A high school has three math teachers. To facilitate students, math teachers are scheduled for duty from Monday to Friday, with two teachers on duty on Monday. If each teacher is on duty for two days a week, then there are ________ possible duty schedules for a week.

Answer: 36


The polynomial \( p(x) \) has all coefficients 0 or 1, and \( p(0) = 1 \). Show that if the complex number \( z \) is a root, then \( |z| \geq \frac{\sqrt{5} - 1}{2} \).

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


Let \( n \) be the product of 3659893456789325678 and 342973489379256. Determine the number of digits of \( n \).

Answer: 34


A circle passing through the vertex \( P \) of triangle \( PQR \) touches side \( QR \) at point \( F \) and intersects sides \( PQ \) and \( PR \) at points \( M \) and \( N \), respectively, different from vertex \( P \). Find the ratio \( QF : FR \) if it is known that the length of side \( PQ \) is 1.5 times the length of side \( PR \), and the ratio \( QM : RN = 1 : 6 \).

Answer: 1/2


The number obtained from the last two nonzero digits of $90!$ is equal to $n$. What is $n$?

Answer: 12


Let  $ u_1$ ,  $ u_2$ ,  $ \ldots$ ,  $ u_{1987}$  be an arithmetic progression with  $ u_1 \equal{} \frac {\pi}{1987}$  and the common difference  $ \frac {\pi}{3974}$ . Evaluate
\[ S \equal{} \sum_{\epsilon_i\in\left\{ \minus{} 1, 1\right\}}\cos\left(\epsilon_1 u_1 \plus{} \epsilon_2 u_2 \plus{} \cdots \plus{} \epsilon_{1987} u_{1987}\right)
\]

Answer: 0


A square-shaped area with a side length of \(10 \text{ km}\) has car routes running along two parallel sides. Within this area, there are 4 observation stations. It is necessary to build connecting roads that consist of segments parallel to the sides of the square so that each observation station can cycle to both car routes. Cycling on the car routes is not allowed.

Prove that it is possible to achieve this with at most \(25 \text{ km}\) of roads, regardless of the locations of the observation stations, but not always with a shorter distance.

Answer: 25


Find the number of solutions to:
\[\sin x = \left(\frac{1}{3}\right)^x\]
on the interval $(0, 50\pi)$.

Answer: 50


The perimeter of triangle $APM$ is $152$, and the angle $PAM$ is a right angle. A circle of radius $19$ with center $O$ on $\overline{AP}$ is drawn so that it is tangent to $\overline{AM}$ and $\overline{PM}$. Given that $OP=m/n$ where $m$ and $n$ are relatively prime positive integers, find $m+n$.

Answer: 98


There are 15 girls in a class of 27 students. The ratio of boys to girls in this class is:

Answer: 4:5


Find the number of ordered quadruples \((a,b,c,d)\) of nonnegative real numbers such that
\[
a^2 + b^2 + c^2 + d^2 = 9,
\]
\[
(a + b + c + d)(a^3 + b^3 + c^3 + d^3) = 81.
\]

Answer: 15


Distribute 5 students into 3 groups: Group A, Group B, and Group C, with the condition that Group A must have at least two people, and Groups B and C must have at least one person each. How many different distribution schemes exist?

Answer: 80


Find the greatest positive integer \( N \) with the following property: there exist integers \( x_{1}, \ldots, x_{N} \) such that \( x_{i}^{2} - x_{i} x_{j} \) is not divisible by 1111 for any \( i \neq j \).

Answer: 1000


A fair coin is tossed 4 times. Calculate the probability of getting at least two consecutive heads.

Answer: \frac{1}{2}


Answer: 729


Determine the smallest positive integer  $a$  for which there exist a prime number  $p$  and a positive integer  $b \ge 2$  such that   $$ \frac{a^p -a}{p}=b^2. $$

Answer: 9


Find the sum of all integer values of \( h \) for which the equation \( ||r+h|-r|-4r = 9|r-3| \) with respect to \( r \) has at most one root.

Answer: -93


Answer: 3^{2^{2018}}-1


Find all solutions in positive integers to \( n^2 = \frac{m^5 - 1}{m - 1} \).

Answer: (n,)=(11,3)


There exists a scalar $k$ such that for any vectors $\mathbf{a},$ $\mathbf{b},$ and $\mathbf{c}$ such that $\mathbf{a} + \mathbf{b} + \mathbf{c} = \mathbf{0},$ the equation
\[k (\mathbf{b} \times \mathbf{a}) + \mathbf{b} \times \mathbf{c} + \mathbf{c} \times \mathbf{a} = \mathbf{0}\]holds.  Find $k.$

Answer: 2


A and B each independently toss a fair coin. A tosses the coin 10 times, and B tosses the coin 11 times. What is the probability that the number of heads B gets is greater than the number of heads A gets?

Answer: 1/2


One student has 6 mathematics books, and another has 8. In how many ways can they exchange three books?

Answer: 1120


Find all the odd positive integers  $n$  such that there are  $n$  odd integers  $x_1, x_2,..., x_n$  such that  $$ x_1^2+x_2^2+...+x_n^2=n^4 $$

Answer: n\equiv1\pmod{8}


Given that the terminal side of angle $\alpha$ passes through point $P$((-$	sqrt{3}$, $m$)), and $\sin \alpha$ = $\frac{\sqrt{3}}{4}$ $m$($m$($m$ne 0)), determine in which quadrant angle $\alpha$ lies, and find the value of $\tan \alpha$.

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


Given the set \( A = \{1,2,3,4\} \), Ander randomly selects a number from \( A \) every second (with replacement). The selection stops when the sum of the last two selected numbers is a prime number. What is the probability that the last number selected is "1"?

Answer: \frac{15}{44}


58 balls of two colors - red and blue - are arranged in a circle. It is known that the number of consecutive triplets of balls with a majority of red balls is equal to the number of triplets with a majority of blue balls. What is the minimum possible number of red balls?

Answer: 20


The shaded region consists of 16 congruent squares. If $PQ = 6$ cm, what is the area of the entire shaded region?

[asy]
for(int i = 0; i < 5; ++i)
{

for(int j = 0; j < 2; ++j)

{

filldraw((i,j)--(i+1,j)--(i+1,j+1)--(i,j+1)--(i,j)--cycle,gray,linewidth(2));

}
}

for(int i = 0; i < 2; ++i)
{

for(int j = 0; j < 5; ++j)

{

filldraw((i,j)--(i+1,j)--(i+1,j+1)--(i,j+1)--(i,j)--cycle,gray,linewidth(2));

}
}

draw((2,2)--(5,2)--(2,5)--(2,2)--cycle,linewidth(2));

label("P",(2,5),N);
label("Q",(5,2),E);
[/asy]

Answer: 32


In triangle \( \triangle ABC \), the angles are \( \angle B = 30^\circ \) and \( \angle A = 90^\circ \). Point \( K \) is marked on side \( AC \), and points \( L \) and \( M \) are marked on side \( BC \) such that \( KL = KM \) (point \( L \) lies on segment \( BM \)).

Find the length of segment \( LM \), given that \( AK = 4 \), \( BL = 31 \), and \( MC = 3 \).

Answer: 14


Suppose $a$, $b$, $c$ are positive integers such that $a+b+c=23$ and $\gcd(a,b)+\gcd(b,c)+\gcd(c,a)=9.$ What is the sum of all possible distinct values of $a^2+b^2+c^2$?

Answer: 438


Given the line y=b intersects with the function f(x)=2x+3 and the function g(x)=ax+ln x (where a is a real constant in the interval [0, 3/2]), find the minimum value of |AB|.

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


One writes 268 numbers around a circle, such that the sum of 20 consectutive numbers is always equal to 75. The number 3, 4 and 9 are written in positions 17, 83 and 144 respectively. Find the number in position 210.

Answer: -1


Is it true that any number divisible by 6 and greater than 1000 can be represented as

$$
n(n+1)(n+2)(n+3)(n+4) - m(m+1)(m+2)
$$

where $m$ and $n$ are natural numbers?

Answer: No


The numbers  $1,2,\ldots,49,50$  are written on the blackboard. Ann performs the following operation: she chooses three arbitrary numbers  $a,b,c$  from the board, replaces them by their sum  $a+b+c$  and writes  $(a+b)(b+c)(c+a)$  to her notebook. Ann performs such operations until only two numbers remain on the board (in total 24 operations). Then she calculates the sum of all  $24$  numbers written in the notebook. Let  $A$  and  $B$  be the maximum and the minimum possible sums that Ann san obtain.
Find the value of  $\frac{A}{B}$ .

*(I. Voronovich)*

Answer: 4


Given squares $ABCD$ and $EFGH$ are congruent, $AB=12$, and $H$ is located at vertex $D$ of square $ABCD$. Calculate the total area of the region in the plane covered by these squares.

Answer: 252


Find all positive integers  $a, b,c$  greater than  $1$ , such that  $ab + 1$  is divisible by  $c, bc + 1$  is divisible by  $a$  and  $ca + 1$  is divisible by  $b$ .

Answer: (2,3,7)


Solve the following equation and provide its root. If the equation has multiple roots, provide their product.

\[ \sqrt{2 x^{2} + 8 x + 1} - x = 3 \]

Answer: -8


There are  $n$  cards on a table in a line, with a positive real written on eachcard. LTF and Sunny are playing a game where they take turns taking away the first or the last card in line. The player that has the bigger sum of all the numberson his cards wins. If LTF goes first, find all  $n$  such that LTF can always prevent Sunny from winning, regardless of the numbers written on the cards.

Answer: n


From point \( A \), tangents \( AB \) and \( AC \) are drawn to a circle with center \( O \). Through point \( X \) on segment \( BC \), a line \( KL \) is drawn perpendicular to \( XO \) (points \( K \) and \( L \) lie on lines \( AB \) and \( AC \)). Prove that \( KX = XL \).

Answer: KX=XL


A metal weight has a mass of 20 kg and is an alloy of four metals. The mass of the first metal in this alloy is one and a half times greater than that of the second, the mass of the second metal is in the ratio $3:4$ to the mass of the third, and the mass of the third metal is in the ratio $5:6$ to the mass of the fourth. Determine the mass of the third metal. Give the answer in kilograms, rounded to hundredths if necessary.

Answer: 4.91


In the plane rectangular coordinate system $xOy$, the parameter equations of the line $l$ are $\left\{\begin{array}{l}x=1+\frac{{\sqrt{2}}}{2}t\\ y=\frac{{\sqrt{2}}}{2}t\end{array}\right.$ (where $t$ is the parameter). Taking the coordinate origin $O$ as the pole and the positive half-axis of the $x$-axis as the polar axis, establishing a polar coordinate system with the same unit length, the polar coordinate equation of the curve $C$ is $ρ=2\sqrt{2}\sin({θ+\frac{π}{4}})$.
$(1)$ Find the general equation of the line $l$ and the rectangular coordinate equation of the curve $C$;
$(2)$ Let point $P(4,3)$, the intersection points of line $l$ and curve $C$ be $A$ and $B$, find the value of $\frac{1}{{|{PA}|}}+\frac{1}{{|{PB}|}}$.

Answer: \frac{{5\sqrt{2}}}{{11}}


If the price of a stamp is 50 cents, what is the maximum number of stamps that could be purchased with $50? Furthermore, if a customer buys more than 80 stamps, they receive a discount of 5 cents per stamp. How many stamps would then be purchased at maximum?

Answer: 111


For all triples \((x, y, z)\) satisfying the system

\[
\left\{\begin{array}{l}
\sqrt{3} \sin x = \tan y \\
2 \sin y = \cot z \\
\sin z = 2 \tan x
\end{array}\right.
\]

find the minimum value of \(\cos x - \cos z\).

Answer: -\frac{7 \sqrt{2}}{6}


Let \( F \) be the number of integral solutions of \( x^{2}+y^{2}+z^{2}+w^{2}=3(x+y+z+w) \). Find the value of \( F \).

Answer: 208


A secret facility is a rectangle measuring $200 \times 300$ meters. There is one guard at each of the four corners of the facility. An intruder approaches the perimeter of the facility from the outside, and all the guards run towards the intruder by the shortest routes along the outer perimeter (while the intruder remains stationary). Three guards cover a total distance of 850 meters to reach the intruder. How many meters did the fourth guard run to reach the intruder?

Answer: 150


From the vertex $B$ of an isosceles triangle $ABC$, a height $BD$ is dropped to its base $AC$. Each of the legs $AB$ and $BC$ of triangle $ABC$ is equal to 8. In triangle $BCD$, a median $DE$ is drawn. A circle is inscribed in triangle $BDE$, touching side $BE$ at point $K$ and side $DE$ at point $M$. Segment $KM$ is equal to 2. Find angle $A$.

Answer: 30


The rectangle shown has length $AC=32$, width $AE=20$, and $B$ and $F$ are midpoints of $\overline{AC}$ and $\overline{AE}$, respectively.  The area of quadrilateral $ABDF$ is

Answer: 320


Given triangle  $ ABC$  of area 1. Let  $ BM$  be the perpendicular from  $ B$  to the bisector of angle  $ C$ . Determine the area of triangle  $ AMC$ .

Answer: \frac{1}{2}


In a bus, there are single and double seats. In the morning, 13 people were sitting in the bus, and there were 9 completely free seats. In the evening, 10 people were sitting in the bus, and there were 6 completely free seats. How many seats are there in the bus?

