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It is given that $2^{333}$ is a 101-digit number whose first digit is 1. How many of the numbers $2^k$, where $1 \leq k \leq 332$, have the first digit as 4?
|
32
|
1/8
|
In triangle $ABC$, angle $A$ is twice angle $B$, angle $C$ is obtuse, and the three side lengths $a, b, c$ are integers. Determine, with proof, the minimum possible perimeter.
|
77
|
3/8
|
The diagram shows a $20 \times 20$ square $ABCD$. The points $E$, $F$, and $G$ are equally spaced on side $BC$. The points $H$, $I$, $J$, and $K$ on side $DA$ are placed so that the triangles $BKE$, $EJF$, $FIG$, and $GHC$ are isosceles. Points $L$ and $M$ are midpoints of the sides $AB$ and $CD$, respectively. Find the total area of the shaded regions.
|
200
|
5/8
|
It is well-known that the $n^{\text{th}}$ triangular number can be given by the formula $\frac{n(n+1)}{2}$. A Pythagorean triple of \textit{square numbers} is an ordered triple $(a,b,c)$ such that $a^2+b^2=c^2$. Let a Pythagorean triple of \textit{triangular numbers} (a PTTN) be an ordered triple of positive integers $(a,b,c)$ such that $a\leq b<c$ and
\[\frac{a(a+1)}{2}+\frac{b(b+1)}{2}=\frac{c(c+1)}{2}.\]
For instance, $(3,5,6)$ is a PTTN ($6+15=21$). Here we call both $a$ and $b$ \textit{legs} of the PTTN. Find the smallest natural number $n$ such that $n$ is a leg of \textit{at least} six distinct PTTNs.
|
14
|
0/8
|
Let $x$ be a positive real number. Define
\[
A = \sum_{k=0}^{\infty} \frac{x^{3k}}{(3k)!}, \quad
B = \sum_{k=0}^{\infty} \frac{x^{3k+1}}{(3k+1)!}, \quad \text{and} \quad
C = \sum_{k=0}^{\infty} \frac{x^{3k+2}}{(3k+2)!}.
\]
Given that $A^3+B^3+C^3 + 8ABC = 2014$, compute $ABC$.
|
183
|
4/8
|
How many ways can you fill a table of size $n \times n$ with integers such that each cell contains the total number of even numbers in its row and column, excluding itself? Two tables are considered different if they differ in at least one cell.
|
1
|
1/8
|
There are $100$ people standing in a line from left to right. Half of them are randomly chosen to face right (with all $\binom{100}{50}$ possible choices being equally likely), and the others face left. Then, while there is a pair of people who are facing each other and have no one between them, the leftmost such pair leaves the line. Compute the expected number of people remaining once this process terminates.
|
50
|
0/8
|
Consider polynomials $P$ of degree $2015$, all of whose coefficients are in the set \{0,1,\dots,2010\}. Call such a polynomial "good" if for every integer $m$, one of the numbers $P(m)-20$, $P(m)-15$, $P(m)-1234$ is divisible by $2011$, and there exist integers $m_{20}, m_{15}, m_{1234}$ such that $P(m_{20})-20$, $P(m_{15})-15$, $P(m_{1234})-1234$ are all multiples of $2011$. Let $N$ be the number of good polynomials. Find the remainder when $N$ is divided by $1000$.
|
460
|
0/8
|
Let $p$ be an odd prime number less than $10^5$. Granite and Pomegranate play a game. First, Granite picks an integer $c \in \{2,3,\dots,p-1\}$. Pomegranate then picks two integers $d$ and $x$, defines $f(t) = ct + d$, and writes $x$ on a sheet of paper. Next, Granite writes $f(x)$ on the paper, Pomegranate writes $f(f(x))$, Granite writes $f(f(f(x)))$, and so on, with the players taking turns writing. The game ends when two numbers appear on the paper whose difference is a multiple of $p$, and the player who wrote the most recent number wins. Find the sum of all $p$ for which Pomegranate has a winning strategy.
|
65819
|
4/8
|
Let $T$ denote the 15-element set $\{10a+b : a,b \in \mathbb{Z}, 1 \le a < b \le 6\}$. Let $S$ be a subset of $T$ in which all six digits $1,2,\ldots ,6$ appear and in which no three elements together use all these six digits. Determine the largest possible size of $S$.
|
9
|
3/8
|
Let $f(x) = x^4 + 14x^3 + 52x^2 + 56x + 16$. Let $z_1, z_2, z_3, z_4$ be the four roots of $f$. Find the smallest possible value of $|z_a z_b + z_c z_d|$ where $\{a, b, c, d\} = \{1, 2, 3, 4\}$.
|
8
|
3/8
|
The incircle $\Gamma$ of a scalene triangle $ABC$ touches $BC$ at $D$, $CA$ at $E$, and $AB$ at $F$. Let $r_A$ be the radius of the circle inside $\triangle ABC$ which is tangent to $\Gamma$ and the sides $AB$ and $AC$. Define $r_B$ and $r_C$ similarly. If $r_A = 16$, $r_B = 25$, and $r_C = 36$, determine the radius of $\Gamma$.
|
74
|
3/8
|
We call $A_1, A_2, \ldots, A_n$ an $n$-division of $A$ if:
1. $A_1 \cap A_2 \cap \cdots \cap A_n = A$,
2. $A_i \cap A_j \neq \emptyset$ for $i \neq j$.
