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There are 47 students in a classroom with seats arranged in 6 rows $ \times$ 8 columns, and the seat in the $ i$ -th row and $ j$ -th column is denoted by $ (i,j).$ Now, an adjustment is made for students’ seats in the new school term. For a student with the original seat $ (i,j),$ if his/her new seat is $ (m,n),$ we say that the student is moved by $ [a, b] \equal{} [i \minus{} m, j \minus{} n]$ and define the position value of the student as $ a\plus{}b.$ Let $ S$ denote the sum of the position values of all the students. Determine the difference between the greatest and smallest possible values of $ S.$ | 24 | 0.625 |
Find all pairs of positive integers $m$ , $n$ such that the $(m+n)$ -digit number \[\underbrace{33\ldots3}_{m}\underbrace{66\ldots 6}_{n}\] is a perfect square. | (1, 1) | 0.75 |
Find the smallest constant $C > 1$ such that the following statement holds: for every integer $n \geq 2$ and sequence of non-integer positive real numbers $a_1, a_2, \dots, a_n$ satisfying $$ \frac{1}{a_1} + \frac{1}{a_2} + \cdots + \frac{1}{a_n} = 1, $$ it's possible to choose positive integers $b_i$ such that
(i) for each $i = 1, 2, \dots, n$ , either $b_i = \lfloor a_i \rfloor$ or $b_i = \lfloor a_i \rfloor + 1$ , and
(ii) we have $$ 1 < \frac{1}{b_1} + \frac{1}{b_2} + \cdots + \frac{1}{b_n} \leq C. $$ (Here $\lfloor \bullet \rfloor$ denotes the floor function, as usual.)
*Merlijn Staps* | C = \frac{3}{2} | 0.125 |
An equilateral triangle with side length $6$ has a square of side length $6$ attached to each of its edges as shown. The distance between the two farthest vertices of this figure (marked $A$ and $B$ in the figure) can be written as $m + \sqrt{n}$ where $m$ and $n$ are positive integers. Find $m + n$ .
[asy]
draw((0,0)--(1,0)--(1/2,sqrt(3)/2)--cycle);
draw((1,0)--(1+sqrt(3)/2,1/2)--(1/2+sqrt(3)/2,1/2+sqrt(3)/2)--(1/2,sqrt(3)/2));
draw((0,0)--(-sqrt(3)/2,1/2)--(-sqrt(3)/2+1/2,1/2+sqrt(3)/2)--(1/2,sqrt(3)/2));
dot((-sqrt(3)/2+1/2,1/2+sqrt(3)/2));
label("A", (-sqrt(3)/2+1/2,1/2+sqrt(3)/2), N);
draw((1,0)--(1,-1)--(0,-1)--(0,0));
dot((1,-1));
label("B", (1,-1), SE);
[/asy] | 114 | 0.375 |
A regular $n$ -gon is inscribed in a unit circle. Compute the product from a fixed vertex to all the other vertices. | n | 0.625 |
A right triangle has perimeter $2008$ , and the area of a circle inscribed in the triangle is $100\pi^3$ . Let $A$ be the area of the triangle. Compute $\lfloor A\rfloor$ . | 31541 | 0.625 |
What is the sum of all primes $p$ such that $7^p - 6^p + 2$ is divisible by 43?
| 3 | 0.75 |
Siva has the following expression, which is missing operations: $$ \frac12 \,\, \_ \,\,\frac14 \,\, \_ \,\, \frac18 \,\, \_ \,\,\frac{1}{16} \,\, \_ \,\,\frac{1}{32}. $$ For each blank, he flips a fair coin: if it comes up heads, he fills it with a plus, and if it comes up tails, he fills it with a minus. Afterwards, he computes the value of the expression. He then repeats the entire process with a new set of coinflips and operations. If the probability that the positive difference between his computed values is greater than $\frac12$ can be expressed as $\frac{a}{b}$ for relatively prime positive integers $a$ , $b$ , then find $a + b$ . | 39 | 0.875 |
Use $ \log_{10} 2 \equal{} 0.301,\ \log_{10} 3 \equal{} 0.477,\ \log_{10} 7 \equal{} 0.845$ to find the value of $ \log_{10} (10!)$ .