Answer: 16


Let $[x]$ denote the greatest integer less than or equal to the real number $x$. Given:
$$
a_{k}=\sum_{i=k^{2}}^{(k+1)^{2}-1} \frac{1}{i} \quad (k=1,2, \cdots)
$$
Find:
$$
\sum_{k=1}^{n}\left(\left[\frac{1}{a_{k}}\right]+\left[\frac{1}{a_{k}}+\frac{1}{2}\right]\right) .
$$

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


In triangle  $ABC$ , let  $I, O, H$  be the incenter, circumcenter and orthocenter, respectively. Suppose that  $AI = 11$  and  $AO = AH = 13$ . Find  $OH$ .

*Proposed by Kevin You*

Answer: 10


A quadrilateral that has consecutive sides of lengths $70,90,130$ and $110$ is inscribed in a circle and also has a circle inscribed in it. The point of tangency of the inscribed circle to the side of length 130 divides that side into segments of length $x$ and $y$. Find $|x-y|$.
$\text{(A) } 12\quad \text{(B) } 13\quad \text{(C) } 14\quad \text{(D) } 15\quad \text{(E) } 16$

Answer: 13


Using each of the eight digits  $1,3,4,5,6,7,8$  and  $9$  exactly once, a three-digit number  $A$ , two two-digit numbers  $B$  and  $C$ ,  $B<C$ , and a one digit number  $D$  are formed. The numbers are such that  $A+D=B+C=143$ . In how many ways can this be done?

Answer: 24


Given an ellipse with its foci on the x-axis and its lower vertex at D(0, -1), the eccentricity of the ellipse is $e = \frac{\sqrt{6}}{3}$. A line L passes through the point P(0, 2).
(Ⅰ) Find the standard equation of the ellipse.
(Ⅱ) If line L is tangent to the ellipse, find the equation of line L.
(Ⅲ) If line L intersects the ellipse at two distinct points M and N, find the maximum area of triangle DMN.

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


A three-stage launch vehicle consists of cylindrical stages. All these cylinders are similar to each other. The length of the middle stage is half the sum of the lengths of the first and third stages. When fueled, the mass of the middle stage is $13 / 6$ times less than the combined mass of the fueled first and third stages. Find the ratio of the lengths of the first and third stages. The masses of the engines, instruments, and the shells of the stages can be neglected.

Answer: \frac{7}{5}


There are 4 college entrance examination candidates entering the school through 2 different intelligent security gates. Each security gate can only allow 1 person to pass at a time. It is required that each security gate must have someone passing through. Then there are ______ different ways for the candidates to enter the school. (Answer in numbers)

Answer: 72


Let $S$ be the set of all polynomials of the form $z^3 + az^2 + bz + c$, where $a$, $b$, and $c$ are integers. Find the number of polynomials in $S$ such that each of its roots $z$ satisfies either $|z| = 20$ or $|z| = 13$.

Answer: 540


A passenger on a train knows that the speed of this train on the current section of the track is 40 km/h. As soon as an oncoming train began to pass by the window, the passenger started a stopwatch and noticed that the oncoming train took 3 seconds to pass by the window. Determine the speed of the oncoming train, given that its length is 75 meters.

Answer: 50\,


In a $29 \times 29$ table, the numbers $1, 2, 3, \ldots, 29$ are written, each number repeated 29 times. It turns out that the sum of the numbers above the main diagonal is three times the sum of the numbers below this diagonal. Find the number written in the central cell of the table.

Answer: 15


Buses travel along a country road, at equal intervals in both directions and at equal speeds. A cyclist, traveling at $16 \, \mathrm{km/h}$, begins counting buses from the moment when two buses meet beside him. He counts the 31st bus approaching from the front and the 15th bus from behind, both meeting the cyclist again.

How fast were the buses traveling?

Answer: 46


Given a convex pentagon, Petya wrote down the sine values of all its angles in his notebook, and Vasya wrote down the cosine values of all its angles. It was found that among the numbers written by Petya, there are not four different values. Can all the numbers written by Vasya be different?

Answer: No


Let  $A=\{a_1,a_2,\cdots,a_n,b_1,b_2,\cdots,b_n\}$  be a set with  $2n$  distinct elements, and  $B_i\subseteq A$  for any  $i=1,2,\cdots,m.$  If  $\bigcup_{i=1}^m B_i=A,$  we say that the ordered  $m-$ tuple  $(B_1,B_2,\cdots,B_m)$  is an ordered  $m-$ covering of  $A.$  If  $(B_1,B_2,\cdots,B_m)$  is an ordered  $m-$ covering of  $A,$  and for any  $i=1,2,\cdots,m$  and any  $j=1,2,\cdots,n,$   $\{a_j,b_j\}$  is not a subset of  $B_i,$  then we say that ordered  $m-$ tuple  $(B_1,B_2,\cdots,B_m)$  is an ordered  $m-$ covering of  $A$  without pairs. Define  $a(m,n)$  as the number of the ordered  $m-$ coverings of  $A,$  and  $b(m,n)$  as the number of the ordered  $m-$ coverings of  $A$  without pairs. $(1)$  Calculate  $a(m,n)$  and  $b(m,n).$  $(2)$  Let  $m\geq2,$  and there is at least one positive integer  $n,$  such that  $\dfrac{a(m,n)}{b(m,n)}\leq2021,$  Determine the greatest possible values of  $m.$

Answer: 26


Let \( a, b, c, d \) be integers such that \( a > b > c > d \geq -2021 \) and
\[ \frac{a+b}{b+c} = \frac{c+d}{d+a} \]
(and \( b+c \neq 0 \neq d+a \)). What is the maximum possible value of \( a \cdot c \)?

Answer: 510050


Answer: 2017!+2018


Given the set $T$ of all positive divisors of $2004^{100}$, determine the maximum possible number of elements in a subset $S$ of $T$ such that no element of $S$ is a multiple of any other element in $S$.

Answer: 10201


Using the digits 0, 1, 2, 3, and 4, how many even numbers can be formed without repeating any digits?

Answer: 163


Given the function $f(x)=e^{x}(ax+b)-x^{2}-4x$, the equation of the tangent line to the curve $y=f(x)$ at the point $(0, f(0))$ is $y=4x+4$.

(1) Find the values of $a$ and $b$;

(2) Determine the intervals of monotonicity for $f(x)$ and find the maximum value of $f(x)$.

Answer: 4(1-e^{-2})


The quadrilateral \(ABCD\) is inscribed in a circle. It is known that \(BC = CD, \angle BCA = 64^\circ,\) and \(\angle ACD = 70^\circ\). A point \(O\) is marked on the segment \(AC\) such that \(\angle ADO = 32^\circ\). What is the measure of the angle \(\angle BOC\) in degrees?

Answer: 58


Bus stop \(B\) is located on a straight highway between stops \(A\) and \(C\). After some time driving from \(A\), the bus finds itself at a point on the highway where the distance to one of the three stops is equal to the sum of the distances to the other two stops. After the same amount of time, the bus again finds itself at a point with this property, and 25 minutes later it arrives at \(B\). How much time does the bus need for the entire journey from \(A\) to \(C\) if its speed is constant and it stops at \(B\) for 5 minutes?

Answer: 180


Three rectangles are placed mutually adjacent and without gaps or overlaps to form a larger rectangle. One of the three rectangles has dimensions 70 by 110. Another of the rectangles has dimensions 40 by 80. What is the maximum perimeter of the third rectangle?

Answer: 300


The secant passing through the intersection points of two circles, one with center \( O_1 \) and radius \( 4 \text{ cm} \), and the other with center \( O_2 \) and radius \( 6 \text{ cm} \), intersects the segment \( O_1O_2 \) at point \( T \). It is known that the length of \( O_1O_2 \) is not less than \( 6 \text{ cm} \). The larger circle intersects the segment \( O_1O_2 \) at point \( A \), while the smaller circle intersects it at point \( B \), with the ratio \( AT : BT = 1 : 2 \). Calculate the length of the segment \( O_1O_2 \).

Answer: 6


Answer: 20302


Suppose there exists a convex \( n \)-gon such that each of its angle measures, in degrees, is an odd prime number. Compute the difference between the largest and smallest possible values of \( n \).

Answer: 356


The function $y=(m^2-m-1)x^{m^2-3m-3}$ is a power function, and it is an increasing function on the interval $(0, +\infty)$. Find the value of $m$.

Answer: -1


If \( f(n, k) \) is the number of ways to divide the set \(\{1,2, \ldots, n\}\) into \(k\) nonempty subsets and \(m\) is a positive integer, find a formula for \(\sum_{k=1}^{n} f(n, k) m(m-1)(m-2) \cdots (m-k+1)\).

Answer: ^n


In the xy-plane, what is the length of the shortest path from $(0,0)$ to $(12,16)$ that does not go inside the circle $(x-6)^{2}+(y-8)^{2}= 25$?

Answer: 10\sqrt{3}+\frac{5\pi}{3}


Let \(a, b, c\) be the sides of a triangle. Prove the inequality \(a^{3} + b^{3} + 3abc > c^{3}\).

Answer: ^3+b^3+3abc>^3


In an equilateral triangle \( ABC \) with side length 1, points \( D \) and \( E \) are taken on sides \( AB \) and \( AC \) respectively such that when the triangle is folded along line segment \( DE \), vertex \( A \) falls exactly on side \( BC \). The minimum length of \( AD \) is \( \qquad \)

Answer: 2\sqrt{3}-3


An ancient Greek was born on January 7, 40 B.C., and died on January 7, 40 A.D. Calculate the number of years he lived.

Answer: 79


Let \( a \) be an integer such that \( |a| \leq 2005 \). Find the number of values of \( a \) for which the system of equations
\[
\begin{cases}
x^2 = y + a, \\
y^2 = x + a
\end{cases}
\]
has integer solutions.

Answer: 90


A frog located at $(x,y)$, with both $x$ and $y$ integers, makes successive jumps of length $5$ and always lands on points with integer coordinates. Suppose that the frog starts at $(0,0)$ and ends at $(1,0)$. What is the smallest possible number of jumps the frog makes?
$\textbf{(A)}\ 2 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 5 \qquad \textbf{(E)}\ 6$

Answer: 3\textbf{(B)}


Answer: 1716


Find the range of real numbers $a$ such that the inequality $a x^{2}-2 a x+2 a-3<0$ satisfies the following conditions respectively:
(1) The solution set is $\mathbf{R}$ (the set of all real numbers).
(2) The solution set is $\varnothing$ (the empty set).
(3) There is at least one real solution.

Answer: (-\infty,3)


Pools $A$ and $B$ are both rectangular cuboids with a length of 3 meters, a width of 2 meters, and a depth of 1.2 meters. Valve 1 is used to fill pool $A$ with water and can fill an empty pool $A$ in 18 minutes. Valve 2 is used to transfer water from pool $A$ to pool $B$, taking 24 minutes to transfer a full pool $A$. If both Valve 1 and Valve 2 are opened simultaneously, how many cubic meters of water will be in pool $B$ when the water depth in pool $A$ is 0.4 meters?

Answer: 7.2


Define \(P(x) =(x-1^2)(x-2^2)\cdots(x-100^2)\). How many integers \(n\) are there such that \(P(n)\leq 0\)?

Answer: 5100


The number \( n \) is such that \( 8d \) is a 100-digit number, and \( 81n - 102 \) is a 102-digit number. What can be the second digit from the beginning of \( n \)?

Answer: 2


Define a positive integer $n$ to be a factorial tail if there is some positive integer $m$ such that the decimal representation of $m!$ ends with exactly $n$ zeroes. How many positive integers less than $2500$ are not factorial tails?

Answer: 499


Choose one digit from the set {0, 2} and two different digits from the set {1, 3, 5} to form a three-digit number without any repeating digits. The total number of such odd three-digit numbers is _________.

Answer: 18


Jack and Jill are going swimming at a pool that is one mile from their house. They leave home simultaneously. Jill rides her bicycle to the pool at a constant speed of $10$ miles per hour. Jack walks to the pool at a constant speed of $4$ miles per hour. How many minutes before Jack does Jill arrive?
$\textbf{(A) }5\qquad\textbf{(B) }6\qquad\textbf{(C) }8\qquad\textbf{(D) }9\qquad \textbf{(E) }10$

Answer: \textbf{(D)}~9


Find the number of subsets \( B \) of the set \( \{1, 2, \ldots, 2005\} \) such that the sum of its elements leaves a remainder of 2006 when divided by 2048.

Answer: 2^{1994}


Given triangle \( \triangle ABC \) with \( AB < AC \), the altitude \( AD \), angle bisector \( AE \), and median \( AF \) are drawn from \( A \), with \( D, E, F \) all lying on \(\overline{BC}\). If \( \angle BAD = 2 \angle DAE = 2 \angle EAF = \angle FAC \), what are all possible values of \( \angle ACB \)?

Answer: 30


Kolya, an excellent student in the 7th-8th grade, found the sum of the digits of all the numbers from 0 to 2012 and added them all together. What number did he get?

Answer: 28077


A solid right prism $PQRSTU$ has a height of 20. Its bases are equilateral triangles with side length 10. Points $M$, $N$, and $O$ are the midpoints of edges $PQ$, $QR$, and $RT$, respectively. Calculate the perimeter of triangle $MNO$.

Answer: 5 + 10\sqrt{5}


In a box, there are 22 kg of cranberries. How, using a single 2-kilogram weight and a two-pan scale, can you measure out 17 kg of cranberries in two weighings?

Answer: 17


The real numbers \( a_0, a_1, \ldots, a_n \) satisfy \( a_0 = a_n = 0 \) and \( a_k = c + \sum_{i=k}^{n-1} (a_{i-k} (a_i + a_{i+1})) \). Show that \( c \leq \frac{1}{4n} \).

Answer: \le\frac{1}{4n}


How many distinct four-digit even numbers can be formed using the digits 0, 1, 2, 3?

Answer: 10


The remainder when 392 is divided by a 2-digit positive integer is \( b \). If \( c \) is the number of such 2-digit positive integers, find the value of \( c \).

