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Let \( O \) be the circumcenter of triangle \( ABC \), and let us draw the reflections of \( O \) with respect to the sides of the triangle. Show that the resulting triangle \( A_1 B_1 C_1 \) is congruent to triangle \( ABC \). Determine the orthocenter of triangle \( A_1 B_1 C_1 \). | O | 1/8 |
Denote by \( f(n) \) the integer obtained by reversing the digits of a positive integer \( n \). Find the greatest integer that is certain to divide \( n^{4} - f(n)^{4} \) regardless of the choice of \( n \). | 99 | 2/8 |
Determine the number of pairs of integers, \((a, b)\), with \(1 \leq a \leq 100\) so that the line with equation \(b=ax-4y\) passes through point \((r, 0)\), where \(r\) is a real number with \(0 \leq r \leq 3\), and passes through point \((s, 4)\), where \(s\) is a real number with \(2 \leq s \leq 4\). | 6595 | 1/8 |
How many integers $n>1$ are there such that $n$ divides $x^{13}-x$ for every positive integer $x$ ? | 31 | 7/8 |
Two circles of radius $r$ are externally tangent to each other and internally tangent to the ellipse $x^2 + 4y^2 = 5$. Find $r$. | \frac{\sqrt{15}}{4} | 7/8 |
Given that the sequence $\{a\_n\}$ is an arithmetic sequence with all non-zero terms, $S\_n$ denotes the sum of its first $n$ terms, and satisfies $a\_n^2 = S\_{2n-1}$ for all positive integers $n$. If the inequality $\frac{λ}{a\_{n+1}} \leqslant \frac{n + 8 \cdot (-1)^n}{2n}$ holds true for any positive integer $n$, determine the maximum value of the real number $λ$. | -\frac{21}{2} | 6/8 |
Given the medians $A A_{1}, B B_{1}$, and $C C_{1}$ of triangle $A B C$, triangle $K M N$ is formed. From the medians $K K_{1}, M M_{1}$, and $N N_{1}$ of triangle $K M N$, triangle $P Q R$ is formed. Prove that the third triangle is similar to the first one and find the similarity ratio. | \frac{3}{4} | 5/8 |
In a convex quadrilateral \(ABCD\) with internal angles \(<180^\circ\), point \(E\) is the intersection of the diagonals. Let \(F_1\) and \(F_2\) be the areas of triangles \(\triangle ABE\) and \(\triangle CDE\) respectively, and let \(F\) be the area of quadrilateral \(ABCD\). Prove that \(\sqrt{F_1} + \sqrt{F_2} \leq \sqrt{F}\). In what case is equality possible? (6 points) | \sqrt{F_1}+\sqrt{F_2}\le\sqrt{F} | 7/8 |
Find a nonzero monic polynomial $P(x)$ with integer coefficients and minimal degree such that $P(1-\sqrt[3]{2}+\sqrt[3]{4})=0$. (A polynomial is called monic if its leading coefficient is 1.) | x^{3}-3x^{2}+9x-9 | 7/8 |
A positive integer is said to be "nefelibata" if, upon taking its last digit and placing it as the first digit, keeping the order of all the remaining digits intact (for example, 312 -> 231), the resulting number is exactly double the original number. Find the smallest possible nefelibata number. | 105263157894736842 | 3/8 |
There are 50,000 employees at an enterprise. For each employee, the sum of the number of their direct supervisors and their direct subordinates is 7. On Monday, each employee issues an order and gives a copy of this order to each of their direct subordinates (if any). Then, every day, each employee takes all orders received the previous day and either distributes copies of them to all their direct subordinates or, if they do not have any subordinates, executes the orders themselves. It turns out that by Friday, no orders are being distributed within the institution. Prove that there are at least 97 supervisors who do not have any higher supervisors. | 97 | 1/8 |
Circle $C$ has its center at $C(5, 5)$ and has a radius of 3 units. Circle $D$ has its center at $D(14, 5)$ and has a radius of 3 units. What is the area of the gray region bound by the circles and the $x$-axis?