Note that you must answer according to the rules:fractional part of $ 0.5$ and higher is rounded up, and everything strictly less than $ 0.5$ is rounded down,
say $ 1.234\longrightarrow 1.23$ . Then find the minimum integer value $ n$ such that $ 10! < 2^{n}$ . | 22 | 0.5 |
There are $2017$ turtles in a room. Every second, two turtles are chosen uniformly at random and combined to form one super-turtle. (Super-turtles are still turtles.) The probability that after $2015$ seconds (meaning when there are only two turtles remaining) there is some turtle that has never been combined with another turtle can be written in the form $\tfrac{p}{q}$ where $p$ and $q$ are relatively prime positive integers. Find $p + q$ . | 1009 | 0.125 |
For a finite graph $G$ , let $f(G)$ be the number of triangles and $g(G)$ the number of tetrahedra formed by edges of $G$ . Find the least constant $c$ such that \[g(G)^3\le c\cdot f(G)^4\] for every graph $G$ .
*Proposed by Marcin Kuczma, Poland* | \frac{3}{32} | 0.75 |
A rectangular piece of paper $ABCD$ has sides of lengths $AB = 1$ , $BC = 2$ . The rectangle is folded in half such that $AD$ coincides with $BC$ and $EF$ is the folding line. Then fold the paper along a line $BM$ such that the corner $A$ falls on line $EF$ . How large, in degrees, is $\angle ABM$ ?
[asy]
size(180); pathpen = rgb(0,0,0.6)+linewidth(1); pointpen = black+linewidth(3); pointfontpen = fontsize(10); pen dd = rgb(0,0,0.6) + linewidth(0.7) + linetype("4 4"), dr = rgb(0.8,0,0), dg = rgb(0,0.6,0), db = rgb(0,0,0.6)+linewidth(1);
pair A=(0,1), B=(0,0), C=(2,0), D=(2,1), E=A/2, F=(2,.5), M=(1/3^.5,1), N=reflect(B,M)*A;
D(B--M--D("N",N,NE)--B--D("C",C,SE)--D("D",D,NE)--M); D(D("M",M,plain.N)--D("A",A,NW)--D("B",B,SW),dd); D(D("E",E,W)--D("F",F,plain.E),dd);
[/asy] | 30^\circ | 0.875 |
The repeating decimals $0.abab\overline{ab}$ and $0.abcabc\overline{abc}$ satisfy
\[0.abab\overline{ab}+0.abcabc\overline{abc}=\frac{33}{37},\]
where $a,b$ , and $c$ are (not necessarily distinct) digits. Find the three-digit number $abc$ . | 447 | 0.875 |
Let $p$ be a prime number such that $p\equiv 1\pmod{4}$ . Determine $\sum_{k=1}^{\frac{p-1}{2}}\left \lbrace \frac{k^2}{p} \right \rbrace$ , where $\{x\}=x-[x]$ . | \frac{p-1}{4} | 0.375 |
In a finite sequence of real numbers the sum of any seven successive terms is negative and the sum of any eleven successive terms is positive. Determine the maximum number of terms in the sequence. | 16 | 0.375 |
**Q.** Find all polynomials $P: \mathbb{R \times R}\to\mathbb{R\times R}$ with real coefficients, such that $$ P(x,y) = P(x+y,x-y), \ \forall\ x,y \in \mathbb{R}. $$ *Proposed by TuZo* | P(x, y) = (a, b) | 0.25 |
Let $m$ and $n$ be positive integers such that $x=m+\sqrt{n}$ is a solution to the equation $x^2-10x+1=\sqrt{x}(x+1)$ . Find $m+n$ . | 55 | 0.875 |
In the plane there are $2020$ points, some of which are black and the rest are green. For every black point, the following applies: *There are exactly two green points that represent the distance $2020$ from that black point.*
Find the smallest possible number of green dots.