Answer: 4


Let  $f(x) = x^3 + 3x - 1$  have roots  $ a, b, c$ . Given that  $\frac{1}{a^3 + b^3}+\frac{1}{b^3 + c^3}+\frac{1}{c^3 + a^3}$  can be written as  $\frac{m}{n}$ , where  $m, n$  are positive integers and  $gcd(m, n) = 1$ , find  $100m + n$ .

Answer: 3989


Given a geometric sequence $\{a_n\}$, where the sum of the first $n$ terms is denoted as $S_n$, and $S_n = a\left(\frac{1}{4}\right)^{n-1} + 6$, find the value of $a$.

Answer: -\frac{3}{2}


In a certain exam, 6 questions are randomly selected from 20 questions. If a student can correctly answer at least 4 of these questions, they pass the exam. If they can correctly answer at least 5 of these questions, they achieve an excellent grade. It is known that a certain student can correctly answer 10 of these questions and that they have already passed the exam. The probability that they achieve an excellent grade is \_\_\_\_\_\_.

Answer: \frac{13}{58}


Given the function \( f(x) = -\frac{1}{2} x^{2} + \frac{13}{2} \) has a minimum value of \( 2a \) and a maximum value of \( 2b \) on the interval \([a, b]\), find the interval \([a, b]\).

Answer: [1,3]or[-2-\sqrt{17},\frac{13}{4}]


ABCD is a square. BDEF is a rhombus with A, E, and F collinear. Find ∠ADE.

Answer: 15


When ethane is mixed with chlorine gas under lighting conditions, determine the maximum number of substances that can be generated.

Answer: 10


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](https://artofproblemsolving.com/wiki/index.php/Odd_integer) and $a_i>a_{i+1}$ if $i$ is [ even](https://artofproblemsolving.com/wiki/index.php/Even_integer). How many snakelike integers between 1000 and 9999 have four distinct digits?

Answer: 882


Let  $f(x) =x^4+14x^3+52x^2+56x+16.$   Let  $z_1, z_2, z_3, z_4$  be the four roots of  $f$ .  Find the smallest possible value of  $|z_az_b+z_cz_d|$  where  $\{a,b,c,d\}=\{1,2,3,4\}$ .

Answer: 8


There are $10$ people standing equally spaced around a circle. Each person knows exactly $3$ of the other $9$ people: the $2$ people standing next to her or him, as well as the person directly across the circle. How many ways are there for the $10$ people to split up into $5$ pairs so that the members of each pair know each other?

Answer: 12


Given a set of 2013 points such that among any three of these points, there are always two whose distance is less than or equal to 1. Show that it is possible to cover 1007 of these points with a disk of radius 1.

Answer: 1007


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

Answer: 8


A polyhedron is called regular if all its faces are regular congruent polygons and all its polyhedral angles are also regular and congruent to each other. Investigate the possibility of constructing regular polyhedra with $n$-sided faces and $m$-sided angles and determine the possible number of their types using the following considerations.

Divide the entire space into $f$ congruent regular $n$-sided angles with a common vertex $S$, with $m$ angles at each edge, and express the magnitudes of their dihedral angles in terms of $n$ and $f$. If the calculated magnitude of the dihedral angle is used to construct a regular $n$-sided pyramid, whose base is a regular $n$-sided polygon (the face of the sought polyhedron), then such congruent pyramids can be assembled into a regular polyhedron (for example, a cube is composed of six regular quadrilateral pyramids with a common vertex at the center of the cube). It can be proven that any regular polyhedron can be obtained through such construction.

Answer: 5


The triangle \(PQR\) is isosceles with \(PR = QR\). Angle \(PRQ = 90^\circ\) and length \(PQ = 2 \text{ cm}\). Two arcs of radius \(1 \text{ cm}\) are drawn inside triangle \(PQR\). One arc has its center at \(P\) and intersects \(PR\) and \(PQ\). The other arc has its center at \(Q\) and intersects \(QR\) and \(PQ\). What is the area of the shaded region, in \(\text{cm}^2\)?

Answer: 1-\frac{\pi}{4}


Let \( x_{1}, x_{2}, x_{3}, x_{4} \) be non-negative real numbers satisfying the equation:
\[
x_{1} + x_{2} + x_{3} + x_{4} = 1
\]
Find the maximum value of \( \sum_{1 \leq i < j \leq 4}(x_{i} + x_{j}) \sqrt{x_{i} x_{j}} \), and determine the values of \( x_{1}, x_{2}, x_{3}, x_{4} \) that achieve this maximum value.

Answer: 3/4


In space, there are 43 points: 3 yellow and 40 red. No four of them lie in the same plane. Can the number of triangles with red vertices, linked with a triangle with yellow vertices, be equal to \( 2023 \)?

A triangle with yellow vertices is linked with a red one if the contour of the red one intersects the part of the plane bounded by the yellow one at exactly one point. Triangles differing by a permutation of vertices are considered the same.

Answer: No


A ticket to a school play cost $x$ dollars, where $x$ is a whole number. A group of 9th graders buys tickets costing a total of $$48$, and a group of 10th graders buys tickets costing a total of $$64$. How many values for $x$ are possible?
$\textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 5$

Answer: \textbf{(E)}\5


How many four-digit numbers are there in which at least one digit occurs more than once?

Answer: 4464


The children brought full baskets of mushrooms from the forest. A total of 289 mushrooms were collected, and each basket contained the same number of mushrooms. How many children were there?

Answer: 17


It is known that ЖЖ + Ж = МЁД. What digit does the product В $\cdot И \cdot H \cdot H \cdot U \cdot \Pi \cdot У \cdot X$ end in (different letters represent different digits, identical letters represent identical digits)?

Answer: 0


The numbers \(a\), \(b\), and \(c\) satisfy the condition

\[
\frac{a}{b-c}+\frac{b}{c-a}+\frac{c}{a-b}=0
\]

Prove that

\[
\frac{a}{(b-c)^{2}}+\frac{b}{(c-a)^{2}}+\frac{c}{(a-b)^{2}}=0
\]

Answer: 0


In the quadrilateral pyramid \( P-ABCD \), the plane \( PAD \perp \) plane \( ABCD \), \(\triangle PAD\) is an equilateral triangle with side length 2, the base \( ABCD \) is a rhombus, and \(\angle BAD = 60^\circ\). Find the measure of the acute dihedral angle between planes \( PAD \) and \( PBC \).

Answer: 45


Answer: 315


Find the derivative \( y_{x}^{\prime} \).

$$
\left\{\begin{array}{l}
x=\sqrt{t-t^{2}}-\operatorname{arctg} \sqrt{\frac{1-t}{t}} \\
y=\sqrt{t}-\sqrt{1-t} \cdot \arcsin \sqrt{t}
\end{array}\right.
$$

Answer: \frac{\sqrt{}\cdot\arcsin\sqrt{}}{2(1-)}


A soccer ball rolls at $4 \mathrm{~m} / \mathrm{s}$ towards Marcos in a direct line from Michael. The ball is $15 \mathrm{~m}$ ahead of Michael who is chasing it at $9 \mathrm{~m} / \mathrm{s}$. Marcos is $30 \mathrm{~m}$ away from the ball and is running towards it at $8 \mathrm{~m} / \mathrm{s}$. Calculate the distance between Michael and Marcos when the ball is touched for the first time by one of them.

Answer: 2.5


Quadrilateral \(ABCD\) is inscribed in a circle. Diagonal \(AC\) is the bisector of angle \(BAD\) and intersects diagonal \(BD\) at point \(K\).

Find \(KC\), if \(BC = 4\) and \(AK = 6\).

Answer: 2


Suppose the quadratic function \( f(x)=a x^{2}+b x+c \) (where \( a, b, c \in \mathbf{R} \) and \( a \neq 0 \)) satisfies the following conditions:
(1) For \( x \in \mathbf{R} \), \( f(x-4)=f(2-x) \), and \( f(x) \geqslant x \);
(2) For \( x \in(0,2) \), \( f(x) \leqslant \left(\frac{x+1}{2}\right)^{2} \);
(3) The minimum value of \( f(x) \) on \( \mathbf{R} \) is 0.

Find the largest real number \( m \) (with \( m > 1 \)) such that there exists \( t \in \mathbf{R} \) such that for all \( x \in [1, m] \), \( f(x+t) \leqslant x \).

Answer: 9


A rectangular piece of paper $A B C D$ is folded and flattened as shown in the diagram, so that triangle $D C F$ falls onto triangle $D E F$, with vertex $E$ precisely landing on side $A B$. Given $\angle 1 = 22^\circ$, find the measure of $\angle 2$.

Answer: 44


For circles \( k_{j} \) centered at \( O \) (where \( j = 1, 2 \)), each has a tangent \( e_{j} \) parallel to a given direction \( i \) that intersects a ray \( E_{j} \) starting from \( O \), and a perpendicular \( f_{j} \) to \( i \) at the point of intersection. Another ray from \( O \) intersects the circle \( k_{j} \) at \( P_{j} \) and meets the line \( f_{j} \), drawn through \( P_{j} \) parallel to \( i \), at \( Q_{j} \). It is required to prove that the line \( Q_{1} Q_{2} \) passes through \( O \).

Answer: Q_{1}Q_{2}passesthroughO


Find the largest natural number in which all the digits are different and each pair of adjacent digits differs by 6 or 7.

Answer: 60718293


Let  $\alpha$  and  $\beta$  be positive integers such that   $\dfrac{43}{197} < \dfrac{ \alpha }{ \beta } < \dfrac{17}{77}$ . Find the minimum possible value of  $\beta$ .

Answer: 32


Given that $a, b \in \mathbb{R}$, and $a^2 + 2ab - 3b^2 = 1$, find the minimum value of $a^2 + b^2$.

Answer: \frac{\sqrt{5} + 1}{4}


Danielle picks a positive integer $1 \leq n \leq 2016$ uniformly at random. What is the probability that \operatorname{gcd}(n, 2015)=1?

Answer: \frac{1441}{2016}


What is the limit of the following sequence?

$$
a_{n}=\frac{\sum_{j=0}^{\infty}\binom{n}{2 j} \cdot 2^{j}}{\sum_{j=0}^{\infty}\binom{n}{2 j+1} \cdot 2^{j}}
$$

Answer: \sqrt{2}


If the area of $\triangle ABC$ is $64$ square units and the geometric mean (mean proportional) between sides $AB$ and $AC$ is $12$ inches, then $\sin A$ is equal to
$\textbf{(A) }\dfrac{\sqrt{3}}{2}\qquad \textbf{(B) }\frac{3}{5}\qquad \textbf{(C) }\frac{4}{5}\qquad \textbf{(D) }\frac{8}{9}\qquad \textbf{(E) }\frac{15}{17}$

Answer: \textbf{(D)}\frac{8}{9}


What is the minimum number of circles required to completely cover a circle with twice the radius?

Answer: 7


Fill in the table with the numbers $0, 1, 2, \cdots, 14, 15$ so that for each row and each column, the remainders when divided by 4 are exactly $0, 1, 2, 3$ each, and the quotients when divided by 4 are also exactly $0, 1, 2, 3$ each, and determine the product of the four numbers in the bottom row of the table.

Answer: 32760


Answer: \frac{9134}{527}


Find the area of the region in the first quadrant \(x>0, y>0\) bounded above by the graph of \(y=\arcsin(x)\) and below by the graph of \(y=\arccos(x)\).

Answer: 2 - \sqrt{2}


Solve the inequality

$$
8 \cdot \frac{|x+3|-|x-5|}{|2x-11|-|2x+7|} - 9 \cdot \frac{|x+3|+|x-5|}{|2x-11|+|2x+7|} \geq -8
$$

Find the sum of its integer solutions that satisfy the condition \( |x| < 90 \).

Answer: 8


An  $m \times n$  chessboard where  $m \le n$  has several black squares such that no two rows have the same pattern. Determine the largest integer  $k$  such that we can always color  $k$  columns red while still no two rows have the same pattern.

Answer: n-+1


Let  $n$  be positive integer. Define a sequence  $\{a_k\}$  by
\[a_1=\frac{1}{n(n+1)},\ a_{k+1}=-\frac{1}{k+n+1}+\frac{n}{k}\sum_{i=1}^k a_i\ \ (k=1,\ 2,\ 3,\ \cdots).\]

(1) Find  $a_2$  and  $a_3$ .

(2) Find the general term  $a_k$ .

(3) Let  $b_n=\sum_{k=1}^n \sqrt{a_k}$ . Prove that  $\lim_{n\to\infty} b_n=\ln 2$ .

50 points

Answer: \ln2


Let \( a \) and \( b \) be positive integers such that \( 79 \mid (a + 77b) \) and \( 77 \mid (a + 79b) \). Find the smallest possible value of \( a + b \).

Answer: 193


Given a convex pentagon \(ABCDE\) with the properties \(AB = BC\), \(CD = DE\), \(\angle ABC = 150^\circ\), \(\angle CDE = 30^\circ\), and \(BD = 2\), find the area of pentagon \(ABCDE\).

Answer: 1


The Evil League of Evil plans to set out from their headquarters at (5,1) to poison two pipes: one along the line \( y = x \) and the other along the line \( x = 7 \). They wish to determine the shortest distance they can travel to visit both pipes and then return to their headquarters.

Answer: 4\sqrt{5}


It is known that the vertex of angle $\theta$ is at the origin of coordinates, its initial side coincides with the positive half-axis of the x-axis, and its terminal side falls on the ray $y= \frac {1}{2}x$ ($x\leq0$).
(Ⅰ) Find the value of $\cos\left( \frac {\pi}{2}+\theta \right)$;
(Ⅱ) If $\cos\left( \alpha+ \frac {\pi}{4} \right)=\sin\theta$, find the value of $\sin\left(2\alpha+ \frac {\pi}{4}\right)$.