Find the smallest positive integer $m$ such that for any $14$-division $A_1, A_2, \ldots, A_{14}$ of $A = \{1, 2, \ldots, m\}$, there exists a set $A_i$ ($1 \leq i \leq 14$) such that there are two elements $a, b$ of $A_i$ where $b < a \leq \frac{4}{3}b$.
|
56
|
0/8
|
There are infinitely many boxes - initially, one of them contains $n$ balls, and all others are empty. In a single move, we take some balls from a non-empty box and put them into an empty box. On a sheet of paper, we write down the product of the resulting number of balls in the two boxes. After some moves, the sum of all numbers on the sheet of paper became $2023$. What is the smallest possible value of $n$?
|
65
|
4/8
|
Suppose $P(x)$ is a monic polynomial of degree $2023$ such that $P(k) = k^{2023}P(1-\frac{1}{k})$ for every positive integer $1 \leq k \leq 2023$. Then $P(-1) = \frac{a}{b}$ where $a$ and $b$ are relatively prime integers. Compute the unique integer $0 \leq n < 2027$ such that $bn-a$ is divisible by the prime $2027$.
|
0
|
1/8
|
10. (12 points) In the $3 \times 3$ table of Figure 1, numbers have already been filled in. Choose a black cell as the starting point. If the number in this black cell and the number in an adjacent white cell undergo one of the four arithmetic operations (addition, subtraction, multiplication, or division, with the larger number always first), and the result is an integer multiple of the number in another adjacent black cell, you can move from this white cell to the next black cell. Each cell must be visited exactly once. (For example, in Figure 2, from 7 through 8 to 5, and the direction of the arrows in Figure 2 is a correct path). Find the correct path in Figure 1. If the first 3 cells in the correct path are filled with $A$, $B$, and $C$ respectively, then the three-digit number $\overline{\mathrm{ABC}}=$ $\qquad$
|
834
|
0/8
|
B3. We have a sheet of graph paper of size $7 \times 7$ squares and an equally large plastic graph stencil, on which some squares are green, and the rest are transparent. If we place the stencil on the sheet of paper so that the sides of the stencil align with the sides of the paper, the grid pattern on the stencil matches the grid pattern on the paper. If we choose any square on the paper and color it red, then we can place the stencil on the paper so that its sides align with the sides of the paper and the red square on the paper is covered by one of the green squares on the stencil (the stencil can be rotated and flipped over as needed). What is the minimum number of green squares on the stencil?
67th Mathematical Competition for High School Students in Slovenia National Competition, April 22, 2023
# Problems for 3rd Year
Time for solving: 180 minutes. Each problem in section A has exactly one correct answer. In section A, we will award two points for the correct answer, and deduct one point for an incorrect answer. Write the answers for section A in the left column of the answer sheet, and leave the right column blank. The committee will consider only what is written on the answer sheets, marked with the contestant's code.
|
10
|
2/8
|
16. Fill in each square in $\square \square+\square \square=\square \square \square$ with a digit from $0, 1, 2, \cdots \cdots, 9$ (digits in the squares can be the same, and the leading digit of any number cannot be 0), so that the equation holds.
There are
|
4860
|
1/8
|
If we walk along the fence from north to south, the distances between its posts are initially the same. From a certain post, the distance decreases to 2.9 meters and remains so until the southern end of the fence. Between the 1st and 16th posts (counted from the north), the distances do not change, and the distance between these two posts is 48 meters. The distance between the 16th and 28th posts is 36 meters.
Which post has different distances from its neighboring posts?
(L. Šimünek)
|
20
|
4/8
|
For an integer $n \ge 3$ we consider a circle with $n$ points on it.
We place a positive integer at each point, where the numbers are not necessary need to be different. Such placement of numbers is called [i]stable [/i] as three numbers next to always have product $n$ each other.
For how many values of $n$ with $3 \le n \le 2020$ is it possible to place numbers in a stable way?
|
680
|
3/8
|
The side lengths $a,b,c$ of a triangle $ABC$ are positive integers. Let:\\
\[T_{n}=(a+b+c)^{2n}-(a-b+c)^{2n}-(a+b-c)^{2n}+(a-b-c)^{2n}\]
for any positive integer $n$. If $\frac{T_{2}}{2T_{1}}=2023$ and $a>b>c$ , determine all possible perimeters of the triangle $ABC$.
|
49
|
2/8
|
3. Given $a, b, c \in \mathbf{R}_{+}$, and $abc=4$. Then the minimum value of the algebraic expression $a^{a+b} b^{3b} c^{c+b}$ is $\qquad$
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly.
|
64
|
5/8
|
15. Let $M$ be a set composed of a finite number of positive integers
$$
\begin{array}{l}
\text { such that, } M=\bigcup_{i=1}^{20} A_{i}=\bigcup_{i=1}^{20} B_{i}, \\
A_{i} \neq \varnothing, B_{i} \neq \varnothing(i=1,2, \cdots, 20),
\end{array}
$$
and satisfies:
(1) For any $1 \leqslant i<j \leqslant 20$,
$$
A_{i} \cap A_{j}=\varnothing, B_{i} \cap B_{j}=\varnothing \text {; }
$$
(2) For any $1 \leqslant i \leqslant 20,1 \leqslant j \leqslant 20$, if $A_{i} \cap B_{j}=\varnothing$, then $\left|A_{i} \cup B_{j}\right| \geqslant 18$.
