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Antal and Béla set off from Hazul to Cegléd on their motorcycles. From the one-fifth point of the journey, Antal turns back for some reason, which causes him to "switch gears," and he manages to increase his speed by a quarter. He immediately sets off again from Hazul. Béla, traveling alone, reduces his speed by a quarter. They eventually meet again and travel the last part of the journey together at a speed of $48 \mathrm{~km} /$ hour, arriving 10 minutes later than planned. What can we calculate from this?
|
40
|
1/8
|
## Problem Statement
Calculate the force with which water presses on a dam, the cross-section of which has the shape of an isosceles trapezoid (see figure). The density of water \(\rho = 1000 \mathrm{kg} / \mathrm{m}^{3}\), and the acceleration due to gravity \(g\) is taken to be \(10 \mathrm{m} / \mathrm{s}^{2}\).
Hint: The pressure at depth \(x\) is \(\rho g x\).
\[
a = 6.6 \mathrm{m}, b = 10.8 \mathrm{m}, h = 4.0 \mathrm{m}
\]
|
640000
|
0/8
|
8. For any real number $x$, $[x]$ represents the greatest integer not exceeding $x$. When $0 \leqslant x \leqslant 100$, the range of $f(x)=[3 x]+[4 x]+[5 x]+[6 x]$ contains $\qquad$ different integers.
|
1201
|
1/8
|
1. The sequence is given as $\frac{1}{2}, \frac{2}{1}, \frac{1}{2}, \frac{3}{1}, \frac{2}{2}, \frac{1}{3}, \frac{4}{1}, \frac{3}{2}, \frac{2}{3}, \frac{1}{4}, \ldots$. Which number stands at the 1997th position?
|
\dfrac{5}{11}
|
1/8
|
[ Inscribed angle, based on the diameter ] Sine Theorem
[ Sine Theorem $\quad$]
Diagonal $B D$ of quadrilateral $A B C D$ is the diameter of the circle circumscribed around this quadrilateral. Find diagonal $A C$, if $B D=2, A B=1, \angle A B D: \angle D B C=4: 3$.
|
\dfrac{\sqrt{6} + \sqrt{2}}{2}
|
1/8
|
LVII OM - I - Problem 4
Participants in a mathematics competition solved six problems, each graded with one of the scores 6, 5, 2, 0. It turned out that
for every pair of participants $ A, B $, there are two problems such that in each of them $ A $ received a different score than $ B $.
Determine the maximum number of participants for which such a situation is possible.
|
1024
|
4/8
|
Bradley is driving at a constant speed. When he passes his school, he notices that in $20$ minutes he will be exactly $\frac14$ of the way to his destination, and in $45$ minutes he will be exactly $\frac13$ of the way to his destination. Find the number of minutes it takes Bradley to reach his destination from the point where he passes his school.
|
245
|
2/8
|
Aileen plays badminton where she and her opponent stand on opposite sides of a net and attempt to bat a birdie back and forth over the net. A player wins a point if their opponent fails to bat the birdie over the net. When Aileen is the server (the first player to try to hit the birdie over the net), she wins a point with probability $\frac{9}{10}$ . Each time Aileen successfully bats the birdie over the net, her opponent, independent of all previous hits, returns the birdie with probability $\frac{3}{4}$ . Each time Aileen bats the birdie, independent of all previous hits, she returns the birdie with probability $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
|
73
|
4/8
|
Let $a, b$, and $c$ be integers simultaneously satisfying the equations $4abc + a + b + c = 2018$ and $ab + bc + ca = -507$. Find $|a| + |b|+ |c|$.
|
46
|
3/8
|
Rectangle $ABCD$ has sides $AB = 10$ and $AD = 7$. Point $G$ lies in the interior of $ABCD$ a distance $2$ from side $\overline{CD}$ and a distance $2$ from side $\overline{BC}$. Points $H, I, J$, and $K$ are located on sides $\overline{BC}, \overline{AB}, \overline{AD}$, and $\overline{CD}$, respectively, so that the path $GHIJKG$ is as short as possible. Then $AJ = \frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
|
2
|
0/8
|
Each of the $48$ faces of eight $1\times 1\times 1$ cubes is randomly painted either blue or green. The probability that these eight cubes can then be assembled into a $2\times 2\times 2$ cube in a way so that its surface is solid green can be written $\frac{p^m}{q^n}$ , where $p$ and $q$ are prime numbers and $m$ and $n$ are positive integers. Find $p + q + m + n$.
|
77
|
0/8
|
Binhao has a fair coin. He writes the number $+1$ on a blackboard. Then he flips the coin. If it comes up heads (H), he writes $+\frac12$ , and otherwise, if he flips tails (T), he writes $-\frac12$ . Then he flips the coin again. If it comes up heads, he writes $+\frac14$ , and otherwise he writes $-\frac14$ . Binhao continues to flip the coin, and on the nth flip, if he flips heads, he writes $+ \frac{1}{2n}$ , and otherwise he writes $- \frac{1}{2n}$ . For example, if Binhao flips HHTHTHT, he writes $1 + \frac12 + \frac14 - \frac18 + \frac{1}{16} -\frac{1}{32} + \frac{1}{64} -\frac{1}{128}$ . The probability that Binhao will generate a series whose sum is greater than $\frac17$ is $\frac{p}{q}$ , where $p$ and $q$ are relatively prime positive integers. Find $p + 10q$.