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


A boy walks from home to school at a speed of \( p \) km/h and returns home along the same route at a speed of 3 km/h. If the average speed for the round trip is \( \frac{24}{q} \) km/h, find \( q \).

Answer: 7


\( n \) is a 17-digit number. \( m \) is derived from \( n \) by taking its decimal digits in the reverse order. Show that at least one digit of \( n + m \) is even.

Answer: 1


Let \( f:[0,1] \rightarrow \mathbb{R} \) be a function satisfying:
1. \( f(x) \geq 0 \) for all \( x \in [0,1] \);
2. \( f(1) = 1 \);
3. \( f(x) + f(y) \leq f(x+y) \) for all \( x, y \) such that \( x+y \in [0,1] \).

Find the smallest constant \( c \) such that \( f(x) \leq c x \) for all \( x \in [0,1] \) and prove your conclusion.

Answer: 2


From a point P outside the parabola \( y^2 = 2px \), two tangents are drawn to the parabola, touching it at points M and N respectively. Let F be the focus of the parabola. Prove that \(\angle PMF = \angle FPN\).

Answer: \anglePMF=\angleFPN


If $A = 3009 \div 3$, $B = A \div 3$, and $Y = A - B$, then what is the value of $Y$?

Answer: 669


In the polynomial \((1-z)^{b_{1}} \cdot (1-z^{2})^{b_{2}} \cdot (1-z^{3})^{b_{3}} \cdots (1-z^{32})^{b_{32}}\), where \(b_{i} (i=1,2, \cdots, 32)\) are positive integers, this polynomial has the following remarkable property: after expanding it and removing the terms with \(z\) of degree higher than 32, exactly \(1-2z\) remains. Determine \(b_{32}\) (the answer can be expressed as the difference of two powers of 2).

Answer: 2^{27}-2^{11}


The average of \( p, q, r, \) and \( s \) is 5. The average of \( p, q, r, s, \) and \( A \) is 8. Find \( A \).

If the lines \( 3x - 2y + 1 = 0 \) and \( Ax + By + 1 = 0 \) are perpendicular, find \( B \).

Answer: 30


Given that \( n \) is a positive integer and \( n \geq 4 \). Find the largest positive integer \( k \) such that there exists a triangle with side lengths all being positive integers not greater than \( n \), and the difference between any two sides (larger minus smaller) is not less than \( k \).

Answer: \lfloor\frac{n-1}{3}\rfloor


Cynthia and Lynnelle are collaborating on a problem set. Over a  $24$ -hour period, Cynthia and Lynnelle each independently pick a random, contiguous  $6$ -hour interval to work on the problem set. Compute the probability that Cynthia and Lynnelle work on the problem set during completely disjoint intervals of time.

Answer: 4/9


$k$  is a fixed positive integer. Let  $a_n$  be the number of maps  $f$  from the subsets of  $\{1, 2, ... , n\}$  to  $\{1, 2, ... , k\}$  such that for all subsets  $A, B$  of  $\{1, 2, ... , n\}$  we have  $f(A \cap B) = \min (f(A), f(B))$ . Find  $\lim_{n \to \infty} \sqrt[n]{a_n}$ .

Answer: k


Determine the common rational root \( k \) of the following polynomial equations which is not integral:
\[45x^4 + ax^3 + bx^2 + cx + 8 = 0\]
\[8x^5 + dx^4 + ex^3 + fx^2 + gx + 45 = 0\]
This root \( k \) is assumed to be a negative non-integer.

Answer: -\frac{1}{3}


On each of the one hundred cards, a different non-zero number is written such that each number equals the square of the sum of all the others.

What are these numbers?

Answer: \frac{1}{99^2}


Given that the midpoint of side $BC$ of triangle $\triangle ABC$ is $D$, point $E$ lies in the plane of $\triangle ABC$, and $\overrightarrow{CD}=3\overrightarrow{CE}-2\overrightarrow{CA}$, if $\overrightarrow{AC}=x\overrightarrow{AB}+y\overrightarrow{BE}$, then determine the value of $x+y$.

Answer: 11


The popular variable-speed bicycle installs gears with different numbers of teeth on the driving and rear axles. By connecting different gear combinations with a chain, several different speeds can be achieved through different transmission ratios.

The "Hope" variable-speed bicycle has three gears on the driving axle with 48, 36, and 24 teeth, respectively, and four gears on the rear axle with 36, 24, 16, and 12 teeth, respectively. Question: How many different speeds does this variable-speed bicycle have in total?

Answer: 8


Five cards are lying on a table as shown.
\[\begin{matrix} & \qquad & \boxed{\tt{P}} & \qquad & \boxed{\tt{Q}} \\  \\ \boxed{\tt{3}} & \qquad & \boxed{\tt{4}} & \qquad & \boxed{\tt{6}} \end{matrix}\]
Each card has a letter on one side and a [whole number](https://artofproblemsolving.com/wiki/index.php/Whole_number) on the other side.  Jane said, "If a vowel is on one side of any card, then an [even number](https://artofproblemsolving.com/wiki/index.php/Even_number) is on the other side."  Mary showed Jane was wrong by turning over one card.  Which card did Mary turn over? (Each card number is the one with the number on it. For example card 4 is the one with 4 on it, not the fourth card from the left/right)
$\text{(A)}\ 3 \qquad \text{(B)}\ 4 \qquad \text{(C)}\ 6 \qquad \text{(D)}\ \text{P} \qquad \text{(E)}\ \text{Q}$

Answer: \textbf{(A)}3


A parallelepiped is inscribed in a sphere of radius $\sqrt{3}$, and its volume is 8. Find the total surface area of the parallelepiped.

Answer: 24


A sequence  $a_1, a_2, a_3, \ldots$  of positive integers satisfies  $a_1 > 5$  and  $a_{n+1} = 5 + 6 + \cdots + a_n$  for all positive integers  $n$ . Determine all prime numbers  $p$  such that, regardless of the value of  $a_1$ , this sequence must contain a multiple of  $p$ .

Answer: 2


The first three numbers of a sequence are \(1, 7, 8\). Every subsequent number is the remainder obtained when the sum of the previous three numbers is divided by 4. Find the sum of the first 2011 numbers in this sequence.

Answer: 3028


Calculate the line integral

$$
\int_{L} \frac{y}{3} d x - 3 x d y + x d z
$$

along the curve \( L \), which is given parametrically by

$$
\begin{cases}
x = 2 \cos t \\
y = 2 \sin t \\
z = 1 - 2 \cos t - 2 \sin t
\end{cases}
\quad \text{for} \quad 0 \leq t \leq \frac{\pi}{2}
$$

Answer: 2 - \frac{13\pi}{3}


The formula for converting a Fahrenheit temperature  $F$  to the corresponding Celsius temperature  $C$  is  $C=\frac{5}{9}(F-32)$ .  An integer Fahrenheit temperature is converted to Celsius and rounded to the nearest integer; the resulting integer Celsius temperature is converted back to Fahrenheit and rounded to the nearest integer.  For how many integer Fahrenheit temperatures  $T$  with  $32 \leq T \leq 1000$  does the original temperature equal the final temperature?

Answer: 539


In the sequence \(\{a_n\}\), let \(S_n = \sum_{i=1}^{n} a_i\) (with \(n \in \mathbb{Z}_+\)), and it is agreed that \(S_0=0\). It is known that
\[
a_k =
\begin{cases}
k, & \text{if } S_{k-1} < k; \\
-k, & \text{if } S_{k-1} \geq k
\end{cases}
\quad (1 \leq k \leq n, k, n \in \mathbb{Z}_+).
\]
Find the largest positive integer \(n\) not exceeding 2019 such that
\[
S_n = 0.
\]

Answer: 1092


Given the function \( f: \mathbf{R} \rightarrow \mathbf{R} \), for any real numbers \( x, y, z \), the inequality \(\frac{1}{3} f(x y) + \frac{1}{3} f(x z) - f(x) f(y z) \geq \frac{1}{9} \) always holds. Find the value of \(\sum_{i=1}^{100} [i f(i)]\), where \([x]\) represents the greatest integer less than or equal to \( x \).

Answer: 1650


Fill the numbers $1, 2, 3, \ldots, 9, 10$ into 10 circles that form a rectangle. The sum of the numbers on each side of the rectangle must be equal. What is the maximum possible sum for each side?

Answer: 22


How many distinct triangles can be drawn using three of the dots below as vertices, where the dots are arranged in a grid of 2 rows and 4 columns?

Answer: 48


There are 5 integers written on the board. By summing them in pairs, the following set of 10 numbers is obtained: 2, 6, 10, 10, 12, 14, 16, 18, 20, 24. Determine which numbers are written on the board and write their product as the answer.

Answer: -3003


Given two moving points \( A \) and \( B \) on the circle \(\odot O: x^{2}+y^{2}=1\), the central angle subtended by arc \(\overparen{A B}\) (in a counterclockwise direction from \(A\) to \(B\)) is \( 90^{\circ} \). The points \( P(2,0) \) and \( Q(0,2) \) are known. The intersection of line \( PA \) with line \( QB \) is point \( M \), and \( M \) is outside the circle \(\odot O\). From point \( M \), two tangent lines are drawn to the circle, touching it at points \( T_{1} \) and \( T_{2} \). Determine the range of values of \(\overrightarrow{MT_{1}} \cdot \overrightarrow{MT_{2}}\).

Answer: [2\sqrt{2}-3,0]


Calculate to five decimal places (i.e., with an accuracy of 0.00001) the product:

\[
\left(1-\frac{1}{10}\right)\left(1-\frac{1}{10^{2}}\right)\left(1-\frac{1}{10^{3}}\right) \ldots\left(1-\frac{1}{10^{99}}\right)
\]

Answer: 0.89001


Calculate the number of five-digit numbers formed from the digits 1, 2, 3, 4, 5 without repeating any digits, and where the digits 1 and 2 are not adjacent to the digit 5.

Answer: 36


Let the prime number \( p \) satisfy \( p \equiv 7 \pmod{8} \). Prove that \( p \) cannot be expressed as the sum of three squares.

Answer: 3


Let \( A \) be a set of \( m \) real numbers with the sum of its elements equal to \( a \); \( B \) be a set of \( n \) real numbers with the sum of its elements equal to \( b \). Define
\[
C = \{\boldsymbol{x} \mid \boldsymbol{x} = (s, t), s \in A, t \in B\}.
\]

Let \(\boldsymbol{x}\) and \(\boldsymbol{y}\) each range over the vectors in \( C \) (allowing \(\boldsymbol{x} = \boldsymbol{y}\)) and compute \( x \cdot y \). Then the sum of these dot products \(\sum_{x y \in C} x \cdot y\) equals (express the result in terms of \( a, b, m, n \)).

Answer: n^2^2+^2b^2


Formulate an equation with integer coefficients, one root of which is $\sqrt[3]{2} + \sqrt[3]{3}$.

Answer: x^9-15x^6-87x^3-125=0


Let  $ n$  and  $ k$  be positive integers with  $ k \geq n$  and  $ k \minus{} n$  an even number. Let  $ 2n$  lamps labelled  $ 1$ ,  $ 2$ , ...,  $ 2n$  be given, each of which can be either *on* or *off*. Initially all the lamps are off. We consider sequences of steps: at each step one of the lamps is switched (from on to off or from off to on).

Let  $ N$  be the number of such sequences consisting of  $ k$  steps and resulting in the state where lamps  $ 1$  through  $ n$  are all on, and lamps  $ n \plus{} 1$  through  $ 2n$  are all off.

Let  $ M$  be number of such sequences consisting of  $ k$  steps, resulting in the state where lamps  $ 1$  through  $ n$  are all on, and lamps  $ n \plus{} 1$  through  $ 2n$  are all off, but where none of the lamps  $ n \plus{} 1$  through  $ 2n$  is ever switched on.

Determine  $ \frac {N}{M}$ .

*Author: Bruno Le Floch and Ilia Smilga, France*

Answer: 2^{k-n}


Which of the following equations does NOT have a solution?
$\text{(A)}\:(x+7)^2=0$
$\text{(B)}\:|-3x|+5=0$
$\text{(C)}\:\sqrt{-x}-2=0$
$\text{(D)}\:\sqrt{x}-8=0$
$\text{(E)}\:|-3x|-4=0$

Answer: \mathrm{(B)}


Given $\theta \in [0, 2\pi]$, if for any $x \in [0, 1]$, the inequality $2x^{2} \sin \theta - 4x(1-x) \cos \theta + 3(1-x)^{2} > 0$ always holds, determine the range of values for $\theta$.

Answer: (\frac{\pi}{6},\pi)


A school wishes to understand the psychological state of learning among its senior students and adopts a systematic sampling method to select 40 students out of 800 for a test. The students are randomly assigned numbers from 1 to 800 and then grouped. In the first group, number 18 is selected through simple random sampling. Among the 40 selected students, those with numbers in the range [1, 200] take test paper A, numbers in the range [201, 560] take test paper B, and the remaining students take test paper C. Calculate the number of students who take test paper C.

Answer: 12


Maxim came up with a new method for dividing numbers by a two-digit number \( N \). To divide an arbitrary number \( A \) by the number \( N \), you need to do the following steps:
1) Divide \( A \) by the sum of the digits of \( N \);
2) Divide \( A \) by the product of the digits of \( N \);
3) Subtract the second result from the first.

For which numbers \( N \) will Maxim’s method give the correct result? (20 points)

Answer: 24


Given that $\triangle ABC$ is an inscribed triangle in the ellipse $\frac{x^{2}}{9}+\frac{y^{2}}{4}=1$, and the line segment $AB$ passes through the point $P(1,0)$, find the maximum area of $\triangle ABC$.

