n}\right\}\) satisfies \(A_{1} = 1\) and \(A_{n+1} = A_{n} + 2n\).

Prove that:
\[ \sum_{k=1}^{n} B_{k} = \frac{n \left(n^{2} + 3n - 1\right)}{9}. \]

Answer: \frac{n(n^2+3n-1)}{9}


Evaluate the infinite sum
$$
\sum_{n=2}^{\infty} \log_{2}\left(\frac{1-\frac{1}{n}}{1-\frac{1}{n+1}}\right).
$$

Answer: -1


Find the non-negative integer-valued functions $f$ defined on the non-negative integers that have the following two properties:

(i) $\quad f(1)>0$;

(ii) $\quad f\left(m^{2}+n^{2}\right)=f^{2}(m)+f^{2}(n)$

for any non-negative integers $m$ and $n$.

Answer: f(n)=n


We color a $98 \times 98$ checkerboard in the style of a chessboard. A move consists of selecting a rectangle made up of small squares and inverting their colors. What is the minimum number of moves required to make the entire checkerboard monochromatic?

Answer: 98


There are 15 stones placed in a line. In how many ways can you mark 5 of these stones so that there are an odd number of stones between any two of the stones you marked?

Answer: 77


Three cones with apex $A$ touch each other externally, with the first two cones being identical and the vertex angle of the third cone equal to $2 \arcsin \frac{1}{4}$. Each of the cones touches internally a fourth cone with apex at point $A$. Find the vertex angle of the first two cones if it is half that of the fourth cone. (The vertex angle of a cone is the angle between its generatrices in an axial section.)

Answer: \frac{\pi}{6}+\arcsin(\frac{1}{4})


Answer: 6.694


$AB$ and $AC$ are tangents to a circle, $\angle BAC = 60^{\circ}$, the length of the broken line $BAC$ is 1. Find the distance between the points of tangency $B$ and $C$.

Answer: \frac{1}{2}


A magician has one hundred cards numbered 1 to 100. He puts them into three boxes, a red one, a white one and a blue one, so that each box contains at least one card. A member of the audience draws two cards from two different boxes and announces the sum of numbers on those cards. Given this information, the magician locates the box from which no card has been drawn.

How many ways are there to put the cards in the three boxes so that the trick works?

Answer: 12


In tetrahedron $S\-(ABC)$, $SA$ is perpendicular to plane $ABC$, $\angle BAC=120^{\circ}$, $SA=AC=2$, $AB=1$, find the surface area of the circumscribed sphere of the tetrahedron.

Answer: \frac{40\pi}{3}


Start with an angle of \(60^\circ\) and bisect it, then bisect the lower \(30^\circ\) angle, then the upper \(15^\circ\) angle, and so on, always alternating between the upper and lower of the previous two angles constructed. This process approaches a limiting line that divides the original \(60^\circ\) angle into two angles. Find the measure (degrees) of the smaller angle.

Answer: 20


The function $f(n)$ is defined on the positive integers such that $f(f(n)) = 2n$ and $f(4n + 1) = 4n + 3$ for all positive integers $n.$  Find $f(1000).$

Answer: 1016


Let $f(n)$ be the integer closest to $\sqrt[4]{n}.$ Find $\sum_{k=1}^{1995}\frac 1{f(k)}.$

Answer: 400


Find all  $t\in \mathbb Z$  such that: exists a function  $f:\mathbb Z^+\to \mathbb Z$  such that: $f(1997)=1998$  $\forall x,y\in \mathbb Z^+ , \text{gcd}(x,y)=d : f(xy)=f(x)+f(y)+tf(d):P(x,y)$

Answer: -1


Let  $f(n)$  denote the largest odd factor of  $n$ , including possibly  $n$ . Determine the value of
\[\frac{f(1)}{1} + \frac{f(2)}{2} + \frac{f(3)}{3} + \cdots + \frac{f(2048)}{2048},\]
rounded to the nearest integer.

Answer: 1365


The lengths of the six edges of a tetrahedron \(ABCD\) are 7, 13, 18, 27, 36, and 41, respectively. If \(AB = 41\), what is the length of \(CD\)?

Answer: 13


Anička has saved 290 coins in her piggy bank, consisting of one-crown coins and two-crown coins. When she uses one-fourth of all her two-crown coins, she assembles the same amount as when she uses one-third of all her one-crown coins.

What is the total amount Anička has saved?

Answer: 406


Dave arrives at an airport which has twelve gates arranged in a straight line with exactly $100$ feet between adjacent gates. His departure gate is assigned at random. After waiting at that gate, Dave is told the departure gate has been changed to a different gate, again at random. Let the probability that Dave walks $400$ feet or less to the new gate be a fraction $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Answer: 52


Answer: 3003


On the radius \(A O\) of a circle with center \(O\), a point \(M\) is chosen. Points \(B\) and \(C\) are chosen on the circle on one side of \(A O\) such that \(\angle A M B = \angle O M C = \alpha\). Find the length of \(B C\) if the radius of the circle is 12 and \(\cos \alpha = \frac{3}{4}\).

Answer: 18


A standard six-sided fair die is rolled four times. The probability that the product of all four numbers rolled is a perfect square is $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Answer: 187


The slopes of lines $l_1$ and $l_2$ are the two roots of the equation $6x^2+x-1=0$, respectively. The angle between lines $l_1$ and $l_2$ is __________.

Answer: \frac{\pi}{4}


Three of the edges of a cube are $\overline{AB}, \overline{BC},$ and $\overline{CD},$ and $\overline{AD}$ is an interior diagonal. Points $P, Q,$ and $R$ are on $\overline{AB}, \overline{BC},$ and $\overline{CD},$ respectively, so that $AP = 5, PB = 15, BQ = 15,$ and $CR = 10.$ What is the area of the polygon that is the intersection of plane $PQR$ and the cube?

Answer: 525


Let the product of the digits of a positive integer \( n \) be denoted by \( a(n) \). Find the positive integer solution to the equation \( n^2 - 17n + 56 = a(n) \).

Answer: 4


A rectangle,  $HOMF$ , has sides  $HO=11$  and  $OM=5$ . A triangle  $\Delta ABC$  has  $H$  as orthocentre,   $O$  as circumcentre,  $M$  be the midpoint of  $BC$ ,  $F$  is the feet of altitude from  $A$ . What is the length of  $BC$  ?

[asy]
unitsize(0.3 cm);
pair F, H, M, O;
F = (0,0);
H = (0,5);
O = (11,5);
M = (11,0);
draw(H--O--M--F--cycle);
label(" $F$ ", F, SW);
label(" $H$ ", H, NW);
label(" $M$ ", M, SE);
label(" $O$ ", O, NE);
[/asy]

Answer: 28


Given a cube with an edge length of 6, there is a regular tetrahedron with an edge length of \( x \) that can be rotated freely within the cube. Find the maximum value of \( x \).

Answer: 6\sqrt{2}


A circle is divided into six equal sections. Each section is to be coloured with a single colour so that three sections are red, one is blue, one is green, and one is yellow. Two circles have the same colouring if one can be rotated to match the other. How many different colourings are there for the circle?

Answer: 20


There are two rows of seats, with 6 seats in the front row and 7 seats in the back row. Arrange seating for 2 people in such a way that these 2 people cannot sit next to each other. Determine the number of different seating arrangements.

Answer: 134


On the side AB of triangle ABC with a $100^{\circ}$ angle at vertex C, points P and Q are taken such that $AP = BC$ and $BQ = AC$. Let M, N, and K be the midpoints of segments AB, CP, and CQ respectively. Find the angle $NMK$.

Answer: 40


Find the maximum value of the expression
\[
(\sqrt{3-\sqrt{2}} \sin x-\sqrt{2(1+\cos 2 x)}-1) \cdot(3+2 \sqrt{7-\sqrt{2}} \cos y-\cos 2 y)
\]
If the answer is not an integer, round it to the nearest whole number.

Answer: 9


Answer: 1


A circle on a plane divides the plane into 2 parts. A circle and a line can divide the plane into a maximum of 4 parts. A circle and 2 lines can divide the plane into a maximum of 8 parts. A circle and 5 lines can divide the plane into a maximum of how many parts?

Answer: 26


In a grain storage facility, the following are the amounts of grain (in tons) that were received or dispatched over a period of 6 days (where "+" indicates received and "-" indicates dispatched):
+26, -32, -15, +34, -38, -20.
(1) After these 6 days, did the amount of grain in the storage increase or decrease? By how much?
(2) After these 6 days, when the manager did the settlement, it was found that there were still 480 tons of grain in storage. How much grain was there in the storage 6 days ago?
(3) If the loading and unloading fee is 5 yuan per ton, how much would the loading and unloading fees be for these 6 days?

Answer: 825


One hundred friends, including Petya and Vasya, live in several cities. Petya found out the distance from his city to the city of each of the remaining 99 friends and summed these 99 numbers. Vasya did the same. Petya obtained 1000 km. What is the largest number Vasya might have obtained? (Consider the cities as points on a plane; if two people live in the same city, the distance between their cities is considered to be zero.)