Find the minimum number of elements in the set $M$ ( $|X|$ denotes the number of elements in the set $X$).
|
180
|
4/8
|
Example 9 For any positive integer $n$, let $x_{n}=\mathrm{C}_{2 n}^{n}$. Find all positive integers $h(h>1)$, such that the sequence of remainders of $\left\{x_{n}\right\}$ modulo $h$ is eventually periodic.
|
2
|
1/8
|
In the universe of Pi Zone, points are labeled with $2 \times 2$ arrays of positive reals. One can teleport from point $M$ to point $M'$ if $M$ can be obtained from $M'$ by multiplying either a row or column by some positive real. For example, one can teleport from $\left( \begin{array}{cc} 1 & 2 \\ 3 & 4 \end{array} \right)$ to $\left( \begin{array}{cc} 1 & 20 \\ 3 & 40 \end{array} \right)$ and then to $\left( \begin{array}{cc} 1 & 20 \\ 6 & 80 \end{array} \right)$.
A [i]tourist attraction[/i] is a point where each of the entries of the associated array is either $1$, $2$, $4$, $8$ or $16$. A company wishes to build a hotel on each of several points so that at least one hotel is accessible from every tourist attraction by teleporting, possibly multiple times. What is the minimum number of hotels necessary?
[i]Proposed by Michael Kural[/i]
|
17
|
3/8
|
In triangle \(ABC\), the altitude \(BM\) is drawn, and a circle is constructed with diameter \(BM\). This circle intersects side \(AB\) at point \(K\) and side \(BC\) at point \(Z\). Find the ratio of the area of triangle \(KZM\) to the area of triangle \(ABC\), given that \(\angle A = \alpha\) and \(\angle C = \beta\).
|
\dfrac{\sin 2\alpha \sin 2\beta}{4}
|
0/8
|
12. A wristwatch is 5 minutes slow per hour; 5.5 hours ago they were set to the correct time. Now, on the clock showing the correct time, it is 1 PM. In how many minutes will the wristwatch show 1 PM?
|
30
|
3/8
|
8. Let the integer sequence $a_{1}, a_{2}, \cdots, a_{10}$ satisfy:
$$
a_{10}=3 a_{1}, a_{2}+a_{8}=2 a_{5} \text {, }
$$
and $a_{i+1} \in\left\{1+a_{i}, 2+a_{i}\right\}(i=1,2, \cdots, 9)$.
Then the number of such sequences is $\qquad$
|
80
|
4/8
|
Arrange $\frac{n(n+1)}{2}$ different numbers randomly into a triangular array. Let $M_k$ be the maximum number in the $k$-th row (from top to bottom). Find the probability that $M_1 < M_2 < \cdots < M_n$.
|
\dfrac{2^n}{(n + 1)!}
|
3/8
|
5. (1996 Beijing Middle School Mathematics Competition) The function $f(x)$ is defined on $\mathbf{N}$, taking values in $\mathbf{N}$, and is a strictly increasing function (if for any $x_{1}, x_{2} \in A$, when $x_{1}<x_{2}$, we have $f\left(x_{1}\right)<f\left(x_{2}\right)$, then $f(x)$ is called a strictly increasing function on $A$), and satisfies the condition $f(f(k))=3 k$. Try to find: the value of $f(1)+f(9)+f(96)$.
|
197
|
0/8
|
12. Let $[x]$ denote the greatest integer not exceeding $x$, and let the natural number $n$ satisfy $\left[\frac{1}{15}\right]+\left[\frac{2}{15}\right]+\left[\frac{3}{15}\right]+\cdots+\left[\frac{\mathrm{n}-1}{15}\right]+\left[\frac{\mathrm{n}}{15}\right]>$ 2011, then what is the minimum value of $n$?
|
253
|
1/8
|
5. Given is a regular pentagon $A B C D E$. Determine the least value of the expression
$$
\frac{P A+P B}{P C+P D+P E},
$$
where $P$ is an arbitrary point lying in the plane of the pentagon $A B C D E$.
|
\sqrt{5} - 2
|
3/8
|
4. A circle touches the sides of an angle at points $A$ and $B$. A point $M$ is chosen on the circle. The distances from $M$ to the sides of the angle are 24 and 6. Find the distance from $M$ to the line $A B$.
|
12
|
4/8
|
[ Sphere inscribed in a pyramid ]
[Volume helps solve the problem.]
