Given a circle of radius $2$, there are many line segments of length $2$ that are tangent to the circle at their midpoints. Find the area of the region consisting of all such line segments.

Answer: \pi


In the quadrilateral \(ABCD\), the opposite sides \(AD\) and \(BC\) are extended and intersect at \(E\), while the opposite sides \(AB\) and \(CD\) are extended and intersect at \(F\). \(O\) is the intersection point of the diagonals. A line \(OQ\) is drawn through \(O\) parallel to \(AB\) and intersects \(EF\) at \(Q\). Prove that \(OG=GQ\).

Answer: OG=GQ


In three-dimensional space, let \( S \) be the region of points \( (x, y, z) \) satisfying \( -1 \leq z \leq 1 \). Let \( S_{1}, S_{2}, \ldots, S_{2022} \) be 2022 independent random rotations of \( S \) about the origin \( (0,0,0) \). The expected volume of the region \( S_{1} \cap S_{2} \cap \cdots \cap S_{2022} \) can be expressed as \( \frac{a \pi}{b} \), for relatively prime positive integers \( a \) and \( b \). Compute \( 100a+b \).

Answer: 271619


Find the equation of the line that contains the point \((1, 0)\), that is of least positive slope, and that does not intersect the curve \( 4x^2 - y^2 - 8x = 12 \).

Answer: 2x-2


Given a natural number \( n \geqslant 5 \), find:
(1) The minimum number of distinct values of \( a_i + a_j \) (where \( 1 \leqslant i < j \leqslant n \)) in an \( n \)-element set \(\left\{a_{1}, a_{2}, \cdots, a_{n}\right\}\).
(2) Determine all \( n \)-element sets that achieve this minimum value.

Answer: 2n-3


A line \( l \) passes through a point \( X \) with barycentric coordinates \( (\alpha: \beta: \gamma) \). Let \( d_{\mathrm{a}}, d_{\mathrm{b}}, d_{\mathrm{c}} \) be the signed distances from vertices \( A, B, C \) to the line \( l \) (points on opposite sides of the line \( l \) have different signs). Prove that \( d_{\mathrm{a}} \alpha + d_{\mathrm{b}} \beta + d_{\mathrm{c}} \gamma = 0 \).

Answer: d_{\mathrm{}}\alpha+d_{\mathrm{b}}\beta+d_{\mathrm{}}\gamma=0


Understanding and trying: When calculating $\left(-4\right)^{2}-\left(-3\right)\times \left(-5\right)$, there are two methods; Method 1, please calculate directly: $\left(-4\right)^{2}-\left(-3\right)\times \left(-5\right)$, Method 2, use letters to represent numbers, transform or calculate the polynomial to complete, let $a=-4$, the original expression $=a^{2}-\left(a+1\right)\left(a-1\right)$, please complete the calculation above; Application: Please calculate $1.35\times 0.35\times 2.7-1.35^{3}-1.35\times 0.35^{2}$ according to Method 2.

Answer: -1.35


When $0.76\overline{204}$ is expressed as a fraction in the form $\frac{x}{999000}$, what is the value of $x$?

Answer: 761280


A circle with center P and radius 4 inches is tangent at D to a circle with center Q, located at a 45-degree angle from P. If point Q is on the smaller circle, what is the area of the shaded region? Express your answer in terms of $\pi$.

Answer: 48\pi


Given a quadrilateral \(ABCD\) with \(AD \parallel BC\), \(BA = AD = DC = \frac{1}{2} BC = a\), and \(E\) as the midpoint of \(BC\). The triangle \(BAE\) is folded along \(AE\) to form \(\triangle B_{1}AE\), such that plane \(B_{1}AE \perp\) plane \(AECD\), and \(F\) is the midpoint of \(B_{1}D\).
(1) Find the volume of the tetrahedron \(B_{1}-AECD\);
(2) Find the sine of the angle between plane \(ADB_{1}\) and plane \(ECB_{1}\).

Answer: \frac{4}{5}


A 5-digit natural number \(abcde\) is called a "\(\pi_1\)" number if and only if it satisfies \(a < b < c\) and \(c > d > e\). Determine the total number of "\(\pi_1\)" numbers among all 5-digit numbers.

Answer: 2142


There are 90 players in a league. Each of the 90 players plays exactly one match against each of the other 89 players. Each match ends with a win for one player and a loss for the other player, or with a tie for both players. Each player earns 1 point for a win, 0 points for a loss, and 0.5 points for a tie. After all matches have been played, the points earned by each player are added up. What is the greatest possible number of players whose total score can be at least 54 points?

Answer: 71


In the regular quadrangular pyramid \(P-ABCD\), \(M\) and \(N\) are the midpoints of \(PA\) and \(PB\) respectively. If the tangent of the dihedral angle between a side face and the base is \(\sqrt{2}\), find the cosine of the angle between skew lines \(DM\) and \(AN\).

Answer: 1/6


Find all functions \( f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+} \) such that
\[
f(x + f(y)) = f(x + y) + f(y)
\]
for all \( x, y \in \mathbb{R}^{+} \). (Symbol \( \mathbb{R}^{+} \) denotes the set of all positive real numbers.)

Answer: f(x)=2x


Cyclic quadrilateral  $ABCD$  has  $AC\perp BD$ ,  $AB+CD=12$ , and  $BC+AD=13$ . FInd the greatest possible area of  $ABCD$ .

Answer: 36


(a) It is said that a random variable \(\xi\) with natural values is distributed according to the discrete Pareto distribution with parameter \(\rho>0\) if

\[ \mathrm{P}(\xi=n)=\frac{1}{\zeta(\rho+1) n^{\rho+1}}, \quad n \in \mathbb{N} \]

where \(\zeta(s)=\sum_{n \geqslant 1} n^{-s}\) is the Riemann zeta function. Show that

\[ \mathrm{E} \xi=\frac{\zeta(\rho)}{\zeta(\rho+1)} \]

(b) Show that

\[ \zeta(2)=\sum_{n \geqslant 1} \frac{1}{n^{2}}=\frac{\pi^{2}}{6} \]

by evaluating the auxiliary integral

\[ \int_{[0,1]^{2}} \frac{dx \, dy}{1-xy} \]

Answer: \frac{\pi^2}{6}


On a table, there are 20 cards numbered from 1 to 20. Each time, Xiao Ming picks out 2 cards such that the number on one card is 2 more than twice the number on the other card. What is the maximum number of cards Xiao Ming can pick?

Answer: 12


Given rectangle ABCD, AB=4, BC=8, points K and L are midpoints of BC and AD, respectively, and point M is the midpoint of KL. What is the area of the quadrilateral formed by the rectangle diagonals and segments KM and LM.

Answer: 16


Let \( OP \) be the diameter of the circle \( \Omega \), and let \( \omega \) be a circle with its center at point \( P \) and a radius smaller than that of \( \Omega \). The circles \( \Omega \) and \( \omega \) intersect at points \( C \) and \( D \). The chord \( OB \) of circle \( \Omega \) intersects the second circle at point \( A \). Find the length of the segment \( AB \) if \( BD \cdot BC = 5 \).

Answer: \sqrt{5}


The numbers from 1 to 8 are placed at the vertices of a cube such that the sum of the numbers at any three vertices lying on the same face is at least 10. What is the minimum possible sum of the numbers at the vertices of one face?

Answer: 16


It is known that the variance of each of the given independent random variables does not exceed 4. Determine the number of such variables for which the probability that the deviation of the arithmetic mean of the random variable from the arithmetic mean of their mathematical expectations by no more than 0.25 exceeds 0.99.

Answer: 6400


Let the function $f(x) = (x-3)^3 + x - 1$, and $\{a_n\}$ be an arithmetic sequence with a non-zero common difference. If $f(a_1) + f(a_2) + \ldots + f(a_7) = 14$, then calculate the value of $a_1 + a_2 + \ldots + a_7$.

Answer: 21


Let $S$ be the set of all positive integer divisors of $100,000.$ How many numbers are the product of two distinct elements of $S?$

Answer: 117


Consider the nonconvex quadrilateral \(ABCD\) with \(\angle C > 180^\circ\). Let the side \(DC\) extended meet \(AB\) at \(F\) and the side \(BC\) extended meet \(AD\) at \(E\). A line intersects the interiors of the sides \(AB, AD, BC, CD\) at points \(K, L, J, I\), respectively. Prove that if \(DI = CF\) and \(BJ = CE\), then \(KJ = IL\).

Answer: KJ=IL


We consider positive integers  $n$  having at least six positive divisors. Let the positive divisors of  $n$  be arranged in a sequence  $(d_i)_{1\le i\le k}$  with   $$ 1=d_1<d_2<\dots <d_k=n\quad (k\ge 6). $$  Find all positive integers  $n$  such that   $$ n=d_5^2+d_6^2. $$

Answer: 500


Given $f(x)=1-2x^{2}$ and $g(x)=x^{2}-2x$, let $F(x) = \begin{cases} f(x), & \text{if } f(x) \geq g(x) \\ g(x), & \text{if } f(x) < g(x) \end{cases}$. Determine the maximum value of $F(x)$.

Answer: \frac{7}{9}


Given a circle of radius $3$, there are multiple line segments of length $6$ that are tangent to the circle at their midpoints. Calculate the area of the region occupied by all such line segments.

Answer: 9\pi


Each vertex of convex pentagon $ABCDE$ is to be assigned a color. There are $6$ colors to choose from, and the ends of each diagonal must have different colors. How many different colorings are possible?

Answer: 3120


The wording is just ever so slightly different, however the problem is identical.

Problem 3. Determine all functions  $f: \mathbb{N} \to \mathbb{N}$  such that  $n^2 + f(n)f(m)$  is a multiple of  $f(n) + m$  for all natural numbers  $m, n$ .

Answer: f(n)=n


A flock of 155 birds sit down on a circle \( C \). Two birds \( P_{i}, P_{j} \) are mutually visible if \( m(P_{i} P_{j}) \leq 10^\circ \). Find the smallest number of mutually visible pairs of birds. (One assumes that a position (point) on \( C \) can be occupied simultaneously by several birds.)

Answer: 270


Find the lowest possible values from the function
\[ f(x) \equal{} x^{2008} \minus{} 2x^{2007} \plus{} 3x^{2006} \minus{} 4x^{2005} \plus{} 5x^{2004} \minus{} \cdots \minus{} 2006x^3 \plus{} 2007x^2 \minus{} 2008x \plus{} 2009\]
for any real numbers  $ x$ .

Answer: 1005


A three-quarter sector of a circle of radius $4$ inches together with its interior can be rolled up to form the lateral surface area of a right circular cone by taping together along the two radii shown. What is the volume of the cone in cubic inches?

Answer: $3 \pi \sqrt{7}$


Find the smallest positive integer that cannot be expressed in the form \(\frac{2^{a}-2^{b}}{2^{c}-2^{d}}\), where \(a, b, c, d\) are non-negative integers.

Answer: 11


Ainsley and Buddy play a game where they repeatedly roll a standard fair six-sided die. Ainsley wins if two multiples of 3 in a row are rolled before a non-multiple of 3 followed by a multiple of 3, and Buddy wins otherwise. If the probability that Ainsley wins is \(\frac{a}{b}\) for relatively prime positive integers \(a\) and \(b\), compute \(100a + b\).

Answer: 109


Find the number of subsets  $ S$  of  $ \{1,2, \dots 63\}$  the sum of whose elements is  $ 2008$ .

Answer: 6


Let $R$ be the set of all possible remainders when a number of the form $2^n$, $n$ a nonnegative integer, is divided by 1000. Let $S$ be the sum of the elements in $R$. Find the remainder when $S$ is divided by 1000.

Answer: 7


Given quadrilateral $\Box FRDS$ with $\triangle FDR$ being a right-angled triangle at point $D$, with side lengths $FD = 3$ inches, $DR = 4$ inches, $FR = 5$ inches, and $FS = 8$ inches, and $\angle RFS = \angle FDR$, find the length of RS.

Answer: \sqrt{89}


Three circles, each of radius $3$, are drawn with centers at $(14, 92)$, $(17, 76)$, and $(19, 84)$. A line passing through $(17,76)$ is such that the total area of the parts of the three circles to one side of the line is equal to the total area of the parts of the three circles to the other side of it. What is the absolute value of the slope of this line?

Answer: 24


Alice writes 1001 letters on a blackboard, each one chosen independently and uniformly at random from the set $S=\{a, b, c\}$. A move consists of erasing two distinct letters from the board and replacing them with the third letter in $S$. What is the probability that Alice can perform a sequence of moves which results in one letter remaining on the blackboard?

Answer: \frac{3-3^{-999}}{4}


In a circle, two perpendicular diameters $AE$ and $BF$ are drawn. A point $C$ is chosen on the arc $EF$. The chords $CA$ and $CB$ intersect the diameters $BF$ and $AE$ at points $P$ and $Q$ respectively. Prove that the area of quadrilateral $APQB$ is equal to the square of the radius of the circle.

Answer: R^2


Let  $d \geq 1$  be an integer that is not the square of an integer. Prove that for every integer  $n \geq 1,$ \[(n \sqrt d +1) \cdot | \sin(n \pi \sqrt d )| \geq 1\]

Answer: (n\sqrt{}+1)\cdot|\sin(n\pi\sqrt{})|\ge1


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


How many whole numbers between $200$ and $500$ contain the digit $3$?