Answer: 99000


Triangle $ABC$ has $AC = 450$ and $BC = 300$. Points $K$ and $L$ are located on $\overline{AC}$ and $\overline{AB}$ respectively so that $AK = CK$, and $\overline{CL}$ is the angle bisector of angle $C$. Let $P$ be the point of intersection of $\overline{BK}$ and $\overline{CL}$, and let $M$ be the point on line $BK$ for which $K$ is the midpoint of $\overline{PM}$. If $AM = 180$, find $LP$.
Diagram
[asy] import markers; defaultpen(fontsize(8)); size(300); pair A=(0,0), B=(30*sqrt(331),0), C, K, L, M, P; C = intersectionpoints(Circle(A,450), Circle(B,300))[0]; K = midpoint(A--C); L = (3*B+2*A)/5; P = extension(B,K,C,L); M = 2*K-P; draw(A--B--C--cycle); draw(C--L);draw(B--M--A); markangle(n=1,radius=15,A,C,L,marker(markinterval(stickframe(n=1),true))); markangle(n=1,radius=15,L,C,B,marker(markinterval(stickframe(n=1),true))); dot(A^^B^^C^^K^^L^^M^^P); label("$A$",A,(-1,-1));label("$B$",B,(1,-1));label("$C$",C,(1,1)); label("$K$",K,(0,2));label("$L$",L,(0,-2));label("$M$",M,(-1,1)); label("$P$",P,(1,1)); label("$180$",(A+M)/2,(-1,0));label("$180$",(P+C)/2,(-1,0));label("$225$",(A+K)/2,(0,2));label("$225$",(K+C)/2,(0,2)); label("$300$",(B+C)/2,(1,1)); [/asy]

Answer: 72


Lily is taking a 30-question, multiple-choice Biology quiz. Each question offers four possible answers. Lily guesses on the last six questions. What is the probability that she will get at least two of these last six questions wrong?

Answer: \frac{4077}{4096}


For all real numbers \( x \) and \( y \) satisfying \( |x| + |y| \leq 1 \), the inequality \(\left|2x - 3y + \frac{3}{2}\right| + |y - 1| + |2y - x - 3| \leq a\) always holds. What is the minimum value of the real number \( a \)?

Answer: \frac{23}{2}


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


On the hyperbola \( xy = 1 \), let \( A_n \) be the point with the abscissa \( \frac{n}{n+1} \) and \( B_n \) the point with the abscissa \( \frac{n+1}{n} \), where \( n \in \mathbf{N}_{+} \). Furthermore, let \( M \) be the point with coordinates \((1,1)\). Let \( P_n (x_n, y_n) \) be the circumcenter of \( \triangle A_n B_n M \). When \( n \rightarrow +\infty \), find the coordinates \((a, b)\) of the limit point of \( P_n \), where \( a = \lim_{n \rightarrow +\infty} x_n \) and \( b = \lim_{n \rightarrow +\infty} y_n \).

Answer: (2,2)


Let \(\mathcal{E}\) be an ellipse with foci \(A\) and \(B\). Suppose there exists a parabola \(\mathcal{P}\) such that:
- \(\mathcal{P}\) passes through \(A\) and \(B\),
- the focus \(F\) of \(\mathcal{P}\) lies on \(\mathcal{E}\),
- the orthocenter \(H\) of \(\triangle FAB\) lies on the directrix of \(\mathcal{P}\).

If the major and minor axes of \(\mathcal{E}\) have lengths 50 and 14, respectively, compute \(AH^{2} + BH^{2}\).

Answer: 2402


Acute angles \( A \) and \( B \) of a triangle satisfy the equation \( \tan A - \frac{1}{\sin 2A} = \tan B \) and \( \cos^2 \frac{B}{2} = \frac{\sqrt{6}}{3} \). Determine the value of \( \sin 2A \).

Answer: \frac{2\sqrt{6} - 3}{3}


Let $x_1< x_2 < x_3$ be the three real roots of the equation $\sqrt{2014} x^3 - 4029x^2 + 2 = 0$. Find $x_2(x_1+x_3)$.

Answer: 2


How many roots does \( \arctan x = x^{2} - 1.6 \) have, where the arctan function is defined in the range \( -\frac{\pi}{2} < \arctan x < \frac{\pi}{2} \)?

Answer: 2


Given a trapezoid \(ABCD\) with \(BC \parallel AD\) and an inscribed circle with center \(O\). The lines \(BO\) and \(CO\) intersect the lower base \(AD\) at points \(M\) and \(N\) respectively. Prove the relationship for the areas \(S_{\text{AON}} + S_{\text{DOM}} + 2S_{\text{NOM}} = \frac{1}{2}S_{\text{ABCD}}\).

Answer: S_{


There are 99 sticks with lengths $1, 2, 3, \ldots, 99$. Is it possible to form the outline of some rectangle with these sticks?

Answer: Yes


An ellipse with equation
\[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\]contains the circles $(x - 1)^2 + y^2 = 1$ and $(x + 1)^2  +y^2 = 1.$  Then the smallest possible area of the ellipse can be expressed in the form $k \pi.$  Find $k.$

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


Cyclic pentagon \( A B C D E \) has a right angle \( \angle A B C = 90^\circ \) and side lengths \( A B = 15 \) and \( B C = 20 \). Supposing that \( A B = D E = E A \), find \( C D \).

Answer: 20


Three boys played a "Word" game in which they each wrote down ten words. For each word a boy wrote, he scored three points if neither of the other boys had the same word; he scored one point if only one of the other boys had the same word. No points were awarded for words which all three boys had. When they added up their scores, they found that they each had different scores. Sam had the smallest score (19 points), and James scored the most. How many points did James score?

Answer: 25


The equation \(\sin x = \frac{x}{2021 \pi}\) has exactly \(n\) solutions. Find \(n\).

Answer: 4043


A crazy physicist has discovered a new particle called an omon. He has a machine, which takes two omons of mass  $a$  and  $b$  and entangles them; this process destroys the omon with mass  $a$ , preserves the one with mass  $b$ , and creates a new omon whose mass is  $\frac 12 (a+b)$ . The physicist can then repeat the process with the two resulting omons, choosing which omon to destroy at every step. The physicist initially has two omons whose masses are distinct positive integers less than  $1000$ . What is the maximum possible number of times he can use his machine without producing an omon whose mass is not an integer?

*Proposed by Michael Kural*

Answer: 9


The lengths of a pair of corresponding medians of two similar triangles are 10cm and 4cm, respectively, and the sum of their perimeters is 140cm. The perimeters of these two triangles are     , and the ratio of their areas is     .

Answer: 25:4


Find all pairs of integers \((x, y)\) that satisfy the equation \( x y = 20 - 3x + y \). For each pair \((x, y)\) found, calculate the product \( x y \). Write down the sum of these products.

Answer: 56


If a number is randomly selected from the set $\left\{ \frac{1}{3}, \frac{1}{4}, 3, 4 \right\}$ and denoted as $a$, and another number is randomly selected from the set $\left\{ -1, 1, -2, 2 \right\}$ and denoted as $b$, then the probability that the graph of the function $f(x) = a^{x} + b$ ($a > 0, a \neq 1$) passes through the third quadrant is ______.

Answer: \frac{3}{8}


Let \( a_{1}, a_{2}, \cdots, a_{n} \) be pairwise distinct positive integers satisfying \( a_{1} + a_{2} + \cdots + a_{n} = 2014 \), where \( n \) is an integer greater than 1. Let \( d \) be the greatest common divisor of \( a_{1}, a_{2}, \cdots, a_{n} \). For all possible values of \( n \) and \( a_{1}, a_{2}, \cdots, a_{n} \) that satisfy the above conditions, find the maximum value of \( n d \).

Answer: 530


How many nine-digit positive integers consist of nine pairwise distinct digits and are divisible by 4950?

Answer: 576


Let  $ABC$  be a right angled triangle with  $\angle A = 90^o$ and  $BC = a$ ,  $AC = b$ ,  $AB = c$ . Let  $d$  be a line passing trough the incenter of triangle and intersecting the sides  $AB$  and  $AC$  in  $P$  and  $Q$ , respectively.
(a) Prove that   $$ b \cdot \left( \frac{PB}{PA}\right)+ c \cdot \left( \frac{QC}{QA}\right) =a $$  (b) Find the minimum of   $$ \left( \frac{PB}{PA}\right)^ 2+\left( \frac{QC}{QA}\right)^ 2 $$

Answer: 1


Triangle  $ABC$  is acute. Equilateral triangles  $ABC',AB'C,A'BC$  are constructed externally to  $ABC$ . Let  $BB'$  and  $CC'$  intersect at  $F$ . Let  $CC'$  intersect  $AB$  at  $C_1$  and  $AA'$  intersect  $BC$  at  $A_1$ , and let  $A_1C_1$  intersect  $AC$  at  $D$ . If  $A'F=23$ ,  $CF=13$ , and  $DF=24$ , find  $BD$ .