Answer: \frac{16\sqrt{2}}{3}


There are 3 boys and 3 girls, making a total of 6 students standing in a row.

(1) If the three girls must stand together, find the total number of different arrangements.
(2) If boy A cannot stand at either end, and among the 3 girls, exactly two girls stand together, find the number of different arrangements.

Answer: 288


Given that the plane vectors $\overrightarrow{\mathrm{a}}$, $\overrightarrow{\mathrm{b}}$, and $\overrightarrow{\mathrm{c}}$ satisfy $|\overrightarrow{\mathrm{a}}|=\frac{|\overrightarrow{\mathrm{b}}|}{2}=|\overrightarrow{\mathrm{c}}|=1$ and $\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}}=1$, find the range of the expression $\left|\overrightarrow{\mathrm{c}}+\frac{\vec{a}}{2}\right| + \frac{1}{2}|\overrightarrow{\mathrm{c}}-\overrightarrow{\mathrm{b}}|$.

Answer: [\sqrt{3},\sqrt{7}]


Three circles, with radii  $p$ ,  $q$  and  $r$  and centres  $A$ ,  $B$  and  $C$  respectively, touch one another externally at points  $D$ ,  $E$  and  $F$ . Prove that the ratio of the areas of  $\triangle DEF$  and  $\triangle ABC$  equals
\[\frac{2pqr}{(p+q)(q+r)(r+p)}.\]

Answer: \frac{2pqr}{(p+q)(q+r)(r+p)}


Choose 5 numbers from the set {1, 2, 3, 4, 5, 6, 7, 8, 9} to form a 5-digit number. This 5-digit number should be divisible by each of the 5 chosen numbers, but not divisible by any of the 4 numbers that were not chosen. What is the smallest possible value of this 5-digit number?

Answer: 14728


Each of the $20$ balls is tossed independently and at random into one of the $5$ bins. Let $p$ be the probability that some bin ends up with $3$ balls, another with $5$ balls, and the other three with $4$ balls each. Let $q$ be the probability that every bin ends up with $4$ balls. What is $\frac{p}{q}$?
$\textbf{(A)}\ 1 \qquad\textbf{(B)}\  4 \qquad\textbf{(C)}\  8 \qquad\textbf{(D)}\  12 \qquad\textbf{(E)}\ 16$

Answer: \textbf{(E)}\16


In the sequence $\{a_n\}$, $a_{n+1} = 2(a_n - n + 3)$, $a_1 = -1$. If the sequence $\{a_n - pn + q\}$ is a geometric sequence, where $p$, $q$ are constants, then $a_{p+q} = \_\_\_\_\_\_\_\_\_\_\_\_\_.$

Answer: 40


Triangle $ABC$ has $AB=21$, $AC=22$ and $BC=20$. Points $D$ and $E$ are located on $\overline{AB}$ and $\overline{AC}$, respectively, such that $\overline{DE}$ is parallel to $\overline{BC}$ and contains the center of the inscribed circle of triangle $ABC$. Then $DE=m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Answer: 923


In a $5 \times 5$ grid, place 6 crosses such that each row and each column contains at least one cross. How many ways can this be done?

Answer: 4200


What is the largest positive integer  $n$  for which there are no *positive* integers  $a,b$  such that  $8a+11b=n$ ?

*2019 CCA Math Bonanza Lightning Round #2.2*

Answer: 88


The base of the pyramid is a parallelogram with adjacent sides of 9 cm and 10 cm, and one of the diagonals measuring 11 cm. The opposite lateral edges are equal, and each of the longer edges is 10.5 cm. Calculate the volume of the pyramid.

Answer: 200


Circles with centers $P, Q$ and $R$, having radii $1, 2$ and $3$, respectively, lie on the same side of line $l$ and are tangent to $l$ at $P', Q'$ and $R'$, respectively, with $Q'$ between $P'$ and $R'$. The circle with center $Q$ is externally tangent to each of the other two circles. What is the area of triangle $PQR$?

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


If one vertex and the two foci of an ellipse form an equilateral triangle, determine the eccentricity of this ellipse.

Answer: \dfrac{1}{2}


Given $2n$ real numbers $a_{1}, a_{2}, \cdots, a_{2n}$ satisfying the condition $\sum_{i=1}^{2n}\left(a_{i+1}-a_{i}\right)^{2}=1$, find the maximum value of $\left(a_{n+1}+a_{n+2}+\cdots+a_{2n}\right)-\left(a_{1}+a_{2}+\cdots+a_{n}\right)$.

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


The product of two consecutive page numbers is $20{,}412$. What is the sum of these two page numbers?

Answer: 285


For any natural number \( n \), define \( n! \) as \( 1 \times 2 \times 3 \times \cdots \times n \). What is the last digit of the result of the expression \( 2014! - 3! \)?
A. 2
B. 4
C. 6
D. 8

Answer: 4


The perimeter of a rhombus is $100 \text{ cm}$, and the sum of its diagonals is $62 \text{ cm}$. What are the lengths of the diagonals?

Answer: 48\,


Answer: 7


Two lines with slopes $\dfrac{1}{2}$ and $2$ intersect at $(2,2)$. What is the area of the triangle enclosed by these two lines and the line $x+y=10  ?$
$\textbf{(A) } 4 \qquad\textbf{(B) } 4\sqrt{2} \qquad\textbf{(C) } 6 \qquad\textbf{(D) } 8 \qquad\textbf{(E) } 6\sqrt{2}$

Answer: \textbf{(C)}6


From a quarter circle with a radius of $12 \text{ cm}$, a semicircle is cut out above one of the bounding radii, using it as its diameter. What is the radius of the largest circle that can be inscribed in the remaining shape?

Answer: 3\,


Let  $ X$  be the set of all positive integers greater than or equal to  $ 8$  and let  $ f: X\rightarrow X$  be a function such that  $ f(x\plus{}y)\equal{}f(xy)$  for all  $ x\ge 4, y\ge 4 .$  if  $ f(8)\equal{}9$ , determine  $ f(9) .$

Answer: 9


The ticket price for a cinema is: 6 yuan per individual ticket, 40 yuan for a group ticket for every 10 people, and students enjoy a 10% discount. A school with 1258 students plans to watch a movie (teachers get in for free). The school should pay the cinema at least ____ yuan.

Answer: 4536


How many integers are there from 0 to 999999 in which no two adjacent digits are the same?

Answer: 597871


A cube with side length 2 has vertices $Q_1, Q_2, Q_3, Q_4, Q_1', Q_2', Q_3',$ and $Q_4'$. Vertices $Q_2$, $Q_3$, and $Q_4$ are adjacent to $Q_1$, and for $1\le i\le 4,$ vertices $Q_i$ and $Q_i'$ are opposite to each other. A regular octahedron has one vertex in each of the segments $\overline{Q_1Q_2}$, $\overline{Q_1Q_3}$, $\overline{Q_1Q_4}$, $\overline{Q_1'Q_2'}$, $\overline{Q_1'Q_3'}$, and $\overline{Q_1'Q_4'}$. The vertices on the segments $\overline{Q_1Q_2}$, $\overline{Q_1Q_3}$, and $\overline{Q_1Q_4}$ are $\frac{2}{3}$ from $Q_1$. Find the side length of the octahedron.

Answer: \frac{4\sqrt{2}}{3}


Let $a$, $b$, and $c$ be positive integers with $a\ge$ $b\ge$ $c$ such that
$a^2-b^2-c^2+ab=2011$ and
$a^2+3b^2+3c^2-3ab-2ac-2bc=-1997$.
What is $a$?

Answer: 253


A ticket contains six digits \(a, b, c, d, e, f\). This ticket is said to be "lucky" if \(a + b + c = d + e + f\). How many lucky tickets are there (including the ticket 000000)?

Answer: 55252


Answer: \frac{17}{27}


Find all monic polynomials  $P(x)=x^{2023}+a_{2022}x^{2022}+\ldots+a_1x+a_0$  with real coefficients such that  $a_{2022}=0$ ,  $P(1)=1$  and all roots of  $P$  are real and less than  $1$ .

Answer: P(x)=x^{2023}


Joel is rolling a 6-sided die. After his first roll, he can choose to re-roll the die up to 2 more times. If he rerolls strategically to maximize the expected value of the final value the die lands on, the expected value of the final value the die lands on can be expressed as  $\frac{m}{n}$ , where  $m$  and  $n$  are relatively prime positive integers. What is  $m+n$ ?

*2021 CCA Math Bonanza Individual Round #8*

Answer: 21


Determine the minimum value of the function $$y = \frac {4x^{2}+2x+5}{x^{2}+x+1}$$ for \(x > 1\).

Answer: \frac{16 - 2\sqrt{7}}{3}


Find all triples  $(m,p,q)$  where  $ m $  is a positive integer and  $ p , q $  are primes.
\[ 2^m p^2 + 1 = q^5 \]

Answer: (1,11,3)


The graph of the function $f(x)$ is shown below. How many values of $x$ satisfy $f(f(x)) = 3$? [asy]
import graph; size(7.4cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-4.4,xmax=5.66,ymin=-1.05,ymax=6.16;

for(int i = -4; i <= 5; ++i) {
	draw((i,-1)--(i,6), dashed+mediumgrey);
}

for(int i = 1; i <= 6; ++i) {
	draw((-4,i)--(5,i), dashed+mediumgrey);
}

Label laxis; laxis.p=fontsize(10);

xaxis("$x$",-4.36,5.56,defaultpen+black,Ticks(laxis,Step=1.0,Size=2,OmitTick(0)),Arrows(6),above=true); yaxis("$y$",-0.92,6.12,defaultpen+black,Ticks(laxis,Step=1.0,Size=2,OmitTick(0)),Arrows(6),above=true); draw((xmin,(-(0)-(-2)*xmin)/-2)--(-1,(-(0)-(-2)*-1)/-2),linewidth(1.2)); draw((-1,1)--(3,5),linewidth(1.2)); draw((3,(-(-16)-(2)*3)/2)--(xmax,(-(-16)-(2)*xmax)/2),linewidth(1.2)); // draw((min,(-(-9)-(0)*xmin)/3)--(xmax,(-(-9)-(0)*xmax)/3),linetype("6pt 6pt"));

label("$f(x)$",(-3.52,4.6),SE*lsf);

//dot((-1,1),ds); dot((3,5),ds); dot((-3,3),ds); dot((1,3),ds); dot((5,3),ds);

dot((-4.32,4.32),ds); dot((5.56,2.44),ds);
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);

[/asy]

Answer: 2


Let \( N \) be the number of permutations of the numbers \( 1, 2, 3, \ldots, 1996 \) in which \( 1 \) is fixed and each number differs from its neighbors by at most \( 2 \). Is \( N \) divisible by \( 3 \)?

Answer: No


Krteček started digging a new tunnel. First, the tunnel went 5 meters to the north, then $23 \mathrm{dm}$ to the west, $150 \mathrm{~cm}$ to the south, $37 \mathrm{dm}$ to the west, $620 \mathrm{~cm}$ to the south, $53 \mathrm{~cm}$ to the east, and $27 \mathrm{dm}$ to the north. How many centimeters does he still need to dig to get to the starting point of the tunnel?

(M. Dillingerová)

Answer: 547\,


The sum of the ages of five people \( A, B, C, D, \) and \( E \) is 256 years. The age difference between any two people is not less than 2 years and not more than 10 years. What is the minimum age of the youngest person among them?

Answer: 46


Given that the integer is a 4-digit positive number with four different digits, the leading digit is not zero, the integer is a multiple of 5, 7 is the largest digit, and the first and last digits of the integer are the same, calculate the number of such integers.

Answer: 30


Fill the first eight positive integers in a $2 \times 4$ table, one number per cell, such that each row's four numbers increase from left to right, and each column's two numbers increase from bottom to top. How many different ways can this be done?

Answer: 14


Flat Albert and his buddy Mike are watching the game on Sunday afternoon. Albert is drinking lemonade from a two-dimensional cup which is an isosceles triangle whose height and base measure 9 cm and 6 cm; the opening of the cup corresponds to the base, which points upwards. Every minute after the game begins, the following takes place: if $n$ minutes have elapsed, Albert stirs his drink vigorously and takes a sip of height $\frac{1}{n^{2}} \mathrm{~cm}$. Shortly afterwards, while Albert is busy watching the game, Mike adds cranberry juice to the cup until it's once again full in an attempt to create Mike's cranberry lemonade. Albert takes sips precisely every minute, and his first sip is exactly one minute after the game begins. After an infinite amount of time, let $A$ denote the amount of cranberry juice that has been poured (in square centimeters). Find the integer nearest $\frac{27}{\pi^{2}} A$.

Answer: 26


Find the smallest number, written using only ones and zeros, that would be divisible by 225.

Answer: 11111111100


Given that an isosceles trapezoid is circumscribed around a circle, find the ratio of the area of the trapezoid to the area of the circle if the distance between the points where the circle touches the non-parallel sides of the trapezoid is related to the radius of the circle as $\sqrt{3}: 1$.

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


Calculate the lengths of arcs of curves described by the equations in the rectangular coordinate system.

$$
y=1-\ln \sin x, \frac{\pi}{3} \leq x \leq \frac{\pi}{2}
$$

Answer: \frac{\ln3}{2}


Given that $S_{n}=\sin \frac{\pi}{7}+\sin \frac{2\pi}{7}+\cdots+\sin \frac{n\pi}{7}$, determine the number of positive values in the sequence $S_{1}$, $S_{2}$, …, $S_{100}$.