Answer: 138


Jia participates in a shooting practice with 4 bullets, among which one is a blank (a "blank" means a bullet without a projectile).
(1) If Jia shoots only once, calculate the probability of the shot being a blank;
(2) If Jia shoots a total of 3 times, calculate the probability of a blank appearing in these three shots;
(3) If an equilateral triangle PQR with a side length of 10 is drawn on the target, and Jia, using live rounds, aims and randomly shoots at the area of triangle PQR, calculate the probability that all bullet holes are more than 1 unit away from the vertices of △PQR (ignoring the size of the bullet holes).

Answer: 1 - \frac{\sqrt{3}\pi}{150}


In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively, and it is given that $a\cos B=(3c-b)\cos A$.
$(1)$ If $a\sin B=2\sqrt{2}$, find $b$;
$(2)$ If $a=2\sqrt{2}$ and the area of $\triangle ABC$ is $\sqrt{2}$, find the perimeter of $\triangle ABC$.

Answer: 4+2\sqrt{2}


There are two types of electronic toy cars, Type I and Type II, each running on the same two circular tracks. Type I completes a lap every 5 minutes, while Type II completes a lap every 3 minutes. At a certain moment, both Type I and Type II cars start their 19th lap simultaneously. How many minutes earlier did the Type I car start running compared to the Type II car?

Answer: 36


Given that  $\tbinom{n}{k}=\tfrac{n!}{k!(n-k)!}$ , the value of   $$ \sum_{n=3}^{10}\frac{\binom{n}{2}}{\binom{n}{3}\binom{n+1}{3}} $$   can be written in the form  $\tfrac{m}{n}$ , where  $m$  and  $n$  are relatively prime positive integers. Compute  $m+n$ .

Answer: 329


Let $X$ be a subset of $\{1,2,3, \ldots, 10000\}$ with the following property: If $a, b \in X, a \neq b$, then $a \cdot b \notin X$. What is the maximal number of elements in $X$?

Answer: 9901


On a circle, there are 25 points marked, which are colored either red or blue. Some points are connected by segments, with each segment having one end blue and the other red. It is known that there do not exist two red points that are connected to the same number of segments. What is the greatest possible number of red points?

Answer: 13


A square is inscribed in another square such that its vertices lie on the sides of the first square, and its sides form angles of $60^{\circ}$ with the sides of the first square. What fraction of the area of the given square is the area of the inscribed square?

Answer: 4 - 2\sqrt{3}


If each of the four numbers $3, 4, 6,$ and $7$ replaces a $\square$, what is the largest possible sum of the fractions shown?

Answer: $\frac{23}{6}$


An equilateral triangle $PQR$ is inscribed in the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1,$ so that $Q$ is at $(0,b),$ and $\overline{PR}$ is parallel to the $x$-axis, as shown below.  Also, foci $F_1$ and $F_2$ lie on sides $\overline{QR}$ and $\overline{PQ},$ respectively.  Find $\frac{PQ}{F_1 F_2}.$

[asy]
unitsize(0.4 cm);

pair A, B, C;
pair[] F;
real a, b, c, s;

a = 5;
b = sqrt(3)/2*5;
c = 5/2;
s = 8;

A = (-s/2,-sqrt(3)/2*(s - 5));
B = (0,b);
C = (s/2,-sqrt(3)/2*(s - 5));
F[1] = (c,0);
F[2] = (-c,0);

draw(yscale(b)*xscale(a)*Circle((0,0),1));
draw(A--B--C--cycle);

label("$P$", A, SW);
label("$Q$", B, N);
label("$R$", C, SE);
dot("$F_1$", F[1], NE);
dot("$F_2$", F[2], NW);
[/asy]

Answer: \frac{8}{5}


In triangle  $\triangle ABC$ , the angle  $\angle BAC$  is less than  $90^o$ . The perpendiculars from  $C$  on  $AB$  and from  $B$  on  $AC$  intersect the circumcircle of  $\triangle ABC$  again at  $D$  and  $E$  respectively. If  $|DE| =|BC|$ , find the measure of the angle  $\angle BAC$ .

Answer: 60


A $5 \times 5$ square consisting of cells was divided into several parts of different areas, each part consisting of an integer number of cells. What is the maximum number of parts that could result from such a division?

Answer: 6


Answer: \frac{1}{3}


Let  $P(x)=x+1$  and  $Q(x)=x^2+1.$  Consider all sequences  $\langle(x_k,y_k)\rangle_{k\in\mathbb{N}}$  such that  $(x_1,y_1)=(1,3)$  and  $(x_{k+1},y_{k+1})$  is either  $(P(x_k), Q(y_k))$  or  $(Q(x_k),P(y_k))$  for each  $k.  $  We say that a positive integer  $n$  is nice if  $x_n=y_n$  holds in at least one of these sequences. Find all nice numbers.

Answer: 3


How many positive integers less than $500$ can be written as the sum of two positive perfect cubes?

Answer: 26


Princeton’s Math Club recently bought a stock for  $\$ 2 $ and sold it for $ \ $9$  thirteen days later. Given that the stock either increases or decreases by  $\$ 1 $ every day and never reached $ \ $0$ , in how many possible ways could the stock have changed during those thirteen days?

Answer: 273


The equation $x^3 - 4x^2 + 5x - \frac{19}{10} = 0$ has real roots $r,$ $s,$ and $t.$  Find the area of the triangle with sides $r,$ $s,$ and $t.$

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


Determine the lowest positive integer n such that following statement is true:
If polynomial with integer coefficients gets value 2 for n different integers,
then it can't take value 4 for any integer.

Answer: 4


Let \( ABC \) be a triangle with \(\angle A = 60^\circ\). Line \(\ell\) intersects segments \( AB \) and \( AC \) and splits triangle \( ABC \) into an equilateral triangle and a quadrilateral. Let \( X \) and \( Y \) be on \(\ell\) such that lines \( BX \) and \( CY \) are perpendicular to \(\ell\). Given that \( AB = 20 \) and \( AC = 22 \), compute \( XY \).

Answer: 21


In how many ways can two distinct squares be chosen from an $8 \times 8$ chessboard such that the midpoint of the line segment connecting their centers is also the center of a square on the board?

Answer: 480


Given: \(\cos 2\alpha = m\). Find \(A = \cos^8 \alpha - \sin^8 \alpha\).

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


Given that $n$ is a positive integer, and $4^7 + 4^n + 4^{1998}$ is a perfect square, then one value of $n$ is.

Answer: 3988


Five positive integers (not necessarily all different) are written on five cards. Boris calculates the sum of the numbers on every pair of cards. He obtains only three different totals: 57, 70, and 83. What is the largest integer on any card?

Answer: 48


In the non-isosceles triangle \( \triangle ABC \), points \( D, E, F \) are the midpoints of \( BC, CA, \) and \( AB \) respectively. Draw a tangent to the incircle of \( \triangle ABC \) at point \( D \) (distinct from line \( BC \)), and let this tangent intersect line \( EF \) at point \( X \). Similarly define points \( Y \) and \( Z \). Prove that points \( X, Y, \) and \( Z \) are collinear.

Answer: X,Y,Z


Given the numbers \( a \) and \( b \), it is known that the system of equations

\[ \begin{cases}
y = x^2 + ax + b \\
x = y^2 + ay + b
\end{cases} \]

has a unique solution. Prove that \( a^2 = 2(a + 2b) - 1 \).

Answer: ^2=2(2b)-1


Five men and nine women stand equally spaced around a circle in random order. The probability that every man stands diametrically opposite a woman is $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$

Answer: 191


A box contains more than one hundred hawthorns. When counting by 3s, there is 1 left over; when counting by 4s, there are 2 left over; when counting by 5s, there are 3 left over; when counting by 6s, there are 4 left over. What is the maximum number of hawthorns in this box?

Answer: 178


A right triangle \(ABC\) is divided by the altitude \(CD\), drawn to the hypotenuse, into two triangles \(BCD\) and \(ACD\). The radii of the circles inscribed in triangles \(BCD\) and \(ACD\) are 4 cm and 3 cm, respectively. Determine the distance between their centers.

Answer: 5\sqrt{2}


In an isosceles triangle \(ABC\), the angle at the vertex \(B\) is \(20^\circ\). Points \(D\) and \(K\) are taken on the sides \(AB\) and \(BC\) respectively such that \(\angle KAC = 50^\circ\) and \(\angle DCA = 60^\circ\). Calculate \(\angle CDK\).

Answer: 30


John has recorded completion times, in seconds, of 100, 108, 112, 104, and 110 for running a 100-meter dash. After another race, he realized his median time dropped to 106 seconds. What was his time, in seconds, for the latest race?

Answer: 104


Hooligan Vasya likes to run on the escalator in the metro. He runs downward twice as fast as upward. If the escalator is not working, it will take Vasya 6 minutes to run up and down. If the escalator is moving downward, it will take Vasya 13.5 minutes to run up and down. How many seconds will it take Vasya to run up and down the escalator if it is moving upward? (The escalator always moves at a constant speed.)

Answer: 324


In triangle \( ABC \), the angle bisector \( AL \) (where \( L \) is on \( BC \)) is drawn. Points \( M \) and \( N \) lie on the other two angle bisectors (or their extensions) such that \( MA=ML \) and \( NA=NL \). Given that \( \angle BAC = 50^\circ \), find the measure of \( \angle MAN \) in degrees.

Answer: 65


Find $7463_{8} - 3254_{8}$. Express your answer first in base $8$, then convert it to base $10$.

Answer: 2183_{10}


Convex quadrilateral $ABCD$ has $AB = 18$, $\angle A = 60^\circ$, and $\overline{AB} \parallel \overline{CD}$. In some order, the lengths of the four sides form an arithmetic progression, and side $\overline{AB}$ is a side of maximum length. The length of another side is $a$. What is the sum of all possible values of $a$?

Answer: 66


Given a hyperbola with asymptotes $2x \pm y=0$, that passes through the intersection of the lines $x+y-3=0$ and $2x-y+3t=0$, where $-2 \leq t \leq 5$. Find the maximum possible length of the real axis of the hyperbola.

Answer: 4\sqrt{3}


Let the set \( S = \{1, 2, \cdots, 15\} \). Define \( A = \{a_{1}, a_{2}, a_{3}\} \) as a subset of \( S \), such that \( (a_{1}, a_{2}, a_{3}) \) satisfies \( 1 \leq a_{1} < a_{2} < a_{3} \leq 15 \) and \( a_{3} - a_{2} \leq 6 \). Find the number of such subsets that satisfy these conditions.

Answer: 371


For a pair of integers \((a, b)(0 < a < b < 1000)\), a set \(S \subseteq \{1, 2, \cdots, 2003\}\) is called a "jump set" for the pair \((a, b)\) if for any pair of elements \(\left(s_{1}, s_{2}\right)\) in \(S\), \(|s_{1}-s_{2}| \notin\{a, b\}\).

Let \(f(a, b)\) denote the maximum number of elements in a jump set for the pair \((a, b)\). Find the maximum and minimum values of \(f(a, b)\).

Answer: 668


If the expression $x^2 + 9x + 20$ can be written as $(x + a)(x + b)$, and the expression $x^2 + 7x - 60$ can be written as $(x + b)(x - c)$, where $a$, $b$, and $c$ are integers. What is the value of $a + b + c$?

Answer: 14


On the leg \( BC \) of the right triangle \( ABC \), a circle is constructed with \( BC \) as its diameter. The circle intersects the hypotenuse \( AB \) at point \( K \). Find \( CK \), given that \( BC = a \) and \( AC = b \).

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


Find all functions \( f: \mathbb{R} \rightarrow \mathbb{R} \) that satisfy \( f\left(x^{2}-y^{2}\right)=(x-y)(f(x)+f(y)) \) for all real numbers \( x \) and \( y \).

Answer: f(x)=kx


Find the coefficient of \(x^9\) in the polynomial expansion of \((1+3x-2x^2)^5\).

Answer: 240


Inside a regular tetrahedron \( ABCD \) with edge length 1, there are two tangent spheres. One sphere is tangent to the three faces passing through point \( A \), and the other sphere is tangent to the three faces passing through point \( B \). What is the minimum value of the sum of their radii?

Answer: \frac{\sqrt{6}-1}{5}


A passenger train departs at 12 noon from $A$ to $B$ at a constant speed of $60 \, \mathrm{km/h}$. At the same time, a freight train is traveling from $B$ to $A$ on the same track at a speed of $40 \, \mathrm{km/h}$. Both trains reach their destinations simultaneously, and at that moment they are three times as far apart from each other as they were one hour before their meeting point. When did the freight train depart from $B$?

Answer: 9:30\,


If the real numbers $m$, $n$, $s$, $t$ are all distinct and satisfy $mn=st$, then $m$, $n$, $s$, $t$ are said to have the property of "quasi-geometric progression." Now, randomly select $4$ different numbers from the $7$ numbers $2$, $4$, $8$, $16$, $32$, $64$, $128$. The probability that these $4$ numbers have the property of "quasi-geometric progression" is ____.

Answer: \frac{13}{35}


Prove that the point with coordinates \( x = \operatorname{ch} t, y = \operatorname{sh} t \) lies on the hyperbola \( x^2 - y^2 = 1 \).

Answer: 1


Two candles of different thickness, each 24 cm long, are lit simultaneously. The thin candle burns out in 4 hours, and the thick candle in 6 hours. After what time will one candle be half the length of the other? It is assumed that each candle burns uniformly, meaning it decreases by the same height at equal intervals of time. The comparison is made between the remaining parts of the candles.