```asy
import olympiad; size(150); defaultpen(linewidth(0.8));
xaxis(0,18,Ticks("%",1.0));
yaxis(0,9,Ticks("%",1.0));
fill((5,5)--(14,5)--(14,0)--(5,0)--cycle,gray(0.7));
filldraw(circle((5,5),3),fillpen=white);
filldraw(circle((14,5),3),fillpen=white);
dot("$C$",(5,5),S); dot("$D$",(14,5),S);
``` | 45 - \frac{9\pi}{2} | 1/8 |
A repunit is a positive integer, all of whose digits are 1s. Let $a_{1}<a_{2}<a_{3}<\ldots$ be a list of all the positive integers that can be expressed as the sum of distinct repunits. Compute $a_{111}$. | 1223456 | 2/8 |
Given that $b$ is an odd multiple of 9, find the greatest common divisor of $8b^2 + 81b + 289$ and $4b + 17$. | 17 | 1/8 |
On graph paper, a square measuring $11 \times 11$ cells is drawn. It is required to mark the centers of some cells in such a way that the center of any other cell lies on the segment connecting any two marked points that are on the same vertical or horizontal line. What is the minimum number of cells that need to be marked? | 22 | 1/8 |
Find a polynomial $ p\left(x\right)$ with real coefficients such that
$ \left(x\plus{}10\right)p\left(2x\right)\equal{}\left(8x\minus{}32\right)p\left(x\plus{}6\right)$
for all real $ x$ and $ p\left(1\right)\equal{}210$. | 2(x + 4)(x - 4)(x - 8) | 1/8 |
The $n^{th}$ derivative of a function $f(x)$ (if it exists) is denoted by $f^{(n)}(x) $ . Let $f(x)=\frac{e^x}{x}$ .
Suppose $f$ is differentiable infinitely many times in $(0,\infty) $ . Then find $\lim_{n \to \infty}\frac{f^{(2n)}1}{(2n)!}$ | 1 | 7/8 |
Given the function $$f(x)=\cos 2x + 2\sqrt{3}\sin x\cos x$$
(1) Find the range of the function $f(x)$ and write down the intervals where $f(x)$ is monotonically increasing;
(2) If $$0 < \theta < \frac{\pi}{6}$$ and $$f(\theta) = \frac{4}{3}$$, calculate the value of $\cos 2\theta$. | \frac{\sqrt{15} + 2}{6} | 1/8 |
On a line, 2022 points are marked such that every two neighboring points are at an equal distance from each other. Half of the points are painted red, and the other half are painted blue. Can the sum of the lengths of all possible segments where the left end is red and the right end is blue be equal to the sum of the lengths of all segments where the left end is blue and the right end is red? (The endpoints of the segments under consideration do not have to be neighboring marked points.) | No | 2/8 |
In the right-angled triangle $ABC$, the midpoint of the leg $CB$ is connected to the incenter of the triangle. This line intersects the leg $CA$ at point $D$. Express the ratio $AD:DC$ as a function of the angle at vertex $A$. | \tan(\frac{\alpha}{2}) | 4/8 |
Integers $x$ and $y$ with $x>y>0$ satisfy $x+y+xy=104$. What is $x$? | 34 | 5/8 |
Janne buys a camera which costs $200.00 without tax. If she pays 15% tax on this purchase, how much tax does she pay?
(A) $30.00
(B) $18.00
(C) $20.00
(D) $15.00
(E) $45.00 | 30 | 1/8 |
Let $S=\{1,2, \ldots 2016\}$, and let $f$ be a randomly chosen bijection from $S$ to itself. Let $n$ be the smallest positive integer such that $f^{(n)}(1)=1$, where $f^{(i)}(x)=f\left(f^{(i-1)}(x)\right)$. What is the expected value of $n$? | \frac{2017}{2} | 6/8 |
Define a monic irreducible polynomial with integral coefficients to be a polynomial with leading coefficient 1 that cannot be factored, and the prime factorization of a polynomial with leading coefficient 1 as the factorization into monic irreducible polynomials. How many not necessarily distinct monic irreducible polynomials are there in the prime factorization of $\left(x^{8}+x^{4}+1\right)\left(x^{8}+x+1\right)$ (for instance, $(x+1)^{2}$ has two prime factors)? | 5 | 1/8 |
Compute the value of the infinite series
$$
\sum_{n=2}^{\infty} \frac{n^{4}+3 n^{2}+10 n+10}{2^{n} \cdot \left(n^{4}+4\right)}
$$ | \frac{11}{10} | 4/8 |
Given an ellipse $M$ with its axes of symmetry being the coordinate axes, and its eccentricity is $\frac{\sqrt{2}}{2}$, and one of its foci is at $(\sqrt{2}, 0)$.