*2017 CCA Math Bonanza Team Round #10*

Answer: 26


Given that the center of circle $C$ lies on the $x$-axis and circle $C$ is tangent to the line $x + \sqrt{3}y + n = 0$ at the point $(\frac{3}{2}, \frac{\sqrt{3}}{2})$, find:

1. The value of $n$ and the equation of circle $C$.
2. If circle $M: x^2 + (y - \sqrt{15})^2 = r^2 (r > 0)$ is tangent to circle $C$, find the length of the chord formed by intersecting circle $M$ with the line $\sqrt{3}x - \sqrt{2}y = 0$.

Answer: 2\sqrt{19}


Find the smallest positive integer \( k \) such that for any \( x \in [0,1] \) and \( n \in \mathbb{N}_{+} \), the inequality \( x^{k} (1-x)^{n} < \frac{1}{(1+n)^{3}} \) is always true.

Answer: 4


Given an equilateral triangle \( ABC \). A point \( K \) is marked on the side \( AB \), and points \( L \) and \( M \) are marked on the side \( BC \) such that \( L \) lies on the segment \( BM \). It is given that \( KL = KM \), \( BL = 2 \), and \( AK = 3 \). Find \( CM \).

Answer: 5


How many terms are in the arithmetic sequence $13$, $16$, $19$, $\dotsc$, $70$, $73$?
$\textbf{(A)}\ 20 \qquad\textbf{(B)} \ 21 \qquad\textbf{(C)} \ 24 \qquad\textbf{(D)} \ 60 \qquad\textbf{(E)} \ 61$

Answer: \textbf{(B)}\21


Let \( N \) be the smallest positive integer for which
\[ x^2 + x + 1 \quad \text{divides} \quad 166 - \sum_{d \mid N, d>0} x^d. \]
Find the remainder when \( N \) is divided by 1000.

Answer: 672


Find the number of natural numbers \( k \) not exceeding 333300 such that \( k^2 - 2k \) is divisible by 303.

Answer: 4400


Consider five-dimensional Cartesian space
$$
\mathbb{R}^{5}=\left\{\left(x_{1}, x_{2}, x_{3}, x_{4}, x_{5}\right) \mid x_{i} \in \mathbb{R}\right\},
$$
and consider the hyperplanes with the following equations:
- \( x_{i} = x_{j} \) for every \( 1 \leq i < j \leq 5 \);
- \( x_{1} + x_{2} + x_{3} + x_{4} + x_{5} = -1 \);
- \( x_{1} + x_{2} + x_{3} + x_{4} + x_{5} = 0 \);
- \( x_{1} + x_{2} + x_{3} + x_{4} + x_{5} = 1 \).

Into how many regions do these hyperplanes divide \( \mathbb{R}^{5} \)?

Answer: 480


In a quadrilateral $MNQP$, there are two non-intersecting circles positioned such that one of them is tangent to the sides $MN$, $NP$, $PQ$, and the other is tangent to the sides $MN$, $MQ$, $PQ$. Points $B$ and $A$ lie on the sides $MN$ and $PQ$ respectively, and the segment $AB$ is tangent to both circles. Find the length of side $MQ$, given that $NP = b$ and the perimeter of quadrilateral $BAQM$ exceeds the perimeter of quadrilateral $ABNP$ by an amount of $2p$.

Answer: p


A ray of light originates from point $A$ and travels in a plane, being reflected $n$ times between lines $AD$ and $CD$ before striking a point $B$ (which may be on $AD$ or $CD$) perpendicularly and retracing its path back to $A$ (At each point of reflection the light makes two equal angles as indicated in the adjoining figure. The figure shows the light path for $n=3$). If $\measuredangle CDA=8^\circ$, what is the largest value $n$ can have?

Answer: 10


For positive integers $x$, let $g(x)$ be the number of blocks of consecutive 1's in the binary expansion of $x$. For example, $g(19)=2$ because $19=10011_{2}$ has a block of one 1 at the beginning and a block of two 1's at the end, and $g(7)=1$ because $7=111_{2}$ only has a single block of three 1's. Compute $g(1)+g(2)+g(3)+\cdots+g(256)$.

Answer: 577


Sabrina has a fair tetrahedral die whose faces are numbered 1, 2, 3, and 4, respectively.  She creates a sequence by rolling the die and recording the number on its bottom face.  However, she discards (without recording) any roll such that appending its number to the sequence would result in two consecutive terms that sum to 5.  Sabrina stops the moment that all four numbers appear in the sequence.  Find the expected (average) number of terms in Sabrina's sequence.

Answer: 10


Cut a 12cm long thin iron wire into three segments with lengths a, b, and c,
(1) Find the maximum volume of the rectangular solid with lengths a, b, and c as its dimensions;
(2) If these three segments each form an equilateral triangle, find the minimum sum of the areas of these three equilateral triangles.

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


In a social event, it was originally planned that each pair of participants would shake hands exactly once. However, 4 individuals left the event after shaking hands twice each. As a result, there were a total of 60 handshakes. Determine the initial number of participants in the event.

Answer: 15


One of the eight faces of a hexagonal prism will be transformed into the base of a new pyramid. Calculate the maximum sum of the number of exterior faces, vertices, and edges of the resultant composite shape (combining the prism and pyramid) when the pyramid is added to each type of face of the prism.

Answer: 50


Four teams play in a tournament in which each team plays exactly one game against each of the other three teams. At the end of each game, either the two teams tie or one team wins and the other team loses. A team is awarded 3 points for a win, 0 points for a loss, and 1 point for a tie. If $S$ is the sum of the points of the four teams after the tournament is complete, which of the following values can $S$ not equal?

Answer: 11


Compute $\arccos (\sin 1.5).$ All functions are in radians.

Answer: \frac{\pi}{2} - 1.5


Given the geometric sequence $\{a_{n}\}$, $a_{2}$ and $a_{18}$ are the two roots of the equation $x^{2}+15x+16=0$, find the value of $a_{3}a_{10}a_{17}$.

Answer: -64


For a string of \(P\) 's and \(Q\) 's, the value is defined to be the product of the positions of the \(P\) 's. For example, the string \(P P Q P Q Q\) has value \(1 \cdot 2 \cdot 4 = 8\).

Also, a string is called antipalindromic if writing it backwards, then turning all the \(P\) 's into \(Q\) 's and vice versa, produces the original string. For example, \(P P Q P Q Q\) is antipalindromic.

There are \(2^{1002}\) antipalindromic strings of length 2004. Find the sum of the reciprocals of their values.

Answer: \frac{2005^{1002}}{2004!}


Draw five lines \( l_1, l_2, \cdots, l_5 \) on a plane such that no two lines are parallel and no three lines pass through the same point.
(1) How many intersection points are there in total among these five lines? How many intersection points are there on each line? How many line segments are there among these five lines?
(2) Considering these line segments as sides, what is the maximum number of isosceles triangles that can be formed? Please briefly explain the reasoning and draw the corresponding diagram.

Answer: 10


At a school cafeteria, Sam wants to buy a lunch consisting of one main dish, one beverage, and one snack. The table below lists Sam's choices in the cafeteria. How many distinct possible lunches can he buy if he avoids pairing Fish and Chips with Soda due to dietary restrictions?

\begin{tabular}{ |c | c | c | }
\hline \textbf{Main Dishes} & \textbf{Beverages}&\textbf{Snacks} \\ \hline
Burger & Soda & Apple Pie \\ \hline
Fish and Chips & Juice & Chocolate Cake \\ \hline
Pasta &  &  \\ \hline
Vegetable Salad &  &  \\ \hline
\end{tabular}

Answer: 14


Find the maximum value of \( k \) for which \( 3^{11} \) can be expressed as the sum of \( k \) consecutive positive integers.
(5th Annual American Mathematics Invitational, 1987)

Answer: 486


A rabbit, a skunk, and a turtle are running a race.

The skunk finishes the race in 6 minutes.

The rabbit runs 3 times as quickly as the skunk.

The rabbit runs 5 times as quickly as the turtle.

How long does the turtle take to finish the race?

Answer: 10


Given that tetrahedron PQRS has edge lengths PQ = 3, PR = 4, PS = 5, QR = 5, QS = √34, and RS = √41, calculate the volume of tetrahedron PQRS.

Answer: 10


Each of the 10 people is either a knight, who always tells the truth, or a liar, who always lies. Each of them picked an integer. Then the first person said, "My number is greater than 1," the second said, "My number is greater than 2," and so on, with the tenth person saying, "My number is greater than 10." After that, all ten, speaking in some order, said one of the following statements: "My number is less than 1," "My number is less than 2," ..., "My number is less than 10" (each person said exactly one of these ten phrases). What is the maximum number of knights that could be among these 10 people?

Answer: 8


Given a function \( f(x) \) defined on \(\mathbf{R}\) with a smallest positive period \( T \), and satisfying \( f(1+x) = f(1-x) \) and \( f(8+x) = f(8-x) \), what is the maximum value of \( T \)?