Answer: 3\,


Let \( f(n) \) denote the number of positive integral solutions of the equation \( 4x + 3y + 2z = n \). Find \( f(2009) - f(2000) \).

Answer: 1000


Given that the sequence ${a_n}$ is an arithmetic sequence with first term $1$ and common difference $2$,
(1) Find the general term formula for ${a_n}$;
(2) Let $b_n = \frac{1}{a_n \cdot a_{n-1}}$. Denote the sum of the first $n$ terms of the sequence ${b_n}$ as $T_n$. Find the minimum value of $T_n$.

Answer: \frac{1}{3}


Let \( S = \{1, 2, 3, 4\} \). An \( n \)-term sequence \( q_1, q_2, \ldots, q_n \) has the following property: For any non-empty subset \( B \) of \( S \), there exists exactly one set of \( |B| \) consecutive terms in the sequence that forms the set \( B \). What is the minimum value of \( n \)?

Answer: 8


A square of perimeter 20 is inscribed in a square of perimeter 28. What is the greatest distance between a vertex of the inner square and a vertex of the outer square?

Answer: \sqrt{65}


Let \( f(x) = a_{0} + a_{1} x + \cdots + a_{n} x^{n} \) be an \( n \)-th degree polynomial with integer coefficients. If \( a_{n} \), \( a_{0} \), and \( f(1) \) are all odd, prove that \( f(x) = 0 \) has no rational roots.

(Note: This problem appeared in the 1987 Singapore Math Olympiad.)

Answer: f(x)=0


For $n$ a positive integer, let $f(n)$ be the quotient obtained when the sum of all positive divisors of $n$ is divided by $n.$ For example, \[f(14)=(1+2+7+14)\div 14=\frac{12}{7}\]
What is $f(768)-f(384)?$
$\textbf{(A)}\ \frac{1}{768} \qquad\textbf{(B)}\ \frac{1}{192} \qquad\textbf{(C)}\ 1 \qquad\textbf{(D)}\ \frac{4}{3} \qquad\textbf{(E)}\ \frac{8}{3}$

Answer: \textbf{(B)}\\frac{1}{192}


In a room, $2/5$ of the people are wearing gloves, and $3/4$ of the people are wearing hats. What is the minimum number of people in the room wearing both a hat and a glove?
$\textbf{(A)}\ 3 \qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 15\qquad\textbf{(E)}\ 20$

Answer: \textbf{(A)}\3


In a \(5 \times 5\) table, numbers (not necessarily integers) are placed such that each number is three times smaller than the number in the cell to its right and twice larger than the number in the cell below it. Find the number in the central cell, given that the sum of all the numbers in the table is 341.

Answer: \frac{36}{11}


If for any positive integer \( m \), the set
$$
\{m, m+1, m+2, \cdots, m+99\}
$$
in any \( n \)-element subset with \( n \geq 3 \), there are always three elements that are pairwise coprime, find the smallest value of \( n \).

Answer: 68


Let \( A, B, C, D, E, F \) be 6 points on a circle in that order. Let \( X \) be the intersection of \( AD \) and \( BE \), \( Y \) is the intersection of \( AD \) and \( CF \), and \( Z \) is the intersection of \( CF \) and \( BE \). \( X \) lies on segments \( BZ \) and \( AY \) and \( Y \) lies on segment \( CZ \). Given that \( AX = 3 \), \( BX = 2 \), \( CY = 4 \), \( DY = 10 \), \( EZ = 16 \), and \( FZ = 12 \), find the perimeter of triangle \( XYZ \).

Answer: \frac{77}{6}


Find the sum of all the real numbers \( x \) that satisfy the equation
\[
\left(3^{x}-27\right)^{2}+\left(5^{x}-625\right)^{2}=\left(3^{x}+5^{x}-652\right)^{2} .
\]

Answer: 7


Let \(\mathbf{N}^{*}\) be the set of positive integers. Define a function \(f\) on \(\mathbf{N}^{*}\) as follows:
1. \(f(1)=1, f(3)=3\);
2. For \(n \in \mathbf{N}^{*}\), the following holds:
\[
\begin{array}{l}
f(2 n) = f(n), \\
f(4 n + 1) = 2 f(2 n + 1) - f(n), \\
f(4 n + 3) = 3 f(2 n + 1) - 2 f(n).
\end{array}
\]

Question: How many \(n \in \mathbf{N}^{*}\) are there such that \(n \leq 1988\) and \(f(n) = n\)?

Answer: 92


China's space station has entered the formal construction phase. The Tianhe core module, Wentian experimental module, and Mengtian experimental module will all dock in 2022, forming a "T" shaped structure. During the construction phase of the Chinese space station, there are 6 astronauts staying in the space station. It is expected that in a certain construction task, 6 astronauts need to work simultaneously in the Tianhe core module, Wentian experimental module, and Mengtian experimental module. Due to space limitations, each module must have at least 1 person and at most 3 people. The total number of different arrangement plans is ______.

Answer: 450


Let \( S = \{1, 2, \ldots, n\} \). \( A \) is an arithmetic progression with at least two terms, a positive common difference, and all terms belonging to \( S \). Adding any other element of \( S \) to \( A \) cannot form another arithmetic progression with the same common difference as \( A \). Find the number of such arithmetic progressions \( A \) (considering sequences with only two terms as arithmetic progressions).

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


For what real numbers α does the series \((\frac{1}{1 \csc(1) - 1})^\alpha + (\frac{1}{2 \csc(1/2) - 1})^\alpha + \cdots + (\frac{1}{n \csc(1/n) - 1})^\alpha + \cdots\) converge?

Answer: \alpha>\frac{1}{2}


Given two intersecting circles with radius $R$, where the distance between their centers is greater than $R$. Prove that $\beta=3 \alpha$.

Answer: \beta=3\alpha


Three non-overlapping regular plane polygons, at least two of which are congruent, all have sides of length $1$. The polygons meet at a point $A$ in such a way that the sum of the three interior angles at $A$ is $360^{\circ}$. Thus the three polygons form a new polygon with $A$ as an interior point. What is the largest possible perimeter that this polygon can have?

Answer: 21


Find the number of pairs of integers \((a, b)\) with \(1 \leq a<b \leq 57\) such that \(a^{2}\) has a smaller remainder than \(b^{2}\) when divided by 57.

Answer: 738


For a natural number \(a\), \(S(a)\) represents the sum of the digits of \(a\) (for example, \(S(123) = 1 + 2 + 3 = 6\)). If the digits of a natural number \(n\) are all different and \(S(3n) = 3S(n)\), what is the maximum value of \(n\)?

Answer: 3210


There are 10 cards, each card is written with two different numbers from $1, 2, 3, 4, 5$, and the numbers on any two cards are not completely the same. These 10 cards are to be placed into five boxes labeled $1, 2, 3, 4, 5$, with the rule that a card with numbers $i, j$ can only be placed in box $i$ or box $j$. A placement is deemed "good" if the number of cards in box 1 is greater than the number of cards in any other box. Find the total number of "good" placements.

Answer: 120


There are three video game systems: the Paystation, the WHAT, and the ZBoz2 \pi, and none of these systems will play games for the other systems. Uncle Riemann has three nephews: Bernoulli, Galois, and Dirac. Bernoulli owns a Paystation and a WHAT, Galois owns a WHAT and a ZBoz2 \pi, and Dirac owns a ZBoz2 \pi and a Paystation. A store sells 4 different games for the Paystation, 6 different games for the WHAT, and 10 different games for the ZBoz2 \pi. Uncle Riemann does not understand the difference between the systems, so he walks into the store and buys 3 random games (not necessarily distinct) and randomly hands them to his nephews. What is the probability that each nephew receives a game he can play?

Answer: \frac{7}{25}


Between locations \( A \) and \( B \) there is a tunnel. A car leaves \( B \) towards \( A \) at 08:16, and a truck leaves \( A \) towards \( B \) at 09:00. It is known that the truck and the car arrive at the ends of the tunnel simultaneously, but the truck leaves the tunnel 2 minutes later than the car does. Given that the car arrives at \( A \) at 10:56, and the truck arrives at \( B \) at 12:20, determine the time at which they simultaneously arrive at the ends of the tunnel.

Answer: 10:00


What is the area of the triangle formed by the lines $y=5$, $y=1+x$, and $y=1-x$?
$\textbf{(A) }4\qquad\textbf{(B) }8\qquad\textbf{(C) }10\qquad\textbf{(D) }12\qquad\textbf{(E) }16$

Answer: \textbf{(E)}\16


Calculate

$$
\left|\begin{array}{rrr}
2 & -1 & 1 \\
3 & 2 & 2 \\
1 & -2 & 1
\end{array}\right|
$$

Answer: 5


Let $x$ and $y$ be positive real numbers. Find the minimum value of
\[\left( x + \frac{1}{y} \right) \left( x + \frac{1}{y} - 1024 \right) + \left( y + \frac{1}{x} \right) \left( y + \frac{1}{x} - 1024 \right).\]

Answer: -524288


If $x =\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\cdots}}}}$, then:
$\textbf{(A)}\ x = 1\qquad\textbf{(B)}\ 0 < x < 1\qquad\textbf{(C)}\ 1 < x < 2\qquad\textbf{(D)}\ x\text{ is infinite}$
$\textbf{(E)}\ x > 2\text{ but finite}$

Answer: \textbf{(C)}\1<x<2


Compute the limit of the function:

\[
\lim _{x \rightarrow 0}\left(\frac{1+\sin x \cdot \cos \alpha x}{1+\sin x \cdot \cos \beta x}\right)^{\operatorname{ctg}^{3} x}
\]

Answer: e^{\frac{\beta^2-\alpha^2}{2}}


Consider the sequence
$1,-2,3,-4,5,-6,\ldots,$
whose $n$th term is $(-1)^{n+1}\cdot n$. What is the average of the first $200$ terms of the sequence?

Answer: -0.5


Find the largest natural number \( n \) for which the system of inequalities

\[ 1 < x < 2, \]
\[ 2 < x^2 < 3, \]
\[ \vdots \]
\[ n < x^n < n+1 \]

has a solution.

Answer: 4


An \((n-1) \times (n-1)\) grid is divided into \((n-1)^2\) cells, and each of the \(n^2\) vertices of these cells is colored either red or blue. Determine the number of different ways to color these vertices such that each cell has exactly two red vertices. Two colorings are considered different if at least one vertex is colored differently.

Answer: 2


Determine all possible values of  $a_1$  for which there exists a sequence  $a_1, a_2, \dots$  of rational numbers satisfying   $$ a_{n+1}^2-a_{n+1}=a_n $$  for all positive integers  $n$ .

Answer: 0,2


Find the smallest positive integer \( n \) such that if the first digit is moved to become the last digit, then the new number is \( \frac{7n}{2} \).

Answer: 153846


You are given a $10 \times 2$ grid of unit squares. Two different squares are adjacent if they share a side. How many ways can one mark exactly nine of the squares so that no two marked squares are adjacent?

Answer: 36


Let \( x \) be a positive real number. What is the maximum value of \( \frac{2022 x^{2} \log (x + 2022)}{(\log (x + 2022))^{3} + 2 x^{3}} \)?

Answer: 674


A cold water tap fills the bathtub in 19 minutes, while a hot water tap fills it in 23 minutes. The hot water tap is turned on. After how many minutes should the cold water tap be turned on so that by the time the bathtub is completely filled, it contains equal amounts of cold and hot water?

Answer: 2


Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ that satisfy $\overrightarrow{a} \cdot (\overrightarrow{a} - 2\overrightarrow{b}) = 3$, and $|\overrightarrow{a}| = 1$, $\overrightarrow{b} = (1,1)$, find the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$.

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


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

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

Answer: 9


Given that there are 5 balls in a pocket, among which there are 2 black balls and 3 white balls, calculate the probability that two randomly drawn balls of the same color are both white.

Answer: \frac{3}{4}


In the sum $K A N$ each letter stands for a different digit.
$$
\frac{+G A}{R O O}
$$

What is the answer to the subtraction $R N$ ?
$$
\underline{-K G}
$$

Answer: 11


The ball $P$ starts at point $A$ and rolls back and forth 8 times. Assuming rolling to the right is positive and to the left is negative, the 8 rolls are recorded as (units: millimeters): $+12, -10, +9, -6, +8.5, -6, +8, -7$.

(1) Find the distance of the ball $P$ from point $A$ when it stops, in millimeters.

(2) If the ball takes 0.02 seconds to roll 1 millimeter, calculate the total time taken for these 8 rolls, in seconds.

Answer: 1.33


Let \( a, b, c, d \) be integers such that the system of equations
\[
\begin{cases}
a x + b y = m, \\
c x + d y = n
\end{cases}
\]
has integer solutions \(x\) and \(y\) for all integers \(m\) and \(n\). Prove that
\[
ad - bc = \pm 1.
\]

Answer: ad-=\1


In $\triangle ABC$, where $AC = BC$, let $D$ be a point on side $AB$. Consider the incircle of $\triangle ACD$ and the excircle of $\triangle BCD$ which is tangent to side $BD$, both having radius $r$. Prove that the altitude of $\triangle ABC$ from its equal sides is equal to $4r$.

Answer: 4r


Yesterday, Sasha cooked soup and added too little salt, so he had to add more salt later. Today he doubled the amount of salt, but still had to add more salt, though only half as much as yesterday. By what factor does Sasha need to increase today's portion of salt so that tomorrow he does not have to add any more salt?