(Walther Janous) | 45 | 0.375 |
Considering all numbers of the form $n = \lfloor \frac{k^3}{2012} \rfloor$ , where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$ , and $k$ ranges from $1$ to $2012$ , how many of these $n$ ’s are distinct?
| 1995 | 0.25 |
A circle radius $320$ is tangent to the inside of a circle radius $1000$ . The smaller circle is tangent to a diameter of the larger circle at a point $P$ . How far is the point $P$ from the outside of the larger circle? | 400 | 0.875 |
Find all positive integers $d$ for which there exist polynomials $P(x)$ and $Q(x)$ with real coefficients such that degree of $P$ equals $d$ and $$ P(x)^2+1=(x^2+1)Q(x)^2. $$ | d | 0.25 |
$ABCD$ is a cyclic quadrilateral inscribed in a circle of radius $5$ , with $AB=6$ , $BC=7$ , $CD=8$ . Find $AD$ . | \sqrt{51} | 0.125 |
For every positive integer $k$ , let $\mathbf{T}_k = (k(k+1), 0)$ , and define $\mathcal{H}_k$ as the homothety centered at $\mathbf{T}_k$ with ratio $\tfrac{1}{2}$ if $k$ is odd and $\tfrac{2}{3}$ is $k$ is even. Suppose $P = (x,y)$ is a point such that $$ (\mathcal{H}_{4} \circ \mathcal{H}_{3} \circ \mathcal{H}_2 \circ \mathcal{H}_1)(P) = (20, 20). $$ What is $x+y$ ?
(A *homothety* $\mathcal{H}$ with nonzero ratio $r$ centered at a point $P$ maps each point $X$ to the point $Y$ on ray $\overrightarrow{PX}$ such that $PY = rPX$ .) | 256 | 0.5 |
Find the total number of solutions to the following system of equations: \[ \begin{cases} a^2\plus{}bc\equiv a\pmod {37} b(a\plus{}d)\equiv b\pmod {37} c(a\plus{}d)\equiv c\pmod{37} bc\plus{}d^2\equiv d\pmod{37} ad\minus{}bc\equiv 1\pmod{37}\end{cases}\] | 1 | 0.375 |
Triangle $ABC$ is right angled at $A$ . The circle with center $A$ and radius $AB$ cuts $BC$ and $AC$ internally at $D$ and $E$ respectively. If $BD = 20$ and $DC = 16$ , determine $AC^2$ . | 936 | 0.5 |
Let the three sides of a triangle be $\ell, m, n$ , respectively, satisfying $\ell>m>n$ and $\left\{\frac{3^\ell}{10^4}\right\}=\left\{\frac{3^m}{10^4}\right\}=\left\{\frac{3^n}{10^4}\right\}$ , where $\{x\}=x-\lfloor{x}\rfloor$ and $\lfloor{x}\rfloor$ denotes the integral part of the number $x$ . Find the minimum perimeter of such a triangle. | 3003 | 0.75 |
Let $m$ be a positive integer, and let $a_0, a_1,\ldots,a_m$ be a sequence of reals such that $a_0=37$ , $a_1=72$ , $a_m=0$ , and \[a_{k+1}=a_{k-1}-\frac{3}{a_k}\] for $k=1,2, \dots, m-1$ . Find $m$ . | 889 | 0.875 |
The isoelectric point of glycine is the pH at which it has zero charge. Its charge is $-\frac13$ at pH $3.55$ , while its charge is $\frac12$ at pH $9.6$ . Charge increases linearly with pH. What is the isoelectric point of glycine? | 5.97 | 0.625 |
There are two round tables with $n{}$ dwarves sitting at each table. Each dwarf has only two friends: his neighbours to the left and to the right. A good wizard wants to seat the dwarves at one round table so that each two neighbours are friends. His magic allows him to make any $2n$ pairs of dwarves into pairs of friends (the dwarves in a pair may be from the same or from different tables). However, he knows that an evil sorcerer will break $n{}$ of those new friendships. For which $n{}$ is the good wizard able to achieve his goal no matter what the evil sorcerer does?