Answer: 14


The equation $x^{2}+2 x=i$ has two complex solutions. Determine the product of their real parts.

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


Let  $n$  be an integer greater than or equal to  $1$ . Find, as a function of  $n$ , the smallest integer  $k\ge 2$  such that, among any  $k$  real numbers, there are necessarily two of which the difference, in absolute value, is either strictly less than  $1 / n$ , either  strictly greater than  $n$ .

Answer: n^2+2


Given that \(a\), \(b\), and \(c\) are real numbers and the equation
\[ x^{3} + a x^{2} + b x + c = 0 \]
has three positive roots, prove that \(2a^{3} + 9c \leq 7ab\), and determine the conditions under which equality holds.

Answer: 2a^3+9c\le7ab


Given an equilateral triangle with side length \( a \), find the segment connecting a vertex of the triangle to a point that divides the opposite side in the ratio \( 2:1 \).

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


In the product

\[
24^{a} \cdot 25^{b} \cdot 26^{c} \cdot 27^{d} \cdot 28^{e} \cdot 29^{f} \cdot 30^{g}
\]

seven numbers \(1, 2, 3, 5, 8, 10, 11\) were assigned to the exponents \(a, b, c, d, e, f, g\) in some order. Find the maximum number of zeros that can appear at the end of the decimal representation of this product.

Answer: 32


In a plane Cartesian coordinate system, the vertices of square $OABC$ are $O(0,0)$, $A(100,0)$, $B(100,100)$, and $C(0,100)$. A grid point $P$ inside square $OABC$ (excluding the boundary and vertices) is called a "good point" if it satisfies the condition $S_{\triangle POA} \times S_{\triangle PBC} = S_{\triangle PAB} \times S_{\triangle POC}$. How many "good points" are there inside square $OABC$?

Answer: 197


Find the smallest four-digit number that is equal to the square of the sum of the numbers formed by its first two digits and its last two digits.

Answer: 2025


Xiao Hua plays a certain game where each round can be played several times freely. Each score in a round is one of the numbers $8$, $a$ (a natural number), or $0$. The total score for a round is the sum of all individual scores in that round. Xiao Hua has achieved the following total scores in some rounds: $103, 104, 105, 106, 107, 108, 109, 110$. It is also known that he cannot achieve a total score of $83$. What is the value of $a$?

Answer: 13


In the number $2016 * * * * 02 *$, you need to replace each of the 5 asterisks with any of the digits $0, 2, 4, 6, 7, 8$ (digits may repeat) so that the resulting 11-digit number is divisible by 6. How many ways can this be done?

Answer: 2160


Find all infinite bounded sequences of natural numbers \(a_1, a_2, a_3, \ldots\), for which, starting from the third term, the following holds:

$$
a_n = \frac{a_{n-1} + a_{n-2}}{(a_{n-1}, a_{n-2})}
$$

Answer: 2


Let \( P(x) \) be a polynomial of degree \( n \) such that for \( k = 1, 2, 4, \ldots, 2^{n} \), we have \( P(k) = \frac{1}{k} \). Find \( P(0) \).

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


Find the value of \(m + n\) where \(m\) and \(n\) are integers defined as follows: The positive difference between the roots of the quadratic equation \(5x^2 - 9x - 14 = 0\) can be expressed as \(\frac{\sqrt{m}}{n}\), with \(m\) not divisible by the square of any prime number.

Answer: 366


Given that $a$ and $b$ are positive numbers, and $a+b=1$, find the minimum value of $\frac{1}{2a} + \frac{1}{b}$.

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


Suppose that $f(x)$ and $g(x)$ are functions which satisfy the equations $f(g(x)) = 2x^2$ and $g(f(x)) = x^4$ for all $x \ge 1$. If $g(4) = 16$, compute $[g(2)]^4$.

Answer: 16


For all real numbers $r$ and $s$, define the mathematical operation $\#$ such that the following conditions apply: $r\ \#\ 0 = r, r\ \#\ s = s\ \#\ r$, and $(r + 1)\ \#\ s = (r\ \#\ s) + s + 1$. What is the value of $11\ \#\ 5$?

Answer: 71


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 circumscribed circle \(\omega\) of triangle \(ABC\). The line \(AC\) intersects the circumscribed circle 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 the segment \(AE\)?

Answer: 1:2


In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively, and $b= \sqrt {2}a$, $\sqrt {3}\cos B= \sqrt {2}\cos A$, $c= \sqrt {3}+1$. Find the area of $\triangle ABC$.

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


Nonnegative real numbers  $p_{1},\ldots,p_{n}$  and  $q_{1},\ldots,q_{n}$  are such that  $p_{1}+\cdots+p_{n}=q_{1}+\cdots+q_{n}$ Among all the matrices with nonnegative entries having  $p_i$  as sum of the  $i$ -th row's entries and  $q_j$  as sum of the  $j$ -th column's entries, find the maximum sum of the entries on the main diagonal.

Answer: \sum_{i=1}^np_i,q_i)


A point is randomly thrown onto the segment [6, 11], and let \( k \) be the resulting value. Find the probability that the roots of the equation \( \left(k^{2}-2k-15\right)x^{2}+(3k-7)x+2=0 \) satisfy the condition \( x_{1} \leq 2x_{2} \).

Answer: 1/3


Alena, Bára, Čeněk, and David bought a tandem bicycle for two. They always go for a ride in pairs. Each person has ridden with every other person at least once, and no one else has ridden the tandem. Alena has gone for a ride eleven times, Bára twenty times, and Čeněk only four times.

Determine the minimum and maximum number of times David could have gone for a ride.

Answer: 29


Determine the remainder of the 81-digit integer \(\overline{111 \cdots 1}\) divided by 81.

Answer: 0


The RSA Factoring Challenge, which ended in 2007, involved factoring extremely large numbers that were the product of two prime numbers. The largest number successfully factored in this challenge was RSA-640, which has 193 decimal digits and carried a prize of $20,000. The next challenge number carried a prize of $30,000, and contains $N$ decimal digits. Your task is to submit a guess for $N`. Only the team(s) that have the closest guess(es) receives points. If $k$ teams all have the closest guesses, then each of them receives $\left\lceil\frac{20}{k}\right\rceil$ points.

Answer: 212


Vojta bought 24 identical square tiles. Each tile had a side length of $40 \mathrm{~cm}$. Vojta wanted to arrange them in front of a cabin to form a rectangular platform with the smallest possible perimeter. What was the perimeter of the tiled rectangle in meters, given that no tile was leftover and Vojta did not cut or break any tiles?

Answer: 8


The game takes place on a $9 \times 9$ grid of squared paper. Two players take turns; the player who starts the game places crosses in free cells while their partner places noughts. When all the cells are filled, the number of rows and columns $K$ in which crosses outnumber noughts, and the number of rows and columns $H$ in which noughts outnumber crosses, is counted. The difference $B=K-H$ is considered the winning score of the player who starts. Find the value of $B$ such that:

1) The first player can ensure a win of at least $B$, regardless of how the second player plays;
2) The second player can always ensure that the first player’s winning score is no more than $B$, regardless of how the first player plays.

Answer: 2


Nine hundred forty-three minus eighty-seven equals what number?

Answer: 856


The sum
\[ \sum_{k=0}^{\infty} \frac{2^{k}}{5^{2^{k}}+1}\]
can be written in the form  $\frac{p}{q}$  where  $p$  and  $q$  are relatively prime positive integers. Find  $p+q$ .

Answer: 5


A circle with diameter $\overline{PQ}$ of length 10 is internally tangent at $P$ to a circle of radius 20. Square $ABCD$ is constructed with $A$ and $B$ on the larger circle, $\overline{CD}$ tangent at $Q$ to the smaller circle, and the smaller circle outside $ABCD$. The length of $\overline{AB}$ can be written in the form $m + \sqrt{n}$, where $m$ and $n$ are integers. Find $m + n$.
 Note: The diagram was not given during the actual contest.

Answer: 312


In the sequence \( \left\{a_{n}\right\}_{n=1}^{+\infty} \), where \( x_{1}=1 \), for any positive integer \( n \), \( x_{n+1}=x_{n}+\left[\sqrt{x_{n}}\right] \), where the symbol \( [x] \) denotes the greatest integer not exceeding \( x \), find the smallest positive integer \( n \) such that \( x_{n} \geqslant 2019 \).

Answer: 95


A positive integer is called "balanced" if it has four digits, and one of these digits is equal to the average of the other three. For example, the number 2631 is balanced because 3 is the average of 2, 6, and 1; 4444 is also balanced because 4 is the average of 4, 4, and 4.

a) Find the three smallest balanced numbers.

b) How many balanced numbers are there that are less than 2014?

Answer: 90


a) In how many ways can a rectangle $8 \times 2$ be divided into $1 \times 2$ rectangles?

b) Imagine and describe a shape that can be divided into $1 \times 2$ rectangles in exactly 555 ways.

Answer: 34


The average lifespan of a motor is 4 years. Estimate from below the probability that this motor will not last more than 20 years.