(Each day, Sasha cooks the same amount of soup.)

Answer: 1.5


Alice and Ali each select a positive integer less than 250. Alice's number is a multiple of 25, and Ali's number is a multiple of 30. What is the probability that they selected the same number? Express your answer as a common fraction.

Answer: \frac{1}{80}


Let $a, b, c$, and $d$ be positive real numbers such that
\[\begin{array}{c@{\hspace{3pt}}c@{\hspace{3pt}}c@{\hspace{3pt}}c@{\hspace{3pt}}c}a^2+b^2&=&c^2+d^2&=&2008,\\ ac&=&bd&=&1000.\end{array}\]
If $S=a+b+c+d$, compute the value of $\lfloor S\rfloor$.

Answer: 126


Given that the positive integer \( n \) does not exceed 2000 and can be expressed as the sum of at least 60 consecutive positive integers, find the number of such \( n \).

Answer: 6


Determine the number of subsets $S$ of $\{1,2,3, \ldots, 10\}$ with the following property: there exist integers $a<b<c$ with $a \in S, b \notin S, c \in S$.

Answer: 968


Madam Mim has a deck of  $52$  cards, stacked in a pile with their backs facing up. Mim separates the small pile consisting of the seven cards on the top of the deck, turns it upside down, and places it at the bottom of the deck. All cards are again in one pile, but not all of them face down; the seven cards at the bottom do, in fact, face up. Mim repeats this move until all cards have their backs facing up again. In total, how many moves did Mim make  $?$

Answer: 52


If each digit of a four-digit natural number $M$ is not $0$, and the five times of the digit in the thousandth place is equal to the sum of the digits in the hundredth, tenth, and unit places, then this four-digit number is called a "modest number". For example, for the four-digit number $2163$, since $5\times 2=1+6+3$, $2163$ is a "modest number". Another example is the four-digit number $3147$, since $5\times 3\neq 1+4+7$, $3147$ is not a "modest number". If the four-digit number $\overline{467x}$ is a "modest number", then $x=$______. If a "modest number" $M=\overline{abcd}$ (where $d$ is even), rearrange the tens and units digits of the "modest number" $M$ to the front of the thousand and hundred digits to form a new four-digit number $M'=\overline{cdab}$. Define $G(M)=\frac{M-M'}{99}$. If $G(M)$ is divisible by $11$, and $\overline{abc}$ is divisible by $3$, then the maximum value of $M$ is ______.

Answer: 3816


Given that the sum of the first $n$ terms of the positive arithmetic geometric sequence ${a_n}$ is $S_n$, if $S_2=3$, $S_4=15$, find the common ratio $q$ and $S_6$.

Answer: 63


The sequence  $\{c_{n}\}$  is determined by the following equation. \[c_{n}=(n+1)\int_{0}^{1}x^{n}\cos \pi x\ dx\ (n=1,\ 2,\ \cdots).\] Let  $\lambda$  be the limit value  $\lim_{n\to\infty}c_{n}.$   Find  $\lim_{n\to\infty}\frac{c_{n+1}-\lambda}{c_{n}-\lambda}.$

Answer: 1


In a regular hexagon divided into six regions, each region is to be planted with ornamental plants, with the condition that the same plant must be used within a single region, and adjacent regions must be planted with different plants. Given that there are 4 different types of plants available, how many different planting schemes are there?

Answer: 732


A point $P$ is chosen at random from the interior of a square $ABCD$. What is the probability that the triangle $ABP$ has a greater area than each of the triangles $BCP$, $CDP$, and $DAP$?

Answer: \frac{1}{4}


A sequence \( a_1, a_2, \cdots, a_n, \cdots \) is defined such that \( S(a_i) \) denotes the sum of all digits of \( a_i \). For example, \( S(22) = 2 + 2 = 4 \). Given \( a_1 = 2017 \), \( a_2 = 22 \), and \( a_n = S(a_{n-1}) + S(a_{n-2}) \), find the value of \( a_{2017} \).

Answer: 10


Given that one air conditioner sells for a 10% profit and the other for a 10% loss, and the two air conditioners have the same selling price, determine the percentage change in the shopping mall's overall revenue.

Answer: 1\%


Determine the length of the interval of solutions of the inequality $a \le 3x + 6 \le b$ where the length of the interval is $15$.

Answer: 45


Let  $S_0=0$  and let  $S_k$  equal  $a_1+2a_2+\ldots+ka_k$  for  $k\geq 1$ .  Define  $a_i$  to be  $1$  if  $S_{i-1}<i$  and  $-1$  if  $S_{i-1}\geq i$ .  What is the largest  $k\leq 2010$  such that  $S_k=0$ ?

Answer: 1092


Find all pairs $(a,\, b)$ of positive integers such that $2a-1$ and $2b+1$ are coprime and $a+b$ divides $4ab+1.$

Answer: (a, a+1)


Given a non-zero real number sequence \( x_{1}, x_{2}, x_{3}, \cdots \) that satisfies
\[
x_{n}=\frac{x_{n-2} x_{n-1}}{2 x_{n-2}-x_{n-1}}, \quad n=3,4,5, \cdots
\]
find the necessary and sufficient condition for \( x_{1} \) and \( x_{2} \) such that the sequence contains infinitely many integers.

Answer: x_1=x_2


Given an arithmetic sequence $\{a_n\}$ with a common difference $d \neq 0$ and the first term $a_1 = d$, the sum of the first $n$ terms of the sequence $\{a_n^2\}$ is $S_n$. A geometric sequence $\{b_n\}$ has a common ratio $q$ less than 1 and consists of rational sine values, with the first term $b_1 = d^2$, and the sum of its first $n$ terms is $T_n$. If $\frac{S_3}{T_3}$ is a positive integer, then find the possible value of $q$.

Answer: \frac{1}{2}


Call a positive integer 'mild' if its base-3 representation never contains the digit 2. How many values of $n(1 \leq n \leq 1000)$ have the property that $n$ and $n^{2}$ are both mild?

Answer: 7


On a  $(4n + 2)\times (4n + 2)$  square grid, a turtle can move between squares sharing a side.The turtle begins in a corner square of the grid and enters each square exactly once, ending in the square where she started. In terms of  $n$ , what is the largest positive integer  $k$  such that there must be a row or column that the turtle has entered at least  $k$  distinct times?

Answer: 2n+2


Let $n$ be the least positive integer greater than $1000$ for which
\[\gcd(63, n+120) =21\quad \text{and} \quad \gcd(n+63, 120)=60.\]What is the sum of the digits of $n$?

Answer: 18


Find $10110_2\times10100_2\div10_2$. Express your answer in base 2.

Answer: 11011100_2


If $a, b$, and $c$ are positive integers, the radicals $\sqrt{a+\frac{b}{c}}$ and $a\sqrt{\frac{b}{c}}$ are equal when and only when:
$\textbf{(A)}\ a=b=c=1\qquad\textbf{(B)}\ a=b\text{ and }c=a=1\qquad\textbf{(C)}\ c=\frac{b(a^2-1)}{a}\\ \textbf{(D)}\ a=b\text{ and }c\text{ is any value}\qquad\textbf{(E)}\ a=b\text{ and }c=a-1$

Answer: \textbf{C}


Find the smallest positive integer \( n \) such that every \( n \)-element subset of \( S = \{1, 2, \ldots, 150\} \) contains 4 numbers that are pairwise coprime (it is known that there are 35 prime numbers in \( S \)).

Answer: 111


Let  $s(a)$  denote the sum of digits of a given positive integer a. The sequence   $a_1, a_2,..., a_n, ...$  of positive integers is such that  $a_{n+1} = a_n+s(a_n)$  for each positive integer  $n$ . Find the greatest possible n for which it is possible to have  $a_n = 2008$ .

Answer: 6


For any integer \( n (n \geqslant 4) \), consider the set \(\{1, 2, \cdots, n\}\) with \( m \) subsets \( A_{1}, A_{2}, \cdots, A_{m} \), where \( A_{1} \) has one element, \( A_{2} \) has two elements, ..., \( A_{m} \) has \( m \) elements, and no subset is contained within another subset. Find the maximum value of \( m \).

Answer: n-2


$\triangle ABC$  has side lengths  $AB=20$ ,  $BC=15$ , and  $CA=7$ . Let the altitudes of  $\triangle ABC$  be  $AD$ ,  $BE$ , and  $CF$ . What is the distance between the orthocenter (intersection of the altitudes) of  $\triangle ABC$  and the incenter of  $\triangle DEF$ ?

Answer: 15


In acute-angled triangle  $ABC$  altitudes  $BE,CF$  meet at  $H$ . A perpendicular line is drawn from  $H$  to  $EF$  and intersects the arc  $BC$  of circumcircle of  $ABC$  (that doesn’t contain  $A$ ) at  $K$ . If  $AK,BC$  meet at  $P$ , prove that  $PK=PH$ .

Answer: PK=PH


Each football ticket costs 50 yuan, and the ticket office does not prepare change. There are `m` people each holding a 50 yuan bill, and `n` people each holding a 100 yuan bill (where `m ≥ n`). Everyone lines up to buy tickets, with each person limited to purchasing one ticket. If everyone can buy their ticket without waiting for change, it is called a successful purchase. Find the probability of a successful purchase.

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


Real numbers \(a, b, c\) and a positive number \(\lambda\) such that \(f(x)=x^{3}+a x^{2}+b x+c\) has three real roots \(x_{1}, x_{2}, x_{3}\), satisfying:
(1) \(x_{2}-x_{1}=\lambda\);
(2) \(x_{3}>\frac{1}{2}\left(x_{1}+x_{2}\right)\).

Find the maximum value of \(\frac{2 a^{3}+27 c-9 a b}{\lambda^{3}}\).

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


A banker found out that among a set of identical-looking coins, one is counterfeit (lighter in weight). He asked an expert to identify the counterfeit coin using a balance scale but stipulated that each coin can be used in weighings no more than twice. What is the maximum number of coins the banker can have so that the expert can definitely identify the counterfeit coin in $n$ weighings?

Answer: 2n^2+1


Given the cryptarithm: ЖАЛО + ЛОЖА = ОСЕНЬ. Identical letters represent identical digits, and different letters represent different digits. Determine the value of the letter А.

Answer: 8


Given that P and Q are points on the graphs of the functions $2x-y+6=0$ and $y=2\ln x+2$ respectively, find the minimum value of the line segment |PQ|.

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


When dividing the polynomial \(x^{1951} - 1\) by \(x^{4} + x^{3} + 2x^{2} + x + 1\), a quotient and a remainder are obtained. Find the coefficient of \(x^{14}\) in the quotient.

Answer: -1


Six boys and six girls stood in a circle, alternating. Each of them wrote a non-zero number in their notebook. It is known that each number written by a boy is equal to the sum of the numbers written by the girls standing next to him, and each number written by a girl is equal to the product of the numbers written by the boys standing next to her. What can the sum of all twelve numbers be?

Answer: 4.5


Given a circle of radius $3$ units, find the area of the region consisting of all line segments of length $6$ units that are tangent to the circle at their midpoints.

Answer: 9\pi


Two diameters and one radius are drawn in a circle of radius 1, dividing the circle into 5 sectors. The largest possible area of the smallest sector can be expressed as $\frac{a}{b} \pi$, where $a, b$ are relatively prime positive integers. Compute $100a+b$.

Answer: 106


Given that the equation \(2x^2 + 2xy + y^2 + 2x + 2y - 4 = 0\) represents an ellipse in the Cartesian coordinate system \(xy\)-plane, what are the coordinates of the center of this ellipse?

Answer: (0,-1)


Sandwiches at Joe's Fast Food cost $$3$ each and sodas cost $$2$ each. How many dollars will it cost to purchase $5$ sandwiches and $8$ sodas?
$\textbf{(A)}\ 31\qquad\textbf{(B)}\ 32\qquad\textbf{(C)}\ 33\qquad\textbf{(D)}\ 34\qquad\textbf{(E)}\ 35$

Answer: \textbf{(A)}\31


Cassidy has string of  $n$  bits, where  $n$  is a positive integer, which initially are all  $0$ s or  $1$ s. Every second, Cassidy may choose to do one of two things:

1. Change the first bit (so the first bit changes from a  $0$  to a  $1$ , or vice versa)
2. Change the first bit after the first  $1$ .

Let  $M$  be the minimum number of such moves it takes to get from  $1\dots 1$  to  $0 \dots 0$  (both of length  $12$ ), and  $N$  the number of starting sequences with  $12$  bits that Cassidy can turn into all  $0$ s. Find  $M + N$ .

Answer: 6826


The edge of cube \( ABCD A_1 B_1 C_1 D_1 \) is 12. Point \( K \) lies on the extension of edge \( BC \) at a distance of 9 from vertex \( C \). Point \( L \) on edge \( AB \) is at a distance of 5 from \( A \). Point \( M \) divides segment \( A_1 C_1 \) in a ratio of 1:3, starting from \( A_1 \). Find the area of the cross-section of the cube by the plane passing through points \( K \), \( L \), and \( M \).

Answer: 156


For real numbers $a,$ $b,$ and $c,$ and a real scalar $\lambda,$ consider the matrix
\[\begin{pmatrix} a + \lambda & b & c \\ b & c + \lambda & a \\ c & a & b + \lambda \end{pmatrix}.\] Determine all possible values of \[\frac{a}{b + c} + \frac{b}{a + c} + \frac{c}{a + b}\] assuming the matrix is not invertible.