Answer: 86


Suppose there are 128 ones written on a blackboard. At each step, you can erase any two numbers \(a\) and \(b\) from the blackboard and write \(ab + 1\). After performing this operation 127 times, only one number is left. Let \(A\) be the maximum possible value of this remaining number. Find the last digit of \(A\).

Answer: 2


Find all natural numbers \( k \) such that \( 3^{k} + 5^{k} \) is a power of a natural number with an exponent \(\geq 2\).

Answer: 1


Let \(a, b, c > 0\) such that \(abc = 1\). Show that
\[ \sum_{cyc} \frac{ab}{ab + a^5 + b^5} \leq 1. \]

Answer: 1


A sphere is tangent to all the edges of the pyramid \( SABC \), specifically the lateral edges \( SA, SB, \) and \( SC \) at the points \( A', B' \), and \( C' \) respectively. Find the volume of the pyramid \( SA'B'C' \), given that \( AB = BC = SB = 5 \) and \( AC = 4 \).

Answer: \frac{2 \sqrt{59}}{15}


Given a point $p$ and a line segment $l$, let $d(p, l)$ be the distance between them. Let $A, B$, and $C$ be points in the plane such that $A B=6, B C=8, A C=10$. What is the area of the region in the $(x, y)$-plane formed by the ordered pairs $(x, y)$ such that there exists a point $P$ inside triangle $A B C$ with $d(P, A B)+x=d(P, B C)+y=d(P, A C)$?

Answer: \frac{288}{5}


Given $f(x) = -4x^2 + 4ax - 4a - a^2$ has a maximum value of $-5$ in the interval $[0, 1]$, find the value of $a$.

Answer: -5


Let \( p > 2 \) be an integer that is not divisible by 3. Prove that there exist integers \( a_1, a_2, \ldots, a_k \) such that

$$
-\frac{p}{2} < a_1 < a_2 < \cdots < a_k < \frac{p}{2}
$$

and the product

$$
\frac{p - a_1}{|a_1|} \cdot \frac{p - a_2}{|a_2|} \cdots \frac{p - a_k}{|a_k|}
$$

is equal to \( 3^m \) for some natural number \( m \).

Answer: 3^


Given an integer  $n\ge\ 3$ , find the least positive integer  $k$ , such that there exists a set  $A$  with  $k$  elements, and  $n$  distinct reals  $x_{1},x_{2},\ldots,x_{n}$  such that  $x_{1}+x_{2}, x_{2}+x_{3},\ldots, x_{n-1}+x_{n}, x_{n}+x_{1}$  all belong to  $A$ .

Answer: 3


A circle can be inscribed in quadrilateral \(ABCD\). Let \(K\) be the intersection point of its diagonals. It is known that \(BC > AB > BK\), \(KC = \sqrt{7} - 1\), the cosine of angle \(KBC\) is \(\frac{\sqrt{7} + 1}{4}\), and the perimeter of triangle \(BKC\) is \(2\sqrt{7} + 4\). Find \(DC\).

Answer: 4


Several energy-saving devices with a total weight of 120 kg were delivered to the factory. It is known that the total weight of the three lightest devices is 31 kg, and the total weight of the three heaviest devices is 41 kg. How many energy-saving devices were delivered to the factory if the weights of any two devices are different?

Answer: 10


Find all solutions $(x, y) \in \mathbb Z^2$ of the equation
\[x^3 - y^3 = 2xy + 8.\]

Answer: (2,0),(0,-2)


Given point $P(-2,0)$ and the parabola $C$: $y^{2}=4x$, the line passing through $P$ intersects $C$ at points $A$ and $B$, where $|PA|= \frac {1}{2}|AB|$. Determine the distance from point $A$ to the focus of parabola $C$.

Answer: \frac{5}{3}


Answer: 2023


Elza draws  $2013$  cities on the map and connects some of them with  $N$  roads. Then Elza and Susy erase cities in turns until just two cities left (first city is to be erased by Elza). If these cities are connected with a road then Elza wins, otherwise Susy wins. Find the smallest  $N$  for which Elza has a winning strategy.

Answer: 1006


Answer: 60^\circ


Find the largest six-digit number in which each digit, starting from the third, is the sum of the two preceding digits.

Answer: 303369


The bottom, side, and front areas of a rectangular box are known. The product of these areas
is equal to:
$\textbf{(A)}\ \text{the volume of the box} \qquad\textbf{(B)}\ \text{the square root of the volume} \qquad\textbf{(C)}\ \text{twice the volume}$
$\textbf{(D)}\ \text{the square of the volume} \qquad\textbf{(E)}\ \text{the cube of the volume}$

Answer: \textbf{(D)}\


Find all integer $n$ such that the following property holds: for any positive real numbers $a,b,c,x,y,z$, with $max(a,b,c,x,y,z)=a$ , $a+b+c=x+y+z$ and $abc=xyz$, the inequality $$a^n+b^n+c^n \ge x^n+y^n+z^n$$ holds.

Answer: n \ge 0


Find the set of values of the parameter \(a\) for which the sum of the cubes of the roots of the equation \(x^{2}-a x+a+2=0\) is equal to -8.

Answer: -2


Given \( k \geq 1 \). Find the maximum real number \( \lambda \) such that for any positive real numbers \( x_i > 0 \) \((i=1,2,\ldots,n)\) satisfying \( \sum_{i=1}^{n} \frac{1}{x_{i}} = 2016 \), the following inequality holds:
$$
\lambda \sum_{i=1}^{n} \frac{x_{i}}{1+x_{i}} \leq \left[\sum_{i=1}^{n} \frac{1}{x_{i}^{k}\left(1+x_{i}\right)}\right]\left(\sum_{i=1}^{n} x_{i}^{k}\right).
$$

Answer: 2016


Answer: 2022 + \frac{1}{2023}


Each of $6$ balls is randomly and independently painted either black or white with equal probability. What is the probability that every ball is different in color from more than half of the other $5$ balls?
$\textbf{(A) } \frac{1}{64}\qquad\textbf{(B) } \frac{1}{6}\qquad\textbf{(C) } \frac{1}{4}\qquad\textbf{(D) } \frac{5}{16}\qquad\textbf{(E) }\frac{1}{2}$

Answer: \textbf{(D)}\frac{5}{16}


Given that there are 25 cities in the County of Maplewood, and the average population per city lies between $6,200$ and $6,800$, estimate the total population of all the cities in the County of Maplewood.

Answer: 162,500


An equilateral triangle of side  $n$  is divided into equilateral triangles of side  $1$ .  Find the greatest possible number of unit segments with endpoints at vertices of the small triangles that can be chosen so that no three of them are sides of a single triangle.

Answer: n(n+1)


A sample has a capacity of $80$. After grouping, the frequency of the second group is $0.15$. Then, the frequency of the second group is ______.

Answer: 12


Mona has 12 match sticks of length 1, and she has to use them to make regular polygons, with each match being a side or a fraction of a side of a polygon, and no two matches overlapping or crossing each other. What is the smallest total area of the polygons Mona can make?

Answer: \sqrt{3}


Prove that for any natural number \( n \), the number \( 1+\left\lfloor(3+\sqrt{5})^{n}\right\rfloor \) is divisible by \( 2^{n} \).

Answer: 1+\lfloor(3+\sqrt{5})^n\rfloor


An infinite geometric series has a first term of $540$ and a sum of $4500$. What is its common ratio, and what is the second term of the series?

Answer: 475.2


Any six points are taken inside or on an equilateral triangle with side length 1. Let $b$ be the smallest possible number with the property that it is always possible to select one pair of points from these six such that the distance between them is equal to or less than $b$. Then $b$ is:

Answer: \frac{1}{2}


Let  $ABC$  be a triangle with centroid  $G$  and  $BC = 3$ . If  $ABC$  is similar to  $GAB$ , compute the area of  $ABC$ .

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


Consider triangle $A B C$ with $\angle A=2 \angle B$. The angle bisectors from $A$ and $C$ intersect at $D$, and the angle bisector from $C$ intersects $\overline{A B}$ at $E$. If $\frac{D E}{D C}=\frac{1}{3}$, compute $\frac{A B}{A C}$.

Answer: \frac{7}{9}


Let \( n \) be the answer to this problem. \( a \) and \( b \) are positive integers satisfying
\[
3a + 5b \equiv 19 \pmod{n+1}
\]
\[
4a + 2b \equiv 25 \pmod{n+1}
\]

Find \( 2a + 6b \).

Answer: 96


There are  $64$  towns in a country and some pairs of towns are connected by roads but we do not know these pairs. We may choose any pair of towns and find out whether they are connected or not. Our aim is to determine whether it is possible to travel from any town to any other by a sequence of roads. Prove that there is no algorithm which enables us to do so in less than  $2016$  questions.

(Proposed by Konstantin Knop)

Answer: 2016


For a positive real number  $a$ , let  $C$  be the cube with vertices at  $(\pm a, \pm a, \pm a)$  and let  $T$  be the tetrahedron with vertices at  $(2a,2a,2a),(2a, -2a, -2a),(-2a, 2a, -2a),(-2a, -2a, -2a)$ . If the intersection of  $T$  and  $C$  has volume  $ka^3$  for some  $k$ , find  $k$ .

Answer: 4/3


Petya and Vasya are playing the following game. Petya chooses a non-negative random value  $\xi$  with expectation  $\mathbb{E} [\xi ] = 1$ , after which Vasya chooses his own value  $\eta$  with expectation  $\mathbb{E} [\eta ] = 1$  without reference to the value of  $\xi$ . For which maximal value  $p$  can Petya choose a value  $\xi$  in such a way that for any choice of Vasya's  $\eta$ , the inequality  $\mathbb{P}[\eta \geq \xi ] \leq p$  holds?

Answer: 1/2


A three-meter gas pipe has rusted in two places. Determine the probability that all three resulting segments can be used as connections to gas stoves, given that regulations require the stove to be no closer than 75 cm to the main gas pipe.

Answer: 1/4


Jennifer plans a profit of 20% on the selling price of an item, and her expenses are 10% of the selling price. There is also a sales tax of 5% on the selling price of the item. The item sells for $10.00. Calculate the rate of markup on cost of this item.

Answer: 53.85\%


Given two positive integers  $n$  and  $k$ , we say that  $k$  is *$n$ -ergetic* if:
However the elements of  $M=\{1,2,\ldots, k\}$  are coloured in red and green, there exist  $n$  not necessarily distinct integers of the same colour whose sum is again an element of  $M$  of the same colour. For each positive integer  $n$ , determine the least  $n$ -ergetic integer, if it exists.

Answer: n^2+n-1


The production of ceramic items consists of 3 consecutive stages: shaping the ceramic item on a potter's wheel for 15 minutes, drying for 10 minutes, and firing for 30 minutes. It is required to produce 75 items. How should 13 workers be distributed between shapers and firers for the 1st and 3rd stages respectively throughout the allocated stage time (workers are not needed for the drying stage) to complete the work in the shortest possible time? Record the shortest time (in minutes) needed to complete all the work as a number without specifying the unit.

Answer: 325


Given a quadratic function \( f(x) = a x^{2} + b x + c \) where \( a, b, c \in \mathbf{R} \) and \( a \neq 0 \), the following conditions are satisfied:
1. \( f(x-4) = f(2-x) \) for all \( x \in \mathbf{R} \), and \( f(x) \geq x \);
2. \( f(x) \leq \left(\frac{x+1}{2}\right)^{2} \) for \( x \in (0, 2) \);
3. The minimum value of \( f(x) \) on \( \mathbf{R} \) is 0.

Find the largest \( m \) (with \( m > 1 \)) such that there exists \( t \in \mathbf{R} \) where \( f(x+t) \leq x \) for all \( x \in [1, m] \).

Answer: 9


In the plane are given  $100$  points, such that no three of them are on the same line. The points are arranged in  $10$  groups, any group containing at least  $3$  points. Any two points in the same group are joined by a segment.
a) Determine which of the possible arrangements in  $10$  such groups is the one giving the minimal numbers of triangles.
b) Prove that there exists an arrangement in such groups where each segment can be coloured with one of three given colours and no triangle has all edges of the same colour.

*Vasile Pop*

Answer: 1200


For all non-negative integers \( m \) and \( n \) (with \( m \geq n \)), the sequence \( a_0, a_1, \cdots \) satisfies
\[ a_{m+n} + a_{m-n} + 2m - 2n - 1 = \frac{1}{2}(a_{2m} + a_{2n}). \]

If \( a_1 = 0 \) and \( a_p = 2019^2 \), find the value of \( p \).

Answer: 2020


There are 1000 numbers arranged in a row, such that any three consecutive numbers satisfy the condition that the middle number is equal to the sum of the numbers before and after it. If the first and second numbers are both 1, what is the sum of these 1000 numbers?
(A) 1000
(B) 1
(C) 0
(D) -1

Answer: 1


Answer: -6


In the right triangle \(ABC\), a circle is constructed with \(AC\) as its diameter. The circle intersects side \(AB\) at point \(E\). Point \(G\) is taken on side \(BC\) such that segment \(AG\) intersects the circle at point \(F\). Given that segments \(EF\) and \(AC\) are parallel, \(BG = 2CG\), and \(AC = 2\sqrt{3}\), find \(GF\).

Answer: 1


For some complex number  $\omega$  with  $|\omega| = 2016$ , there is some real  $\lambda>1$  such that  $\omega, \omega^{2},$  and  $\lambda \omega$  form an equilateral triangle in the complex plane. Then,  $\lambda$  can be written in the form  $\tfrac{a + \sqrt{b}}{c}$ , where  $a,b,$  and  $c$  are positive integers and  $b$  is squarefree. Compute  $\sqrt{a+b+c}$ .

