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3,500 |
Higher Secondary P4
If the fraction $\dfrac{a}{b}$ is greater than $\dfrac{31}{17}$ in the least amount while $b<17$ , find $\dfrac{a}{b}$ .
|
\frac{11}{6}
| 1.5625 |
3,501 |
What is the biggest shadow that a cube of side length $1$ can have, with the sun at its peak?
Note: "The biggest shadow of a figure with the sun at its peak" is understood to be the biggest possible area of the orthogonal projection of the figure on a plane.
|
\sqrt{3}
| 0 |
3,502 |
Numbers $1,\frac12,\frac13,\ldots,\frac1{2001}$ are written on a blackboard. A student erases two numbers $x,y$ and writes down the number $x+y+xy$ instead. Determine the number that will be written on the board after $2000$ such operations.
|
2001
| 94.53125 |
3,503 |
$A,B,C$ and $D$ are points on the parabola $y = x^2$ such that $AB$ and $CD$ intersect on the $y$ -axis. Determine the $x$ -coordinate of $D$ in terms of the $x$ -coordinates of $A,B$ and $C$ , which are $a, b$ and $c$ respectively.
|
\frac{ab}{c}
| 95.3125 |
3,504 |
Find all positive integers $n$ for which both $837 + n$ and $837 - n$ are cubes of positive integers.
|
494
| 57.03125 |
3,505 |
Find all natural numbers $n$ such that $n$ , $n^2+10$ , $n^2-2$ , $n^3+6$ , and $n^5+36$ are all prime numbers.
|
n = 7
| 0 |
3,506 |
Let $V$ be the set of all continuous functions $f\colon [0,1]\to \mathbb{R}$ , differentiable on $(0,1)$ , with the property that $f(0)=0$ and $f(1)=1$ . Determine all $\alpha \in \mathbb{R}$ such that for every $f\in V$ , there exists some $\xi \in (0,1)$ such that
\[f(\xi)+\alpha = f'(\xi)\]
|
\frac{1}{e - 1}
| 17.96875 |
3,507 |
Let $r_1$ , $r_2$ , $\ldots$ , $r_{20}$ be the roots of the polynomial $x^{20}-7x^3+1$ . If \[\dfrac{1}{r_1^2+1}+\dfrac{1}{r_2^2+1}+\cdots+\dfrac{1}{r_{20}^2+1}\] can be written in the form $\tfrac mn$ where $m$ and $n$ are positive coprime integers, find $m+n$ .
|
240
| 1.5625 |
3,508 |
Find the largest integer $ n$ satisfying the following conditions:
(i) $ n^2$ can be expressed as the difference of two consecutive cubes;
(ii) $ 2n\plus{}79$ is a perfect square.
|
181
| 92.96875 |
3,509 |
Find the largest positive integer $N$ so that the number of integers in the set $\{1,2,\dots,N\}$ which are divisible by 3 is equal to the number of integers which are divisible by 5 or 7 (or both).
|
65
| 82.03125 |
3,510 |
We color all vertexs of a convex polygon with $10$ vertexs by $2$ colors: red and blue $($ each vertex is colored by $1$ color $).$ How many ways to color all the vertexs such that there are no $2$ adjacent vertex that are both colored red?
|
123
| 0 |
3,511 |
Find the smallest positive real $k$ satisfying the following condition: for any given four DIFFERENT real numbers $a,b,c,d$ , which are not less than $k$ , there exists a permutation $(p,q,r,s)$ of $(a,b,c,d)$ , such that the equation $(x^{2}+px+q)(x^{2}+rx+s)=0$ has four different real roots.
|
4
| 48.4375 |
3,512 |
Determine the least possible value of the natural number $n$ such that $n!$ ends in exactly $1987$ zeros.
<details><summary>Note</summary>Note. Here (and generally in MathLinks) natural numbers supposed to be positive.</details>
|
7960
| 98.4375 |
3,513 |
Square $CASH$ and regular pentagon $MONEY$ are both inscribed in a circle. Given that they do not share a vertex, how many intersections do these two polygons have?
|
8
| 64.0625 |
3,514 |
Suppose that $a_1 = 1$ , and that for all $n \ge 2$ , $a_n = a_{n-1} + 2a_{n-2} + 3a_{n-3} + \ldots + (n-1)a_1.$ Suppose furthermore that $b_n = a_1 + a_2 + \ldots + a_n$ for all $n$ . If $b_1 + b_2 + b_3 + \ldots + b_{2021} = a_k$ for some $k$ , find $k$ .
*Proposed by Andrew Wu*
|
2022
| 72.65625 |
3,515 |
Let $m$ be a positive integer less than $2015$ . Suppose that the remainder when $2015$ is divided by $m$ is $n$ . Compute the largest possible value of $n$ .