Answer: \frac{3}{2}


A cubic tank with rigid walls, with an edge length of \( a = 1 \) m, is divided by a thin, light, undeformable piston that moves vertically without friction. Initially, the tank is completely filled with water having a mass of \( M = 1000 \) kg. How much neon mass should be pumped into the space under the piston, so that the piston moves a distance of \( \Delta h = 0.5 \) mm? The temperatures of neon and water are the same, constant, and equal to \( t = 32^{\circ} \mathrm{C} \). The molar mass of neon is \( \mu = 20 \) g/mol, the value of the universal gas constant is \( R = 8.31 \) J/(K·mol), and the acceleration due to gravity is \( g = 10 \ \text{m} / \text{c}^{2} \). The compressibility of water (relative volume change under isothermal pressure increase) is \( \varepsilon = 5 \cdot 10^{-10} \ \text{Pa}^{-1} \).

Answer: 4\,


Given a sequence $\{a_n\}$ that satisfies: $a_{n+2}=\begin{cases} 2a_{n}+1 & (n=2k-1, k\in \mathbb{N}^*) \\ (-1)^{\frac{n}{2}} \cdot n & (n=2k, k\in \mathbb{N}^*) \end{cases}$, with $a_{1}=1$ and $a_{2}=2$, determine the maximum value of $n$ for which $S_n \leqslant 2046$.

Answer: 19


Xiao Gang goes to buy milk and finds that it's on special offer that day: each bag costs 2.5 yuan, and there is a "buy two, get one free" promotion. Xiao Gang has 30 yuan. What is the maximum number of bags of milk he can buy?

Answer: 18


Given $f(x)=9^{x}-2×3^{x}+4$, where $x\in\[-1,2\]$:
1. Let $t=3^{x}$, with $x\in\[-1,2\}$, find the maximum and minimum values of $t$.
2. Find the maximum and minimum values of $f(x)$.

Answer: 67


Find the number of ordered pairs of integers $(a, b) \in\{1,2, \ldots, 35\}^{2}$ (not necessarily distinct) such that $a x+b$ is a "quadratic residue modulo $x^{2}+1$ and 35 ", i.e. there exists a polynomial $f(x)$ with integer coefficients such that either of the following equivalent conditions holds: - there exist polynomials $P, Q$ with integer coefficients such that $f(x)^{2}-(a x+b)=\left(x^{2}+1\right) P(x)+35 Q(x)$ - or more conceptually, the remainder when (the polynomial) $f(x)^{2}-(a x+b)$ is divided by (the polynomial) $x^{2}+1$ is a polynomial with (integer) coefficients all divisible by 35 .

Answer: 225


Find the minimum of \( f(x, y) = (x - y)^2 + \left( \sqrt{2 - x^2} - \frac{9}{y} \right)^2 \) in the half-infinite strip \( 0 < x < \sqrt{2}, y > 0 \).

Answer: 8


There are 7 volunteers to be arranged to participate in the promotional activities for the Shanghai World Expo on Saturday and Sunday, with 6 people participating each day and 3 people arranged for each day. There are __________ different arrangements possible (answer with a number).

Answer: 140


Shelly writes down a vector $v=(a, b, c, d)$, where $0<a<b<c<d$ are integers. Let $\sigma(v)$ denote the set of 24 vectors whose coordinates are $a, b, c$, and $d$ in some order. For instance, $\sigma(v)$ contains $(b, c, d, a)$. Shelly notes that there are 3 vectors in $\sigma(v)$ whose sum is of the form $(s, s, s, s)$ for some $s$. What is the smallest possible value of $d$?

Answer: 6


Regular octagon $ABCDEFGH$ has area $n$. Let $m$ be the area of quadrilateral $ACEG$. What is $\frac{m}{n}?$

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


The range of the function
$$
f(x)=\frac{\sin x-1}{\sqrt{3-2 \cos x-2 \sin x}} \quad (0 \leqslant x \leqslant 2 \pi)
$$
is .

Answer: [-1,0]


An isosceles trapezoid is circumscribed around a circle. The longer base of the trapezoid is $16$, and one of the base angles is $\arcsin(.8)$. Find the area of the trapezoid.
$\textbf{(A)}\ 72\qquad \textbf{(B)}\ 75\qquad \textbf{(C)}\ 80\qquad \textbf{(D)}\ 90\qquad \textbf{(E)}\ \text{not uniquely determined}$

Answer: 80


Find all the functions  $ f: \mathbb{N}\rightarrow \mathbb{N}$  such that
\[ 3f(f(f(n))) \plus{} 2f(f(n)) \plus{} f(n) \equal{} 6n, \quad \forall n\in \mathbb{N}.\]

Answer: f(n)=n


In the diagram, $COB$ is a sector of a circle with $\angle COB=45^\circ.$ $OZ$ is drawn perpendicular to $CB$ and intersects $CB$ at $W.$ What is the length of $WZ$? Assume the radius of the circle is 10 units.

Answer: 10 - 5 \sqrt{2+\sqrt{2}}


From the numbers 1, 2, 3, 4, 5, a five-digit number is formed with digits not repeating. What is the probability of randomly selecting a five-digit number $\overline{abcde}$ that satisfies the condition "$a < b > c < d > e$"?

Answer: 2/15


Define binary operations $\diamondsuit$ and $\heartsuit$ by $a \diamondsuit b = a^{\log_{7}(b)}$ and $a \heartsuit b = a^{\frac{1}{\log_{7}(b)}}$ for all real numbers $a$ and $b$ for which these expressions are defined. The sequence $(a_n)$ is defined recursively by $a_3 = 3 \heartsuit 2$ and $a_n = (n \heartsuit (n-1)) \diamondsuit a_{n-1}$ for all integers $n \geq 4$. To the nearest integer, what is $\log_{7}(a_{2019})$?

Answer: 11


A stack of $2000$ cards is labelled with the integers from $1$ to $2000,$ with different integers on different cards. The cards in the stack are not in numerical order. The top card is removed from the stack and placed on the table, and the next card is moved to the bottom of the stack. The new top card is removed from the stack and placed on the table, to the right of the card already there, and the next card in the stack is moved to the bottom of the stack. The process - placing the top card to the right of the cards already on the table and moving the next card in the stack to the bottom of the stack - is repeated until all cards are on the table. It is found that, reading from left to right, the labels on the cards are now in ascending order: $1,2,3,\ldots,1999,2000.$ In the original stack of cards, how many cards were above the card labeled $1999$?

Answer: 927


Find the least  $n\in N$  such that among any  $n$  rays in space sharing a common origin there exist two which form an acute angle.

Answer: 7


For \(x, y, z \geq 1\), find the minimum value of the expression

$$
A = \frac{\sqrt{3 x^{4} + y} + \sqrt{3 y^{4} + z} + \sqrt{3 z^{4} + x} - 3}{x y + y z + z x}
$$

Answer: 1


Two circles have radii 13 and 30, and their centers are 41 units apart. The line through the centers of the two circles intersects the smaller circle at two points; let \( A \) be the one outside the larger circle. Suppose \( B \) is a point on the smaller circle and \( C \) a point on the larger circle such that \( B \) is the midpoint of \( AC \). Compute the distance \( AC \).

Answer: 12\sqrt{13}


Let \( M \) be a finite set of real numbers with the following property: From every three different elements of \( M \), two can always be selected whose sum lies in \( M \). What is the maximum number of elements \( M \) can have?

Answer: 7


Given the sequence \(\{a_n\}\):
\[1,1,2,1,2,3,\cdots, 1,2,\cdots, n,\cdots\]

Let \(S_n\) denote the sum of the first \(n\) terms of the sequence \(\{a_n\}\). Find all positive real number pairs \((\alpha, \beta)\) such that
\[
\lim_{n \rightarrow +\infty} \frac{S_n}{n^\alpha} = \beta.
\]

Answer: (\frac{3}{2},\frac{\sqrt{2}}{3})


In the cells of a $9 \times 9$ square, zeros and ones are placed so that in any shape consisting of four cells in a $\square$ pattern, the sum of the numbers is odd. (The shape can be rotated and flipped.) What is the maximum number of ones that can be placed in such an arrangement?

Answer: 65


For a nonnegative integer $n$, let $r_9(n)$ be the remainder when $n$ is divided by $9$. Consider all nonnegative integers $n$ for which $r_9(7n) \leq 5$. Find the $15^{\text{th}}$ entry in an ordered list of all such $n$.

Answer: 21


In the right triangle \( KLM \), a circle is constructed on the leg \( KM \) as the diameter, which intersects the hypotenuse \( KL \) at point \( G \). A tangent to the circle is drawn through point \( G \), intersecting the leg \( ML \) at point \( F \). Find the length of \( FL \), given that \( KG = 5 \) and \( LG = 4 \).

Answer: 3


Given a $4 \times 4$ square grid partitioned into $16$ unit squares, each of which is painted white or black with a probability of $\frac{1}{2}$, determine the probability that the grid is entirely black after a $90^{\circ}$ clockwise rotation and any white square landing in a position previously occupied by a black square is repainted black.

Answer: \frac{1}{65536}


Let \( P \) and \( Q \) be points on line \( l \) with \( PQ = 12 \). Two circles, \( \omega \) and \( \Omega \), are both tangent to \( l \) at \( P \) and are externally tangent to each other. A line through \( Q \) intersects \( \omega \) at \( A \) and \( B \), with \( A \) closer to \( Q \) than \( B \), such that \( AB = 10 \). Similarly, another line through \( Q \) intersects \( \Omega \) at \( C \) and \( D \), with \( C \) closer to \( Q \) than \( D \), such that \( CD = 7 \). Find the ratio \( AD / BC \).

Answer: \frac{8}{9}


Given that the general term of the sequence $\{a_{n}\}$ is ${a}_{n}=97-3n(n∈{N}^{*})$, find the value of $n$ for which the sum of the first $n$ terms of the sequence $\{{a}_{n}{a}_{n+1}{a}_{n+2}\}(n∈{N}^{*})$ reaches its maximum value.

Answer: 32


For a given positive integer \( k \), we call an integer \( n \) a \( k \)-number if both of the following conditions are satisfied:
1. The integer \( n \) is the product of two positive integers which differ by \( k \).
2. The integer \( n \) is \( k \) less than a square number.
Find all \( k \) such that there are infinitely many \( k \)-numbers.

Answer: 4


In an opaque bag, there are three balls, each labeled with the numbers $-1$, $0$, and $\frac{1}{3}$, respectively. These balls are identical except for the numbers on them. Now, a ball is randomly drawn from the bag, and the number on it is denoted as $m$. After putting the ball back and mixing them, another ball is drawn, and the number on it is denoted as $n$. The probability that the quadratic function $y=x^{2}+mx+n$ does not pass through the fourth quadrant is ______.

Answer: \frac{5}{9}


Circle $\omega$ is inscribed in rhombus $H M_{1} M_{2} T$ so that $\omega$ is tangent to $\overline{H M_{1}}$ at $A, \overline{M_{1} M_{2}}$ at $I, \overline{M_{2} T}$ at $M$, and $\overline{T H}$ at $E$. Given that the area of $H M_{1} M_{2} T$ is 1440 and the area of $E M T$ is 405 , find the area of $A I M E$.

Answer: 540


In a WeChat group, five members simultaneously grab for four red envelopes, each person can grab at most one, and all red envelopes are claimed. Among the four red envelopes, there are two containing 2 yuan and two containing 3 yuan. Determine the number of scenarios in which both members A and B have grabbed a red envelope.

Answer: 18


In a convex 2002-gon, several diagonals were drawn such that they do not intersect inside the 2002-gon. As a result, the 2002-gon was divided into 2000 triangles. Could it happen that exactly half of these triangles have all their sides as diagonals of this 2002-gon?

Answer: No


Let \( f(x) = A\left(x^{2}-2x\right)e^{x} - e^{x} + 1 \). For any \( x \leq 0 \), it holds that \( f(x) \geq 0 \). Determine the range of the real number \( A \).

Answer: [-\frac{1}{2},+\infty)


Let  $1=d_1<d_2<d_3<\dots<d_k=n$  be the divisors of  $n$ . Find all values of  $n$  such that  $n=d_2^2+d_3^3$ .

Answer: 68


On the game show $\text{\emph{Wheel of Fortune II}}$, you observe a spinner with the labels ["Bankrupt", "$\$700$", "$\$900$", "$\$200$", "$\$3000$", "$\$800$"]. Given that each region has equal area, determine the probability of earning exactly $\$2400$ in your first three spins.

Answer: \frac{1}{36}


A student travels from his university to his hometown, a distance of 150 miles, in a sedan that averages 25 miles per gallon. For the return trip, he drives his friend's truck, which averages only 15 miles per gallon. Additionally, they make a detour of 50 miles at an average of 10 miles per gallon in the truck. Calculate the average fuel efficiency for the entire trip.

Answer: 16.67


The inscribed circle of triangle $DEF$ is tangent to $\overline{DE}$ at point $P$ and its radius is $13$. Given that $DP = 17$ and $PE = 31$, and the tangent from vertex $F$ to the circle is $20$, determine the perimeter of triangle $DEF$.

Answer: 136


A line through the points $(5, -12)$ and $(k, 23)$ is parallel to the line $4x + 6y = 12$. What is the value of $k$?

Answer: -47.5


In a circle we enscribe a regular  $2001$ -gon and inside it a regular  $667$ -gon with shared vertices.