In a triangular pyramid, two opposite edges are equal to 12 and 4, and the other edges are equal to 7. A sphere is inscribed in the pyramid. Find the distance from the center of the sphere to the edge equal to 12.
|
\dfrac{39 - 3\sqrt{65}}{8}
|
3/8
|
5. Given point $A$ is a point inside the unit circle with center $O$, satisfying $|\overrightarrow{O A}|=\frac{1}{2}, B, C$ are any two points on the unit circle $O$, then the range of $\overrightarrow{A C} \cdot \overrightarrow{B C}$ is $\qquad$ .
|
\left[ -\dfrac{1}{8}, 3 \right]
|
4/8
|
Let $ABCD$ be a quadrilateral with $AD = 20$ and $BC = 13$. The area of $\triangle ABC$ is $338$ and the area of $\triangle DBC$ is $212$. Compute the smallest possible perimeter of $ABCD$.
[i]Proposed by Evan Chen[/i]
|
118
|
1/8
|
Determine the number of positive integers that, in their base \( n \) representation, have all different digits and every digit except the leftmost one differs from some digit to its left by \( \pm 1 \). Express the answer as a simple function of \( n \) and provide a proof of your result.
|
2^{n+1} - 2n - 2
|
1/8
|
Thirty numbers are arranged on a circle in such a way that each number equals the absolute difference of its two neighbors. Given that the sum of the numbers is $2000$, determine the numbers.
|
0
|
4/8
|
3. Let $\mathbb{R}_{+}$ be the set of positive real numbers. Find all functions $f: \mathbb{R}_{+}^{3} \rightarrow \mathbb{R}_{+}$ such that for all positive real numbers $x, y, z$ and $k$ the following three conditions hold:
(i) $x f(x, y, z)=z f(z, y, x)$;
(ii) $f\left(x, k y, k^{2} z\right)=k f(x, y, z)$;
(iii) $f(1, k, k+1)=k+1$.
(Great Britain)
|
\dfrac{y + \sqrt{y^2 + 4xz}}{2x}
|
0/8
|
Problem 10.7. Nikita schematically drew the graph of the quadratic polynomial $y=a x^{2}+b x+c$. It turned out that $A B=C D=1$. Consider four numbers $-a, b, c$ and the discriminant of the quadratic polynomial. It is known that three of them are equal in some order to $1 / 4, -1, -3 / 2$. Find what the fourth number is.
|
-\dfrac{1}{2}
|
0/8
|
$\mathrm{Az} r_{1}$ and $r_{2}$ radius circles touch each other externally. The segment of their common external tangent that lies between the points of tangency is rotated around the line connecting the centers of the circles. Express the area of the frustum of the cone generated by this rotation in terms of $r_{1}$ and $r_{2}$.
|
4\pi r_1 r_2
|
3/8
|
1. As shown in Figure 14, quadrilateral $A B C D$ is a square, and an equilateral triangle $\triangle A B E$ is constructed outward from side $A B$. $C E$ intersects $B D$ at point $F$. Then $\angle A F D=$ $\qquad$ degrees.
|
60
|
5/8
|
6. All natural numbers, the sum of the digits of each of which is equal to 5, were arranged in ascending order. What number is in the 125th place
|
41000
|
3/8
|
7.85 Given the following $5 \times 5$ number table
$$
\left(\begin{array}{ccccc}
11 & 17 & 25 & 19 & 16 \\
24 & 10 & 13 & 15 & 3 \\
12 & 5 & 14 & 2 & 18 \\
23 & 4 & 1 & 8 & 22 \\
6 & 20 & 7 & 21 & 9
\end{array}\right)
$$
Try to select 5 elements from this number table such that no two of them are in the same row or column, and make the smallest one as large as possible. Prove the correctness of your selected answer.
|
15
|
3/8
|
9. Given $z \in \mathbf{C}$, if the equation $4 x^{2}-8 z x+4 \mathrm{i}+3=0$ (where $\mathrm{i}$ is the imaginary unit) has real roots, then the minimum value of the modulus of the complex number $z$ $|z|$ is $\qquad$ .
|
1
|
2/8
|
6. Let $k$ be a real number, in the Cartesian coordinate system $x O y$ there are two point sets $A=\left\{(x, y) \mid x^{2}+y^{2}=\right.$ $2(x+y)\}$ and $B=\{(x, y) \mid k x-y+k+3 \geqslant 0\}$. If $A \cap B$ is a singleton set, then the value of $k$ is $\qquad$
|
-2 - \sqrt{3}
|
2/8
|
Joseane's calculator is malfunctioning: each digit pressed appears doubled on the screen. The addition, subtraction, multiplication, and division operation keys work normally and cannot be pressed consecutively twice. For example, a permissible sequence of operations is pressing $2 \rightarrow \times \rightarrow 3$, which produces the number $4 \cdot 6=24$.
a) How can she make 80 appear by pressing 3 keys?
b) How can she make 50 appear by pressing 3 number keys and two operation keys alternately?