$(1)$ Find the equation of the ellipse $M$;
$(2)$ Suppose a line $l$ intersects the ellipse $M$ at points $A$ and $B$, and a parallelogram $OAPB$ is formed with $OA$ and $OB$ as adjacent sides, where point $P$ is on the ellipse $M$ and $O$ is the origin. Find the minimum distance from point $O$ to line $l$. | \frac{\sqrt{2}}{2} | 7/8 |
Consider the polynomials $P(x) = x^{6} - x^{5} - x^{3} - x^{2} - x$ and $Q(x) = x^{4} - x^{3} - x^{2} - 1.$ Given that $z_{1},z_{2},z_{3},$ and $z_{4}$ are the roots of $Q(x) = 0,$ find $P(z_{1}) + P(z_{2}) + P(z_{3}) + P(z_{4}).$ | 6 | 7/8 |
Find the smallest positive number $\lambda$ such that for any triangle with side lengths $a, b, c$, given $a \geqslant \frac{b+c}{3}$, it holds that
$$
a c + b c - c^{2} \leqslant \lambda\left(a^{2} + b^{2} + 3 c^{2} + 2 a b - 4 b c\right).
$$ | \frac{2\sqrt{2} + 1}{7} | 1/8 |
Given the function $f(x)=2\sin (\\omega x)$, where $\\omega > 0$.
(1) When $ \\omega =1$, determine the even-odd property of the function $F(x)=f(x)+f(x+\\dfrac{\\mathrm{ }\\!\\!\\pi\\!\\!{ }}{2})$ and explain the reason.
(2) When $ \\omega =2$, the graph of the function $y=f(x)$ is translated to the left by $ \\dfrac{\\mathrm{ }\\!\\!\\pi\\!\\!{ }}{6}$ unit, and then translated upward by 1 unit to obtain the graph of the function $y=g(x)$. Find all possible values of the number of zeros of $y=g(x)$ in the interval $[a,a+10π]$ for any $a∈R$. | 20 | 3/8 |
Given vectors $\overrightarrow{a}=(1, \sqrt {3})$ and $\overrightarrow{b}=(-2,2 \sqrt {3})$, calculate the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$. | \dfrac {\pi}{3} | 1/8 |
Let \( a_{1}, a_{2}, \ldots \) be an arithmetic sequence and \( b_{1}, b_{2}, \ldots \) be a geometric sequence. Suppose that \( a_{1} b_{1}=20 \), \( a_{2} b_{2}=19 \), and \( a_{3} b_{3}=14 \). Find the greatest possible value of \( a_{4} b_{4} \). | \frac{37}{4} | 7/8 |
Calculate the lengths of the arcs of the curves given by the equations in polar coordinates.
$$
\rho = 8(1 - \cos \varphi), \quad -\frac{2 \pi}{3} \leq \varphi \leq 0
$$ | 16 | 6/8 |
There is a table of numbers with 20 rows and 15 columns. Let \( A_{1}, \ldots, A_{20} \) be the sums of the numbers in each row, and \( B_{1}, \ldots, B_{15} \) be the sums of the numbers in each column.
a) Is it possible that \( A_{1}=\cdots=A_{20}=B_{1}=\cdots=B_{15} \)?