|
153
|
2/8
|
In the diagram below, points $D, E$, and $F$ are on the inside of equilateral $\vartriangle ABC$ such that $D$ is on $\overline{AE}, E$ is on $\overline{CF}, F$ is on $\overline{BD}$, and the triangles $\vartriangle AEC, \vartriangle BDA$, and $\vartriangle CFB$ are congruent. Given that $AB = 10$ and $DE = 6$, the perimeter of $\vartriangle BDA$ is $\frac{a+b\sqrt{c}}{d}$, where $a, b, c$, and $d$ are positive integers, $b$ and $d$ are relatively prime, and $c$ is not divisible by the square of any prime. Find $a + b + c + d$.
[img]https://cdn.artofproblemsolving.com/attachments/8/6/98da82fc1c26fa13883a47ba6d45a015622b20.png[/img]
|
308
|
0/8
|
6. Point $M$ lies on the edge $A B$ of the cube $A B C D A_{1} B_{1} C_{1} D_{1}$. A rectangle $M N L K$ is inscribed in the square $A B C D$ such that one of its vertices is point $M$, and the other three are located on different sides of the square base. The rectangle $M_{1} N_{1} L_{1} K_{1}$ is the orthogonal projection of the rectangle $M N L K$ onto the plane of the upper base $A_{1} B_{1} C_{1} D_{1}$. The plane of the quadrilateral $M K_{1} L_{1} N$ forms an angle $\alpha$ with the plane of the base of the cube, where $\cos \alpha=\sqrt{\frac{2}{11}}$. Find the ratio $A M: M B$
|
\dfrac{1}{2}
|
0/8
|
1. 202 There are three piles of stones. Each time, A moves one stone from one pile to another, and A can receive payment from B after each move, the amount of which equals the difference between the number of stones in the pile where the stone is placed and the number of stones in the pile where the stone is taken from. If this difference is negative, A must return this amount of money to B (if A has no money to return, it can be temporarily owed). At a certain moment, all the stones are back in their original piles. Try to find the maximum possible amount of money A could have earned up to this moment.
|
0
|
1/8
|
53. As shown in the figure, on the side $AB$ of rectangle $ABCD$, there is a point $E$, and on side $BC$, there is a point $F$. Connecting $CE$ and $DF$ intersects at point $G$. If the area of $\triangle CGF$ is $2$, the area of $\triangle EGF$ is $3$, and the area of the rectangle is $30$, then the area of $\triangle BEF$ is . $\qquad$
|
\dfrac{35}{8}
|
2/8
|
169. In a chess tournament, two 7th-grade students and a certain number of 8th-grade students participated. Each participant played one game with every other participant. The two 7th-graders together scored 8 points, and all the 8th-graders scored the same number of points (in the tournament, each participant earns 1 point for a win and $1 / 2$ point for a draw). How many 8th-graders participated in the tournament?
|
7
|
5/8
|
The sum of two natural numbers and their greatest common divisor is equal to their least common multiple. Determine the ratio of the two numbers.
Translating the text as requested, while maintaining the original formatting and line breaks.
|
\dfrac{3}{2}
|
3/8
|
31. In the following figure, $A D=A B, \angle D A B=\angle D C B=\angle A E C=90^{\circ}$ and $A E=5$. Find the area of the quadrangle $A B C D$.
|
25
|
0/8
|
3 ( A square with an area of 24 has a rectangle inscribed in it such that one vertex of the rectangle lies on each side of the square. The sides of the rectangle are in the ratio $1: 3$.
Find the area of the rectangle.
#
|
9
|
4/8
|
4. The random variable $a$ is uniformly distributed on the interval $[-2 ; 1]$. Find the probability that all roots of the quadratic equation $a x^{2}-x-4 a+1=0$ in absolute value exceed 1.
|
\dfrac{7}{9}
|
2/8
|
I am part of a Math Club, where I have the same number of male colleagues as female colleagues. When a boy is absent, three quarters of the team are girls. Am I a man or a woman? How many women and how many men are there in the club?
|
3
|
1/8
|
5. As shown in Figure 1, trapezoid $ABCD (AB \parallel CD \parallel y$-axis, $|AB| > |CD|)$ is inscribed in the ellipse.
$$
\begin{array}{r}
\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 \\
(a > b > 0), E
\end{array}
$$
is the intersection of diagonals $AC$ and $BD$. Let $|AB| = m, |CD| = n, |OE| = d$. Then the maximum value of $\frac{m-n}{d}$ is $\qquad$
|
\dfrac{2b}{a}
|
2/8
|
9.3. A circle of radius $R$ touches the base $A C$ of an isosceles triangle $A B C$ at its midpoint and intersects side $A B$ at points $P$ and $Q$, and side $C B$ at points $S$ and $T$. The circumcircles of triangles $S Q B$ and $P T B$ intersect at points $B$ and $X$. Find the distance from point $X$ to the base of triangle $A B C$.