c) What is the minimum number of keys she must press to obtain the number 23?
|
4 \times 5
|
0/8
|
3. Circles $\omega_{1}$ and $\omega_{2}$ with centers $O_{1}$ and $O_{2}$ respectively intersect at point $B$. The extension of segment $O_{2} B$ beyond point $B$ intersects circle $\omega_{1}$ at point $K$, and the extension of segment $O_{1} B$ beyond point $B$ intersects circle $\omega_{2}$ at point $L$. The line passing through point $B$ parallel to $K L$ intersects circles $\omega_{1}$ and $\omega_{2}$ again at points $A$ and $C$ respectively. The rays $A K$ and $C L$ intersect at point $N$. Find the angle between the lines $O_{1} N$ and $O_{2} B$.
|
90^\circ
|
3/8
|
Example 7 Let $S=\{1,2,3,4\}$. An $n$-term sequence: $q_{1}, q_{2}, \cdots, q_{n}$ has the following property: For any non-empty subset $B$ of $S$ (the number of elements in $B$ is denoted by $|B|$), there are adjacent $|B|$ terms in the sequence that exactly form the set $B$. Find the minimum value of $n$.
(1997, Shanghai High School Mathematics Competition)
|
8
|
2/8
|
Let $n, m$ be positive integers such that
\[n(4n+1)=m(5m+1)\]
(a) Show that the difference $n-m$ is a perfect square of a positive integer.
(b) Find a pair of positive integers $(n, m)$ which satisfies the above relation.
Additional part (not asked in the TST): Find all such pairs $(n,m)$.
|
(38, 34)
|
5/8
|
Assume $A,B,C$ are three collinear points that $B \in [AC]$. Suppose $AA'$ and $BB'$
are to parrallel lines that $A'$, $B'$ and $C$ are not collinear. Suppose $O_1$ is circumcenter of circle passing through $A$, $A'$ and $C$. Also $O_2$ is circumcenter of circle passing through $B$, $B'$ and $C$. If area of $A'CB'$ is equal to area of $O_1CO_2$, then find all possible values for $\angle CAA'$
|
30^\circ
|
3/8
|
6. In square $A B C D$, there is a point $M$, point $P$ is on $A B$, point $Q$ is on $C D$, and $\angle A P M=150^{\circ}$, $\angle M Q C=35^{\circ}$, then $\angle P M Q=$ $\qquad$ $\circ$.
|
65
|
4/8
|
Let $E F G H$ be a convex quadrilateral, and $K, L, M, N$ be the midpoints of segments $E F, F G, G H, H E$ respectively; $O$ be the intersection point of segments $K M$ and $L N$. It is known that $\angle L O M = 90^{\circ}$, $K M = 3 L N$, and the area of quadrilateral $K L M N$ is $S$. Find the diagonals of quadrilateral $E F G H$.
|
2\sqrt{\dfrac{5S}{3}}
|
3/8
|
Let \( G \) be a graph with \( n \) vertices such that \( G \) contains no triangle (cycle of length 3). Prove that \( G \) has at most \( \left\lfloor \frac{n^2}{4} \right\rfloor \) edges. Show that this result is optimal.
Bonus: Show that in a graph having \( k \) edges, there are at most \( \left\lfloor \frac{\sqrt{2}}{3} k^{3/2} \right\rfloor \) triangles.
|
\left\lfloor \frac{n^2}{4} \right\rfloor
|
3/8
|
## Problem 4.
An encyclopedia consists of 2000 numbered volumes. The volumes are stacked in order with number 1 on top and 2000 in the bottom. One may perform two operations with the stack:
(i) For $n$ even, one may take the top $n$ volumes and put them in the bottom of the stack without changing the order.
(ii) For $n$ odd, one may take the top $n$ volumes, turn the order around and put them on top of the stack again.
How many different permutations of the volumes can be obtained by using these two operations repeatedly?
|
(1000!)^2
|
0/8
|
10.373 In a triangle with sides 6, 10, and 12 cm, a circle is inscribed. A tangent is drawn to the circle such that it intersects the two larger sides. Find the perimeter of the cut-off triangle.
|
16
|
0/8
|
1. Using the digits $4,5,6,7,8$ and 9, a six-digit number was written. Zoran, Darko, and Nikola guessed the number. Zoran: 574698, Darko: 786945, Nikola: 456789. It turned out that Zoran correctly guessed the positions of three digits. Darko also guessed the positions of three digits, and Nikola guessed the position of only one digit. Determine the six-digit number.
|
576948
|
2/8
|
6. Let $A$ and $B$ be non-empty subsets of the set $\{1,2, \cdots, 10\}$, and the smallest element in set $A$ is not less than the largest element in set $B$. Then the number of such pairs $(A, B)$ is.
|
9217
|
4/8
|
Let $n$ be a positive integer. Now, a frog starts jumping from the origin of the number line and makes $2^{n}-1$ jumps. The process satisfies the following conditions:
(1) The frog will jump to each point in the set $\left\{1,2,3, \cdots, 2^{n}-1\right\}$ exactly once, without missing any.