*Proposed by Michael Ren*
|
1007
| 84.375 |
3,516 |
Determine all functions $f:\mathbb R\to\mathbb R$ such that equality $$ f(x + y + yf(x)) = f(x) + f(y) + xf(y) $$ holds for all real numbers $x$ , $y$ .
Proposed by Athanasios Kontogeorgis
|
f(x) \equiv x
| 0 |
3,517 |
Ryan is messing with Brice’s coin. He weights the coin such that it comes up on one side twice as frequently as the other, and he chooses whether to weight heads or tails more with equal probability. Brice flips his modified coin twice and it lands up heads both times. The probability that the coin lands up heads on the next flip can be expressed in the form $\tfrac{p}{q}$ for positive integers $p, q$ satisfying $\gcd(p, q) = 1$ , what is $p + q$ ?
|
8
| 89.84375 |
3,518 |
Today there are $2^n$ species on the planet Kerbin, all of which are exactly n steps from an original species. In an evolutionary step, One species split into exactly two new species and died out in the process. There were already $2^n-1$ species in the past, which are no longer present today can be found, but are only documented by fossils. The famous space pioneer Jebediah Kerman once suggested reducing the biodiversity of a planet by doing this to measure how closely two species are on average related, with also already extinct species should be taken into account. The degree of relationship is measured two types, of course, by how many evolutionary steps before or back you have to do at least one to get from one to the other. What is the biodiversity of the planet Kerbin?
|
2
| 2.34375 |
3,519 |
Determine the sum of absolute values for the complex roots of $ 20 x^8 \plus{} 7i x^7 \minus{}7ix \plus{} 20.$
|
8
| 52.34375 |
3,520 |
A container is shaped like a square-based pyramid where the base has side length $23$ centimeters and the height is $120$ centimeters. The container is open at the base of the pyramid and stands in an open field with its vertex pointing down. One afternoon $5$ centimeters of rain falls in the open field partially filling the previously empty container. Find the depth in centimeters of the rainwater in the bottom of the container after the rain.
|
60
| 48.4375 |
3,521 |
Show that the number $r(n)$ of representations of $n$ as a sum of two squares has $\pi$ as arithmetic mean, that is \[\lim_{n \to \infty}\frac{1}{n}\sum^{n}_{m=1}r(m) = \pi.\]
|
\pi
| 98.4375 |
3,522 |
Suppose the polynomial $f(x) = x^{2014}$ is equal to $f(x) =\sum^{2014}_{k=0} a_k {x \choose k}$ for some real numbers $a_0,... , a_{2014}$ . Find the largest integer $m$ such that $2^m$ divides $a_{2013}$ .
|
2004
| 22.65625 |
3,523 |
We only know that the password of a safe consists of $7$ different digits. The safe will open if we enter $7$ different digits, and one of them matches the corresponding digit of the password. Can we open this safe in less than $7$ attempts?
*(5 points for Juniors and 4 points for Seniors)*
|
6
| 1.5625 |
3,524 |
Hello Everyone,
i'm trying to make a strong marathon for number theory .. which will be in Pre-Olympiad level
Please if you write any problem don't forget to indicate its number and if you write a solution please indicate for what problem also to prevent the confusion that happens in some marathons.
it will be preferred to write the source of the problem.
please , show detailed solutions , and please post some challenging Pre-Olympiad problems..
remember ,, different approaches are most welcome :)
now .. let's get our hands dirty :lol: :
let $ f(n)$ denote the sum of the digits of $ n$ . Now let $ N \equal{} {4444^{4444}}$ .
Find $ f\left( {f\left( {f\left( N \right)} \right)} \right)$ .
|
7
| 99.21875 |
3,525 |
A room is built in the shape of the region between two semicircles with the same center and parallel diameters. The farthest distance between two points with a clear line of sight is $12$ m. What is the area (in $m^2$ ) of the room?
|
18\pi
| 65.625 |
3,526 |
Find all functions $f : \mathbb{N} \to \mathbb{N}$ satisfying the following conditions:
- For every $n \in \mathbb{N}$ , $f^{(n)}(n) = n$ . (Here $f^{(1)} = f$ and $f^{(k)} = f^{(k-1)} \circ f$ .)