|
R
|
2/8
|
Three circles $C_i$ are given in the plane: $C_1$ has diameter $AB$ of length $1$ ; $C_2$ is concentric and has diameter $k$ ( $1 < k < 3$ ); $C_3$ has center $A$ and diameter $2k$ . We regard $k$ as fixed. Now consider all straight line segments $XY$ which have one endpoint $X$ on $C_2$ , one endpoint $Y$ on $C_3$ , and contain the point $B$ . For what ratio $XB/BY$ will the segment $XY$ have minimal length?
|
1
|
2/8
|
5. (10 points) The grid in the figure is composed of 6 identical small squares. Color 4 of the small squares gray, requiring that each row and each column has a colored small square. Two colored grids are considered the same if they are identical after rotation. How many different coloring methods are there? $\qquad$ kinds.
|
7
|
2/8
|
11.1 Figure 1 has $a$ rectangles, find the value of $a$.
I1.2 Given that 7 divides 111111 . If $b$ is the remainder when $\underbrace{111111 \ldots 111111}_{a \text {-times }}$ is divided by 7 , find the value of $b$.
I1.3 If $c$ is the remainder of $\left\lfloor(b-2)^{4 b^{2}}+(b-1)^{2 b^{2}}+b^{b^{2}}\right\rfloor$ divided by 3 , find the value of $c$.
I1.4 If $|x+1|+|y-1|+|z|=c$, find the value of $d=x^{2}+y^{2}+z^{2}$.
|
2
|
4/8
|
28th Putnam 1967 Problem A6 a i and b i are reals such that a 1 b 2 ≠ a 2 b 1 . What is the maximum number of possible 4-tuples (sign x 1 , sign x 2 , sign x 3 , sign x 4 ) for which all x i are non-zero and x i is a simultaneous solution of a 1 x 1 + a 2 x 2 + a 3 x 3 + a 4 x 4 = 0 and b 1 x 1 + b 2 x 2 + b 3 x 3 + b 4 x 4 = 0. Find necessary and sufficient conditions on a i and b i for this maximum to be achieved.
|
8
|
3/8
|
2. Find natural numbers $n$ such that for all positive numbers $a, b, c$, satisfying the inequality
$$
n(a b+b c+c a)>5\left(a^{2}+b^{2}+c^{2}\right)
$$
there exists a triangle with sides $a, b, c$.
|
6
|
4/8
|
13. Given $f(x)=\frac{x^{2}}{x^{2}-100 x+5000}$, find the value of $f(1)+f(2)+\cdots+f(100)$.
|
101
|
4/8
|
12. For a regular tetrahedron \(ABCD\), construct its circumscribed sphere, where \(OC_1\) is the diameter of the sphere. Then the angle between the line \(AC_1\) and the plane \(BCD\) is \(\qquad\).
|
\arcsin \frac{\sqrt{3}}{3}
|
0/8
|
9. (2004 Shanghai High School Competition Problem) The sequence $\left\{a_{n}\right\}$ satisfies $(n-1) a_{n+1}=(n+1) a_{n}-2(n-1), n=$ $1,2, \cdots$ and $a_{100}=10098$. Find the general term formula of the sequence $\left\{a_{n}\right\}$.
|
n^2 + n - 2
|
4/8
|
12. Let $A, B$ be subsets of the set $X=\{1,2, \cdots, n\}$. If every number in $A$ is strictly greater than all the numbers in $B$, then the ordered subset pair $(A, B)$ is called "good". Find the number of "good" subset pairs of $X$.
|
(n + 2) \cdot 2^{n-1}
|
3/8
|
6.1. On a plane, 55 points are marked - the vertices of a certain regular 54-gon and its center. Petya wants to paint a triplet of the marked points in red so that the painted points are the vertices of some equilateral triangle. In how many ways can Petya do this?
|
72
|
1/8
|
Example 14 (IMO-29 Problem) Let $\mathbf{N}$ be the set of positive integers, and define the function $f$ on $\mathbf{N}$ as follows:
(i) $f(1)=1, f(3)=3$;
(ii) For $n \in \mathbf{N}$, we have $f(2 n)=f(n), f(4 n+1)=2 f(2 n+1)-f(n), f(4 n+3)=$ $3 f(2 n+1)-2 f(n)$.