*Mikhail Svyatlovskiy* | n | 0.25 |
Find all functions $f$ from the reals into the reals such that \[ f(ab) = f(a+b) \] for all irrational $a, b$ . | f(x) = c | 0.375 |
For a positive integer $n$ , there is a school with $2n$ people. For a set $X$ of students in this school, if any two students in $X$ know each other, we call $X$ *well-formed*. If the maximum number of students in a well-formed set is no more than $n$ , find the maximum number of well-formed set.
Here, an empty set and a set with one student is regarded as well-formed as well. | 3^n | 0.5 |
The intersection of two squares with perimeter $8$ is a rectangle with diagonal length $1$ . Given that the distance between the centers of the two squares is $2$ , the perimeter of the rectangle can be expressed as $P$ . Find $10P$ . | 25 | 0.125 |
In a competition there are $18$ teams and in each round $18$ teams are divided into $9$ pairs where the $9$ matches are played coincidentally. There are $17$ rounds, so that each pair of teams play each other exactly once. After $n$ rounds, there always exists $4$ teams such that there was exactly one match played between these teams in those $n$ rounds. Find the maximum value of $n$ . | 7 | 0.125 |
In a room there are $144$ people. They are joined by $n$ other people who are each carrying $k$ coins. When these coins are shared among all $n + 144$ people, each person has $2$ of these coins. Find the minimum possible value of $2n + k$ . | 50 | 0.875 |
Point D is from AC of triangle ABC so that 2AD=DC. Let DE be perpendicular to BC and AE
intersects BD at F. It is known that triangle BEF is equilateral. Find <ADB? | 90^\circ | 0.125 |
Let $S_n$ be the sum of the reciprocals of the non-zero digits of the integers from 1 to $10^n$ inclusive. Find the smallest positive integer $n$ for which $S_n$ is an integer. | 63 | 0.625 |
A terrain ( $ABCD$ ) has a rectangular trapezoidal shape. The angle in $A$ measures $90^o$ . $AB$ measures $30$ m, $AD$ measures $20$ m and $DC$ measures 45 m. This land must be divided into two areas of the same area, drawing a parallel to the $AD$ side . At what distance from $D$ do we have to draw the parallel?
 | 18.75 \text{ m} | 0.375 |
For the NEMO, Kevin needs to compute the product
\[
9 \times 99 \times 999 \times \cdots \times 999999999.
\]
Kevin takes exactly $ab$ seconds to multiply an $a$ -digit integer by a $b$ -digit integer. Compute the minimum number of seconds necessary for Kevin to evaluate the expression together by performing eight such multiplications.
*Proposed by Evan Chen* | 870 | 0.375 |
We define the function $f(x,y)=x^3+(y-4)x^2+(y^2-4y+4)x+(y^3-4y^2+4y)$ . Then choose any distinct $a, b, c \in \mathbb{R}$ such that the following holds: $f(a,b)=f(b,c)=f(c,a)$ . Over all such choices of $a, b, c$ , what is the maximum value achieved by
\[\min(a^4 - 4a^3 + 4a^2, b^4 - 4b^3 + 4b^2, c^4 - 4c^3 + 4c^2)?\] | 1 | 0.5 |
A sequence of vertices $v_1,v_2,\ldots,v_k$ in a graph, where $v_i=v_j$ only if $i=j$ and $k$ can be any positive integer, is called a $\textit{cycle}$ if $v_1$ is attached by an edge to $v_2$ , $v_2$ to $v_3$ , and so on to $v_k$ connected to $v_1$ . Rotations and reflections are distinct: $A,B,C$ is distinct from $A,C,B$ and $B,C,A$ . Supposed a simple graph $G$ has $2013$ vertices and $3013$ edges. What is the minimal number of cycles possible in $G$ ? | 1001 | 0.875 |
Gus has to make a list of $250$ positive integers, not necessarily distinct, such that each number is equal to the number of numbers in the list that are different from it. For example, if $15$ is a number from the list so the list contains $15$ numbers other than $15$ . Determine the maximum number of distinct numbers the Gus list can contain. | 21 | 0.375 |
At a certain grocery store, cookies may be bought in boxes of $10$ or $21.$ What is the minimum positive number of cookies that must be bought so that the cookies may be split evenly among $13$ people?