- For every $m, n \in \mathbb{N}$ , $\lvert f(mn) - f(m) f(n) \rvert < 2017$ .
|
f(n) = n
| 100 |
3,527 |
Given a square $ABCD,$ with $AB=1$ mark the midpoints $M$ and $N$ of $AB$ and $BC,$ respectively. A lasar beam shot from $M$ to $N,$ and the beam reflects of $BC,CD,DA,$ and comes back to $M.$ This path encloses a smaller area inside square $ABCD.$ Find this area.
|
\frac{1}{2}
| 32.8125 |
3,528 |
Albert has a very large bag of candies and he wants to share all of it with his friends. At first, he splits the candies evenly amongst his $20$ friends and himself and he finds that there are five left over. Ante arrives, and they redistribute the candies evenly again. This time, there are three left over. If the bag contains over $500$ candies, what is the fewest number of candies the bag can contain?
|
509
| 91.40625 |
3,529 |
If $x_{1}, x_{2},\ldots ,x_{n}$ are positive real numbers with $x_{1}^2+x_2^{2}+\ldots +x_{n}^{2}=1$ , find the minimum value of $\sum_{i=1}^{n}\frac{x_{i}^{5}}{x_{1}+x_{2}+\ldots +x_{n}-x_{i}}$ .
|
\frac{1}{n(n-1)}
| 89.84375 |
3,530 |
Determine all positive integers $n$ such that $n$ divides $\phi(n)^{d(n)}+1$ but $d(n)^5$ does not divide $n^{\phi(n)}-1$ .
|
n = 2
| 0 |
3,531 |
Daniel has a (mostly) standard deck of 54 cards, consisting of 4 suits each containing the ranks 1 to 13 as well as 2 jokers. Daniel plays the following game: He shuffles the deck uniformly randomly and then takes all of the cards that end up strictly between the two jokers. He then sums up the ranks of all the cards he has taken and calls that his score.
Let $p$ be the probability that his score is a multiple of 13. There exists relatively prime positive integers $a$ and $b,$ with $b$ as small as possible, such that $|p - a/b| < 10^{-10}.$ What is $a/b?$ *Proposed by Dilhan Salgado, Daniel Li*
|
\frac{77}{689}
| 0 |
3,532 |
Let $I$ be the set of points $(x,y)$ in the Cartesian plane such that $$ x>\left(\frac{y^4}{9}+2015\right)^{1/4} $$ Let $f(r)$ denote the area of the intersection of $I$ and the disk $x^2+y^2\le r^2$ of radius $r>0$ centered at the origin $(0,0)$ . Determine the minimum possible real number $L$ such that $f(r)<Lr^2$ for all $r>0$ .
|
\frac{\pi}{3}
| 0.78125 |
3,533 |
There is number $N$ on the board. Every minute Ivan makes next operation: takes any number $a$ written on the board,
erases it, then writes all divisors of $a$ except $a$ ( Can be same numbers on the board). After some time on the board there are $N^2$ numbers.
For which $N$ is it possible?
|
N = 1
| 0 |
3,534 |
Let $F$ be the family of all sets of positive integers with $2010$ elements that satisfy the following condition:
The difference between any two of its elements is never the same as the difference of any other two of its elements. Let $f$ be a function defined from $F$ to the positive integers such that $f(K)$ is the biggest element of $K \in F$ . Determine the least value of $f(K)$ .
|
4040100
| 2.34375 |
3,535 |
Let $ABC$ be a triangle with $AB=5$ , $BC=6$ , $CA=7$ . Let $D$ be a point on ray $AB$ beyond $B$ such that $BD=7$ , $E$ be a point on ray $BC$ beyond $C$ such that $CE=5$ , and $F$ be a point on ray $CA$ beyond $A$ such that $AF=6$ . Compute the area of the circumcircle of $DEF$ .
*Proposed by James Lin.*
|
\frac{251}{3} \pi
| 0 |
3,536 |
You drop a 7 cm long piece of mechanical pencil lead on the floor. A bully takes the lead and breaks it at a random point into two pieces. A piece of lead is unusable if it is 2 cm or shorter. If the expected value of the number of usable pieces afterwards is $\frac{m}n$ for relatively prime positive integers $m$ and $n$ , compute $100m + n$ .
*Proposed by Aaron Lin*
|
1007
| 59.375 |
3,537 |
Let $n(n\geq2)$ be a natural number and $a_1,a_2,...,a_n$ natural positive real numbers. Determine the least possible value of the expression $$ E_n=\frac{(1+a_1)\cdot(a_1+a_2)\cdot(a_2+a_3)\cdot...\cdot(a_{n-1}+a_n)\cdot(a_n+3^{n+1})} {a_1\cdot a_2\cdot a_3\cdot...\cdot a_n} $$
|
4^{n+1}
| 0.78125 |
3,538 |
Show that we cannot form more than $4096$ binary sequences of length $24$ so that any two differ in at least $8$ positions.
|
4096
| 98.4375 |
3,539 |
Determine all non-constant monic polynomials $f(x)$ with integer coefficients for which there exists a natural number $M$ such that for all $n \geq M$ , $f(n)$ divides $f(2^n) - 2^{f(n)}$
*Proposed by Anant Mudgal*
|
f(x) = x
| 93.75 |
3,540 |
Find all pairs $\left(m,n\right)$ of positive integers, with $m,n\geq2$ , such that $a^n-1$ is divisible by $m$ for each $a\in \left\{1,2,3,\ldots,n\right\}$ .