b) If the equalities in part (a) hold, what is the value of the sum \( A_{1} + \cdots + A_{20} + B_{1} + \cdots + B_{15} \)? | 0 | 2/8 |
Calculate $(-2)^{23} + 2^{(2^4+5^2-7^2)}$. | -8388607.99609375 | 1/8 |
Let $ABCD$ be an inscribed trapezoid such that the sides $[AB]$ and $[CD]$ are parallel. If $m(\widehat{AOD})=60^\circ$ and the altitude of the trapezoid is $10$ , what is the area of the trapezoid? | 100\sqrt{3} | 2/8 |
For what single digit $n$ does 91 divide the 9-digit number $12345 n 789$? | 7 | 7/8 |
Given the hyperbola $C\_1$: $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 (a > b > 0)$ with left and right foci $F\_1$ and $F\_2$, respectively, and hyperbola $C\_2$: $\frac{x^2}{16} - \frac{y^2}{4} = 1$, determine the length of the major axis of hyperbola $C\_1$ given that point $M$ lies on one of the asymptotes of hyperbola $C\_1$, $OM \perp MF\_2$, and the area of $\triangle OMF\_2$ is $16$. | 16 | 1/8 |
Two congruent squares, $ABCD$ and $JKLM$, each have side lengths of 12 units. Square $JKLM$ is placed such that its center coincides with vertex $C$ of square $ABCD$. Determine the area of the region covered by these two squares in the plane. | 216 | 2/8 |
Each person in their heart silently remembers two non-zero numbers. Calculate the square of the sum of these two numbers, and record the result as "Sum". Calculate the square of the difference of these two numbers, and record the result as "Diff". Then calculate the product of these two numbers, and record the result as "Prod." Use "Sum", "Diff", and "Prod" to compute the following expression:
\[
\left(\frac{\text{Sum} - \text{Diff}}{\text{Prod}}\right)^{2} = ?
\] | 16 | 6/8 |
Yesterday Han drove 1 hour longer than Ian at an average speed 5 miles per hour faster than Ian. Jan drove 2 hours longer than Ian at an average speed 10 miles per hour faster than Ian. Han drove 70 miles more than Ian. How many more miles did Jan drive than Ian?
$\mathrm{(A)}\ 120\qquad\mathrm{(B)}\ 130\qquad\mathrm{(C)}\ 140\qquad\mathrm{(D)}\ 150\qquad\mathrm{(E)}\ 160$ | \mathrm{(D)} | 1/8 |
In the quadrilateral \(ABCD\), \(AB = 1\), \(BC = 2\), \(CD = \sqrt{3}\), \(\angle ABC = 120^\circ\), and \(\angle BCD = 90^\circ\). What is the exact length of side \(AD\)? | \sqrt{7} | 7/8 |
Let \( p(x) \) be a polynomial of degree \( 3n \) such that \( p(0)=p(3)=\cdots=p(3n)=2 \), \( p(1)=p(4)=\cdots=p(3n-2)=1 \), and \( p(2)=p(5)=\cdots=p(3n-1)=0 \). Given that \( p(3n+1)=730 \), find \( n \). | 4 | 1/8 |
Consider polynomial functions $ax^2 -bx +c$ with integer coefficients which have two distinct zeros in the open interval $(0,1).$ Exhibit with proof the least positive integer value of $a$ for which such a polynomial exists. | 5 | 5/8 |
Let \( M = \{1, 2, \cdots, 65\} \) and \( A \subseteq M \) be a subset. If \( |A| = 33 \) and there exist \( x, y \in A \) with \( x < y \) such that \( x \mid y \), then \( A \) is called a "good set". Find the largest \( a \in M \) such that every 33-element subset containing \( a \) is a good set. | 21 | 1/8 |
The ratio of the volume of a cone to the volume of a sphere inscribed in it is $k$. Find the angle between the slant height and the base of the cone and the permissible values of $k$. | k\ge2 | 2/8 |
Compute the number of tuples \(\left(a_{0}, a_{1}, a_{2}, a_{3}, a_{4}, a_{5}\right)\) of (not necessarily positive) integers such that \(a_{i} \leq i\) for all \(0 \leq i \leq 5\) and
\[ a_{0} + a_{1} + \cdots + a_{5} = 6. \] | 2002 | 5/8 |
Using equal-length toothpicks to form a rectangular diagram as shown, if the length of the rectangle is 20 toothpicks long and the width is 10 toothpicks long, how many toothpicks are used? | 430 | 4/8 |
An isosceles trapezoid \( ABCD \) with bases \( BC \) and \( AD \) is such that \( \angle ADC = 2 \angle CAD = 82^\circ \). Inside the trapezoid, a point \( T \) is chosen such that \( CT = CD \) and \( AT = TD \). Find \( \angle TCD \). Give the answer in degrees. | 38 | 1/8 |
On bookshelf A, there are 4 English books and 2 Chinese books, while on bookshelf B, there are 2 English books and 3 Chinese books.