*Author: Ray Li* | 52 | 0.875 |
An infinite sheet of paper is divided into equal squares, some of which are colored red. In each $2\times3$ rectangle, there are exactly two red squares. Now consider an arbitrary $9\times11$ rectangle. How many red squares does it contain? (The sides of all considered rectangles go along the grid lines.) | 33 | 0.25 |
Find all functions $f:\mathbb{N} \to \mathbb{N}$ such that for all $m,n\in \mathbb{N}$ :
- $f(2)=2$ ,
- $f(mn)=f(m)f(n)$ ,
- $f(n+1)>f(n)$ .
| f(n) = n | 0.875 |
Define the *hotel elevator cubic*as the unique cubic polynomial $P$ for which $P(11) = 11$ , $P(12) = 12$ , $P(13) = 14$ , $P(14) = 15$ . What is $P(15)$ ?
*Proposed by Evan Chen* | 13 | 0.5 |
Four positive integers $a, b, c, d$ satisfy the condition: $a < b < c < d$ . For what smallest possible value of $d$ could the following condition be true: the arithmetic mean of numbers $a, b, c$ is twice smaller than the arithmetic mean of numbers $a, b, c, d$ ? | 10 | 0.5 |
Find all real numbers $x$ that satisfy the equation $$ \frac{x-2020}{1}+\frac{x-2019}{2}+\cdots+\frac{x-2000}{21}=\frac{x-1}{2020}+\frac{x-2}{2019}+\cdots+\frac{x-21}{2000}, $$ and simplify your answer(s) as much as possible. Justify your solution. | x = 2021 | 0.75 |
The polynomial $P$ is a quadratic with integer coefficients. For every positive integer $n$ , the integers $P(n)$ and $P(P(n))$ are relatively prime to $n$ . If $P(3) = 89$ , what is the value of $P(10)$ ? | 859 | 0.75 |
Let $k$ be a real number such that the product of real roots of the equation $$ X^4 + 2X^3 + (2 + 2k)X^2 + (1 + 2k)X + 2k = 0 $$ is $-2013$ . Find the sum of the squares of these real roots. | 4027 | 0.25 |
Find all natural numbers $n> 1$ for which the following applies:
The sum of the number $n$ and its second largest divisor is $2013$ .
(R. Henner, Vienna) | n = 1342 | 0.875 |
A student did not notice multiplication sign between two three-digit numbers and wrote it as a six-digit number. Result was 7 times more that it should be. Find these numbers.