|
(m, n) = (p, p-1)
| 0 |
3,541 |
Numbers $a, b$ and $c$ are positive integers and $\frac{1}{a}+\frac{1}{b}+\frac{ 1}{c}< 1.$ Show that \[\frac{1}{a}+\frac{1}{b}+\frac{ 1}{c}\leq \frac{41}{42}.\]
|
\frac{41}{42}
| 95.3125 |
3,542 |
For any positive integer, if the number of $2$ 's in its digits is greater than the number of $3$ 's in its digits, we call that is a **good** number. And if the number of $3$ 's in its digits is more than the number of $2$ 's in its digits, we call that is a **bad** number. For example, there are two $2$ 's and one $3$ in the number $2023$ , so $2023$ is a good number. But in the number $123$ , the number of $2$ and $3$ are both one, so $123$ is neither a good number nor a bad number.
Find the difference of numbers of good numbers and bad numbers among the positive integer not greater than $2023$ .
|
22
| 98.4375 |
3,543 |
The plane is partitioned into congruent regular hexagons. Of these hexagons, some $1998$ are marked. Show that one can select $666$ of the marked hexagons in such a way that no two of them share a vertex.
|
666
| 94.53125 |
3,544 |
Triangle $ABC$ has $\angle{A}=90^{\circ}$ , $AB=2$ , and $AC=4$ . Circle $\omega_1$ has center $C$ and radius $CA$ , while circle $\omega_2$ has center $B$ and radius $BA$ . The two circles intersect at $E$ , different from point $A$ . Point $M$ is on $\omega_2$ and in the interior of $ABC$ , such that $BM$ is parallel to $EC$ . Suppose $EM$ intersects $\omega_1$ at point $K$ and $AM$ intersects $\omega_1$ at point $Z$ . What is the area of quadrilateral $ZEBK$ ?
|
20
| 0.78125 |
3,545 |
In square $ABCD$ with side length $2$ , let $M$ be the midpoint of $AB$ . Let $N$ be a point on $AD$ such that $AN = 2ND$ . Let point $P$ be the intersection of segment $MN$ and diagonal $AC$ . Find the area of triangle $BPM$ .
*Proposed by Jacob Xu*
|
\frac{2}{7}
| 58.59375 |
3,546 |
In a rectangle $ABCD$ , two segments $EG$ and $FH$ divide it into four smaller rectangles. $BH$ intersects $EG$ at $X$ , $CX$ intersects $HF$ and $Y$ , $DY$ intersects $EG$ at $Z$ . Given that $AH=4$ , $HD=6$ , $AE=4$ , and $EB=5$ , find the area of quadrilateral $HXYZ$ .
|
8
| 2.34375 |
3,547 |
Suppose $(a,b)$ is an ordered pair of integers such that the three numbers $a$ , $b$ , and $ab$ form an arithmetic progression, in that order. Find the sum of all possible values of $a$ .
*Proposed by Nathan Xiong*
|
8
| 73.4375 |
3,548 |
Show that the solution set of the inequality
\[ \sum^{70}_{k \equal{} 1} \frac {k}{x \minus{} k} \geq \frac {5}{4}
\]
is a union of disjoint intervals, the sum of whose length is 1988.
|
1988
| 78.90625 |
3,549 |
Let $ABC$ be an equilateral triangle with side length $2$ , and let $M$ be the midpoint of $\overline{BC}$ . Points $X$ and $Y$ are placed on $AB$ and $AC$ respectively such that $\triangle XMY$ is an isosceles right triangle with a right angle at $M$ . What is the length of $\overline{XY}$ ?
|
3 - \sqrt{3}
| 3.90625 |
3,550 |
Triangle $ABC$ satisfies $AB=104$ , $BC=112$ , and $CA=120$ . Let $\omega$ and $\omega_A$ denote the incircle and $A$ -excircle of $\triangle ABC$ , respectively. There exists a unique circle $\Omega$ passing through $A$ which is internally tangent to $\omega$ and externally tangent to $\omega_A$ . Compute the radius of $\Omega$ .
|
49
| 0 |
3,551 |
In parallelogram $ABCD,$ let $O$ be the intersection of diagonals $\overline{AC}$ and $\overline{BD}.$ Angles $CAB$ and $DBC$ are each twice as large as angle $DBA,$ and angle $ACB$ is $r$ times as large as angle $AOB.$ Find the greatest integer that does not exceed $1000r.$
|
777
| 0 |
3,552 |
Let $\mathcal{A}$ be the set of finite sequences of positive integers $a_1,a_2,\dots,a_k$ such that $|a_n-a_{n-1}|=a_{n-2}$ for all $3\leqslant n\leqslant k$ . If $a_1=a_2=1$ , and $k=18$ , determine the number of elements of $\mathcal{A}$ .