Prove that the surface in the  $2001$ -gon but not in the  $667$ -gon is of the form  $k.sin^3\left(\frac{\pi}{2001}\right).cos^3\left(\frac{\pi}{2001}\right)$  with  $k$  a positive integer. Find  $k$ .

Answer: 10672


A regular triangle \( \triangle ABC \) with side length 3 is divided into three equal parts on each side. Parallel lines to each side are drawn through these division points, creating a total of 10 intersection points within and on the edges of the triangle, known as grid points. Determine the minimum number \( n \) of grid points that must be selected such that among these \( n \) points, there are always three points that form an isosceles triangle (including equilateral triangles).

Answer: 5


Define the *bigness*of a rectangular prism to be the sum of its volume, its surface area, and the lengths of all of its edges. Find the least integer  $N$  for which there exists a rectangular prism with integer side lengths and *bigness* $N$  and another one with integer side lengths and *bigness* $N + 1$ .

Answer: 55


$A$  and  $B$  are two points in the plane  $\alpha$ , and line  $r$  passes through points  $A, B$ . There are  $n$  distinct points  $P_1, P_2, \ldots, P_n$  in one of the half-plane divided by line  $r$ . Prove that there are at least  $\sqrt n$  distinct values among the distances  $AP_1, AP_2, \ldots, AP_n, BP_1, BP_2, \ldots, BP_n.$

Answer: \sqrt{n}


Given that $x$ is a root of the equation $x^{2}+x-6=0$, simplify $\frac{x-1}{\frac{2}{{x-1}}-1}$ and find its value.

Answer: \frac{8}{3}


(1) Prove: If $n$ is a natural number greater than 1, then $2^{n}-1$ cannot be divisible by $n$.

(2) Find all prime numbers $n$ such that $2^{n}+1$ is divisible by $n^{2}$.

Answer: 3


The pentagon \(ABCDE\) is inscribed around a circle. The angles at its vertices \(A\), \(C\), and \(E\) are \(100^{\circ}\). Find the angle \(ACE\).

Answer: 40


An exam paper contains 4 multiple-choice questions, each with three options (A), (B), and (C). Several students take the exam, and it is found that for any group of 3 students, there is always at least one question where their answers are all different. Furthermore, each student answers every question. What is the maximum number of students who can take the exam?

Answer: 9


Two adjacent faces of a tetrahedron, representing equilateral triangles with side length 3, form a dihedral angle of 30 degrees. The tetrahedron rotates around the common edge of these faces. Find the maximum area of the projection of the rotating tetrahedron onto a plane containing this edge.

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


For which values of \( a \) do the quadratic trinomials \( x^{2} + a x + 1 \) and \( x^{2} + x + a \) have a common root?

Answer: -2


12 Smurfs are sitting around a circular table, and each Smurf dislikes the 2 Smurfs adjacent to him but does not dislike the other 9 Smurfs. Papa Smurf wants to form a team of 5 Smurfs to rescue Smurfette, who was captured by Gargamel. The team must be made up of Smurfs who do not dislike one another. How many ways are there to form such a team?

Answer: 36


Given a sequence $ \{a_n\} $ with the first term $ \dfrac{3}{5} $, and the sequence $ \{a_n\} $ satisfies $ a_{n+1} = 2 - \dfrac{1}{a_n} $, calculate $ a_{2018} $.

Answer: \dfrac{4031}{4029}


In the Cartesian coordinate plane \(xOy\), given that points \(A\) and \(B\) lie on the hyperbola \(C: 2x^{2} + 4x - y^{2} = 0\), and \(\triangle ABC\) is an isosceles right triangle with \(O\) as the right-angle vertex. Find the total number of such \(\triangle OAB\).

Answer: 3


Anna and Berta play a game in which they take turns in removing marbles from a table. Anna takes the first turn. At the beginning of a turn there are n ≥ 1 marbles on the table, then the player whose turn is removes k marbles, where k ≥ 1 either is an even number with  $k \le \frac{n}{2}$  or an odd number with  $ \frac{n}{2}\le k \le n$ . A player wins the game if she removes the last marble from the table.
Determine the smallest number  $N\ge100000$  which Berta has wining strategy.

*proposed by Gerhard Woeginger*

Answer: 131070


Let \( ABC \) be a triangle. The midpoints of the sides \( BC \), \( AC \), and \( AB \) are denoted by \( D \), \( E \), and \( F \) respectively.

The two medians \( AD \) and \( BE \) are perpendicular to each other and their lengths are \(\overline{AD} = 18\) and \(\overline{BE} = 13.5\).

Calculate the length of the third median \( CF \) of this triangle.

Answer: 22.5


A housewife saved $2.50 in buying a dress on sale. If she spent $25 for the dress, she saved about:

Answer: 9 \%


Egor wrote a number on the board and encoded it according to the rules of letter puzzles (different letters correspond to different digits, the same letters to the same digits). The result was the word "ГВАТЕМАЛА". How many different numbers could Egor have originally written if his number was divisible by 30?

Answer: 21600


When you simplify $\left[ \sqrt [3]{\sqrt [6]{a^9}} \right]^4\left[ \sqrt [6]{\sqrt [3]{a^9}} \right]^4$, the result is:
$\textbf{(A)}\ a^{16} \qquad\textbf{(B)}\ a^{12} \qquad\textbf{(C)}\ a^8 \qquad\textbf{(D)}\ a^4 \qquad\textbf{(E)}\ a^2$

Answer: \textbf{(D)}\^4


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


Let α and β be real numbers. Find the least possible value of
$$
(2 \cos \alpha + 5 \sin \beta - 8)^2 + (2 \sin \alpha + 5 \cos β - 15)^2.
$$

Answer: 100


Select 2 different numbers from 1, 3, 5, and 3 different numbers from 2, 4, 6, 8 to form a five-digit number, and determine the total number of even numbers among these five-digit numbers.

Answer: 864


Let \( P_{0}, P_{1}, P_{2}, \cdots, P_{n} \) be \( n+1 \) points in a plane, with a minimum distance \( d \) between any two points (\( d > 0 \)). Prove that \(\left|P_{0} P_{1}\right| \cdot \left|P_{0} P_{2}\right| \cdots \left|P_{0} P_{n}\right| > \left(\frac{d}{3}\right)^{n} \sqrt{(n+1)!}\).

Answer: |P_0P_1|\cdot|P_0P_2|\cdots|P_0P_n|>(\frac{}{3})^{n}\sqrt{(n+1)!}


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 intercepted by these curves on the line $y=2x$.

Answer: 8\sqrt{5}


Given a sequence \(\{a_{n}\}\) that satisfies

\[
\begin{array}{l}
a_{0}=1, a_{1}=5, \\
a_{n}=\frac{2 a_{n-1}^{2}-3 a_{n-1}-9}{2 a_{n-2}} \quad (n \geqslant 2),
\end{array}
\]

use mathematical induction to prove that \(a_{n}=2^{n+2}-3\).

Answer: a_n=2^{n+2}-3


In the diagram, $P$ is on $RS$ so that $QP$ bisects $\angle SQR$.  Also, $PQ=PR$, $\angle RSQ=2y^\circ$, and $\angle RPQ=3y^\circ$.  What is the measure, in degrees, of $\angle RPQ$? [asy]
// C14
import olympiad;
size(7cm);

real x = 50; real y = 20;

pair q = (1, 0);
pair r = (0, 0);
pair p = intersectionpoints((10 * dir(x))--r, q--(shift(q) * 10 * dir(180 - x)))[0];
pair s = intersectionpoints(r--(r + 10 * (p - r)), 10 * dir(180 - 2 * x)--q)[0];

// Draw lines
draw(p--s--q--p--r--q);

// Label points
label("$R$", r, SW);
label("$Q$", q, SE);
label("$S$", s, N);
label("$P$", p, NW);

// Label angles
label("$x^\circ$", q, 2 * W + 2 * NW);
label("$x^\circ$", q, 4 * N + 2 * NW);
label("$2y^\circ$", s, 5 * S + 4 * SW);
label("$3y^\circ$", p, 4 * S);

// Tick marks
add(pathticks(r--p, 2, spacing=0.6, s=2));
add(pathticks(p--q, 2, spacing=0.6, s=2));
[/asy]

Answer: 108


Let $A = (1,0)$ and $B = (5,4).$  Let $P$ be a point on the parabola $y^2 = 4x.$  Find the smallest possible value of $AP + BP.$

Answer: 6


Determine the primes  $p$  for which the numbers  $2\lfloor p/k\rfloor - 1, \ k = 1,2,\ldots, p,$  are all quadratic residues modulo  $p.$ *Vlad Matei*

Answer: 2


Four elevators in a skyscraper, differing in color (red, blue, green, and yellow), move in different directions at different but constant speeds. An observer timed the events as follows: At the 36th second, the red elevator caught up with the blue one (moving in the same direction). At the 42nd second, the red elevator passed by the green one (moving in opposite directions). At the 48th second, the red elevator passed by the yellow one. At the 51st second, the yellow elevator passed by the blue one. At the 54th second, the yellow elevator caught up with the green one. At what second from the start will the green elevator pass by the blue one, assuming the elevators did not stop or change direction during the observation period?

Answer: 46


Let positive integers  $p,q$  with  $\gcd(p,q)=1$  such as  $p+q^2=(n^2+1)p^2+q$ . If the parameter  $n$  is a positive integer, find all possible couples  $(p,q)$ .

Answer: (p,q)=(n+1,n^2+n+1)


For rational numbers $a$ and $b$, define the operation "$\otimes$" as $a \otimes b = ab - a - b - 2$.
(1) Calculate the value of $(-2) \otimes 3$;
(2) Compare the size of $4 \otimes (-2)$ and $(-2) \otimes 4$.

Answer: -12


Let \(A\) and \(B\) be the foci of a hyperbola, and \(C\) be a point on the hyperbola. It is known that the side lengths of \(\triangle ABC\) form an arithmetic sequence and \(\angle ACB = 120^{\circ}\). Find the eccentricity of the hyperbola. 🟪

Answer: \frac{7}{2}


A factory makes chocolate bars. Five boxes, labelled $V, W, X, Y, Z$, are each packed with 20 bars. Each of the bars in three of the boxes has a mass of 100 g. Each of the bars in the other two boxes has a mass of 90 g. One bar is taken from box $V$, two bars are taken from box $W$, four bars are taken from box $X$, eight bars are taken from box $Y$, and sixteen bars are taken from box $Z$. The total mass of these bars taken from the boxes is 2920 g. Which boxes contain the 90 g bars?

Answer: W \text{ and } Z


Consider an equilateral triangle \(ABC\), where \(AB = BC = CA = 2011\). Let \(P\) be a point inside \(\triangle ABC\). Draw line segments passing through \(P\) such that \(DE \parallel BC\), \(FG \parallel CA\), and \(HI \parallel AB\). Suppose \(DE : FG : HI = 8 : 7 : 10\). Find \(DE + FG + HI\).

Answer: 4022


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


Given the polynomial $f(x)=3x^{9}+3x^{6}+5x^{4}+x^{3}+7x^{2}+3x+1$, calculate the value of $v_{5}$ when $x=3$ using Horner's method.

Answer: 761


Let \( f(x) = x^2 + px + q \). It is known that the inequality \( |f(x)| > \frac{1}{2} \) has no solutions on the interval \([3, 5]\). Find \(\underbrace{f(f(\ldots f}_{2017}\left(\frac{7+\sqrt{15}}{2}\right)) \ldots)\). Round the answer to hundredths if necessary.

Answer: 1.56


Let $S=\{1,2, \ldots, 9\}$. Compute the number of functions $f: S \rightarrow S$ such that, for all $s \in S, f(f(f(s)))=s$ and $f(s)-s$ is not divisible by 3.

Answer: 288


Find all one-to-one mappings  $f:\mathbb{N}\to\mathbb{N}$  such that for all positive integers  $n$  the following relation holds:
\[ f(f(n)) \leq \frac {n+f(n)} 2 . \]

Answer: f(n)=n


A square with sides of length $1$ is divided into two congruent trapezoids and a pentagon, which have equal areas, by joining the center of the square with points on three of the sides, as shown. Find $x$, the length of the longer parallel side of each trapezoid.

Answer: \frac{5}{6}


Given numbers \(x_{1}, \ldots, x_{n} \in (0,1)\), find the maximum value of the expression

\[
A = \frac{\sqrt{1-x_{1}} + \ldots + \sqrt{1-x_{n}}}{\sqrt{\frac{1}{x_{1}} + \ldots + \frac{1}{x_{n}}}}
\]

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


Given an arithmetic-geometric sequence $\{ a_{n} \}$ that satisfies $a\_1 + a\_3 = 10$, $a\_2 + a\_4 = 5$, find the maximum value of the product $a\_1 a\_2 \ldots a\_n$.

Answer: 64


Let's call a ten-digit number interesting if it is divisible by 11111 and all its digits are different. How many interesting numbers exist?

Answer: 3456


Let  $ABC$  be a triangle with circumcenter  $O$ . Point  $D, E, F$  are chosen on sides  $AB, BC$  and  $AC$ , respectively, such that  $ADEF$  is a rhombus. The circumcircles of  $BDE$  and  $CFE$  intersect  $AE$  at  $P$  and  $Q$  respectively. Show that  $OP=OQ$ .

*Proposed by Ariel García*

Answer: OP=OQ


Given the function \( f(x) = -x^2 + x + m + 2 \), if the inequality \( f(x) \geq |x| \) has exactly one integer in its solution set, determine the range of the real number \( m \).