*(2 points)* | (143, 143) | 0.375 |
Suppose that each of $n$ people knows exactly one piece of information and all $n$ pieces are different. Every time person $A$ phones person $B$ , $A$ tells $B$ everything he knows, while tells $A$ nothing. What is the minimum of phone calls between pairs of people needed for everyone to know everything? | 2n - 2 | 0.5 |
Let $ S(n) $ be the sum of the squares of the positive integers less than and coprime to $ n $ . For example, $ S(5) = 1^2 + 2^2 + 3^2 + 4^2 $ , but $ S(4) = 1^2 + 3^2 $ . Let $ p = 2^7 - 1 = 127 $ and $ q = 2^5 - 1 = 31 $ be primes. The quantity $ S(pq) $ can be written in the form $$ \frac{p^2q^2}{6}\left(a - \frac{b}{c} \right) $$ where $ a $ , $ b $ , and $ c $ are positive integers, with $ b $ and $ c $ coprime and $ b < c $ . Find $ a $ . | 7561 | 0.125 |
When a function $f(x)$ is differentiated $n$ times ,the function we get id denoted $f^n(x)$ .If $f(x)=\dfrac {e^x}{x}$ .Find the value of
\[\lim_{n \to \infty} \dfrac {f^ {2n}(1)}{(2n)!}\] | 1 | 0.5 |
Determine the smallest integer $k > 1$ such that there exist $k$ distinct primes whose squares sum to a power of $2$ . | k = 5 | 0.75 |
Let $S$ be the sum of the base 10 logarithms of all the proper divisors of 1000000. What is the integer nearest to $S$ ? | 141 | 0.75 |
Two players play a game on a pile of $n$ beans. On each player's turn, they may take exactly $1$ , $4$ , or $7$ beans from the pile. One player goes first, and then the players alternate until somebody wins. A player wins when they take the last bean from the pile. For how many $n$ between $2014$ and $2050$ (inclusive) does the second player win? | 14 | 0.5 |
**Q11.** Let be given a sequense $a_1=5, \; a_2=8$ and $a_{n+1}=a_n+3a_{n-1}, \qquad n=1,2,3,...$ Calculate the greatest common divisor of $a_{2011}$ and $a_{2012}$ . | 1 | 0.875 |
It is given that there exists a unique triple of positive primes $(p,q,r)$ such that $p<q<r$ and \[\dfrac{p^3+q^3+r^3}{p+q+r} = 249.\] Find $r$ . | r = 19 | 0.625 |
Let $p$ prime and $m$ a positive integer. Determine all pairs $( p,m)$ satisfying the equation: $ p(p+m)+p=(m+1)^3$ | (2, 1) | 0.5 |
$p(x)$ is the cubic $x^3 - 3x^2 + 5x$ . If $h$ is a real root of $p(x) = 1$ and $k$ is a real root of $p(x) = 5$ , find $h + k$ . | h + k = 2 | 0.875 |
A crazy physicist has discovered a new particle called an omon. He has a machine, which takes two omons of mass $a$ and $b$ and entangles them; this process destroys the omon with mass $a$ , preserves the one with mass $b$ , and creates a new omon whose mass is $\frac 12 (a+b)$ . The physicist can then repeat the process with the two resulting omons, choosing which omon to destroy at every step. The physicist initially has two omons whose masses are distinct positive integers less than $1000$ . What is the maximum possible number of times he can use his machine without producing an omon whose mass is not an integer?
*Proposed by Michael Kural* | 9 | 0.625 |
A mustache is created by taking the set of points $(x, y)$ in the $xy$ -coordinate plane that satisfy $4 + 4 \cos(\pi x/24) \le y \le 6 + 6\cos(\pi x/24)$ and $-24 \le x \le 24$ . What is the area of the mustache? | 96 | 0.875 |
Let $\{a_n\}_{n=1}^{\infty}$ and $\{b_n\}_{n=1}^{\infty}$ be sequences of integers such that $a_1 = 20$ , $b_1 = 15$ , and for $n \ge 1$ , \[\left\{\begin{aligned} a_{n+1}&=a_n^2-b_n^2, b_{n+1}&=2a_nb_n-b_n^2 \end{aligned}\right.\] Let $G = a_{10}^2-a_{10}b_{10}+b_{10}^2$ . Determine the number of positive integer factors of $G$ .