|
1597
| 3.90625 |
3,553 |
Find the least number of elements of a finite set $A$ such that there exists a function $f : \left\{1,2,3,\ldots \right\}\rightarrow A$ with the property: if $i$ and $j$ are positive integers and $i-j$ is a prime number, then $f(i)$ and $f(j)$ are distinct elements of $A$ .
|
4
| 25 |
3,554 |
Given are sheets and the numbers $00, 01, \ldots, 99$ are written on them. We must put them in boxes $000, 001, \ldots, 999$ so that the number on the sheet is the number on the box with one digit erased. What is the minimum number of boxes we need in order to put all the sheets?
|
34
| 14.84375 |
3,555 |
The numbers $1,2,\ldots,49,50$ are written on the blackboard. Ann performs the following operation: she chooses three arbitrary numbers $a,b,c$ from the board, replaces them by their sum $a+b+c$ and writes $(a+b)(b+c)(c+a)$ to her notebook. Ann performs such operations until only two numbers remain on the board (in total 24 operations). Then she calculates the sum of all $24$ numbers written in the notebook. Let $A$ and $B$ be the maximum and the minimum possible sums that Ann san obtain.
Find the value of $\frac{A}{B}$ .
*(I. Voronovich)*
|
4
| 0 |
3,556 |
Find the smallest natural number nonzero n so that it exists in real numbers $x_1, x_2,..., x_n$ which simultaneously check the conditions:
1) $x_i \in [1/2 , 2]$ , $i = 1, 2,... , n$
2) $x_1+x_2+...+x_n \ge \frac{7n}{6}$ 3) $\frac{1}{x_1}+\frac{1}{x_2}+...+\frac{1}{x_n}\ge \frac{4n}{3}$
|
9
| 20.3125 |
3,557 |
Let the sequence $\{x_n\}$ be defined by $x_1 \in \{5, 7\}$ and, for $k \ge 1, x_{k+1} \in \{5^{x_k} , 7^{x_k} \}$ . For example, the possible values of $x_3$ are $5^{5^5}, 5^{5^7}, 5^{7^5}, 5^{7^7}, 7^{5^5}, 7^{5^7}, 7^{7^5}$ , and $7^{7^7}$ . Determine the sum of all possible values for the last two digits of $x_{2012}$ .
|
75
| 35.15625 |
3,558 |
Find all sequences of integer $x_1,x_2,..,x_n,...$ such that $ij$ divides $x_i+x_j$ for any distinct positive integer $i$ , $j$ .
|
x_i = 0
| 35.9375 |
3,559 |
Juan chooses a five-digit positive integer. Maria erases the ones digit and gets a four-digit number. The sum of this four-digit number and the original five-digit number is $52,713$ . What can the sum of the five digits of the original number be?
|
23
| 67.96875 |
3,560 |
A permutation $(a_1, a_2, a_3, \dots, a_{100})$ of $(1, 2, 3, \dots, 100)$ is chosen at random. Denote by $p$ the probability that $a_{2i} > a_{2i - 1}$ for all $i \in \{1, 2, 3, \dots, 50\}$ . Compute the number of ordered pairs of positive integers $(a, b)$ satisfying $\textstyle\frac{1}{a^b} = p$ .
*Proposed by Aaron Lin*
|
6
| 60.15625 |
3,561 |
Given real numbers $b_0,b_1,\ldots, b_{2019}$ with $b_{2019}\neq 0$ , let $z_1,z_2,\ldots, z_{2019}$ be the roots in the complex plane of the polynomial
\[
P(z) = \sum_{k=0}^{2019}b_kz^k.
\]
Let $\mu = (|z_1|+ \cdots + |z_{2019}|)/2019$ be the average of the distances from $z_1,z_2,\ldots, z_{2019}$ to the origin. Determine the largest constant $M$ such that $\mu\geq M$ for all choices of $b_0,b_1,\ldots, b_{2019}$ that satisfy
\[
1\leq b_0 < b_1 < b_2 < \cdots < b_{2019} \leq 2019.