Answer: [-2,-1)


Given a positive integer  $n,$  let  $s(n)$  denote the sum of the digits of  $n.$  Compute the largest positive integer  $n$  such that  $n = s(n)^2 + 2s(n) - 2.$

Answer: 397


Three numbers have a mean (average) of 7. The mode of these three numbers is 9. What is the smallest of these three numbers?
(A) 1
(B) 2
(C) 3
(D) 4
(E) 5

Answer: 3


Andrei lives near a market and during the summer holidays often helped one of the traders early in the morning to arrange fruits on the stall. For this, the trader gave Andrei a discount on his favorite apples of $10 \%$. But autumn has come, and the price of apples has increased by $10 \%$. Although Andrei has gone back to school and stopped helping the trader, the $10 \%$ discount for him remained. What will be Andrei's monthly spending on apples now, considering he buys 2 kilograms per month? Before the price increase, apples at the market cost 50 rubles per kg for all customers. (Provide the answer as an integer, without spaces or units of measurement.)

Answer: 99


Determine the number of digits in the value of $2^{15} \times 5^{12} - 10^5$.

Answer: 12


Calculate the value of $v_4$ for the polynomial $f(x) = 12 + 35x - 8x^2 + 79x^3 + 6x^4 + 5x^5 + 3x^6$ using the Horner's method when $x = -4$.

Answer: 220


Consider the set of all rectangles with a given area  $S$ .
Find the largest value o   $ M = \frac{16-p}{p^2+2p}$  where  $p$  is the perimeter of the rectangle.

Answer: \frac{4-\sqrt{S}}{4S+2\sqrt{S}}


Two points $B$ and $C$ are in a plane. Let $S$ be the set of all points $A$ in the plane for which $\triangle ABC$ has area $1.$ Which of the following describes $S?$
$\textbf{(A) } \text{two parallel lines} \qquad\textbf{(B) } \text{a parabola} \qquad\textbf{(C) } \text{a circle} \qquad\textbf{(D) } \text{a line segment} \qquad\textbf{(E) } \text{two points}$

Answer: \mathrm{(A)\}


A kitten bites off a quarter of a sausage from one end, then a puppy bites off a third of the remaining piece from the opposite end, then the kitten again bites off a quarter from its end, and the puppy bites off a third from its end, and so on. You need to tie a string around the sausage in advance to ensure that no one eats the string. In what ratio should the sausage be divided?

Answer: 1:1


Mrs. Toad has a class of 2017 students, with unhappiness levels $1,2, \ldots, 2017$ respectively. Today in class, there is a group project and Mrs. Toad wants to split the class in exactly 15 groups. The unhappiness level of a group is the average unhappiness of its members, and the unhappiness of the class is the sum of the unhappiness of all 15 groups. What's the minimum unhappiness of the class Mrs. Toad can achieve by splitting the class into 15 groups?

Answer: 1121


For the quadratic trinomial \( a x^{2} + b x + c \) where \( a > 100 \), what is the maximum number of integer values for \( x \) such that the absolute value of the quadratic trinomial does not exceed 50?

Answer: 2


Amy and Ben need to eat 1000 total carrots and 1000 total muffins. The muffins can not be eaten until all the carrots are eaten. Furthermore, Amy can not eat a muffin within 5 minutes of eating a carrot and neither can Ben. If Amy eats 40 carrots per minute and 70 muffins per minute and Ben eats 60 carrots per minute and 30 muffins per minute, what is the minimum number of minutes it will take them to finish the food?

Answer: 23.5


In the diagram, each of \( \triangle W X Z \) and \( \triangle X Y Z \) is an isosceles right-angled triangle. The length of \( W X \) is \( 6 \sqrt{2} \). The perimeter of quadrilateral \( W X Y Z \) is closest to

Answer: 23


Petya's bank account contains 500 dollars. The bank allows only two types of transactions: withdrawing 300 dollars or adding 198 dollars.

What is the maximum amount Petya can withdraw from the account if he does not have any other money?

Answer: 498


Two people are flipping a coin: one flipped it 10 times, and the other flipped it 11 times. Find the probability that the second person gets more heads than the first person.

Answer: 1/2


Let sets \( A \) and \( B \) satisfy:
\[
A \cup B = \{1, 2, \cdots, 10\}, \quad A \cap B = \varnothing
\]
If the number of elements in set \( A \) is not an element of \( A \) and the number of elements in set \( B \) is not an element of \( B \), find the total number of different sets \( A \) that satisfy the conditions.

Answer: 186


Let  $ABC$  be triangle such that  $|AB| = 5$ ,  $|BC| = 9$  and  $|AC| = 8$ . The angle bisector of  $\widehat{BCA}$  meets  $BA$  at  $X$  and the angle bisector of  $\widehat{CAB}$  meets  $BC$  at  $Y$ . Let  $Z$  be the intersection of lines  $XY$  and  $AC$ . What is  $|AZ|$ ? $
\textbf{a)}\ \sqrt{104}
\qquad\textbf{b)}\ \sqrt{145}
\qquad\textbf{c)}\ \sqrt{89}
\qquad\textbf{d)}\ 9
\qquad\textbf{e)}\ 10
$

Answer: 10


Given the sequence \(0, 1, 1, 2, 3, 5, 8, \cdots\), starting from the 3rd term, each term is the sum of the previous two terms. In the first 100,000,001 terms of this sequence, will any term have the last 4 digits all be 0?
(The 9th Moscow Olympiad problem)

Answer: Yes


Let $\triangle XYZ$ be a right triangle with $Y$ as the right angle. A circle with diameter $YZ$ intersects side $XZ$ at point $W$. Given that $XW = 2$ and $YW = 3$, find the length of $ZW$.

Answer: \frac{9}{2}


In $\triangle ABC$, the sides opposite to $\angle A$, $\angle B$, and $\angle C$ are $a$, $b$, and $c$ respectively. If $\angle B = \angle C$ and $7a^2 + b^2 + c^2 = 4\sqrt{3}$, find the maximum value of the area of $\triangle ABC$.

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


Given a random variable $\xi$ that follows the normal distribution $N(0, \sigma^2)$, if $P(\xi > 2) = 0.023$, calculate $P(-2 \leq \xi \leq 2)$.

Answer: 0.954


Due to a snow and ice disaster, a citrus orchard suffered severe damage. To address this, experts proposed a rescue plan for the fruit trees, which needs to be implemented over two years and is independent each year. The plan estimates that in the first year, the probability of the citrus yield recovering to 1.0, 0.9, and 0.8 times the pre-disaster level is 0.2, 0.4, and 0.4, respectively. In the second year, the probability of the citrus yield reaching 1.5, 1.25, and 1.0 times the first year's yield is 0.3, 0.3, and 0.4, respectively. Calculate the probability that the citrus yield will exactly reach the pre-disaster level after two years.

Answer: 0.2


Points $A=(6,13)$ and $B=(12,11)$ lie on circle $\omega$ in the plane. Suppose that the tangent lines to $\omega$ at $A$ and $B$ intersect at a point on the $x$-axis. What is the area of $\omega$?

Answer: \frac{85\pi}{8}


Determine all pairs of positive integers $(m,n)$ such that $(1+x^n+x^{2n}+\cdots+x^{mn})$ is divisible by $(1+x+x^2+\cdots+x^{m})$ .

Answer: \(\gcd(m+1, n) = 1\)


Using the digits $0$ to $9$ exactly once, form several composite numbers. What is the minimum sum of these composite numbers?

Answer: 99


Given $l_{1}$: $ρ \sin (θ- \frac{π}{3})= \sqrt {3}$, $l_{2}$: $ \begin{cases} x=-t \\ y= \sqrt {3}t \end{cases}(t$ is a parameter), find the polar coordinates of the intersection point $P$ of $l_{1}$ and $l_{2}$. Additionally, points $A$, $B$, and $C$ are on the ellipse $\frac{x^{2}}{4}+y^{2}=1$. $O$ is the coordinate origin, and $∠AOB=∠BOC=∠COA=120^{\circ}$, find the value of $\frac{1}{|OA|^{2}}+ \frac{1}{|OB|^{2}}+ \frac{1}{|OC|^{2}}$.

Answer: \frac{15}{8}


Find all functions \( f: \mathbb{Z}_{+} \rightarrow \mathbb{Z}_{+} \) such that for any positive integer \( n \), the following equation holds:
\[ f(f(f(n))) + f(f(n)) + f(n) = 3n. \]

Answer: f(n)=n


Let $A B C$ be a right triangle with $\angle A=90^{\circ}$. Let $D$ be the midpoint of $A B$ and let $E$ be a point on segment $A C$ such that $A D=A E$. Let $B E$ meet $C D$ at $F$. If $\angle B F C=135^{\circ}$, determine $B C / A B$.

Answer: \frac{\sqrt{13}}{2}


The length of a rectangular yard exceeds twice its width by 30 feet, and the perimeter of the yard is 700 feet. What is the area of the yard in square feet?

Answer: 25955.56


In $\Delta ABC$, $AC = BC$, $m\angle DCB = 40^{\circ}$, and $CD \parallel AB$. What is the number of degrees in $m\angle ECD$?

[asy] pair A,B,C,D,E; B = dir(-40); A = dir(-140); D = (.5,0); E = .4 * dir(40);
draw(C--B--A--E,EndArrow); draw(C--D,EndArrow);
label("$A$",A,W); label("$C$",C,NW);label("$B$",B,E);label("$D$",D,E);label("$E$",E,E);
[/asy]

Answer: 40


Given a connected simple graph \( G \) with a known number of edges \( e \), where each vertex has some number of pieces placed on it (each piece can only be placed on one vertex of \( G \)). The only operation allowed is when a vertex \( v \) has a number of pieces not less than the number of its adjacent vertices \( d \), you can choose \( d \) pieces from \( v \) and distribute them to the adjacent vertices such that each adjacent vertex gets one piece. If every vertex in \( G \) has a number of pieces less than the number of its adjacent vertices, no operations can be performed.

Find the minimum value of \( m \) such that there exists an initial placement of the pieces with a total of \( m \) pieces, allowing you to perform infinitely many operations starting from this placement.

Answer: e


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

Answer: 60718293


Ten spheres of identical radius are stacked in the shape of a triangular pyramid such that each sphere touches at least three others. Find the radius of a sphere inscribed in the center of the pyramid of spheres, which touches six identical spheres, if the radius of the sphere into which the pyramid is inscribed is equal to $\sqrt{6}+1$.

Answer: \sqrt{2}-1


Find all natural numbers \( n \) for which \( 2^n + n^{2016} \) is a prime number.

Answer: 1


Find the best approximation of  $\sqrt{3}$  by a rational number with denominator less than or equal to  $15$

Answer: \frac{26}{15}


The height and the median of a triangle, drawn from one of its vertices, are distinct and form equal angles with the sides emanating from the same vertex. Determine the radius of the circumscribed circle if the median is equal to \( m \).

Answer: m


Consider the quadratic polynomial \( P(x) = a x^2 + b x + c \) which has distinct positive roots. Vasya wrote four numbers on the board: the roots of \( P(x) \) and the roots of the polynomial \( Q(x) = c x^2 + b x + a \). What is the smallest integer value that the sum of these four numbers can have?

Answer: 5


The chromatic number of the (infinite) plane, denoted by \(\chi\), is the smallest number of colors with which we can color the points on the plane in such a way that no two points of the same color are one unit apart.

Answer: 4\le\chi\le7


S is a set of real numbers which is closed under multiplication. \( S = A \cup B \), and \( A \cap B = \emptyset \). If \( a, b, c \in A \), then \( abc \in A \). Similarly, if \( a, b, c \in B \), then \( abc \in B \). Show that at least one of \( A \) or \( B \) is closed under multiplication.

Answer: 1


Gauss is a famous German mathematician, known as the "Prince of Mathematics". There are 110 achievements named after his name "Gauss". For $x\in R$, let $[x]$ represent the largest integer not greater than $x$, and let $\{x\}=x-[x]$ represent the non-negative fractional part of $x$. Then, $y=[x]$ is called the Gauss function. It is known that the sequence $\{a_{n}\}$ satisfies: $a_{1}=\sqrt{3}$, $a_{n+1}=[a_{n}]+\frac{1}{\{a_{n}\}}$, $(n∈N^{*})$, then $a_{2019}=$\_\_\_\_\_\_\_\_.

Answer: 3027+ \sqrt{3}


A company buys an assortment of 150 pens from a catalog for \$15.00. Shipping costs an additional \$5.50. Furthermore, they receive a 10% discount on the total price due to a special promotion. What is the average cost, in cents, for each pen?

Answer: 12


A castle has the shape of a heptagon, with a watchtower at each vertex. Each of the seven walls of the castle is guarded by the watchmen in the towers located at the ends of that wall. What is the minimum number of watchmen that need to be placed in the towers so that each wall is guarded by at least seven watchmen?

Answer: 25


Let  $n$  be a positive integer  $\geq 2$  . Consider a  $n$  by  $n$  grid with all entries  $1$ . Define an operation on a square to be changing the signs of all squares adjacent to it but not the sign of its own. Find all  $n$  such that it is possible after a finite sequence of operations to reach a  $n$  by  $n$  grid with all entries  $-1$

Answer: n


Let the set \( S = \{1, 2, \cdots, n\} \). Let \( A \) be an arithmetic sequence with a positive common difference that contains at least two elements, where every element of \( A \) is in \( S \). Additionally, after adding any other elements from \( S \) to \( A \), it is no longer an arithmetic sequence with the same common difference as \( A \). Determine the number of such sequences \( A \). (Note that sequences with only two elements are considered arithmetic sequences.)