Find all $n$ such that $n \leqslant 1988$ and $f(n)=n$.
|
92
|
1/8
|
1. (10 points). Two scientific and production enterprises supply the market with substrates for growing orchids. In the substrate "Orchid-1," pine bark is three times more than sand; peat is twice as much as sand. In the substrate "Orchid-2," bark is half as much as peat; sand is one and a half times more than peat. In what ratio should the substrates be taken so that in the new, mixed composition, bark, peat, and sand are present in equal amounts.
|
1:1
|
5/8
|
In the Cartesian coordinate system $x 0 y$, there are points $A(3,3)$, $B(-2,1)$, $C(1,-2)$, and $T$ represents the set of all points inside and on the sides (including vertices) of $\triangle ABC$. Try to find the range of values for the bivariate function
$$
f(x, y)=\max \{2 x+y, \min \{3 x-y, 4 x+y\}\}
$$
(points $(x, y) \in T$).
|
[-3, 9]
|
5/8
|
10. Select 1004 numbers from $1,2, \cdots, 2008$ such that their total sum is 1009000, and the sum of any two of these 1004 numbers is not equal to 2009. Then the sum of the squares of these 1004 numbers is $\qquad$
Reference formula:
$$
1^{2}+2^{2}+\cdots+n^{2}=\frac{1}{6} n(n+1)(2 n+1) \text {. }
$$
|
1351373940
|
1/8
|
6. Let $\left(x_{1}, x_{2}, \cdots, x_{20}\right)$ be a permutation of $(1,2, \cdots, 20)$, and satisfy $\sum_{i=1}^{20}\left(\left|x_{i}-i\right|+\left|x_{i}+i\right|\right)=620$, then the number of such permutations is $\qquad$.
|
(10!)^2
|
3/8
|
4. Seven natives from several tribes are sitting in a circle by the fire. Each one says to the neighbor on their left: “Among the other five, there are no members of my tribe.” It is known that the natives lie to foreigners and tell the truth to their own. How many tribes are represented around the fire?
|
3
|
2/8
|
3. In $\triangle A B C$, it is known that $A B=12, A C=8, B C$ $=13, \angle A$'s angle bisector intersects the medians $B E$ and $C F$ at points $M$ and $N$, respectively. Let the centroid of $\triangle A B C$ be $G$. Then
$$
\frac{S_{\triangle G M N}}{S_{\triangle A B C}}=
$$
$\qquad$ .
|
\dfrac{1}{168}
|
4/8
|
Example 14 (1998 Shanghai High School Mathematics Competition) As shown in Figure 5-13, it is known that on the parabola $y=$ $x^{2}$, there are three vertices $A, B, C$ of a square. Find the minimum value of the area of such a square.
|
2
|
3/8
|
A cross-section of the pond's upper opening is a circle with a diameter of $100 \mathrm{~m}$, and plane sections through the center of this circle are equilateral triangles. Let $A$ and $B$ be two opposite points on the upper opening, $C$ the deepest point of the pond, and $H$ the midpoint of $B C$. What is the length of the shortest path connecting $A$ to $H$?
|
50\sqrt{5}
|
2/8
|
The functions \( f(x) \) and \( g(x) \) are differentiable on the interval \([a, b]\). Prove that if \( f(a) = g(a) \) and \( f^{\prime}(x) > g^{\prime}(x) \) for any point \( x \) in the interval \( (a, b) \), then \( f(x) > g(x) \) for any point \( x \) in the interval \( (a, b) \).
|
f(x) > g(x) \text{ for all } x \in (a, b)
|
3/8
|
3. In a non-isosceles triangle $ABC$, the bisector $AD$ intersects the circumcircle of the triangle at point $P$. Point $I$ is the incenter of triangle $ABC$. It turns out that $ID = DP$. Find the ratio $AI: ID$. (20 points)
|
2
|
2/8
|
Let \(\left\{x_{n}\right\}_{n>1}\) be the sequence of positive integers defined by \(x_{1}=2\) and \(x_{n+1} = 2x_{n}^{3} + x_{n}\) for all \(n \geq 1\). Determine the greatest power of 5 that divides the number \(x_{2014}^{2} + 1\).
|
5^{2014}
|
0/8
|
Problem 3. Given an equilateral triangle with an area of $6 \mathrm{~cm}^{2}$. On each of its sides, an equilateral triangle is constructed. Calculate the area of the resulting figure.
|
24
|
4/8
|
\left.\begin{array}{l}{[\quad \text { Law of Sines }} \\ {\left[\begin{array}{l}\text { Angle between a tangent and a chord }]\end{array}\right]}\end{array}\right]
In triangle $ABC$, angle $\angle B$ is equal to $\frac{\pi}{6}$. A circle with a radius of 2 cm is drawn through points $A$ and $B$, tangent to line $AC$ at point $A$. A circle with a radius of 3 cm is drawn through points $B$ and $C$, tangent to line $AC$ at point $C$. Find the length of side $AC$.
|
\sqrt{6}
|
2/8
|
1. From 37 coins arranged in a row, 9 are "tails" up, and 28 are "heads" up. In one step, any 20 coins are flipped. Is it possible after several steps for all coins to be "tails"? "heads"? In how few steps is this possible?
|
2
|
1/8
|
10. On a plane, 2011 points are marked. We will call a pair of marked points $A$ and $B$ isolated if all other points are strictly outside the circle constructed on $A B$ as its diameter. What is the smallest number of isolated pairs that can exist?
|
2010
|
1/8
|
A finite set of integers is called bad if its elements add up to 2010. A finite set of integers is a Benelux-set if none of its subsets is bad. Determine the smallest integer $n$ such that the set $\{502,503,504, \ldots, 2009\}$ can be partitioned into $n$ Benelux-sets.