Answer: 0.8


A square sheet contains 1000 points, with any three points, including the vertices of the square, not being collinear. Connect some of these points and the vertices of the square with line segments to divide the entire square into smaller triangles (using the connected line segments and square edges as sides, and ensuring that no two segments, except at endpoints, share common points). How many line segments are connected in total? How many triangles are formed in total?

Answer: 2002


The line $l: x - 2y + 2 = 0$ passes through the left focus F1 and a vertex B of an ellipse. Find the eccentricity of the ellipse.

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


In $\triangle ABC$, $\angle C = 90^{\circ}$ and $AB = 12$. Squares $ABXY$ and $ACWZ$ are constructed outside of the triangle. The points $X, Y, Z$, and $W$ lie on a circle. What is the perimeter of the triangle?

Answer: 12 + 12\sqrt{2}


Given that Ron mistakenly reversed the digits of the two-digit number $a$, and the product of $a$ and $b$ was mistakenly calculated as $221$, determine the correct product of $a$ and $b$.

Answer: 527


For any \( x \in \left[-\frac{\pi}{6}, \frac{\pi}{2}\right] \), the inequality \( \sin^2 x + a \sin x + a + 3 \geq 0 \) always holds. Find the range of the real number \( a \).

Answer: [-2,+\infty)


In triangle \(ABC\), points \(P\), \(Q\), and \(R\) lie on sides \(BC\), \(CA\), and \(AB\), respectively. Denote the circumcircles of triangles \(\triangle AQR\), \(\triangle BRP\), and \(\triangle CPQ\) as \(\Gamma_A\), \(\Gamma_B\), and \(\Gamma_C\) respectively. It is known that line segment \(AP\) intersects circles \(\Gamma_A\), \(\Gamma_B\), and \(\Gamma_C\) at points \(X\), \(Y\), and \(Z\) respectively. Prove that \(\frac{YX}{XZ} = \frac{BP}{PC}\).

Answer: \frac{YX}{XZ}=\frac{BP}{PC}


There are  $2012$  backgammon checkers (stones, pieces) with one side is black and the other side is white.
These checkers are arranged into a line such that no two consequtive checkers are in same color. At each move, we are chosing two checkers. And we are turning upside down of the two checkers and all of the checkers between the two. At least how many moves are required to make all checkers same color?

Answer: 1006


On the extension of side $AD$ of rectangle $ABCD$ beyond point $D$, point $E$ is taken such that $DE = 0.5 AD$ and $\angle BEC = 30^\circ$.

Find the ratio of the sides of rectangle $ABCD$.

Answer: \sqrt{3}/2


Let  $\sigma(n)$  be the number of positive divisors of  $n$ , and let  $\operatorname{rad} n$  be the product of the distinct prime divisors of  $n$ . By convention,  $\operatorname{rad} 1 = 1$ .  Find the greatest integer not exceeding \[ 100\left(\sum_{n=1}^{\infty}\frac{\sigma(n)\sigma(n \operatorname{rad} n)}{n^2\sigma(\operatorname{rad} n)}\right)^{\frac{1}{3}}. \]*Proposed by Michael Kural*

Answer: 164


In the decimal notation of 13 different numbers, only one digit \( N \) is used and no other digits are used. Can the sum of these numbers be equal to 8,900,098?

Answer: No


Let $\triangle A_0B_0C_0$ be a triangle whose angle measures are exactly $59.999^\circ$, $60^\circ$, and $60.001^\circ$. For each positive integer $n$, define $A_n$ to be the foot of the altitude from $A_{n-1}$ to line $B_{n-1}C_{n-1}$. Likewise, define $B_n$ to be the foot of the altitude from $B_{n-1}$ to line $A_{n-1}C_{n-1}$, and $C_n$ to be the foot of the altitude from $C_{n-1}$ to line $A_{n-1}B_{n-1}$. What is the least positive integer $n$ for which $\triangle A_nB_nC_n$ is obtuse?

Answer: 15


Suppose that  $f : \mathbb{R}^+\to\mathbb R$  satisfies the equation  $$ f(a+b+c+d) = f(a)+f(b)+f(c)+f(d) $$  for all  $a,b,c,d$  that are the four sides of some tangential quadrilateral. Show that  $f(x+y)=f(x)+f(y)$  for all  $x,y\in\mathbb{R}^+$ .

Answer: f(x+y)=f(x)+f(y)


Let \(O, O_{1}, O_{2}, O_{3}, O_{4}\) be points such that \(O_{1}, O, O_{3}\) and \(O_{2}, O, O_{4}\) are collinear in that order, \(O O_{1}= 1\), \(O O_{2}=2\), \(O O_{3}=\sqrt{2}\), \(O O_{4}=2\), and \(\measuredangle O_{1} O O_{2}=45^{\circ}\). Let \(\omega_{1}, \omega_{2}, \omega_{3}, \omega_{4}\) be the circles with respective centers \(O_{1}, O_{2}, O_{3}, O_{4}\) that go through \(O\). Let \(A\) be the intersection of \(\omega_{1}\) and \(\omega_{2}\), \(B\) be the intersection of \(\omega_{2}\) and \(\omega_{3}\), \(C\) be the intersection of \(\omega_{3}\) and \(\omega_{4}\), and \(D\) be the intersection of \(\omega_{4}\) and \(\omega_{1}\), with \(A, B, C, D\) all distinct from \(O\). What is the largest possible area of a convex quadrilateral \(P_{1} P_{2} P_{3} P_{4}\) such that \(P_{i}\) lies on \(O_{i}\) and that \(A, B, C, D\) all lie on its perimeter?

Answer: 8+4\sqrt{2}


Each pack of hot dogs contains 10 hot dogs (no buns), but each pack of hot dog buns contains 8 buns.  Phil buys a number of these packs for a barbecue.  After the barbecue, Phil finds that he has 4 hot dogs left over.  What is the SECOND smallest number of packs of hot dogs he could have bought?

Answer: 6


In unit square $A B C D$, points $E, F, G$ are chosen on side $B C, C D, D A$ respectively such that $A E$ is perpendicular to $E F$ and $E F$ is perpendicular to $F G$. Given that $G A=\frac{404}{1331}$, find all possible values of the length of $B E$.

Answer: \frac{9}{11}


Given the function $f(x)=\ln (ax+1)+ \frac {x^{3}}{3}-x^{2}-ax(a∈R)$,
(1) Find the range of values for the real number $a$ such that $y=f(x)$ is an increasing function on $[4,+∞)$;
(2) When $a\geqslant \frac {3 \sqrt {2}}{2}$, let $g(x)=\ln [x^{2}(ax+1)]+ \frac {x^{3}}{3}-3ax-f(x)(x > 0)$ and its two extreme points $x_{1}$, $x_{2}(x_{1} < x_{2})$ are exactly the zeros of $φ(x)=\ln x-cx^{2}-bx$, find the minimum value of $y=(x_{1}-x_{2})φ′( \frac {x_{1}+x_{2}}{2})$.

Answer: \ln 2- \frac {2}{3}


For any positive integer $n,$ let $\langle n \rangle$ denote the closest integer to $\sqrt{n}.$  Evaluate
\[\sum_{n = 1}^\infty \frac{2^{\langle n \rangle} + 2^{-\langle n \rangle}}{2^n}.\]

Answer: 3


A fat coin is one which, when tossed, has a \(\frac{2}{5}\) probability of being heads, \(\frac{2}{5}\) of being tails, and \(\frac{1}{5}\) of landing on its edge. Mr. Fat starts at 0 on the real line. Every minute, he tosses a fat coin. If it's heads, he moves left, decreasing his coordinate by 1; if it's tails, he moves right, increasing his coordinate by 1. If the coin lands on its edge, he moves back to 0. If Mr. Fat does this indefinitely, what fraction of his time will he spend at 0?

Answer: \frac{1}{3}


Given a sequence \( a_1, a_2, a_3, \ldots, a_n \) of non-zero integers such that the sum of any 7 consecutive terms is positive and the sum of any 11 consecutive terms is negative, what is the largest possible value for \( n \)?

Answer: 16


Two right cones each have base radius 4 and height 3, such that the apex of each cone is the center of the base of the other cone. Find the surface area of the union of the cones.

Answer: 62\pi


Let  $a,b,c$  be positive real numbers such that  $a+b+c=3$ . Determine, with certainty, the largest possible value of the expression   $$  \frac{a}{a^3+b^2+c}+\frac{b}{b^3+c^2+a}+\frac{c}{c^3+a^2+b} $$

Answer: 1


Determine the heaviest weight that can be obtained using a combination of 2 lb and 4 lb, and 12 lb weights with a maximum of two weights used at a time.

Answer: 16


The measure of angle $ACB$ is 70 degrees. If ray $CA$ is rotated 600 degrees about point $C$ in a clockwise direction, what will be the positive measure of the new obtuse angle $ACB$, in degrees?