\]
|
2019^{-2019^{-1}}
| 0 |
3,562 |
Find all sets $X$ consisting of at least two positive integers such that for every two elements $m,n\in X$ , where $n>m$ , there exists an element $k\in X$ such that $n=mk^2$ .
|
X = \{a, a^3\}
| 0 |
3,563 |
Suppose $f:\mathbb{R} \to \mathbb{R}$ is a function given by $$ f(x) =\begin{cases} 1 & \mbox{if} \ x=1 e^{(x^{10}-1)}+(x-1)^2\sin\frac1{x-1} & \mbox{if} \ x\neq 1\end{cases} $$ (a) Find $f'(1)$ (b) Evaluate $\displaystyle \lim_{u\to\infty} \left[100u-u\sum_{k=1}^{100} f\left(1+\frac{k}{u}\right)\right]$ .
|
-50500
| 58.59375 |
3,564 |
Find all positive integers $n$ such that the number $$ n^6 + 5n^3 + 4n + 116 $$ is the product of two or more consecutive numbers.
|
n = 3
| 0 |
3,565 |
Let $S$ be the set of positive integer divisors of $20^9.$ Three numbers are chosen independently and at random from the set $S$ and labeled $a_1,a_2,$ and $a_3$ in the order they are chosen. The probability that both $a_1$ divides $a_2$ and $a_2$ divides $a_3$ is $\frac mn,$ where $m$ and $n$ are relatively prime positive integers. Find $m.$
|
77
| 67.1875 |
3,566 |
You are trying to maximize a function of the form $f(x, y, z) = ax + by + cz$ , where $a$ , $b$ , and $c$ are constants. You know that $f(3, 1, 1) > f(2, 1, 1)$ , $f(2, 2, 3) > f(2, 3, 4)$ , and $f(3, 3, 4) > f(3, 3, 3)$ . For $-5 \le x,y,z \le 5$ , what value of $(x,y,z)$ maximizes the value of $f(x, y, z)$ ? Give your answer as an ordered triple.
|
(5, -5, 5)
| 99.21875 |
3,567 |
Suppose that $x$ and $y$ are real numbers that satisfy the system of equations $2^x-2^y=1$ $4^x-4^y=\frac{5}{3}$ Determine $x-y$
|
2
| 94.53125 |
3,568 |
Let $A$ be the area of the locus of points $z$ in the complex plane that satisfy $|z+12+9i| \leq 15$ . Compute $\lfloor A\rfloor$ .
|
706
| 99.21875 |
3,569 |
Find the largest integer $n$ for which $2^n$ divides \[ \binom 21 \binom 42 \binom 63 \dots \binom {128}{64}. \]*Proposed by Evan Chen*
|
193
| 81.25 |
3,570 |
Find all pairs of natural numbers $(k, m)$ such that for any natural $n{}$ the product\[(n+m)(n+2m)\cdots(n+km)\]is divisible by $k!{}$ .
*Proposed by P. Kozhevnikov*
|
(k, m)
| 35.15625 |
3,571 |
Polynomial $ P(t)$ is such that for all real $ x$ ,
\[ P(\sin x) \plus{} P(\cos x) \equal{} 1.
\]
What can be the degree of this polynomial?
|
0
| 84.375 |
3,572 |
Find the smallest positive constant $c$ satisfying: For any simple graph $G=G(V,E)$ , if $|E|\geq c|V|$ , then $G$ contains $2$ cycles with no common vertex, and one of them contains a chord.
Note: The cycle of graph $G(V,E)$ is a set of distinct vertices ${v_1,v_2...,v_n}\subseteq V$ , $v_iv_{i+1}\in E$ for all $1\leq i\leq n$ $(n\geq 3, v_{n+1}=v_1)$ ; a cycle containing a chord is the cycle ${v_1,v_2...,v_n}$ , such that there exist $i,j, 1< i-j< n-1$ , satisfying $v_iv_j\in E$ .
|
4
| 0 |
3,573 |
How many solutions of the equation $\tan x = \tan \tan x$ are on the interval $0 \le x \le \tan^{-1} 942$ ? (Here $\tan^{-1}$ means the inverse tangent function, sometimes written $\arctan$ .)
|
300
| 19.53125 |
3,574 |
There are $n+1$ containers arranged in a circle. One container has $n$ stones, the others are empty. A move is to choose two containers $A$ and $B$ , take a stone from $A$ and put it in one of the containers adjacent to $B$ , and to take a stone from $B$ and put it in one of the containers adjacent to $A$ . We can take $A = B$ . For which $n$ is it possible by series of moves to end up with one stone in each container except that which originally held $n$ stones.
|
n
| 96.875 |
3,575 |
Find the least positive integer which is a multiple of $13$ and all its digits are the same.