Answer: 4032


In triangle \( ABC \), the angle bisector \( AL \) is drawn. Points \( E \) and \( D \) are marked on segments \( AB \) and \( BL \), respectively, such that \( DL = LC \) and \( ED \parallel AC \). Find the length of segment \( ED \), given \( AE = 15 \) and \( AC = 12 \).

Answer: 3


If a positive integer \( n \) makes the equation \( x^{3} + y^{3} = z^{n} \) have a positive integer solution \( (x, y, z) \), then \( n \) is called a "good number." How many good numbers are there that do not exceed 2,019?

Answer: 1346


Let \( S = \{1, 2, 3, \ldots, 48, 49\} \). What is the maximum value of \( n \) such that it is possible to select \( n \) numbers from \( S \) and arrange them in a circle in such a way that the product of any two adjacent numbers in the circle is less than 100?

Answer: 18


Given \( n \) points on a plane, with no three points collinear (\( n > 4 \)), any two points can be connected to form \( \mathrm{C}_{n}^{4} \) quadrilaterals. Prove that at least 20% of these quadrilaterals are convex.

Answer: 20


Given that there are \( C \) integers that satisfy the equation \( |x-2| + |x+1| = B \), find the value of \( C \).

Answer: 4


Azar, Carl, Jon, and Sergey are the four players left in a singles tennis tournament. They are randomly assigned opponents in the semifinal matches, and the winners of those matches play each other in the final match to determine the winner of the tournament. When Azar plays Carl, Azar will win the match with probability $\frac23$. When either Azar or Carl plays either Jon or Sergey, Azar or Carl will win the match with probability $\frac34$. Assume that outcomes of different matches are independent. The probability that Carl will win the tournament is $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.

Answer: 125


Seven people of seven diﬀerent ages are attending a meeting. The seven people leave the meeting one at a
time in random order. Given that the youngest person leaves the meeting sometime before the oldest
person leaves the meeting, the probability that the third, fourth, and ﬁfth people to leave the meeting do so in order of their ages (youngest to oldest) is  $\frac{m}{n}$  , where m and n are relatively prime positive integers. Find  $m + n$ .

Answer: 25


Solve the system

$$
\left\{\begin{array}{l}
x^{3}+3 y^{3}=11 \\
x^{2} y+x y^{2}=6
\end{array}\right.
$$

Calculate the values of the expression $\frac{x_{k}}{y_{k}}$ for each solution $\left(x_{k}, y_{k}\right)$ of the system and find the smallest among them. If necessary, round your answer to two decimal places.

Answer: -1.31


A rental company owns 100 cars. When the monthly rent for each car is 3000 yuan, all of them can be rented out. For every 50 yuan increase in the monthly rent per car, there will be one more car that is not rented out. The maintenance cost for each rented car is 150 yuan per month, and for each car not rented out, the maintenance cost is 50 yuan per month.
(1) How many cars can be rented out when the monthly rent for each car is 3600 yuan?
(2) At what monthly rent per car will the rental company's monthly revenue be maximized? What is the maximum monthly revenue?

Answer: 307050


It is known that  $a^{2005} + b^{2005}$  can be expressed as the polynomial of  $a + b$  and  $ab$ . Find the coefficients' sum of this polynomial.

Answer: 1


In the tetrahedron \( ABCD \), \( AC = 8 \), \( AB = CD = 7 \), \( BC = AD = 5 \), and \( BD = 6 \). Given a point \( P \) on \( AC \), find the minimum value of \( BP + PD \).

Answer: 2\sqrt{21}


ABC is a triangle. The point X on the side AB is such that AX/XB = 4/5. The point Y on the segment CX is such that CY = 2 XY, and the point Z on the ray CA is such that ∠CXZ + ∠B = 180°. Also ∠XYZ = 45°. Show that the angles of triangle ABC are determined and find the smallest angle.

Answer: 45


Two 4 by 4 squares are randomly placed on an 8 by 8 chessboard so that their sides lie along the grid lines of the board. What is the probability that the two squares overlap?

Answer: \frac{529}{625}


If the equation $\tan \alpha + \tan \beta + \tan \gamma = \tan \alpha \tan \beta \tan \gamma$ holds, what algebraic relationship should $\alpha, \beta, \gamma$ satisfy?

Answer: \alpha+\beta+\gamma=k\pi


Given the family of curves
$$
2(2 \sin \theta - \cos \theta + 3) x^{2} - (8 \sin \theta + \cos \theta + 1) y = 0,
$$
where $\theta$ is a parameter. Find the maximum length of the chord that these curves cut on the line $y = 2 x$.

Answer: 8\sqrt{5}


The sequence \( a_{0}, a_{1}, \cdots, a_{n} \) satisfies:
\[
a_{0}=\sqrt{3}, \quad a_{n+1}=[a_{n}]+\frac{1}{\{a_{n}\}}
\]
where \( [x] \) denotes the greatest integer less than or equal to the real number \( x \), and \( \{x\}=x-[x] \). Find the value of \( a_{2016} \).

Answer: 3024 + \sqrt{3}


The number of teeth in three meshed gears $A$, $B$, and $C$ are $x$, $y$, and $z$, respectively. (The teeth on all gears are the same size and regularly spaced.) The angular speeds, in revolutions per minutes of $A$, $B$, and $C$ are in the proportion
$\text{(A)} \ x: y: z ~~\text{(B)} \ z: y: x ~~ \text{(C)} \ y: z: x~~ \text{(D)} \ yz: xz: xy ~~ \text{(E)} \ xz: yx: zy$

Answer: D:yz:xz:xy


Let  $\theta=\frac{2\pi}{2015}$ , and suppose the product \[\prod_{k=0}^{1439}\left(\cos(2^k\theta)-\frac{1}{2}\right)\] can be expressed in the form  $\frac{b}{2^a}$ , where  $a$  is a non-negative integer and  $b$  is an odd integer (not necessarily positive). Find  $a+b$ .

*2017 CCA Math Bonanza Tiebreaker Round #3*

Answer: 1441


Given that \( n \geq 2 \) is a positive integer, \( a \) is a real number such that \( 0 < a < \frac{n+1}{n-1} \), and the complex number \( z \) satisfies the equation \( z^{n+1} - a z^n + a z - 1 = 0 \), prove that \( |z| = 1 \).

Answer: |z|=1


Among the 100 natural numbers from 1 to 100, how many numbers can be represented as \( m \cdot n + m + n \) where \( m \) and \( n \) are natural numbers?

Answer: 74


In $\triangle ABC$, if $a + c = 2b$, then find the value of $\tan \frac{A}{2} \cdot \tan \frac{C}{2}$.

Answer: 1/3


Prove that a convex polyhedron cannot be covered by three polyhedra, each homothetic to it with a coefficient $k$, where $0 < k < 1$.

Answer: 3


Let \( A, B, C \) be positive integers such that the number \( 1212017ABC \) is divisible by 45. Find the difference between the largest and the smallest possible values of the two-digit number \( AB \).

Answer: 85


Given a sequence of positive numbers $\left\{a_{n}\right\}$ whose first $n$ terms sum to $S_{n}$, and $S_{n}=\frac{1}{2}\left(a_{n}+\frac{1}{a_{n}}\right)$, find $a_{n}$.

Answer: \sqrt{n}-\sqrt{n-1}


How many pairs of natural numbers \((a, b)\) exist such that the number \(5a - 3\) is divisible by \(b\) and the number \(5b - 1\) is divisible by \(a\)? Enter the number of such pairs in the given field.

Answer: 18


Given circle $M$: $(x+1)^{2}+y^{2}=1$, and circle $N$: $(x-1)^{2}+y^{2}=9$, a moving circle $P$ is externally tangent to circle $M$ and internally tangent to circle $N$. The trajectory of the center of circle $P$ is curve $C$.
$(1)$ Find the equation of $C$.
$(2)$ Let $l$ be a line tangent to both circle $P$ and circle $M$, and $l$ intersects curve $C$ at points $A$ and $B$. When the radius of circle $P$ is the longest, find $|AB|$.

Answer: \dfrac {18}{7}


The area of the orthogonal projection of a circle with a radius of 1 on the plane $\alpha$ is 1. Find the length of the orthogonal projection of this circle on a line perpendicular to the plane $\alpha$.

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


An integer consists of 7 different digits, and is a multiple of each of its digits.

What digits are in this nubmer?

Answer: 1,2,3,6,7,8,9


For $\pi \le \theta < 2\pi$, let
\begin{align*} P &= \frac12\cos\theta - \frac14\sin 2\theta - \frac18\cos 3\theta + \frac{1}{16}\sin 4\theta + \frac{1}{32} \cos 5\theta - \frac{1}{64} \sin 6\theta - \frac{1}{128} \cos 7\theta + \cdots \end{align*}
 and
\begin{align*} Q &= 1 - \frac12\sin\theta -\frac14\cos 2\theta + \frac18 \sin 3\theta + \frac{1}{16}\cos 4\theta - \frac{1}{32}\sin 5\theta - \frac{1}{64}\cos 6\theta +\frac{1}{128}\sin 7\theta + \cdots \end{align*}
 so that $\frac{P}{Q} = \frac{2\sqrt2}{7}$. Then $\sin\theta = -\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Answer: 36


In rectangle $PQRS$, $PQ = 8$ and $QR = 4$. Points $T$ and $U$ are on $\overline{RS}$ such that $RT = 2$ and $SU = 3$. Lines $PT$ and $QU$ intersect at $V$. Find the area of $\triangle PVQ$. [asy]
pair P,Q,R,S,V,T,U;
P=(0,0);
Q=(8,0);
R=(8,4);
S=(0,4);
T=(2,4);
U=(6,4);
V=(3.2,6);
draw(P--Q--R--S--cycle,linewidth(0.7));
draw(P--T--V--cycle,linewidth(0.7));
label("$P$",P,SW);
label("$Q$",Q,SE);
label("$R$",R,NE);
label("$S$",S,NW);
label("$T$",T,S);
label("$U$",U,S);
label("$V$",V,N);
label("2",(1,4),N);
label("3",(7,4),N);
label("4",(8,2),E);
label("8",(4,0),S);
[/asy]

Answer: 32


Sergey and Lena have several chocolate bars, each weighing no more than 100 grams. No matter how they divide these chocolate bars, one of them will have a total weight of chocolate bars that does not exceed 100 grams. What is the maximum total weight that all the chocolate bars can have?

Answer: 300


Given that the graph of a power function passes through the points $(2,16)$ and $(\frac{1}{2},m)$, find the value of $m$.

Answer: \frac{1}{16}


On the side \( BC \) of triangle \( ABC \), point \( A_1 \) is taken such that \( BA_1 : A_1C = 2:1 \). In what ratio does median \( CC_1 \) divide segment \( AA_1 \)?

Answer: 3:1


Three points are chosen inside a unit cube uniformly and independently at random. What is the probability that there exists a cube with side length $\frac{1}{2}$ and edges parallel to those of the unit cube that contains all three points?

Answer: \frac{1}{8}


Let \(ABCDE\) be a convex pentagon such that \(AB = AE = CD = BC + DE = 1\) and \(\angle ABC = \angle DEA = 90^\circ\). Calculate the area of \(ABCDE\).

Answer: 1


Which of the following is the sum of three consecutive integers?
(A) 17
(B) 11
(C) 25
(D) 21
(E) 8

Answer: 21


Let \( n \geqslant 3 \). Do there exist \( n \) points, not lying on a single line, such that the pairwise distances between them are irrational, and the areas of all triangles formed by these points are rational?

Answer: Yes


In the sequence $\{{a_{n}}\}$, the adjacent terms ${a_{n}}$ and ${a_{n+1}}$ are the roots of the equation ${x^{2}}+3nx+{{b_{n}}=0}$. Given that ${{a_{10}}=-17}$, find the value of ${{b_{51}}}$.

Answer: 5840


Let \( n \) be a positive integer and let \( a_{1}, \ldots, a_{n} \) be real numbers satisfying \( 0 \leq a_{i} \leq 1 \) for \( i=1, \ldots, n \). Prove the inequality
$$
\left(1-a_{1}^{n}\right)\left(1-a_{2}^{n}\right) \cdots\left(1-a_{n}^{n}\right) \leq\left(1-a_{1} a_{2} \cdots a_{n}\right)^{n} .
$$

Answer: (1-a_1^n)(1-a_2^n)\cdots(1-a_n^n)\le(1-a_1a_2\cdotsa_n)^n


It can be shown that there exists a unique polynomial $P$ in two variables such that for all positive integers $m$ and $n$, $$P(m, n)=\sum_{i=1}^{m} \sum_{j=1}^{n}(i+j)^{7}$$ Compute $P(3,-3)$.

Answer: -2445


Given that $f(\alpha) = \cos\alpha \sqrt{\frac{\cot\alpha - \cos\alpha}{\cot\alpha + \cos\alpha}} + \sin\alpha \sqrt{\frac{\tan\alpha - \sin\alpha}{\tan\alpha + \sin\alpha}}$, and $\alpha$ is an angle in the second quadrant.
(1) Simplify $f(\alpha)$.
(2) If $f(-\alpha) = \frac{1}{5}$, find the value of $\frac{1}{\tan\alpha} - \frac{1}{\cot\alpha}$.

Answer: -\frac{7}{12}


There are 5 people standing in a row. If person A is not at the left end and not adjacent to person B, determine the number of possible arrangements.

Answer: 54


At the beginning of the year in a certain school, the number of female students was 51 less than the number of male students. During the year, 19 boys and 41 girls left the school, resulting in the number of girls at the end of the year being 4% less in terms of the total percentage compared to the beginning of the year. How many male and female students were there at the beginning of the year?

Answer: 187


Three cards are dealt at random from a standard deck of 52 cards. What is the probability that the first card is a $\clubsuit$, the second card is a $\heartsuit$, and the third card is a king?