(2) Each time the frog jumps, it can choose a step length from the set $\left\{2^{0}, 2^{1}, 2^{2}, \cdots\right\}$, and it can jump either left or right.
Let $T$ be the reciprocal sum of the step lengths of the frog. When $n=2024$, the minimum value of $T$ is \_\_\_\_\_.
|
2024
|
2/8
|
4. A $10 \times 10$ table is filled with numbers 1 and -1 such that the sum of the numbers in each row, except one, is equal to zero, and the sum of the numbers in each column, except one, is also equal to zero. Determine the maximum possible sum of all the numbers in the table.
(Patrik Bak)
|
10
|
0/8
|
Example 3. As shown in Figure 3, the equilateral triangles $\triangle A B C$ and $\triangle A_{1} B_{1} C_{1}$ have their sides $A C$ and $A_{1} C_{1}$ bisected at $O$. Then $A A_{1}: B B_{1}=$ $\qquad$
|
1 : \sqrt{3}
|
4/8
|
[ The ratio in which the bisector divides the side.] Pythagorean Theorem (direct and inverse). $\quad]$
In a right triangle $ABC$, the bisector $AP$ of the acute angle $A$ is divided by the center $O$ of the inscribed circle in the ratio
$AO: OP=(\sqrt{3}+1):(\sqrt{3}-1)$. Find the acute angles of the triangle.
|
30^\circ
|
0/8
|
8. Given that $a_{1}, a_{2}, a_{3}, a_{4}, a_{5}$ is a permutation of $1,2,3,4,5$, and satisfies $\left|a_{i}-a_{i+1}\right| \neq 1(i=1,2,3,4)$. Then the number of permutations $a_{1}, a_{2}, a_{3}, a_{4}, a_{5}$ that meet the condition is $\qquad$.
|
14
|
5/8
|
# 3. Clone 1
A rope was divided into 19 equal parts and arranged in a snake-like pattern. After that, a cut was made along the dotted line. The rope split into 20 pieces: the longest of them is 8 meters, and the shortest is 2 meters. What was the length of the rope before it was cut? Express your answer in meters.
|
152
|
2/8
|
Assume that all angles of a triangle \(ABC\) are acute. Let \(D\) and \(E\) be points on the sides \(AC\) and \(BC\) of the triangle such that \(A, B, D,\) and \(E\) lie on the same circle. Further suppose the circle through \(D, E,\) and \(C\) intersects the side \(AB\) in two points \(X\) and \(Y\). Show that the midpoint of \(XY\) is the foot of the altitude from \(C\) to \(AB\).
|
M = H
|
1/8
|
Let $(x, y, z, t) \in\left(\mathbb{N}^{*}\right)^{4}$ be a quadruple satisfying $x+y=z+t$ and $2xy=zt$, with $x \geq y$. Find the largest value of $m$ such that $m \leq x / y$ for sure.
|
3 + 2\sqrt{2}
|
0/8
|
Numbers 1, 2, 3, ..., 10 are written in a circle in some order. Petya calculated the sums of all triplets of neighboring numbers and wrote the smallest of these sums on the board. What is the largest number that could have been written on the board?
|
15
|
0/8
|
Five integers were written on the board. By summing them in pairs, the following ten numbers were obtained:
$$
0, 2, 4, 4, 6, 8, 9, 11, 13, 15
$$
What are the five numbers that were written on the board? Is it possible to obtain the following ten numbers by summing five integers in this way:
$$
12, 13, 14, 15, 16, 16, 17, 17, 18, 20?
$$
|
\text{No}
|
0/8
|
Out of 24 matchsticks of the same length, use some of them to form 6 triangles in a plane where each side of the equilateral triangle is one matchstick. Then, use the remaining matchsticks to form squares in the plane where each side of the square is one matchstick. What is the maximum number of such squares that can be formed?
|
4
|
1/8
|
A regular triangular prism $ABC A_1B_1C_1$ is inscribed in a sphere, with base $ABC$ and lateral edges $AA_1, BB_1, CC_1$. Segment $CD$ is a diameter of this sphere, and point $K$ is the midpoint of edge $AA_1$. Find the volume of the prism if $CK = 2 \sqrt{6}$ and $DK = 4$.
|
36
|
5/8
|
Initially, there are three cards in a box, each with numbers $1, 2, 3$. Each time, two cards are drawn from the box, and their sum is written on another blank card, then one card is put back into the box. After performing this operation 5 times, all cards except the last one have been drawn at least once but not more than twice. What is the maximum number on the card in the box at this point?
|
28
|
0/8
|
A $7 \times 7$ grid is colored black and white. If the number of columns with fewer black squares than white squares is $m$, and the number of rows with more black squares than white squares is $n$, find the maximum value of $m + n$.
|
12
|
5/8
|
Let $z$ be a complex number such that $z^{23} = 1$ and $z \neq 1.$ Find
\[\sum_{n = 0}^{22} \frac{1}{1 + z^n + z^{2n}}.\]The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.