(A partition of a set $S$ into $n$ subsets is a collection of $n$ pairwise disjoint subsets of $S$, the union of which equals $S$.)
|
2
|
0/8
|
3. In a $2 \times 2019$ grid, each cell is filled with a real number, where the 2019 real numbers in the top row are all different, and the bottom row is also filled with these 2019 real numbers, but in a different order from the top row. In each of the 2019 columns, two different real numbers are written, and the sum of the two numbers in each column is a rational number. How many irrational numbers can appear at most in the first row?
|
2016
|
3/8
|
5. Person A and Person B start walking towards each other at a constant speed from points $A$ and $B$ respectively, and they meet for the first time at a point 700 meters from $A$; then they continue to walk, with A reaching $B$ and B reaching $A$, and both immediately turning back, meeting for the second time at a point 400 meters from $B$. Then the distance between $A$ and $B$ is meters.
|
1700
|
4/8
|
## Task B-1.4.
Points $F, G$, and $H$ lie on the side $\overline{A B}$ of triangle $A B C$. Point $F$ is between points $A$ and $G$, and point $H$ is between points $G$ and $B$. The measure of angle $C A B$ is $5^{\circ}$, and $|B H|=|B C|,|H G|=$ $|H C|,|G F|=|G C|,|F A|=|F C|$. What is the measure of angle $A B C$?
|
100
|
1/8
|
Three. (50 points) Let $S=\{1,2, \cdots, 100\}$. Find the largest integer $k$, such that the set $S$ has $k$ distinct non-empty subsets with the property: for any two different subsets among these $k$ subsets, if their intersection is non-empty, then the smallest element in their intersection is different from the largest elements in both of these subsets.
---
The translation maintains the original text's formatting and structure.
|
2^{99} - 1
|
1/8
|
11.207 Two cubes with an edge equal to $a$ have a common segment connecting the centers of two opposite faces, but one cube is rotated by $45^{\circ}$ relative to the other. Find the volume of the common part of these cubes.
|
2a^3(\sqrt{2} - 1)
|
1/8
|
On an \( n \times n \) chessboard, place \( n^2 - 1 \) pieces (with \( n \geq 3 \)). The pieces are numbered as follows:
$$
(1,1), \cdots, (1, n), (2,1), \cdots, (2, n), \cdots, (n, 1), \cdots, (n, n-1).
$$
The chessboard is said to be in the "standard state" if the piece numbered \( (i, j) \) is placed exactly in the \( i \)-th row and \( j \)-th column, i.e., the \( n \)-th row and \( n \)-th column is empty. Now, randomly place the \( n^2 - 1 \) pieces on the chessboard such that each cell contains exactly one piece. In each step, you can move a piece that is adjacent to the empty cell into the empty cell (two cells are adjacent if they share a common edge).
Question: Is it possible, from any initial placement, to achieve the standard state after a finite number of moves? Prove your conclusion.
|
\text{No}
|
4/8
|
Example 5. As shown in the figure, on a semicircle with center $C$ and diameter $M N$, there are two distinct points $A$ and $B$. $P$ is on $C N$, and $\angle C A P=\angle C B P=10^{\circ}, \angle M C A=40^{\circ}$. Find $\angle B C N$. (34th American Competition Problem)
|
20^\circ
|
3/8
|
15. The sequence of positive integers $\left\{a_{n}\right\}$ satisfies
$$
\begin{array}{l}
a_{2}=15, \\
2+\frac{4}{a_{n}+1}<\frac{a_{n}}{a_{n}-4 n+2}+\frac{a_{n}}{a_{n+1}-4 n-2} \\
<2+\frac{4}{a_{n}-1} .
\end{array}
$$
Calculate: $S_{n}=\frac{1}{a_{1}}+\frac{1}{a_{2}}+\cdots+\frac{1}{a_{n}}$.
|
\dfrac{n}{2n + 1}
|
5/8
|
The average age of grandpa, grandma, and their five grandchildren is 26 years. The average age of the grandchildren themselves is 7 years. Grandma is one year younger than grandpa.
How old is grandma?