$(Ⅰ)$ Without replacement, 2 books are taken from bookshelf A, one at a time. Find the probability of getting an English book on the first draw and still getting an English book on the second draw.
$(Ⅱ)$ First, 2 books are randomly taken from bookshelf B and placed on bookshelf A. Then, 2 books are randomly taken from bookshelf A. Find the probability of getting 2 English books from bookshelf A. | \frac{93}{280} | 7/8 |
Given $$\alpha \in \left( \frac{5}{4}\pi, \frac{3}{2}\pi \right)$$ and it satisfies $$\tan\alpha + \frac{1}{\tan\alpha} = 8$$, then $\sin\alpha\cos\alpha = \_\_\_\_\_\_$; $\sin\alpha - \cos\alpha = \_\_\_\_\_\_$. | -\frac{\sqrt{3}}{2} | 7/8 |
Sasha has $10$ cards with numbers $1, 2, 4, 8,\ldots, 512$ . He writes the number $0$ on the board and invites Dima to play a game. Dima tells the integer $0 < p < 10, p$ can vary from round to round. Sasha chooses $p$ cards before which he puts a “ $+$ ” sign, and before the other cards he puts a “ $-$ " sign. The obtained number is calculated and added to the number on the board. Find the greatest absolute value of the number on the board Dima can get on the board after several rounds regardless Sasha’s moves. | 1023 | 1/8 |
There are $N$ points marked on a plane. Any three of them form a triangle whose angles are expressible in degrees as natural numbers. What is the maximum possible $N$ for which this is possible? | 180 | 7/8 |
A student, Ellie, was supposed to calculate $x-y-z$, but due to a misunderstanding, she computed $x-(y+z)$ and obtained 18. The actual answer should have been 6. What is the value of $x-y$? | 12 | 7/8 |
Define a "digitized number" as a ten-digit number $a_0a_1\ldots a_9$ such that for $k=0,1,\ldots, 9$ , $a_k$ is equal to the number of times the digit $k$ occurs in the number. Find the sum of all digitized numbers. | 6210001000 | 1/8 |
What is the largest four-digit negative integer congruent to $2 \pmod{17}$? | -1001 | 6/8 |
Let \( P \) be a polynomial with integer coefficients such that \( P(0)+P(90)=2018 \). Find the least possible value for \( |P(20)+P(70)| \). | 782 | 1/8 |
A cake has a shape of triangle with sides $19,20$ and $21$ . It is allowed to cut it it with a line into two pieces and put them on a round plate such that pieces don't overlap each other and don't stick out of the plate. What is the minimal diameter of the plate? | 21 | 3/8 |
Given the function $f(x)=2\sin ωx (ω > 0)$, if the minimum value of $f(x)$ in the interval $\left[ -\frac{\pi}{3}, \frac{\pi}{4} \right]$ is $-2$, find the minimum value of $ω$. | \frac{3}{2} | 7/8 |
In an arithmetic sequence $\{a + n b \mid n = 1, 2, \cdots\}$ that contains an infinite geometric sequence, find the necessary and sufficient conditions for the real numbers $a$ and $b$ (where $a, b \neq 0$). | \frac{}{b}\in\mathbb{Q} | 1/8 |
Three numbers, whose sum is 114, are, on one hand, three consecutive terms of a geometric progression, and on the other hand, the first, fourth, and twenty-fifth terms of an arithmetic progression, respectively. Find these numbers. | 2,14,98 | 1/8 |
A hotel has three types of rooms available: a triple room, a double room, and a single room (each type of room can only accommodate the corresponding number of people). There are 4 adult men and 2 little boys looking for accommodation. The little boys should not stay in a room by themselves (they must be accompanied by an adult). If all three rooms are occupied, the number of different arrangements for accommodation is $\boxed{36}$. | 36 | 5/8 |
You are given a positive integer $k$ and not necessarily distinct positive integers $a_1, a_2 , a_3 , \ldots,
a_k$ . It turned out that for any coloring of all positive integers from $1$ to $2021$ in one of the $k$ colors so that there are exactly $a_1$ numbers of the first color, $a_2$ numbers of the second color, $\ldots$ , and $a_k$ numbers of the $k$ -th color, there is always a number $x \in \{1, 2, \ldots, 2021\}$ , such that the total number of numbers colored in the same color as $x$ is exactly $x$ . What are the possible values of $k$ ?