|
49
|
3/8
|
Rational numbers $a$ and $b$ are chosen at random among all rational numbers in the interval $[0,2)$ that can be written as fractions $\frac{n}{d}$ where $n$ and $d$ are integers with $1 \le d \le 5$. What is the probability that \[(\text{cos}(a\pi)+i\text{sin}(b\pi))^4\]is a real number?The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.
|
31
|
0/8
|
The equation \[2000x^6+100x^5+10x^3+x-2=0\]has two real roots. Compute the square of the difference between them.The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.
|
561
|
4/8
|
Consider a cubic polynomial \( x^3 - sx^2 + px - q \) whose roots \( r_1, r_2, \) and \( r_3 \) satisfy \( r_1 + r_2 + r_3 = r_1^2 + r_2^2 + r_3^2 \). Calculate the maximum possible value of \( \frac{1}{r_1^3} + \frac{1}{r_2^3} + \frac{1}{r_3^3} \).
|
3
|
1/8
|
A line contains the points $(7, -3)$ and $(3, 1)$. Where does this line intersect the $x$-axis?
|
(4, 0)
|
4/8
|
When $600_{10}$ is expressed in a certain base, it has four digits in the form ABBA, where A and B are different digits. What is this base?
|
7
|
2/8
|
A framed artwork consists of a painting surrounded by a uniform frame. The outer dimensions of the framed artwork are 90 cm by 120 cm, and the width of the frame is 15 cm. What is the area of the painting inside the frame?
|
5400 \text{ cm}^2
|
0/8
|
The isosceles triangle and the rectangle shown here have the same area in square units. What is the height of the triangle, $h$, in terms of the length $l$ and width $w$ of the rectangle?
|
2w
|
4/8
|
Two triangles, $\triangle ABC$ and $\triangle DEF$, are drawn such that $\triangle ABC \sim \triangle DEF$. The lengths of segments $BC = 8$ cm and $AB = 10$ cm in $\triangle ABC$, and $DE = 24$ cm in $\triangle DEF$. If angle $BAC = 90^\circ$ and angle $EDF = 90^\circ$, what is the length of segment $EF$?
|
19.2 \text{ cm}
|
1/8
|
In a public park, a circular arrangement is designed as follows: at the center, there is a circular water fountain surrounded by a ring-shaped flower garden. Outside the garden, there is a walking path and then another inner walking path adjacent to the garden but inside it. The water fountain has a diameter of 12 feet. The flower garden ring is 9 feet wide, the outer walking path is 7 feet wide, and the inner walking path is 3 feet wide. Calculate the diameter of the outermost boundary of the outer walking path.
|
50
|
2/8
|
A triangle has sides of length 10 cm, 13 cm, and 7 cm. Find the perimeter of the triangle and also calculate its area using Heron's formula.
|
20\sqrt{3} \text{ cm}^2
|
0/8
|
A right triangle has one leg of 15 inches and the hypotenuse of 34 inches. A second triangle, which is similar to the first, has a hypotenuse that is twice the length of the first triangle’s hypotenuse. What is the length of the second triangle's shortest leg, if the longer leg of the second triangle is 33% larger than the corresponding leg of the first triangle?
|
30 \text{ inches}
|
5/8
|
The sum of the lengths of all the edges of a rectangular prism is 88 cm. The prism's length is twice its width, and its height is half of its width. Find the volume of the rectangular prism.
|
V = \frac{85184}{343} \text{ cm}^3
|
0/8
|
The planet Zorion travels in an elliptical orbit around its sun, which is positioned at one focus of the ellipse. Zorion's closest approach to its sun is 3 astronomical units (AU), and its furthest point is 15 AU away. If the entire orbit of Zorion is tilted by 30 degrees from its original plane, calculate the distance from Zorion to its sun when Zorion is exactly halfway through its orbit from perigee to apogee.
|
9 \text{ AU}
|
0/8
|
Elmo makes $N$ sandwiches for a school event. For each sandwich, he uses $B$ dollops of peanut butter at $3$ cents each and $J$ spoonfuls of jelly at $7$ cents each. The total cost for the peanut butter and jelly to make all the sandwiches is $\$3.78$. Assume that $B$, $J$, and $N$ are positive integers with $N>1$. What is the cost, in dollars, of the jelly Elmo uses to make the sandwiches?
|
\$2.94
|
3/8
|
7.37. What is the maximum number of planes of symmetry that a spatial figure consisting of three pairwise non-parallel lines can have?
Symmetry with respect to a line $l$ is a transformation of space that maps a point $X$ to a point $X^{\prime}$ such that the line $l$ passes through the midpoint of the segment $X X^{\prime}$ and is perpendicular to it. This transformation is also called axial symmetry, and $l$ is called the axis of symmetry:
|
9
|
3/8
|
Task 3.