*(Adapted from Gazeta Matematică 1/1982, Florin Nicolăită)*
|
111111
| 97.65625 |
3,576 |
Let $f(x)=\frac{ax+b}{cx+d}$ $F_n(x)=f(f(f...f(x)...))$ (with $n\ f's$ )
Suppose that $f(0) \not =0$ , $f(f(0)) \not = 0$ , and for some $n$ we have $F_n(0)=0$ ,
show that $F_n(x)=x$ (for any valid x).
|
F_n(x) = x
| 92.96875 |
3,577 |
Let $n$ be an integer. We consider $s (n)$ , the sum of the $2001$ powers of $n$ with the exponents $0$ to $2000$ . So $s (n) = \sum_{k=0}^{2000}n ^k$ . What is the unit digit of $s (n)$ in the decimal system?
|
1
| 75.78125 |
3,578 |
Find the sum of all the real values of x satisfying $(x+\frac{1}{x}-17)^2$ $= x + \frac{1}{x} + 17.$
|
35
| 92.1875 |
3,579 |
A Yule log is shaped like a right cylinder with height $10$ and diameter $5$ . Freya cuts it parallel to its bases into $9$ right cylindrical slices. After Freya cut it, the combined surface area of the slices of the Yule log increased by $a\pi$ . Compute $a$ .
|
100
| 69.53125 |
3,580 |
A meeting is held at a round table. It is known that 7 women have a woman on their right side, and 12 women have a man on their right side. It is also known that 75% of the men have a woman on their right side. How many people are sitting at the round table?
|
35
| 45.3125 |
3,581 |
We say an integer $n$ is naoish if $n \geq 90$ and the second-to-last digit of $n$ (in decimal notation) is equal to $9$ . For example, $10798$ , $1999$ and $90$ are naoish, whereas $9900$ , $2009$ and $9$ are not. Nino expresses 2020 as a sum:
\[
2020=n_{1}+n_{2}+\ldots+n_{k}
\]
where each of the $n_{j}$ is naoish.
What is the smallest positive number $k$ for which Nino can do this?
|
8
| 2.34375 |
3,582 |
Let $n$ be a positive integer. We are given a $3n \times 3n$ board whose unit squares are colored in black and white in such way that starting with the top left square, every third diagonal is colored in black and the rest of the board is in white. In one move, one can take a $2 \times 2$ square and change the color of all its squares in such way that white squares become orange, orange ones become black and black ones become white. Find all $n$ for which, using a finite number of moves, we can make all the squares which were initially black white, and all squares which were initially white black.
Proposed by *Boris Stanković and Marko Dimitrić, Bosnia and Herzegovina*
|
n
| 73.4375 |
3,583 |
$P(x)$ is a polynomial with real coefficients such that $P(a_1) = 0, P(a_{i+1}) = a_i$ ( $i = 1, 2,\ldots$ ) where $\{a_i\}_{i=1,2,\ldots}$ is an infinite sequence of distinct natural numbers. Determine the possible values of degree of $P(x)$ .
|
\deg P = 1
| 0 |
3,584 |
Square $ABCD$ is divided into four rectangles by $EF$ and $GH$ . $EF$ is parallel to $AB$ and $GH$ parallel to $BC$ . $\angle BAF = 18^\circ$ . $EF$ and $GH$ meet at point $P$ . The area of rectangle $PFCH$ is twice that of rectangle $AGPE$ . Given that the value of $\angle FAH$ in degrees is $x$ , find the nearest integer to $x$ .
[asy]
size(100); defaultpen(linewidth(0.7)+fontsize(10));
pair D2(pair P) {
dot(P,linewidth(3)); return P;
}
// NOTE: I've tampered with the angles to make the diagram not-to-scale. The correct numbers should be 72 instead of 76, and 45 instead of 55.
pair A=(0,1), B=(0,0), C=(1,0), D=(1,1), F=intersectionpoints(A--A+2*dir(-76),B--C)[0], H=intersectionpoints(A--A+2*dir(-76+55),D--C)[0], E=F+(0,1), G=H-(1,0), P=intersectionpoints(E--F,G--H)[0];
draw(A--B--C--D--cycle);
draw(F--A--H); draw(E--F); draw(G--H);
label(" $A$ ",D2(A),NW);
label(" $B$ ",D2(B),SW);
label(" $C$ ",D2(C),SE);
label(" $D$ ",D2(D),NE);
label(" $E$ ",D2(E),plain.N);
label(" $F$ ",D2(F),S);
label(" $G$ ",D2(G),W);
label(" $H$ ",D2(H),plain.E);
label(" $P$ ",D2(P),SE);
[/asy]
|
45^\circ
| 37.5 |
3,585 |
In equality $$ 1 * 2 * 3 * 4 * 5 * ... * 60 * 61 * 62 = 2023 $$ Instead of each asterisk, you need to put one of the signs “+” (plus), “-” (minus), “•” (multiply) so that the equality becomes true. What is the smallest number of "•" characters that can be used?