*Proposed by Arsenii Nikolaiev* | 2021 | 1/8 |
Given that Kira needs to store 25 files onto disks, each with 2.0 MB of space, where 5 files take up 0.6 MB each, 10 files take up 1.0 MB each, and the rest take up 0.3 MB each, determine the minimum number of disks needed to store all 25 files. | 10 | 1/8 |
In the plane $Oxy$, the coordinates of point $A$ are defined by the equation $5a^{2} + 12ax + 4ay + 8x^{2} + 8xy + 4y^{2} = 0$, and a parabola with vertex at point $B$ is defined by the equation $ax^{2} - 2a^{2}x - ay + a^{3} + 4 = 0$. Find all values of the parameter $a$ for which points $A$ and $B$ lie on opposite sides of the line $y - 3x = 4$ (points $A$ and $B$ do not lie on this line). | (-\infty,-2)\cup(0,\frac{2}{3})\cup(\frac{8}{7},+\infty) | 1/8 |
Let \( a_{1}, \ldots, a_{n} \in \{-1, 1\} \) such that \( a_{1} a_{2} + a_{2} a_{3} + \cdots + a_{n-1} a_{n} + a_{n} a_{1} = 0 \). Prove that \( 4 \mid n \). | 4\midn | 1/8 |
Find the $2019$ th strictly positive integer $n$ such that $\binom{2n}{n}$ is not divisible by $5$ . | 37805 | 7/8 |
A strip of width 1 is being tightly arranged with non-overlapping rectangular plates of width 1 and lengths \( a_1, a_2, a_3, \ldots \) (where \( a_1 \neq 1 \)). Starting from the second rectangular plate, each plate is similar but not congruent to the rectangle formed by the previously placed plates. After placing the first \( n \) rectangular plates, let \( S_n \) denote the length of the strip covered. Does there exist a real number that \( S_n \) does not exceed? Prove your conclusion. | Nosuchrealexists | 1/8 |
Let $P(x) = x^3 + ax^2 + bx + 1$ be a polynomial with real coefficients and three real roots $\rho_1$ , $\rho_2$ , $\rho_3$ such that $|\rho_1| < |\rho_2| < |\rho_3|$ . Let $A$ be the point where the graph of $P(x)$ intersects $yy'$ and the point $B(\rho_1, 0)$ , $C(\rho_2, 0)$ , $D(\rho_3, 0)$ . If the circumcircle of $\vartriangle ABD$ intersects $yy'$ for a second time at $E$ , find the minimum value of the length of the segment $EC$ and the polynomials for which this is attained.
*Brazitikos Silouanos, Greece* | \sqrt{2} | 4/8 |
Let \( M \) be a set of \( n \) points on a plane, satisfying the following conditions:
1. There are 7 points in \( M \) that are the vertices of a convex heptagon.
2. For any 5 points in \( M \), if these 5 points are the vertices of a convex pentagon, then the interior of this convex pentagon contains at least one point from \( M \).
Find the minimum value of \( n \). | 11 | 2/8 |
Given $\tan (\alpha-\beta)= \frac {1}{2}$, $\tan \beta=- \frac {1}{7}$, and $\alpha$, $\beta\in(0,\pi)$, find the value of $2\alpha-\beta$. | - \frac {3\pi}{4} | 5/8 |
Positive numbers $x, y, z$ satisfy $x^2+y^2+z^2+xy+yz+zy \le 1$ .