(L. Hozová)
Hint. How old are all the grandchildren together?
|
73
|
4/8
|
11.4. We will call a natural number interesting if it is the product of exactly two (distinct or equal) prime numbers. What is the greatest number of consecutive numbers, all of which are interesting
|
3
|
3/8
|
Given a circle on a plane with center \( O \) and a point \( P \) outside the circle. Construct a regular \( n \)-sided polygon inscribed in the circle, and denote its vertices as \( A_{1}, A_{2}, \ldots, A_{n} \). Show that
$$
\lim _{n \rightarrow \infty} \sqrt[n]{\overline{P A}_{1} \cdot P A_{2} \cdot \ldots \cdot \overline{P A}_{n}}=\overline{P O}.
$$
|
\overline{PO}
|
0/8
|
8. Let $P_{n}(k)$ denote the number of permutations of $\{1,2, \cdots, n\}$ with $k$ fixed points. Let $a_{t}=\sum_{k=0}^{n} k^{t} P_{n}(k)$. Then
$$
\begin{array}{l}
a_{5}-10 a_{4}+35 a_{3}-50 a_{2}+25 a_{1}-2 a_{0} \\
=
\end{array}
$$
|
-n!
|
1/8
|
## Task A-3.7.
Let $A B C D$ be a parallelogram such that $|A B|=4,|A D|=3$, and the measure of the angle at vertex $A$ is $60^{\circ}$. Circle $k_{1}$ touches sides $\overline{A B}$ and $\overline{A D}$, while circle $k_{2}$ touches sides $\overline{C B}$ and $\overline{C D}$.
Circles $k_{1}$ and $k_{2}$ are congruent and touch each other externally. Determine the length of the radius of these circles.
|
\dfrac{7\sqrt{3} - 6}{6}
|
5/8
|
In triangle $ABC$, the angle $C$ is obtuse, $m(\angle A) = 2m(\angle B)$, and the side lengths are integers. Find the smallest possible perimeter of this triangle.
|
77
|
4/8
|
Milo rolls five fair dice, which have 4, 6, 8, 12, and 20 sides respectively, and each one is labeled $1$ to $n$ for the appropriate $n$. How many distinct ways can they roll a full house (three of one number and two of another)? The same numbers appearing on different dice are considered distinct full houses, so $(1,1,1,2,2)$ and $(2,2,1,1,1)$ would both be counted.
|
248
|
0/8
|
Dexter's Laboratory has $2024$ robots, each with a program set up by Dexter. One day, his naughty sister Dee Dee intrudes and writes an integer in the set $\{1, 2, \dots, 113\}$ on each robot's forehead. Each robot detects the numbers on all other robots' foreheads and guesses its own number based on its program, individually and simultaneously.
Find the largest positive integer $k$ such that Dexter can set up the programs so that, no matter how the numbers are distributed, there are always at least $k$ robots who guess their numbers correctly.
|
17
|
3/8
|
The sequence $x_1, x_2, x_3, \ldots$ is defined by $x_1 = 2022$ and $x_{n+1} = 7x_n + 5$ for all positive integers $n$. Determine the maximum positive integer $m$ such that \[ \frac{x_n(x_n - 1)(x_n - 2) \cdots (x_n - m + 1)}{m!} \] is never a multiple of $7$ for any positive integer $n$.
|
404
|
0/8
|
The expression below has six empty boxes. Each box is to be filled in with a number from $1$ to $6$, where all six numbers are used exactly once, and then the expression is evaluated. What is the maximum possible final result that can be achieved?
$$\frac{\frac{\square}{\square} + \frac{\square}{\square}}{\frac{\square}{\square}}$$
|
23
|
5/8
|
Bradley is driving at a constant speed. When he passes his school, he notices that in $20$ minutes he will be exactly $\frac{1}{4}$ of the way to his destination, and in $45$ minutes he will be exactly $\frac{1}{3}$ of the way to his destination. Find the number of minutes it takes Bradley to reach his destination from the point where he passes his school.
|
245
|
4/8
|
Five members of the Lexington Math Team are sitting around a table. Each flips a fair coin. Given that the probability that three consecutive members flip heads is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, find $m+n$.
|
43
|
3/8
|
The Matini company released a special album with the flags of the $12$ countries that compete in the CONCACAM Mathematics Cup. Each postcard envelope contains two flags chosen randomly. Determine the minimum number of envelopes that need to be opened so that the probability of having a repeated flag is $50\%$.
|
3
|
5/8
|
Say a positive integer $n$ is radioactive if one of its prime factors is strictly greater than $\sqrt{n}$. For example, $2012 = 2^2 \cdot 503$, $2013 = 3 \cdot 11 \cdot 61$, and $2014 = 2 \cdot 19 \cdot 53$ are all radioactive, but $2015 = 5 \cdot 13 \cdot 31$ is not. How many radioactive numbers have all prime factors less than $30$?
|
119
|
4/8
|
For a positive integer $n$, let $f(n)$ be the sum of the positive integers that divide at least one of the nonzero base $10$ digits of $n$. For example, $f(96) = 1 + 2 + 3 + 6 + 9 = 21$. Find the largest positive integer $n$ such that for all positive integers $k$, there is some positive integer $a$ such that $f^k(a) = n$, where $f^k(a)$ denotes $f$ applied $k$ times to $a$.