Answer: 170


If \( n \) is a positive integer, and \( a(n) \) is the smallest positive integer such that \( (a(n))! \) is divisible by \( n \), find all positive integers \( n \) such that \( \frac{a(n)}{n} = \frac{2}{3} \). (2003 German Math Olympiad)

Answer: 9


We place a coin in one of three identical boxes and then put these three boxes into two drawers such that each drawer contains at least one box. What is the probability that if someone randomly selects a drawer and then takes a box out from that drawer, they will find the coin in that box,

a) if the box containing the coin is alone in one drawer?

b) if the box containing the coin is in a drawer with one other box?

c) if the 3 boxes are randomly placed into the two drawers without knowing which box contains the coin?

Answer: \frac{1}{3}


Let \( n \) be any integer \(\geq 2\). Prove that \(\sum \frac{1}{pq} = \frac{1}{2}\), where the summation is over all integers \( p, q \) which satisfy \(0 < p < q \leq n\), \( p + q > n \), and \(\gcd(p, q) = 1\).

Answer: \frac{1}{2}


In $\triangle ABC$, $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$ respectively, and it is given that $4\cos C \cdot \sin^2 \frac{C}{2} + \cos 2C = 0$.

(1) If $\tan A = 2\tan B$, find the value of $\sin(A-B)$;

(2) If $3ab = 25 - c^2$, find the maximum area of $\triangle ABC$.

Answer: \frac{25\sqrt{3}}{16}


Compute the sum of all positive integers \( n < 1000 \) for which \( a_{n} \) is a rational number, given the sequence \( \{a_{n}\} \) defined by \( a_{1} = 1 \) and for all integers \( n \geq 1 \):
$$
a_{n+1} = \frac{a_{n} \sqrt{n^{2} + n}}{\sqrt{n^{2} + n + 2 a_{n}^{2}}}.
$$

Answer: 131


The four points of a plane determine six distances. Prove that the ratio of the largest of these distances to the smallest cannot be less than $\sqrt{2}$.

Answer: \sqrt{2}


In trapezoid $PQRS$, it is known that $\angle PQR=90^{\circ}$, $\angle QRS<90^{\circ}$, diagonal $SQ$ is 24 and is the angle bisector of $\angle S$, and the distance from vertex $R$ to line $QS$ is 16. Find the area of the trapezoid PQRS.

Answer: \frac{8256}{25}


Given the function $f(x)=\sin(\omega x+\varphi)$ is monotonically increasing on the interval ($\frac{π}{6}$,$\frac{{2π}}{3}$), and the lines $x=\frac{π}{6}$ and $x=\frac{{2π}}{3}$ are the two symmetric axes of the graph of the function $y=f(x)$, evaluate the value of $f(-\frac{{5π}}{{12}})$.

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


A school provides apples, bananas, and pears as snacks for students. Each student must choose at least one fruit, but can select multiple. The statistics show that 70% of the students choose apples, 40% choose bananas, and 30% choose pears. What is the maximum percentage of students who chose all three fruits?

Answer: 20


The lateral edges of a triangular pyramid are pairwise perpendicular, and the sides of the base are $\sqrt{61}, \sqrt{52}$, and $\sqrt{41}$. The center of a sphere that touches all the lateral faces lies on the base of the pyramid. Find the radius of this sphere.

Answer: \frac{60}{37}


Find the largest positive integer  $n$  such that for each prime  $p$  with  $2<p<n$  the difference  $n-p$  is also prime.

Answer: 10


There are two cylinders whose volume ratio is $5: 8$. Their lateral surfaces can be unfolded into identical rectangles. If the length and width of this rectangle are both increased by 6, its area increases by 114. What is the area of this rectangle?

Answer: 40


An isosceles trapezoid is circumscribed around a circle with radius $r$. The legs of the trapezoid form an angle $\alpha$ with the shorter base. Find the radius of the circle circumscribed around the trapezoid.

Answer: \frac{r\sqrt{1+\sin^2\alpha}}{\sin^2\alpha}


In each cell of a $1987 \times 1987$ square table, a number not exceeding 1 in absolute value is written. In any $2 \times 2$ square within this table, the sum of the numbers is 0. Prove that the sum of all the numbers in the table does not exceed 1987.

Answer: 1987


Points \( A, B, C \) lie on a circle centered at \( O \) with radius 7. The perpendicular bisector of \( AB \) meets the segment \( BC \) at \( P \) and the extension of \( AC \) at \( Q \). Determine the value of \( OP \cdot OQ \).

Answer: 49


The triangle  $ABC$  is isosceles with  $AB=BC$ . The point F on the side  $[BC]$  and the point  $D$  on the side  $AC$  are the feets of the the internals bisectors drawn from  $A$  and altitude drawn from  $B$  respectively so that  $AF=2BD$ . Fine the measure of the angle  $ABC$ .

Answer: 36


In a jewelry store display, there are 15 diamonds. Next to them are labels indicating their weights, written as $1, 2, \ldots, 15$ carats. The shopkeeper has a balance scale and four weights with masses $1, 2, 4,$ and $8$ carats. A customer is allowed only one type of weighing: place one of the diamonds on one side of the scale and the weights on the other side to verify that the weight indicated on the corresponding label is correct. However, for each weight taken, the customer must pay the shopkeeper 100 coins. If a weight is removed from the scale and not used in the next weighing, the shopkeeper keeps it. What is the minimum amount that needs to be paid to verify the weights of all the diamonds?

[8 points]

(A. V. Gribalko)

Answer: 800


Find how many integer values of \( x \) are there such that \( \lceil{\sqrt{x}}\rceil=18 \)?

Answer: 35


Three sportsmen called Primus, Secundus, and Tertius take part in a race every day. Primus wears the number '1' on his shirt, Secundus wears '2', and Tertius wears '3'.

On Saturday, Primus wins, Secundus is second, and Tertius is third. Using their shirt numbers this result is recorded as '123'.

On Sunday, Primus starts the race in the lead with Secundus in second. During Sunday's race:
- Primus and Secundus change places exactly 9 times,
- Secundus and Tertius change places exactly 10 times,
- Primus and Tertius change places exactly 11 times.

How will Sunday's result be recorded?

Answer: 231


Prove that from the equality

$$
\frac{x(y+z-x)}{\log _{a} x}=\frac{y(x+z-y)}{\log _{a} y}=\frac{z(x+y-z)}{\log _{a} z}
$$

it follows that

$$
x^{y} \cdot y^{x}=z^{x} \cdot x^{z}=y^{z} \cdot z^{y}
$$

Answer: x^y\cdoty^x=z^x\cdotx^z=y^z\cdotz^y


In the famous book "Algorithm for Direct Calculation" by the Chinese mathematician Cheng Dawei of the Ming Dynasty, there is a well-known math problem:

"One hundred mantou for one hundred monks, three big monks have no dispute, three small monks share one, how many big and small monks are there?"

Answer: 75


Let triangle  $ABC$  with incenter  $I$  and circumcircle  $\Gamma$  satisfy  $AB = 6\sqrt{3}, BC = 14,$  and  $CA = 22$ . Construct points  $P$  and  $Q$  on rays  $BA$  and  $CA$  such that  $BP = CQ = 14$ . Lines  $PI$  and  $QI$  meet the tangents from  $B$  and  $C$  to  $\Gamma$ , respectively, at points  $X$  and  $Y$ . If  $XY$  can be expressed as  $a\sqrt{b} - c$  for positive integers  $a,b,c$  with  $c$  squarefree, find  $a + b + c$ .

*Proposed by Andrew Wu*

Answer: 31


In rectangle \(ABCD\), \(AB = 2\) and \(AD = 1\). Let \(P\) be a moving point on side \(DC\) (including points \(D\) and \(C\)), and \(Q\) be a moving point on the extension of \(CB\) (including point \(B\)). The points \(P\) and \(Q\) satisfy \(|\overrightarrow{DP}| = |\overrightarrow{BQ}|\). What is the minimum value of the dot product \(\overrightarrow{PA} \cdot \overrightarrow{PQ}\)?

Answer: 3/4


A group of friends went for a morning jog around a lake. During the jog, one by one, they realized they had overestimated their strength and switched from running to walking. One of the friends calculated that he ran one-eighth of the total distance the entire group ran and walked one-tenth of the total distance they walked. How many people were there in the group?

Answer: 9


Let \( f(x) \) be an odd function defined on \( \mathbf{R} \) such that \( f(x) = x^2 \) for \( x \geq 0 \). If the inequality \( f(x+a) \geq 2 f(x) \) holds for any \( x \in [a, a+2] \), determine the range of the real number \( a \).

Answer: [\sqrt{2},+\infty)


Find all positive integers  $n$  such that equation   $$ 3a^2-b^2=2018^n $$   has a solution in integers  $a$  and  $b$ .

Answer: n


Seven standard dice are glued together to make a solid. The pairs of faces of the dice that are glued together have the same number of dots on them. How many dots are on the surface of the solid?

Answer: 105


For how many integer values of \( k \) is \( \sqrt{200 - \sqrt{k}} \) also an integer?