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


The sum \( b_{7} + b_{6} + \ldots + b_{2019} \) of the terms of a geometric progression \(\left\{b_{n}\right\}\), where \( b_{n} > 0 \), equals 27, and the sum of their reciprocals \(\frac{1}{b_{7}} + \frac{1}{b_{6}} + \ldots + \frac{1}{b_{2019}} \) equals 3. Find the product \( b_{7} \cdot b_{6} \cdot \ldots \cdot b_{2019} \).

Answer: 3^{2013}


Timur and Alexander are counting the trees growing around the house. Both move in the same direction, but they start counting from different trees. What is the number of trees growing around the house if the tree that Timur called the 12th, Alexander counted as $33-m$, and the tree that Timur called the $105-m$, Alexander counted as the 8th?

Answer: 76


Given a set \( A = \{0, 1, 2, \cdots, 9\} \), and a family of non-empty subsets \( B_1, B_2, \cdots, B_j \) of \( A \), where for \( i \neq j \), \(\left|B_i \cap B_j\right| \leqslant 2\), determine the maximum value of \( k \).

Answer: 175


A box contains $5$ chips, numbered $1$, $2$, $3$, $4$, and $5$. Chips are drawn randomly one at a time without replacement until the sum of the values drawn exceeds $4$. What is the probability that $3$ draws are required?
$\textbf{(A)} \frac{1}{15} \qquad \textbf{(B)} \frac{1}{10} \qquad \textbf{(C)} \frac{1}{6} \qquad \textbf{(D)} \frac{1}{5} \qquad \textbf{(E)} \frac{1}{4}$

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


Answer: 40


In counting $n$ colored balls, some red and some black, it was found that $49$ of the first $50$ counted were red.
Thereafter, $7$ out of every $8$ counted were red. If, in all, $90$ % or more of the balls counted were red, the maximum value of $n$ is:
$\textbf{(A)}\ 225 \qquad \textbf{(B)}\ 210 \qquad \textbf{(C)}\ 200 \qquad \textbf{(D)}\ 180 \qquad \textbf{(E)}\ 175$

Answer: \textbf{(B)}\210


Let  $a,b,c,x,y,$  and  $z$  be complex numbers such that \[a=\dfrac{b+c}{x-2},\qquad b=\dfrac{c+a}{y-2},\qquad c=\dfrac{a+b}{z-2}.\] If  $xy+yz+xz=67$  and  $x+y+z=2010$ , find the value of  $xyz$ .

Answer: -5892


Answer: -4321


Sides $AB$, $BC$, and $CD$ of (simple*) quadrilateral $ABCD$ have lengths $4$, $5$, and $20$, respectively.
If vertex angles $B$ and $C$ are obtuse and $\sin C = - \cos B = \frac{3}{5}$, then side $AD$ has length
A polygon is called “simple” if it is not self intersecting.

Answer: 25


Let  $1\le k\le n$  be integers. At most how many  $k$ -element subsets can we select from  $\{1,2,\dots,n\}$  such that for any two selected subsets, one of the subsets consists of the  $k$  smallest elements of their union?

Answer: n-k+1


In $\triangle ABC$, $\angle A = 60^{\circ}$ and $AB > AC$. Point $O$ is the circumcenter. The two altitudes $BE$ and $CF$ intersect at point $H$. Points $M$ and $N$ are on segments $BH$ and $HF$ respectively, and satisfy $BM = CN$. Find the value of $\frac{MH + NH}{OH}$.

Answer: \sqrt{3}


Given a point $P$ on the ellipse $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 (a > b > 0)$ with foci $F_{1}$, $F_{2}$, find the eccentricity of the ellipse given that $\overrightarrow{PF_{1}} \cdot \overrightarrow{PF_{2}}=0$ and $\tan \angle PF_{1}F_{2}= \frac{1}{2}$.

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


Let the sides opposite to the internal angles $A$, $B$, and $C$ of triangle $\triangle ABC$ be $a$, $b$, and $c$ respectively. Given that $\sin B - \sin C = \sin (A-C)$.
$(1)$ Find the value of $A$.
$(2)$ If $AB=2$, $AC=5$, and the medians $AM$ and $BN$ on sides $BC$ and $AC$ intersect at point $P$, find the cosine value of $\angle MPN$.

Answer: \frac{4\sqrt{91}}{91}


There are five students, A, B, C, D, and E, arranged to participate in the volunteer services for the Shanghai World Expo. Each student is assigned one of four jobs: translator, guide, etiquette, or driver. Each job must be filled by at least one person. Students A and B cannot drive but can do the other three jobs, while students C, D, and E are capable of doing all four jobs. The number of different arrangements for these tasks is _________.

Answer: 108


In the rectangular coordinate system xOy, the parametric equation of line l is $$\begin{cases} x=1+t \\ y=-3+t \end{cases}$$ (where t is the parameter), and the polar coordinate system is established with the origin O as the pole and the positive semi-axis of the x-axis as the polar axis. The polar equation of curve C is ρ=6cosθ.
(I) Find the general equation of line l and the rectangular coordinate equation of curve C.
(II) If line l intersects curve C at points A and B, find the area of triangle ABC.

Answer: \frac { \sqrt {17}}{2}


When $\frac{1}{10101}$ is expressed as a decimal, what is the sum of the first 100 digits after the decimal point?

Answer: 450


Given two distinct geometric progressions with first terms both equal to 1, and the sum of their common ratios equal to 3, find the sum of the fifth terms of these progressions if the sum of the sixth terms is 573. If the answer is ambiguous, provide the sum of all possible values of the required quantity.

Answer: 161


The value of $\log_5{3125}$ is between which consecutive integers $c$ and $d$? Find $c+d$.

Answer: 11


$\triangle ABC$  is isosceles  $AB = AC$ .  $P$  is a point inside  $\triangle ABC$  such that $\angle BCP = 30$  and  $\angle APB = 150$  and  $\angle CAP = 39$ . Find  $\angle BAP$ .

Answer: 13


Solve the system of equations:
\[
\left\{
\begin{array}{l}
x + 3y + 14 \leq 0, \\
x^4 + 2x^2y^2 + y^4 + 64 - 20x^2 - 20y^2 = 8xy.
\end{array}
\right.
\]

Answer: (-2,-4)


Simplify and then evaluate the expression:
$$( \frac {x}{x-1}- \frac {x}{x^{2}-1})÷ \frac {x^{2}-x}{x^{2}-2x+1}$$
where $$x= \sqrt {2}-1$$

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


Given that the function $f(x)$ is an odd function defined on $\mathbb{R}$, and when $x > 0$, $f(x) = -x^{2} + 4x$.
- (Ⅰ) Find the analytical expression of the function $f(x)$.
- (Ⅱ) Find the minimum value of the function $f(x)$ on the interval $\left[-2,a\right]$ where $\left(a > -2\right)$.

Answer: -4


Let $O$ be the origin. Determine the scalar $m$ such that for any points $A, B, C, D$, if the equation
\[4 \overrightarrow{OA} - 3 \overrightarrow{OB} + 6 \overrightarrow{OC} + m \overrightarrow{OD} = \mathbf{0}\]
holds, then the points $A, B, C$, and $D$ are coplanar.

Answer: -7


Given that $7^{-1} \equiv 55 \pmod{78}$, find $49^{-1} \pmod{78}$, as a residue modulo 78. (Give an answer between 0 and 77, inclusive.)

Answer: 61


What is the smallest positive integer $n$ such that $\frac{n}{n+110}$ is equal to a terminating decimal?

Answer: 15


If the equation $\sin^{2} x + \sin 2x = m + 2\cos^{2} x$ has real solutions, what is the range of the real number $m$?

Answer: [\frac{-1-\sqrt{13}}{2},\frac{-1+\sqrt{13}}{2}]


Evaluate the expression: $\left(2\left(3\left(2\left(3\left(2\left(3 \times (2+1) \times 2\right)+2\right)\times 2\right)+2\right)\times 2\right)+2\right)$.

Answer: 5498


Fifty numbers have an average of 76. Forty of these numbers have an average of 80. What is the average of the other ten numbers?
(A) 60
(B) 4
(C) 72
(D) 40
(E) 78

Answer: 60


Bob Barker went back to school for a PhD in math, and decided to raise the intellectual level of The Price is Right by having contestants guess how many objects exist of a certain type, without going over. The number of points you will get is the percentage of the correct answer, divided by 10, with no points for going over (i.e. a maximum of 10 points). Let's see the first object for our contestants...a table of shape (5,4,3,2,1) is an arrangement of the integers 1 through 15 with five numbers in the top row, four in the next, three in the next, two in the next, and one in the last, such that each row and each column is increasing (from left to right, and top to bottom, respectively). For instance: \begin{tabular}{lcccc} 1 & 2 & 3 & 4 & 5 \\ 6 & 7 & 8 & 9 & \\ 10 & 11 & 12 & & \\ 13 & 14 & & & \\ 15 & & & & \end{tabular} is one table. How many tables are there?

Answer: 292864


Answer: \sqrt[3]{\frac{19}{2}} - 2


Find all functions $f:(0,\infty)\rightarrow (0,\infty)$ such that for any $x,y\in (0,\infty)$, $$xf(x^2)f(f(y)) + f(yf(x)) = f(xy) \left(f(f(x^2)) + f(f(y^2))\right).$$

Answer: f(x) = \frac{1}{x}


On a semicircle of diameter  $AB$  and center  $C$ , consider variable points  $M$  and  $N$  such that  $MC \perp NC$ . The circumcircle of triangle  $MNC$  intersects  $AB$  for the second time at  $P$ . Prove that  $\frac{|PM-PN|}{PC}$   constant and find its value.

Answer: \sqrt{2}


From the set $\{1, 2, \cdots, 100\}$, if 55 different numbers are randomly chosen, is it certain that there must be two numbers among them such that their difference is equal to: (a) 9; (b) 11?

Answer: No


In a rectangular array of points, with 5 rows and $N$ columns, the points are numbered consecutively from left to right beginning with the top row. Thus the top row is numbered 1 through $N,$ the second row is numbered $N + 1$ through $2N,$ and so forth. Five points, $P_1, P_2, P_3, P_4,$ and $P_5,$ are selected so that each $P_i$ is in row $i.$ Let $x_i$ be the number associated with $P_i.$ Now renumber the array consecutively from top to bottom, beginning with the first column. Let $y_i$ be the number associated with $P_i$ after the renumbering. It is found that $x_1 = y_2,$ $x_2 = y_1,$ $x_3 = y_4,$ $x_4 = y_5,$ and $x_5 = y_3.$ Find the smallest possible value of $N.$

Answer: 149


Given a sequence $\{a_{n}\}$, where $a_{1}=1$, $a_{2}=2$, and $a_{n+2}-3 a_{n+1}+2 a_{n}=1$. Let $T_{n}$ be the sum of the first $n$ terms of the sequence $\left\{\frac{8}{a_{n}}\right\}$. Then, find $\left[T_{2023}\right]$. (The symbol $[x]$ denotes the greatest integer less than or equal to $x$)

Answer: 14


In the vertices of a regular 300-gon, the numbers from 1 to 300 are arranged in some order, each number appearing exactly once. It turns out that for each number \(a\), there are as many numbers smaller than \(a\) among the 15 closest numbers to it clockwise as there are among the 15 closest numbers to it counterclockwise. A number that is larger than all 30 numbers closest to it is called "huge." What is the smallest possible number of huge numbers?

Answer: 10


Find the following differences:

a) $\Delta n^{2}$

b) $\Delta n(n-1)$

c) $\Delta n^{\mathrm{k}}$

d) $\Delta C_{\mathrm{n}}^{\mathrm{k}}$

Find the sequence $\left\{a_{n}\right\}$ such that $\Delta a_{n}=n^{2}$. Using the result from the previous problem, derive the formula for the sum $1^{2}+2^{2}+3^{2}+\ldots+n^{2}$.

Answer: \frac{n(n+1)(2n+1)}{6}


There are 5 different lines on a plane, forming $m$ intersection points. How many different values can $m$ take?

Answer: 9


Kanga labelled the vertices of a square-based pyramid using \(1, 2, 3, 4,\) and \(5\) once each. For each face, Kanga calculated the sum of the numbers on its vertices. Four of these sums equaled \(7, 8, 9,\) and \(10\). What is the sum for the fifth face?

Answer: 13


Mad scientist Kyouma writes  $N$  positive integers on a board. Each second, he chooses two numbers  $x, y$  written on the board with  $x > y$ , and writes the number  $x^2-y^2$  on the board. After some time, he sends the list of all the numbers on the board to Christina. She notices that all the numbers from 1 to 1000 are present on the list. Aid Christina in finding the minimum possible value of N.

Answer: 252


If in the formula $C =\frac{en}{R+nr}$, where $e$, $n$, $R$ and $r$ are all positive, $n$ is increased while $e$, $R$ and $r$ are kept constant, then $C$:
$\textbf{(A)}\ \text{Increases}\qquad\textbf{(B)}\ \text{Decreases}\qquad\textbf{(C)}\ \text{Remains constant}\qquad\textbf{(D)}\ \text{Increases and then decreases}\qquad\\ \textbf{(E)}\ \text{Decreases and then increases}$

Answer: \textbf{(A)}\