Prove that $\big( \frac{1}{x}-1\big) \big( \frac{1}{y}-1\big)\big( \frac{1}{z}-1\big) \ge 9 \sqrt6 -19$ . | 9\sqrt{6}-19 | 6/8 |
Let \( n \) be a positive integer greater than 3, such that \((n, 3) = 1\). Find the value of \(\prod_{k=1}^{m}\left(1+2 \cos \frac{2 a_{k} \pi}{n}\right)\), where \(a_{1}, a_{2}, \cdots, a_{m}\) are all positive integers less than or equal to \( n \) that are relatively prime to \( n \). | 1 | 5/8 |
It is currently $3\!:\!00\!:\!00 \text{ p.m.}$ What time will it be in $6666$ seconds? (Enter the time in the format "HH:MM:SS", without including "am" or "pm".) | 4\!:\!51\!:\!06 \text{ p.m.} | 4/8 |
Let $n$ be a positive integer. Find all $n \times n$ real matrices $A$ with only real eigenvalues satisfying $$A+A^{k}=A^{T}$$ for some integer $k \geq n$. | A = 0 | 7/8 |
In how many ways can the sequence $1,2,3,4,5$ be rearranged so that no three consecutive terms are increasing and no three consecutive terms are decreasing?
$\textbf{(A)} ~10\qquad\textbf{(B)} ~18\qquad\textbf{(C)} ~24 \qquad\textbf{(D)} ~32 \qquad\textbf{(E)} ~44$ | \textbf{(D)}~32 | 1/8 |
The median \(AD\) of an acute-angled triangle \(ABC\) is 5. The orthogonal projections of this median onto the sides \(AB\) and \(AC\) are 4 and \(2\sqrt{5}\), respectively. Find the side \(BC\). | 2 \sqrt{10} | 7/8 |
Altitudes $\overline{AP}$ and $\overline{BQ}$ of an acute triangle $\triangle ABC$ intersect at point $H$. If $HP=5$ while $HQ=2$, then calculate $(BP)(PC)-(AQ)(QC)$. [asy]
size(150); defaultpen(linewidth(0.8));
pair B = (0,0), C = (3,0), A = (2,2), P = foot(A,B,C), Q = foot(B,A,C),H = intersectionpoint(B--Q,A--P);
draw(A--B--C--cycle);
draw(A--P^^B--Q);
label("$A$",A,N); label("$B$",B,W); label("$C$",C,E); label("$P$",P,S); label("$Q$",Q,E); label("$H$",H,NW);
[/asy] | 21 | 1/8 |
Given a bicycle's front tire lasts for 5000km and the rear tire lasts for 3000km, determine the maximum distance the bicycle can travel if the tires are swapped reasonably during use. | 3750 | 7/8 |
There are 300 non-zero integers written in a circle such that each number is greater than the product of the three numbers that follow it in the clockwise direction. What is the maximum number of positive numbers that can be among these 300 written numbers? | 200 | 1/8 |
Let $R$ be a set of nine distinct integers. Six of the elements are $2$, $3$, $4$, $6$, $9$, and $14$. What is the number of possible values of the median of $R$? | 7 | 5/8 |
When the two-digit integer \( XX \), with equal digits, is multiplied by the one-digit integer \( X \), the result is the three-digit integer \( PXQ \). What is the greatest possible value of \( PXQ \) if \( PXQ \) must start with \( P \) and end with \( X \)? | 396 | 7/8 |
How many three-digit whole numbers have at least one 7 or at least one 9 as digits? | 452 | 6/8 |
A bitstring of length $\ell$ is a sequence of $\ell$ $0$ 's or $1$ 's in a row. How many bitstrings of length $2014$ have at least $2012$ consecutive $0$ 's or $1$ 's? | 16 | 1/8 |
An isosceles right triangle with side lengths in the ratio 1:1:\(\sqrt{2}\) is inscribed in a circle with a radius of \(\sqrt{2}\). What is the area of the triangle and the circumference of the circle? | 2\pi\sqrt{2} | 1/8 |
Rotate a square with a side length of 1 around a line that contains one of its sides. The lateral surface area of the resulting solid is \_\_\_\_\_\_. | 2\pi | 7/8 |
Calculate the car's average miles-per-gallon for the entire trip given that the odometer readings are $34,500, 34,800, 35,250$, and the gas tank was filled with $8, 10, 15$ gallons of gasoline. | 22.7 | 1/8 |
Let's call any natural number "very prime" if any number of consecutive digits (in particular, a digit or number itself) is a prime number. For example, $23$ and $37$ are "very prime" numbers, but $237$ and $357$ are not. Find the largest "prime" number (with justification!). | 373 | 4/8 |
If $\{a_1,a_2,a_3,\ldots,a_n\}$ is a set of real numbers, indexed so that $a_1 < a_2 < a_3 < \cdots < a_n,$ its complex power sum is defined to be $a_1i + a_2i^2+ a_3i^3 + \cdots + a_ni^n,$ where $i^2 = - 1.$ Let $S_n$ be the sum of the complex power sums of all nonempty subsets of $\{1,2,\ldots,n\}.$ Given that $S_8 = - 176 - 64i$ and $S_9 = p + qi,$ where $p$ and $q$ are integers, find $|p| + |q|.$ | 368 | 4/8 |
In a certain land, all Arogs are Brafs, all Crups are Brafs, all Dramps are Arogs, and all Crups are Dramps. Which of the following statements is implied by these facts?