Answer: \frac{13}{2550}


A gives B as many cents as B has and C as many cents as C has. Similarly, B then gives A and C as many cents as each then has. C, similarly, then gives A and B as many cents as each then has. After this, each person gives half of what they have to each other person. If each finally has 24 cents, calculate the number of cents A starts with.

Answer: 24


Let $a, b$ and $c$ be positive real numbers such that $$\begin{aligned} a^{2}+a b+b^{2} & =9 \\ b^{2}+b c+c^{2} & =52 \\ c^{2}+c a+a^{2} & =49 \end{aligned}$$ Compute the value of $\frac{49 b^{2}-33 b c+9 c^{2}}{a^{2}}$.

Answer: 52


Using the vertices of a single rectangular solid (cuboid), how many different pyramids can be formed?

Answer: 106


Select two distinct numbers a, b from the set {0,1,2,3,4,5,6} to form a complex number a+bi, and determine the total number of such complex numbers with imaginary parts.

Answer: 36


If a pot can hold 2 cakes at a time and it takes 5 minutes to cook both sides of a cake, calculate the minimum time it will take to cook 3 cakes thoroughly.

Answer: 15


Find all three-digit numbers that are five times the product of their digits.

Answer: 175


Let $c$ be a complex number.  Suppose there exist distinct complex numbers $r$, $s$, and $t$ such that for every complex number $z$, we have
\[
  (z - r)(z - s)(z - t) = (z - cr)(z - cs)(z - ct).
\]Compute the number of distinct possible values of $c$.

Answer: 4


Find the rightmost non-zero digit of the expansion of (20)(13!).

Answer: 6


Let \(a\), \(b\), \(c\), and \(d\) be distinct positive integers such that \(a+b\), \(a+c\), and \(a+d\) are all odd and are all squares. Let \(L\) be the least possible value of \(a + b + c + d\). What is the value of \(10L\)?

Answer: 670


Given two non-collinear vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfying $|\overrightarrow{a}|=|\overrightarrow{b}|$, and $\overrightarrow{a}\perp(\overrightarrow{a}-2\overrightarrow{b})$, the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is ______.

Answer: \frac{\pi}{3}


The sum of the house numbers on one side of a street from corner to corner is 117. What is the house number of the fifth house from the beginning of this section?

Answer: 13


Answer: (wx+yz-1)^2\ge(w^2+y^2-1)(x^2+z^2-1)


The quadrangular pyramid \(P-ABCD\) has a base \(ABCD\) in the shape of a rhombus with a vertex angle of \(60^\circ\). Each lateral face forms an angle of \(60^\circ\) with the base. There is a point \(M\) inside the pyramid such that the distances from \(M\) to the base and each of the lateral faces are all equal to 1. What is the volume of the pyramid?

Answer: 8\sqrt{3}


Find all pairs of positive integers \((m, n)\) such that
$$
m+n-\frac{3mn}{m+n}=\frac{2011}{3}.
$$

Answer: (1144,377)or(377,1144)


Square $ABCD$ has sides of length 1. Points $E$ and $F$ are on $\overline{BC}$ and $\overline{CD},$ respectively, so that $\triangle AEF$ is equilateral. A square with vertex $B$ has sides that are parallel to those of $ABCD$ and a vertex on $\overline{AE}.$ The length of a side of this smaller square is $\frac{a-\sqrt{b}}{c},$ where $a, b,$ and $c$ are positive integers and $b$ is not divisible by the square of any prime. Find $a+b+c.$

Answer: 12


In a competition with a total of 4 multiple-choice questions, each question has three choices: $A, B,$ and $C$. For any three contestants participating, there is at least one question for which their answers are all different. What is the maximum number of participants allowed in the competition?

Answer: 9


In a circle, a quadrilateral \(ABCD\) is inscribed such that its diagonals are perpendicular to each other and intersect at point \(E\). A line passing through point \(E\) and perpendicular to \(BC\) intersects side \(AD\) at point \(M\). Prove that \(EM\) is the median of triangle \(AED\) and find its length, given that \(AB = 7\), \(CE = 3\), and \(\angle ADB = \alpha\).

Answer: \frac{\sqrt{49-9\tan^2\alpha}}{2\sin\alpha}


Let  $f(x,y)$  be a function defined for all pairs of nonnegative integers  $(x, y),$  such that  $f(0,k)=f(k,0)=2^k$  and \[f(a,b)+f(a+1,b+1)=f(a+1,b)+f(a,b+1)\] for all nonnegative integers  $a, b.$  Determine the number of positive integers  $n\leq2016$  for which there exist two nonnegative integers  $a, b$  such that  $f(a,b)=n$ .

*Proposed by Michael Ren*

Answer: 65


Alyosha wrote five integers on the board - the coefficients and roots of a quadratic polynomial. Borya erased one of them. The remaining numbers are $2, 3, 4, -5$. Restore the erased number.

Answer: -30


Given that the coefficients $p$ and $q$ are integers and the roots $\alpha_{1}$ and $\alpha_{2}$ are irrational, a quadratic trinomial $x^{2} + px + q$ is called an irrational quadratic trinomial. Determine the minimum sum of the absolute values of the roots among all irrational quadratic trinomials.

Answer: \sqrt{5}


Factory A and Factory B both produce the same type of clothing. Factory A produces 2700 sets of clothing per month, with the time ratio of producing tops to pants being 2:1. Factory B produces 3600 sets of clothing per month, with the time ratio of producing tops to pants being 3:2. If the two factories work together for one month, what is the maximum number of sets of clothing they can produce?

Answer: 6700


In trapezoid \(A B C D\), the base \(A D\) is four times larger than \(B C\). A line passing through the midpoint of diagonal \(B D\) and parallel to \(A B\) intersects side \(C D\) at point \(K\). Find the ratio \(D K : K C\).

Answer: 2:1


Call a positive integer $n$ weird if $n$ does not divide $(n-2)$!. Determine the number of weird numbers between 2 and 100 inclusive.

Answer: 26


Given the hexadecimal system, determine the product of $A$ and $B$.

Answer: 6E


There is a target on the wall consisting of five zones: a central circle (bullseye) and four colored rings. The width of each ring is equal to the radius of the bullseye. It is known that the number of points awarded for hitting each zone is inversely proportional to the probability of hitting that zone, and the bullseye is worth 315 points. How many points is hitting the blue (penultimate) zone worth?

Answer: 45


An integer $n$ is said to be square-free if the only perfect square that divides $n$ is $1^2$. How many positive odd integers greater than 1 and less than $100$ are square-free?

Answer: 40


The product of the digits of the positive integer \( N \) is 20. One of the following could not be the product of the digits of \( N+1 \). Which is it?
A 24
B 25
C 30
D 35
E 40

Answer: 35


Find all integers \( n \) for which \( x^2 - x + n \) divides \( x^{13} + x + 90 \).

Answer: 2


Two circles are drawn, each centered at the origin with radii 1 and 2. Determine the value of \( k \) so that the shaded region above the \( x \)-axis, below the line \( y = k x \), and between the two circles has an area of 2.

Answer: \tan(\frac{4}{3})


Quadrilateral  $ABCD$  satisfies  $AB = 8, BC = 5, CD = 17, DA = 10$ . Let  $E$  be the intersection of  $AC$  and  $BD$ . Suppose  $BE : ED = 1 : 2$ . Find the area of  $ABCD$ .

Answer: 60


Answer: y


If $N$, written in base $2$, is $11000$, the integer immediately preceding $N$, written in base $2$, is:

Answer: 10111


In the pyramid \(ABCD\), the lengths of edges \(AB\) and \(CD\) are each 4, and the lengths of all other edges are each 3. A sphere is inscribed in this pyramid. Find the volume of the pyramid whose vertices are the points of tangency of the sphere with the faces of the pyramid \(ABCD\).

Answer: \frac{8}{375}


Given that $21^{-1} \equiv 17 \pmod{53}$, find $32^{-1} \pmod{53}$, as a residue modulo 53. (Give a number between 0 and 52, inclusive.)

Answer: 36


Let \( u_1, \ldots, u_{2019} \) be real numbers satisfying

\[ u_1 + \ldots + u_{2019} = 0 \text{ and } u_1^2 + \ldots + u_{2019}^2 = 1 \]

Denote \( a = \max \left(u_1, \ldots, u_{2019}\right) \) and \( b = \min \left(u_1, \ldots, u_{2019}\right) \). Show that

\[ ab \leq -\frac{1}{2019} \]

Answer: \le-\frac{1}{2019}


Given an ellipse $E$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ($a > b > 0$) whose left focus $F_1$ coincides with the focus of the parabola $y^2 = -4x$, and the eccentricity of ellipse $E$ is $\frac{\sqrt{2}}{2}$. A line $l$ with a non-zero slope passes through point $M(m,0)$ ($m > \frac{3}{4}$) and intersects the ellipse $E$ at points $A$ and $B$. Point $P(\frac{5}{4},0)$ is given, and $\overrightarrow{PA} \cdot \overrightarrow{PB}$ is a constant.
- (Ⅰ) Find the equation of the ellipse $E$.
- (Ⅱ) Find the maximum area of $\triangle OAB$.

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


Determine the largest value of \(x\) for which
\[
\left| x^2 - 11x + 24 \right| + \left| 2x^2 + 6x - 56 \right| = \left| x^2 + 17x - 80 \right|
. \]

Answer: 8


Find the range of variation of the similarity ratio of triangles with side lengths $x, y, z$ and $y, z, p$. In the answer, specify the closest integers between which the found range is located.

Answer: 02


Determine the integers $a$ and $b$ such that $a^{4}+(a+b)^{4}+b^{4}$ is a perfect square.

Answer: 0,0


Let $a,$ $b,$ $c$ be positive real numbers.  Find the smallest possible value of
\[6a^3 + 9b^3 + 32c^3 + \frac{1}{4abc}.\]

Answer: 6


An isosceles triangle and a rectangle have the same area. The base of the triangle is equal to the width of the rectangle, and this dimension is 10 units. The length of the rectangle is twice its width. What is the height of the triangle, $h$, in terms of the dimensions of the rectangle?

Answer: 40


Find all functions \( f: \mathbf{R} \rightarrow \mathbf{R} \) such that for all \( x, y \in \mathbf{R} \), the following holds:
\[ f(1 + xy) - f(x + y) = f(x) f(y), \]
and \( f(-1) \neq 0 \).

Answer: f(x)=x-1


Let $p=2^{24036583}-1$, the largest prime currently known. For how many positive integers $c$ do the quadratics \pm x^{2} \pm p x \pm c all have rational roots?

Answer: 0


The graph shows the constant rate at which Suzanna rides her bike. If she rides a total of a half an hour at the same speed, how many miles would she have ridden?

Answer: 6


In tetrahedron \(ABCD\), it is known that \(\angle ADB = \angle BDC = \angle CDA = 60^\circ\), \(AD = BD = 3\), and \(CD = 2\). Find the radius of the circumscribed sphere of tetrahedron \(ABCD\).

Answer: \sqrt{3}


Define \(\varphi^{k}(n)\) as the number of positive integers that are less than or equal to \(n / k\) and relatively prime to \(n\). Find \(\varphi^{2001}\left(2002^{2} - 1\right)\). (Hint: \(\varphi(2003) = 2002\).)

Answer: 1233


For distinct complex numbers $z_1,z_2,\dots,z_{673}$, the polynomial \[(x-z_1)^3(x-z_2)^3 \cdots (x-z_{673})^3\]can be expressed as $x^{2019} + 20x^{2018} + 19x^{2017}+g(x)$, where $g(x)$ is a polynomial with complex coefficients and with degree at most $2016$. The sum $\left| \sum_{1 \le j <k \le 673} z_jz_k \right|$ can be expressed in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Answer: 352


Let the operation $x*y$ be defined as $x*y = (x+1)(y+1)$. The operation $x^{*2}$ is defined as $x^{*2} = x*x$. Calculate the value of the polynomial $3*(x^{*2}) - 2*x + 1$ when $x=2$.

Answer: 32


In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and it is given that $a\cos B - b\cos A = \frac{1}{2}c$. When $\tan(A-B)$ takes its maximum value, the value of angle $B$ is \_\_\_\_\_\_.

Answer: \frac{\pi}{6}


A deck of playing cards is laid out on a table (e.g., in a row). Another deck of cards is placed on top of each card. Some cards may match. Find:

a) the expected number of matches;

b) the variance of the number of matches.

Answer: 1


Let the complex number \( z = \cos \frac{2\pi}{13} + i \sin \frac{2\pi}{13} \). Find the value of \( \left(z^{-12} + z^{-11} + z^{-10}\right)\left(z^{3} + 1\right)\left(z^{6} + 1\right) \).

Answer: -1


In $\triangle ABC$, prove:
$$
\sin A \sin \left(\frac{A}{2}+B\right)=\sin B \sin \left(\frac{B}{2}+A\right) \Leftrightarrow a=b .
$$

Answer: b


$25$  checkers are placed on  $25$  leftmost squares of  $1 \times N$  board. Checker can either move to the empty adjacent square to its right or jump over adjacent right checker to the next square if it is empty. Moves to the left are not allowed. Find minimal  $N$  such that all the checkers could be placed in the row of  $25$  successive squares but in the reverse order.

Answer: 50


Suppose the graph of \( y=g(x) \) includes the points \( (1,4), (2,6), \) and \( (3,2) \).

Based only on this information, there are two points that must be on the graph of \( y=g(g(x)) \). If we call these points \( (a,b) \) and \( (c,d) \), what is \( ab + cd \)?

Answer: 20


How many even four-digit numbers can be formed using the digits $0, 1, 2, 3, 4, 5$?

Answer: 540