Answer: 15


Let $p$, $q$, $r$, $s$, and $t$ be distinct integers such that $(8-p)(8-q)(8-r)(8-s)(8-t) = -120$. Calculate the sum $p+q+r+s+t$.

Answer: 27


In a language, there are three letters - Ш, У, and Я. A word is defined as a sequence of 100 letters, exactly 40 of which are vowels (У or Я), and the remaining 60 are the letter Ш. What is the maximum number of words that can be chosen so that for any two selected words, there is at least one position among the hundred where different vowels stand simultaneously?

Answer: 2^{40}


Show that for \( x > 0 \) the inequalities

\[
\frac{x}{1+x^{2}} \varphi(x) < 1 - \Phi(x) < \frac{\varphi(x)}{x}
\]

are valid, where \(\varphi(x) = \frac{1}{\sqrt{2\pi}} e^{-x^{2} / 2}\) and \(\Phi(x) = \int_{-\infty}^{x} \varphi(y) \, dy\).

Answer: \frac{x}{1+x^2}\varphi(x)<1-\Phi(x)<\frac{\varphi(x)}{x}


Find the maximal constant  $ M$ , such that for arbitrary integer  $ n\geq 3,$  there exist two sequences of positive real number  $ a_{1},a_{2},\cdots,a_{n},$  and  $ b_{1},b_{2},\cdots,b_{n},$  satisfying
(1): $ \sum_{k \equal{} 1}^{n}b_{k} \equal{} 1,2b_{k}\geq b_{k \minus{} 1} \plus{} b_{k \plus{} 1},k \equal{} 2,3,\cdots,n \minus{} 1;$
(2): $ a_{k}^2\leq 1 \plus{} \sum_{i \equal{} 1}^{k}a_{i}b_{i},k \equal{} 1,2,3,\cdots,n, a_{n}\equiv M$ .

Answer: \frac{3}{2}


Four points \( B, A, E, L \) are on a straight line. \( G \) is a point off the line such that \(\angle B A G = 120^\circ\) and \(\angle G E L = 80^\circ\). If the reflex angle at \( G \) is \( x^\circ \), then \( x \) equals:

Answer: 340


Exactly one side of a tetrahedron is of length greater than 1. Show that its volume is less than or equal to \( \frac{1}{8} \).

Answer: \frac{1}{8}


Given the curve $C$ represented by the equation $\sqrt {x^{2}+2 \sqrt {7}x+y^{2}+7}+ \sqrt {x^{2}-2 \sqrt {7}x+y^{2}+7}=8$, find the distance from the origin to the line determined by two distinct points on the curve $C$.

Answer: \dfrac {12}{5}


A circle inscribed in triangle \( ABC \) touches side \( AB \) at point \( M \), and \( AM = 1 \), \( BM = 4 \). Find \( CM \) given that \( \angle BAC = 120^\circ \).

Answer: \sqrt{273}


Find all angles $\theta,$ $0 \le \theta \le 2 \pi,$ with the following property: For all real numbers $x,$ $0 \le x \le 1,$
\[x^2 \cos \theta - x(1 - x) + (1 - x)^2 \sin \theta > 0.\]

Answer: \left( \frac{\pi}{12}, \frac{5 \pi}{12} \right)


Given real numbers $x$ and $y$ satisfying $x^{2}+2y^{2}-2xy=4$, find the maximum value of $xy$.

Answer: 2\sqrt{2} + 2


At Clover View Junior High, one half of the students go home on the school bus.  One fourth go home by automobile.  One tenth go home on their bicycles.  The rest walk home.  What fractional part of the students walk home?
$\text{(A)}\ \dfrac{1}{16} \qquad \text{(B)}\ \dfrac{3}{20} \qquad \text{(C)}\ \dfrac{1}{3} \qquad \text{(D)}\ \dfrac{17}{20} \qquad \text{(E)}\ \dfrac{9}{10}$

Answer: (B)\\frac{3}{20}


Given the ellipse $C$: $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (a > b > 0)$ with an eccentricity of $\frac{\sqrt{2}}{2}$, and it passes through point $M(-2, 0)$.
(I) Find the standard equation of ellipse $C$;
(II) Let line $l$ with a slope of $1$ intersect ellipse $C$ at points $A(x\_1, y\_1)$ and $B(x\_2, y\_2)$. Connect $MA$ and $MB$, then extend them to intersect line $x = 4$ at points $P$ and $Q$. Let $y\_P$ and $y\_Q$ be the y-coordinates of points $P$ and $Q$ respectively, and $\frac{1}{y\_1} + \frac{1}{y\_2} = \frac{1}{y\_P} + \frac{1}{y\_Q}$. Find the area of triangle $ABM$.

Answer: S = \sqrt{10}


Seryozha cut a square birthday cake weighing 900 g with two straight cuts parallel to one pair of sides and two cuts parallel to the other pair of sides, resulting in 9 rectangular pieces. Prove that Petya can choose three pieces of cake that do not share sides and their total weight is at least 300 g.

Answer: 3


In the equation "Xiwangbei jiushi hao $\times$ 8 = Jiushihao Xiwangbei $\times$ 5", different Chinese characters represent different digits. The six-digit even number represented by "Xiwangbei jiushi hao" is ____.

Answer: 256410


Bob buys four burgers and three sodas for $\$5.00$, and Carol buys three burgers and four sodas for $\$5.40$. How many cents does a soda cost?

Answer: 94


The school plans to set up two computer labs, each equipped with one teacher's computer and several student computers. In a standard lab, the teacher's computer costs 8000 yuan, and each student computer costs 3500 yuan; in an advanced lab, the teacher's computer costs 11500 yuan, and each student computer costs 7000 yuan. It is known that the total investment for purchasing computers in both labs is equal and is between 200,000 yuan and 210,000 yuan. How many student computers should be prepared for each lab?

Answer: 27


Let $P$ be a point on the circle circumscribing square $ABCD$ that satisfies $PA \cdot PC = 56$ and $PB \cdot PD = 90.$ Find the area of $ABCD.$

Answer: 106


Given that every high school in the town of Pythagoras sent a team of 3 students to a math contest, and Andrea's score was the median among all students, and hers was the highest score on her team, and Andrea's teammates Beth and Carla placed 40th and 75th, respectively, calculate the number of schools in the town.

Answer: 25


King Gvidon had 5 sons. Among his descendants, 100 each had exactly 3 sons, and the rest died childless.

How many descendants did King Gvidon have?

Answer: 305


In a rectangular coordinate system, a circle centered at the point $(1,0)$ with radius $r$ intersects the parabola $y^2 = x$ at four points $A$, $B$, $C$, and $D$. If the intersection point $F$ of diagonals $AC$ and $BD$ is exactly the focus of the parabola, determine $r$.

Answer: \frac{\sqrt{15}}{4}


Let  $n{}$  be a natural number. The playing field for a "Master Sudoku" is composed of the  $n(n+1)/2$  cells located on or below the main diagonal of an  $n\times n$  square. A teacher secretly selects  $n{}$  cells of the playing field and tells his student

- the number of selected cells on each row, and
- that there is one selected cell on each column.
The teacher's selected cells form a Master Sudoku if his student can determine them with the given information. How many Master Sudokus are there?

*Proposed by T. Amdeberkhan, M. Ruby and F. Petrov*

Answer: 2^{n-1}


How many different ways are there to rearrange the letters in the word 'BRILLIANT' so that no two adjacent letters are the same after the rearrangement?

Answer: 55440


Compute the number of ways to assemble 2 red unit cubes and 25 white unit cubes into a \(3 \times 3 \times 3\) cube such that the red cubes are visible on exactly 4 faces of the larger cube. (Rotations and reflections are considered distinct.)

Answer: 114


Given \( f(x) = |1 - 2x| \) for \( x \in [0, 1] \), find the number of solutions to the equation \( f(f(f(x))) = \frac{1}{2} x \).

Answer: 8


Let \(a, b, c\) be integers. Define \(f(x) = ax^2 + bx + c\). Suppose there exist pairwise distinct integers \(u, v, w\) such that \(f(u) = 0\), \(f(v) = 0\), and \(f(w) = 2\). Find the maximum possible value of the discriminant \(b^2 - 4ac\) of \(f\).

Answer: 16


Find all functions $f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$, such that $f(x+f(x)+f(y))=2f(x)+y$ for all positive reals $x,y$.

Answer: f(x) = x


Vasya constructed a robot, but due to some programming mistake, the robot makes random movements, which still somehow lead it to the target point.

A move is a shift of one kilometer either north (N), east (E), south (S), or west (W).

The robot reached its target in 22 moves represented by the sequence:

EENENNWNENESEEEEESSSWN.

How many moves would the robot need if it did not make any unnecessary movements?

Answer: 10


Given that a match between two people is played with a best-of-five-games format, where the winner is the first to win three games, and that the probability of person A winning a game is $\dfrac{2}{3}$, calculate the probability that person A wins with a score of $3:1$.