|
2
| 4.6875 |
3,586 |
Let $\{\epsilon_n\}^\infty_{n=1}$ be a sequence of positive reals with $\lim\limits_{n\rightarrow+\infty}\epsilon_n = 0$ . Find \[ \lim\limits_{n\rightarrow\infty}\dfrac{1}{n}\sum\limits^{n}_{k=1}\ln\left(\dfrac{k}{n}+\epsilon_n\right) \]
|
-1
| 97.65625 |
3,587 |
Find the functions $f:\mathbb{Z}\times \mathbb{Z}\to\mathbb{R}$ such that
a) $f(x,y)\cdot f(y,z) \cdot f(z,x) = 1$ for all integers $x,y,z$ ;
b) $f(x+1,x)=2$ for all integers $x$ .
|
f(x,y) = 2^{x-y}
| 30.46875 |
3,588 |
Gabriela found an encyclopedia with $2023$ pages, numbered from $1$ to $2023$ . She noticed that the pages formed only by even digits have a blue mark, and that every three pages since page two have a red mark. How many pages of the encyclopedia have both colors?
|
44
| 78.90625 |
3,589 |
Let the medians of the triangle $ABC$ meet at $G$ . Let $D$ and $E$ be different points on the line $BC$ such that $DC=CE=AB$ , and let $P$ and $Q$ be points on the segments $BD$ and $BE$ , respectively, such that $2BP=PD$ and $2BQ=QE$ . Determine $\angle PGQ$ .
|
90^\circ
| 83.59375 |
3,590 |
In an acute triangle $ABC$ , the segment $CD$ is an altitude and $H$ is the orthocentre. Given that the circumcentre of the triangle lies on the line containing the bisector of the angle $DHB$ , determine all possible values of $\angle CAB$ .
|
60^\circ
| 15.625 |
3,591 |
Let $\Delta ABC$ be an acute-angled triangle and let $H$ be its orthocentre. Let $G_1, G_2$ and $G_3$ be the centroids of the triangles $\Delta HBC , \Delta HCA$ and $\Delta HAB$ respectively. If the area of $\Delta G_1G_2G_3$ is $7$ units, what is the area of $\Delta ABC $ ?
|
63
| 96.875 |
3,592 |
Determine all integers $ n > 3$ for which there exist $ n$ points $ A_{1},\cdots ,A_{n}$ in the plane, no three collinear, and real numbers $ r_{1},\cdots ,r_{n}$ such that for $ 1\leq i < j < k\leq n$ , the area of $ \triangle A_{i}A_{j}A_{k}$ is $ r_{i} \plus{} r_{j} \plus{} r_{k}$ .
|
n = 4
| 0 |
3,593 |
In an exotic country, the National Bank issues coins that can take any value in the interval $[0, 1]$ . Find the smallest constant $c > 0$ such that the following holds, no matter the situation in that country:
*Any citizen of the exotic country that has a finite number of coins, with a total value of no more than $1000$ , can split those coins into $100$ boxes, such that the total value inside each box is at most $c$ .*
|
\frac{1000}{91}
| 0 |
3,594 |
Let $f(x)=x^2-2x+1.$ For some constant $k, f(x+k) = x^2+2x+1$ for all real numbers $x.$ Determine the value of $k.$
|
2
| 99.21875 |
3,595 |
$n$ is a fixed natural number. Find the least $k$ such that for every set $A$ of $k$ natural numbers, there exists a subset of $A$ with an even number of elements which the sum of it's members is divisible by $n$ .
|
k = 2n
| 0 |
3,596 |
For some complex number $\omega$ with $|\omega| = 2016$ , there is some real $\lambda>1$ such that $\omega, \omega^{2},$ and $\lambda \omega$ form an equilateral triangle in the complex plane. Then, $\lambda$ can be written in the form $\tfrac{a + \sqrt{b}}{c}$ , where $a,b,$ and $c$ are positive integers and $b$ is squarefree. Compute $\sqrt{a+b+c}$ .
|
4032
| 13.28125 |
3,597 |
Let $S=\{1,2,3,\ldots,280\}$ . Find the smallest integer $n$ such that each $n$ -element subset of $S$ contains five numbers which are pairwise relatively prime.
|
217
| 2.34375 |
3,598 |
Anders is solving a math problem, and he encounters the expression $\sqrt{15!}$ . He attempts to simplify this radical as $a\sqrt{b}$ where $a$ and $b$ are positive integers. The sum of all possible values of $ab$ can be expressed in the form $q\cdot 15!$ for some rational number $q$ . Find $q$ .
|
4
| 0 |
3,599 |
let $p$ and $q=p+2$ be twin primes. consider the diophantine equation $(+)$ given by
$n!+pq^2=(mp)^2$ $m\geq1$ , $n\geq1$ i. if $m=p$ ,find the value of $p$ .
ii. how many solution quadruple $(p,q,m,n)$ does $(+)$ have ?
|
1
| 73.4375 |
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