## Maximum 10 points
In the Country of Wonders, a pre-election campaign is being held for the position of the best tea lover, in which the Mad Hatter, March Hare, and Dormouse are participating. According to a survey, $20 \%$ of the residents plan to vote for the Mad Hatter, $25 \%$ for the March Hare, and $30 \%$ for the Dormouse. The rest of the residents are undecided. Determine the smallest percentage of the undecided voters that the Mad Hatter must attract to ensure he does not lose to the March Hare and the Dormouse (under any distribution of votes), knowing that each of the undecided voters will vote for one of the candidates. The winner is determined by a simple majority of votes. Justify your answer.
|
70
|
1/8
|
3. If $x \in(-1,1)$, then
$$
f(x)=x^{2}-a x+\frac{a}{2}
$$
is always positive, the range of the real number $a$ is $\qquad$ $ـ$.
|
(0,2]
|
4/8
|
Emília's tree grows according to the following rule: two weeks after a branch appears, that branch produces a new branch every week, and the original branch continues to grow. After five weeks, the tree has five branches, as shown in the figure. How many branches, including the main branch, will the tree have at the end of eight weeks?
|
21
|
2/8
|
8. A thin beam of light falls normally on a plane-parallel glass plate. Behind the plate, at some distance from it, stands an ideal mirror (its reflection coefficient is equal to one). The plane of the mirror is parallel to the plate. It is known that the intensity of the beam that has passed through this system is 256 times less than the intensity of the incident beam. The reflection coefficient at the glass-air boundary is assumed to be constant regardless of the direction of the beam. Neglect absorption and scattering of light in air and glass. Find the reflection coefficient at the glass-air boundary under these conditions. (10 points)
|
0.75
|
0/8
|
13. Ring Line (from 8th grade. 3 points). On weekdays, the Absent-Minded Scientist travels to work on the ring line of the Moscow metro from the station "Taganskaya" to the station "Kievskaya", and back in the evening (see the diagram).
Upon entering the station, the Scientist boards the first train that arrives. It is known that trains run at approximately equal intervals in both directions, and that the train traveling on the northern route (through "Belorusskaya") takes 17 minutes to travel from "Kievskaya" to "Taganskaya" or vice versa, while the train on the southern route (through "Paveletskaya") takes 11 minutes.
Out of habit, the Scientist always calculates everything. Once, he calculated that over many years of observation:
- the train traveling counterclockwise arrives at "Kievskaya" on average 1 minute and 15 seconds after the train traveling clockwise arrives at the same station. The same is true for "Taganskaya";
- the average travel time from home to work is 1 minute less than the travel time from work to home.
Find the expected interval between trains traveling in the same direction.
|
3
|
3/8
|
Problem 2. A square canvas is divided into 100 identical squares, arranged in 10 rows and 10 columns. We have 10 cards, numbered differently with digits from 0 to 9. On the canvas, we need to place two cards, whose sum is 10, in squares located on different rows and columns. Determine the number of possibilities for placing these cards.
|
32400
|
3/8
|
6. Given that $a$ is a real number, and for any $k \in$ $[-1,1]$, when $x \in(0,6]$, we have
$$
6 \ln x+x^{2}-8 x+a \leqslant k x \text {. }
$$
Then the maximum value of $a$ is $\qquad$ .
|
6 - 6 \ln 6
|
4/8
|
2. A stack of A4 sheets was folded in half and folded in two (resulting in an A5 booklet). After that, the pages of the resulting booklet were renumbered: $1,2,3, \ldots$ It turned out that the sum of the numbers on one of the sheets was 74. How many sheets were in the stack?
|
9
|
1/8
|
5. A regular triangle with a side length of 2 is divided into four smaller triangles by its median lines. Except for the central small triangle, three other small triangles are used as bases to construct three regular tetrahedrons of height 2 on the same side. There is a sphere that is tangent to the plane of the original triangle and also tangent to one side face of each tetrahedron. What is the radius of this small sphere?
|
\dfrac{1}{3}
|
0/8
|
Integer $n>2$ is given. Find the biggest integer $d$, for which holds, that from any set $S$ consisting of $n$ integers, we can find three different (but not necesarilly disjoint) nonempty subsets, such that sum of elements of each of them is divisible by $d$.
|
n-1
|
2/8
|
A square is inscribed in a triangle such that two of its vertices lie on the base, and the other two vertices lie on the sides of the triangle. Prove that the side of the square is less than $2r$ but greater than $\sqrt{2}r$, where $r$ is the radius of the circle inscribed in the triangle.
|
\sqrt{2}r < s < 2r
|
0/8
|
6. Let $F_{1}$ and $F_{2}$ be the left and right foci of the hyperbola $C: \frac{x^{2}}{4}-\frac{y^{2}}{5}=1$, respectively. Point $P$ is on the right branch of the hyperbola $C$, and the excenter of $\triangle P F_{1} F_{2}$ opposite to $\angle P F_{1} F_{2}$ is $I$. The line $P I$ intersects the $x$-axis at point $Q$. Then
$$
\frac{|P Q|}{|P I|}+\frac{\left|F_{1} Q\right|}{\left|F_{1} P\right|}=
$$
$\qquad$
|
4
|
1/8
|
Subsets and Splits
Filtered Answers A-D
Retrieves 100 rows where the answer is a single letter from A to D, providing basic filtering of the dataset.