Answer: \dfrac{8}{27}


Given \( a=\underset{2016 \uparrow}{55 \cdots 5} \), determine the remainder when \( a \) is divided by 84.

Answer: 63


The lateral face of a regular quadrilateral pyramid forms a $45^{\circ}$ angle with the base plane. Find the angle between the adjacent lateral faces.

Answer: 120


Let $S$ be the set of all positive rational numbers $r$ such that when the two numbers $r$ and $55r$ are written as fractions in lowest terms, the sum of the numerator and denominator of one fraction is the same as the sum of the numerator and denominator of the other fraction. The sum of all the elements of $S$ can be expressed in the form $\frac{p}{q},$ where $p$ and $q$ are relatively prime positive integers. Find $p+q.$

Answer: 719


Given the function $f(x) = \cos x \cdot \sin \left( x + \frac{\pi}{3} \right) - \sqrt{3}\cos^2 x + \frac{\sqrt{3}}{4}, x \in \mathbb{R}$,

$(1)$ Find the smallest positive period of $f(x)$;

$(2)$ Find the maximum and minimum values of $f(x)$ on the closed interval $\left[ -\frac{\pi}{4}, \frac{\pi}{4} \right]$.

Answer: -\frac{1}{2}


An integer is called snakelike if its decimal representation $a_1a_2a_3\cdots a_k$ satisfies $a_i<a_{i+1}$ if $i$ is odd and $a_i>a_{i+1}$ if $i$ is even. How many snakelike integers between 1000 and 9999 have four distinct digits?

Answer: 882


Let  $D$  be a point in the interior of  $\triangle{ABC}$  such that  $AB=ab$ ,  $AC=ac$ ,  $BC=bc$ ,  $AD=ad$ ,  $BD=bd$ ,  $CD=cd$ . Show that  $\angle{ABD}+\angle{ACD}=60^{\circ}$ .

Source: 2016 Singapore Mathematical Olympiad (Open) Round 2, Problem 1

Answer: 60


The sequence $\{a_n\}$ satisfies $a_n+a_{n+1}=n^2+(-1)^n$. Find the value of $a_{101}-a_1$.

Answer: 5150


The quiz participants were asked four questions: 90 participants answered the first question correctly, 50 answered the second correctly, 40 answered the third correctly, and 20 answered the fourth correctly. No one was able to answer more than two questions correctly. What is the minimum number of participants in the quiz under these conditions?

Answer: 100


On the coordinate plane, the points \(A(0, 2)\), \(B(1, 7)\), \(C(10, 7)\), and \(D(7, 1)\) are given. Find the area of the pentagon \(A B C D E\), where \(E\) is the intersection point of the lines \(A C\) and \(B D\).

Answer: 36


Answer: 90


Let \( n \) be a fixed integer with \( n \geq 2 \). Determine the minimal constant \( c \) such that the inequality

\[ \sum_{1 \leq i < j \leq n} x_i x_j (x_i^2 + x_j^2) \leq c \left( \sum_{i=1}^{n} x_i \right)^4 \]

holds for all non-negative real numbers \( x_1, x_2, \ldots, x_n \). Additionally, determine the necessary and sufficient conditions for which equality holds.

Answer: \frac{1}{8}


What three-digit number with units digit 4 and hundreds digit 5 is divisible by 8 and has an even tens digit?

Answer: 544


Given a sphere, a cylinder with a square axial section, and a cone. The cylinder and cone have identical bases, and their heights equal the diameter of the sphere. How do the volumes of the cylinder, sphere, and cone compare to each other?

Answer: 3 : 2 : 1


In 2019, our county built 4 million square meters of new housing, of which 2.5 million square meters are mid-to-low-priced houses. It is expected that in the coming years, the average annual increase in the area of new housing in our county will be 8% higher than the previous year. In addition, the area of mid-to-low-priced houses built each year will increase by 500,000 square meters compared to the previous year. So, by the end of which year:

$(1)$ The cumulative area of mid-to-low-priced houses built in our county over the years (with 2019 as the first cumulative year) will first exceed 22.5 million square meters?

$(2)$ The proportion of the area of mid-to-low-priced houses built in that year to the total area of housing built in that year will first exceed 85%? (Reference data: $1.08^{4}\approx 1.36$, $1.08^{5}\approx 1.47$)

Answer: 2024


Calculate the expression

$$
\frac{x_{1}}{x_{2}}+\frac{x_{2}}{x_{1}}
$$

if $x_{1}$ and $x_{2}$ are the roots of the equation

$$
x^{2}+px+q=0.

Answer: \frac{p^2-2q}{q}


Let the sequence \( u_{0}, u_{1}, u_{2}, \cdots \) be defined as follows:
$$
\begin{array}{l}
u_{0}=2, u_{1}=\frac{5}{2}, \\
u_{n+1}=u_{n}\left(u_{n-1}^{2}-2\right)-u_{1}, \quad n=1,2, \cdots
\end{array}
$$

Prove that \( \left\lfloor u_{n} \right\rfloor = 2^{\frac{2^{n}-(-1)^{n}}{3}}, \quad n=1,2, \cdots \)
where \( \lfloor x \rfloor \) denotes the greatest integer less than or equal to \( x \).

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


All of Marcy's marbles are blue, red, green, or yellow. One third of her marbles are blue, one fourth of them are red, and six of them are green. What is the smallest number of yellow marbles that Macy could have?
$\textbf{(A) }1\qquad\textbf{(B) }2\qquad\textbf{(C) }3\qquad\textbf{(D) }4\qquad\textbf{(E) }5$

Answer: \textbf{(D)}4


Regular octagon $C H I L D R E N$ has area 1. Find the area of pentagon $C H I L D$.

Answer: \frac{1}{2}


Two snails crossed the surface of a planet shaped like a torus, leaving trails behind them. One snail crawled along the outer equator, and the other along a helical line. Into how many parts did the snails' trails divide the surface of the planet? (It is sufficient to write only the answer.)

Answer: 3


$P(x)=ax^2+bx+c$ has exactly $1$ different real root where $a,b,c$ are real numbers. If $P(P(P(x)))$ has exactly $3$ different real roots, what is the minimum possible value of $abc$ ?

Answer: -2


We are fitting new tires on both wheels of a motorcycle. A tire is considered completely worn out if it has run $15000 \mathrm{~km}$ on the rear wheel or $25000 \mathrm{~km}$ on the front wheel. What is the maximum distance the motorcycle can travel until the tires are completely worn out, if we exchange the front tire with the rear tire at the appropriate time?

Answer: 18750


Let  $ABC$  be a right triangle with  $\angle ACB = 90^{\circ}$  and centroid  $G$ . The circumcircle  $k_1$  of triangle  $AGC$  and the circumcircle  $k_2$  of triangle  $BGC$  intersect  $AB$  at  $P$  and  $Q$ , respectively. The perpendiculars from  $P$  and  $Q$  respectively to  $AC$  and  $BC$  intersect  $k_1$  and  $k_2$  at  $X$  and  $Y$ . Determine the value of  $\frac{CX \cdot CY}{AB^2}$ .

Answer: \frac{4}{9}


Ninety-four bricks, each measuring  $4''\times10''\times19'',$  are to stacked one on top of another to form a tower 94 bricks tall.  Each brick can be oriented so it contribues  $4''$  or  $10''$  or  $19''$  to the total height of the tower.  How many differnt tower heights can be achieved using all 94 of the bricks?

Answer: 465


Let \( x_{1}, x_{2}, \cdots, x_{n} \) and \( a_{1}, a_{2}, \cdots, a_{n} (n>2) \) be sequences of arbitrary real numbers satisfying the following conditions:
1. \( x_{1}+x_{2}+\cdots+x_{n}=0 \);
2. \( \left|x_{1}\right|+\left|x_{2}\right|+\cdots+\left|x_{n}\right|=1 \);
3. \( a_{1} \geqslant a_{2} \geqslant \cdots \geqslant a_{n} \).

To ensure that the inequality \( \mid a_{1} x_{1}+a_{2} x_{2}+\cdots+a_{n} x_{n} \mid \leqslant A\left(a_{1}-a_{n}\right) \) holds, find the minimum value of \( A \).

Answer: \frac{1}{2}


Choose $4-4$: Parameter Equation Lecture. In the plane rectangular coordinate system $xOy$, with $O$ as the pole and the non-negative half-axis of $x$ as the polar axis, establish a polar coordinate system. The polar coordinates of point $P$ are $(2\sqrt{3}, \dfrac{\pi}{6})$. The polar coordinate equation of curve $C$ is $\rho ^{2}+2\sqrt{3}\rho \sin \theta = 1$.

(Ⅰ) Write down the rectangular coordinates of point $P$ and the general equation of curve $C.

(Ⅱ) If $Q$ is a moving point on $C$, find the minimum value of the distance from the midpoint $M$ of $PQ$ to the line $l: \left\{\begin{array}{l}x=3+2t\\y=-2+t\end{array}\right.$ (where $t$ is a parameter).

Answer: \dfrac{11\sqrt{5}}{10} - 1