$\textbf{(A) } \text{All Dramps are Brafs and are Crups.}\\ \textbf{(B) } \text{All Brafs are Crups and are Dramps.}\\ \textbf{(C) } \text{All Arogs are Crups and are Dramps.}\\ \textbf{(D) } \text{All Crups are Arogs and are Brafs.}\\ \textbf{(E) } \text{All Arogs are Dramps and some Arogs may not be Crups.}$ | \textbf{(D)} | 1/8 |
Let \(ABCDEFGH\) be a rectangular cuboid. How many acute-angled triangles are formed by joining any three vertices of the cuboid? | 8 | 1/8 |
Let $p$ be an odd prime of the form $p=4n+1$ . [list=a][*] Show that $n$ is a quadratic residue $\pmod{p}$ . [*] Calculate the value $n^{n}$ $\pmod{p}$ . [/list] | 1 | 6/8 |
A company gathered at a meeting. Let's call a person sociable if, in this company, they have at least 20 acquaintances, with at least two of those acquaintances knowing each other. Let's call a person shy if, in this company, they have at least 20 non-acquaintances, with at least two of those non-acquaintances not knowing each other. It turned out that in the gathered company, there are neither sociable nor shy people. What is the maximum number of people that can be in this company? | 40 | 1/8 |
In triangle \(ABC\), it is given that \(|AB| = |BC|\) and \(\angle ABC = 20^\circ\). Point \(M\) is taken on \(AB\) such that \(\angle MCA = 60^\circ\), and point \(N\) is taken on \(CB\) such that \(\angle NAC = 50^\circ\). Find \(\angle NMA\). | 30 | 1/8 |
The inclination angle $\alpha$ of the line $l: \sqrt{3}x+3y+1=0$ is $\tan^{-1}\left( -\frac{\sqrt{3}}{3} \right)$. Calculate the value of the angle $\alpha$. | \frac{5\pi}{6} | 6/8 |
A point inside a triangle is connected to the vertices by three line segments. What is the maximum number of these line segments that can be equal to the opposite side? | 1 | 1/8 |
How many different rectangles with sides parallel to the grid can be formed by connecting four of the dots in a $5\times 5$ square array of dots? | 100 | 7/8 |
A positive integer $n$ is loose if it has six positive divisors and satisfies the property that any two positive divisors $a<b$ of $n$ satisfy $b \geq 2 a$. Compute the sum of all loose positive integers less than 100. | 512 | 2/8 |
Given that \( a \) and \( d \) are non-negative numbers, \( b \) and \( c \) are positive numbers, and \( b+c \geq a+d \), find the minimum value of the following expression:
\[
\frac{b}{c+d} + \frac{c}{a+b}.
\] | \sqrt{2}-\frac{1}{2} | 1/8 |
Find the differentials of the functions:
1) \( y = x^3 - 3^x \);
2) \( F(\varphi) = \cos \frac{\varphi}{3} + \sin \frac{3}{\varphi} \);
3) \( z = \ln \left(1 + e^{10x}\right) + \operatorname{arcctg} e^{5x} \); calculate \(\left.dz\right|_{x=0; dx=0.1}\). | 0.25 | 7/8 |
How many positive integers \(N\) possess the property that exactly one of the numbers \(N\) and \((N+20)\) is a 4-digit number? | 40 | 7/8 |
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