|
15
|
2/8
|
The diagram shows a regular pentagon $ABCDE$ and a square $ABFG$. Find the degree measure of $\angle FAD$.
|
18
|
0/8
|
On equilateral $\triangle{ABC}$, point D lies on BC a distance 1 from B, point E lies on AC a distance 1 from C, and point F lies on AB a distance 1 from A. Segments AD, BE, and CF intersect in pairs at points G, H, and J, which are the vertices of another equilateral triangle. The area of $\triangle{ABC}$ is twice the area of $\triangle{GHJ}$. The side length of $\triangle{ABC}$ can be written as $\frac{r+\sqrt{s}}{t}$, where r, s, and t are relatively prime positive integers. Find $r + s + t$.
|
30
|
3/8
|
Let $\sigma(n)$ represent the number of positive divisors of $n$. Define $s(n)$ as the number of positive divisors of $n+1$ such that for every divisor $a$, $a-1$ is also a divisor of $n$. Find the maximum value of $2s(n) - \sigma(n)$.
|
2
|
3/8
|
The sequence of integers $a_1, a_2, \dots$ is defined as follows:
- $a_1 = 1$.
- For $n > 1$, $a_{n+1}$ is the smallest integer greater than $a_n$ such that $a_i + a_j \neq 3a_k$ for any $i, j, k$ from $\{1, 2, \dots, n+1\}$, where $i, j, k$ are not necessarily different.
Define $a_{2004}$.
|
4507
|
0/8
|
Let $a_1$, $a_2$, $a_3$, $a_4$, and $a_5$ be real numbers satisfying
\begin{align*}
a_1a_2 + a_2a_3 + a_3a_4 + a_4a_5 + a_5a_1 & = 20,\\
a_1a_3 + a_2a_4 + a_3a_5 + a_4a_1 + a_5a_2 & = 22.
\end{align*}
Then the smallest possible value of $a_1^2 + a_2^2 + a_3^2 + a_4^2 + a_5^2$ can be expressed as $m + \sqrt{n}$, where $m$ and $n$ are positive integers. Compute $100m + n$.
|
2105
|
0/8
|
2. Let point $P$ be inside the face $A B C$ of a regular tetrahedron $A B C D$ with edge length 2, and its distances to the faces $D A B$, $D B C$, and $D C A$ form an arithmetic sequence. Then the distance from point $P$ to the face $D B C$ is $\qquad$
|
\dfrac{2\sqrt{6}}{9}
|
4/8
|
Father played chess with uncle. For a won game, the winner received 8 crowns from the opponent, and for a draw, nobody got anything. Uncle won four times, there were five draws, and in the end, father earned 24 crowns.
How many games did father play with uncle?
(M. Volfová)
|
16
|
4/8
|
9. In $\triangle A B C$, $\angle A<\angle B<\angle C$, $\frac{\sin A+\sin B+\sin C}{\cos A+\cos B+\cos C}=\sqrt{3}$.
Then $\angle B=$
|
\dfrac{\pi}{3}
|
0/8
|
5. In a $7 \times 7$ unit square grid, there are 64 grid points, and there are many squares with these grid points as vertices. How many different values can the areas of these squares have?
(21st Jiangsu Province Junior High School Mathematics Competition)
|
18
|
2/8
|
2、The following is a calendar for May of a certain year. Using a $2 \times 2$ box that can frame four numbers (excluding Chinese characters), the number of different ways to frame four numbers is $\qquad$.
---
Note: The original text includes a placeholder for the answer (indicated by $\qquad$). This has been preserved in the translation.
|
20
|
2/8
|
5. Call two vertices of a simple polygon "visible" to each other if and only if they are adjacent or the line segment connecting them lies entirely inside the polygon (except for the endpoints which lie on the boundary). If there exists a simple polygon with $n$ vertices, where each vertex is visible to exactly four other vertices, find all possible values of the positive integer $n$.
Note: A simple polygon is one that has no holes and does not intersect itself.
|
5
|
3/8
|
2. Initially, the numbers 2, 3, and 4 are written on the board. Every minute, Anton erases the numbers written on the board and writes down their pairwise sums instead. After an hour, three enormous numbers are written on the board. What are their last digits? List them in any order, separated by semicolons.
Example of answer format:
$1 ; 2 ; 3$
|
7 ; 8 ; 9
|
4/8
|
12. (18 points) The inverse function of $f(x)$ is $y=\frac{x}{1+x}$, $g_{n}(x)+\frac{1}{f_{n}(x)}=0$, let $f_{1}(x)=f(x)$, and for $n>1\left(n \in \mathbf{N}_{+}\right)$, $f_{n}(x)=f_{n-1}\left(f_{n-1}(x)\right)$. Find the analytical expression for $g_{n}(x)\left(n \in \mathbf{N}_{+}\right)$.
|
2^{n-1} - \dfrac{1}{x}
|
3/8
|
Given a natural number $n$, if the tuple $(x_1,x_2,\ldots,x_k)$ satisfies
$$2\mid x_1,x_2,\ldots,x_k$$
$$x_1+x_2+\ldots+x_k=n$$
then we say that it's an [i]even partition[/i]. We define [i]odd partition[/i] in a similar way. Determine all $n$ such that the number of even partitions is equal to the number of odd partitions.
|
2
|
1/8
|
Example 6 In the complete quadrilateral CFBEGA, the line containing diagonal $CE$ intersects the circumcircle of $\triangle ABC$ at point $D$. The circle passing through point $D$ and tangent to $FG$ at point $E$ intersects $AB$ at point $M$. Given $\frac{AM}{AB}=t$. Find $\frac{GE}{EF}$ (expressed in terms of $t$).
|
\dfrac{t}{1 - t}
|
2/8
|
17. In $\triangle A B C, A C>A B$, the internal angle bisector of $\angle A$ meets $B C$ at $D$, and $E$ is the foot of the perpendicular from $B$ onto $A D$. Suppose $A B=5, B E=4$ and $A E=3$. Find the value of the expression $\left(\frac{A C+A B}{A C-A B}\right) E D$.
|
3
|
5/8
|
Consider the polynomial \(x^5 + bx^4 + cx^3 + dx^2 + ex + f = 0\) with integer coefficients. Determine all possible values for \(m\), which represents the exact number of integer roots of the polynomial, counting multiplicity.
|
0, 1, 2, 3, 4, 5
|
0/8
|
A square and a right triangle are adjacent to each other, with each having one side on the $x$-axis. The lower right vertex of the square and the lower left vertex of the triangle are at $(12, 0)$. They share one vertex, which is the top right corner of the square and the hypotenuse-end of the triangle. The side of the square and the base of the triangle on the x-axis each equal $12$ units. A segment is drawn from the top left vertex of the square to the farthest vertex of the triangle, creating a shaded area. Calculate the area of this shaded region.
|
36 \text{ square units}
|
2/8
|
Consider a truncated pyramid formed by slicing the top off a regular pyramid. The larger base now has 6 vertices, and the smaller top base is a regular hexagon. The ant starts at the top vertex of the truncated pyramid, walks randomly to one of the six adjacent vertices on the larger base, and then from there, it walks to one of the six adjacent vertices on the smaller top base. What is the probability that this final vertex is the initially started top vertex? Express your answer as a common fraction.
|
\frac{1}{6}
|
3/8
|
The planet Zephyr moves in an elliptical orbit with its star at one of the foci. At its nearest point (perihelion), it is 3 astronomical units (AU) from the star, while at its furthest point (aphelion) it is 15 AU away. Determine the distance of Zephyr from its star when it is exactly halfway through its orbit.
|
9 \text{ AU}
|
5/8
|
Let $f : \mathbb{R} \to \mathbb{R}$ be a function such that
\[f(f(x) + y) = f(x^2 - y) + 2cf(x)y\]
for all real numbers $x$ and $y$, where $c$ is a constant. Determine all possible values of $f(2)$, find the sum of these values, and calculate the product of the number of possible values and their sum.
|
8
|
4/8
|
Joel selected an acute angle $x$ (strictly between 0 and 90 degrees) and wrote $\sin x$, $\tan x$, and $\sec x$ on three different cards. He then distributed these cards to three students: Isabelle, Edgar, and Sam, one card to each. After sharing the values on their cards with each other, only Isabelle could accurately identify which function produced the value on her card. Compute the sum of all possible values that Joel wrote on Isabelle's card.
|
1
|
4/8
|
The planet Zytron orbits in an elliptical path around its star with the star located at one of the foci of the ellipse. Zytron’s closest approach to its star (perihelion) is 3 astronomical units (AU), and its furthest point (aphelion) is 8 AU away. How far is Zytron from its star when it is exactly halfway in its orbit along the major axis?
|
5.5 \text{ AU}
|
3/8
|
A quadrilateral with one pair of parallel sides (trapezoid) has its consecutive angles in an arithmetic sequence. The sum of the smallest and largest angles equals $200^{\circ}$. If the second-smallest angle is $70^{\circ}$, find the measure of the third-largest angle.
|
130^\circ
|
2/8
|
Utilizing the twelve-letter alphabet of the Rotokas of Papua New Guinea (A, E, G, I, K, O, P, R, S, T, U, V), imagine the scenario for generating six-letter license plates. How many valid six-letter license plates can be formed that start with either A or E, end with R, cannot contain the letter V, and have no repeating letters?
|
6048
|
5/8
|
Alice throws a fair eight-sided die each morning. If Alice throws a perfect square number, she chooses porridge for breakfast. For any prime numbers, she chooses toast. If she throws an 8, she will throw the die again. How many times can Alice be expected to throw her die in a leap year?
|
\frac{2928}{7}
|
2/8
|
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