*Proposed by Michael Ren* | 525825 | 0.625 |
Let $ n\geq 3 $ be an integer and let $ x_1,x_2,\ldots,x_{n-1} $ be nonnegative integers such that
\begin{eqnarray*} \ x_1 + x_2 + \cdots + x_{n-1} &=& n x_1 + 2x_2 + \cdots + (n-1)x_{n-1} &=& 2n-2. \end{eqnarray*}
Find the minimal value of $ F(x_1,x_2,\ldots,x_n) = \sum_{k=1}^{n-1} k(2n-k)x_k $ . | 3n(n-1) | 0.5 |
Let $a_1, a_2, a_3, a_4$ be integers with distinct absolute values. In the coordinate plane, let $A_1=(a_1,a_1^2)$ , $A_2=(a_2,a_2^2)$ , $A_3=(a_3,a_3^2)$ and $A_4=(a_4,a_4^2)$ . Assume that lines $A_1A_2$ and $A_3A_4$ intersect on the $y$ -axis at an acute angle of $\theta$ . The maximum possible value for $\tan \theta$ can be expressed in the form $\dfrac mn$ for relatively prime positive integers $m$ and $n$ . Find $100m+n$ .
*Proposed by James Lin* | 503 | 0.375 |
Consider the sequence $$ 1,7,8,49,50,56,57,343\ldots $$ which consists of sums of distinct powers of $7$ , that is, $7^0$ , $7^1$ , $7^0+7^1$ , $7^2$ , $\ldots$ in increasing order. At what position will $16856$ occur in this sequence? | 36 | 0.625 |
Compute the maximum integer value of $k$ such that $2^k$ divides $3^{2n+3}+40n-27$ for any positive integer $n$ . | 6 | 0.75 |
Find the greatest positive real number $M$ such that for all positive real sequence $(a_n)$ and for all real number $m < M$ , it is possible to find some index $n \ge 1$ that satisfies the inequality $a_1 + a_2 + a_3 + ...+ a_n +a_{n+1} > m a_n$ . | M = 4 | 0.625 |
Find the number of pairs of positive integers $a$ and $b$ such that $a\leq 100\,000$ , $b\leq 100\,000$ , and $$ \frac{a^3-b}{a^3+b}=\frac{b^2-a^2}{b^2+a^2}. $$ | 10 | 0.625 |
Suppose that $m$ and $n$ are relatively prime positive integers with $A = \tfrac mn$ , where
\[ A = \frac{2+4+6+\dots+2014}{1+3+5+\dots+2013} - \frac{1+3+5+\dots+2013}{2+4+6+\dots+2014}. \] Find $m$ . In other words, find the numerator of $A$ when $A$ is written as a fraction in simplest form.
*Proposed by Evan Chen* | 2015 | 0.75 |
Each person stands on a whole number on the number line from $0$ to $2022$ . In each turn, two people are selected by a distance of at least $2$ . These go towards each other by $1$ . When no more such moves are possible, the process ends.
Show that this process always ends after a finite number of moves, and determine all possible configurations where people can end up standing. (whereby is for each configuration is only of interest how many people stand at each number.)
*(Birgit Vera Schmidt)*
<details><summary>original wording</summary>Bei jeder ganzen Zahl auf dem Zahlenstrahl von 0 bis 2022 steht zu Beginn eine Person.
In jedem Zug werden zwei Personen mit Abstand mindestens 2 ausgewählt. Diese gehen jeweils um 1 aufeinander zu. Wenn kein solcher Zug mehr möglich ist, endet der Vorgang.
Man zeige, dass dieser Vorgang immer nach endlich vielen Zügen endet, und bestimme alle möglichen Konfigurationen, wo die Personen am Ende stehen können. (Dabei ist für jede Konfiguration nur von Interesse, wie viele Personen bei jeder Zahl stehen.)</details> | 1011 | 0.125 |
A sequence $a_1, a_2, \ldots$ satisfies $a_1 = \dfrac 52$ and $a_{n + 1} = {a_n}^2 - 2$ for all $n \ge 1.$ Let $M$ be the integer which is closest to $a_{2023}.$ The last digit of $M$ equals $$ \mathrm a. ~ 0\qquad \mathrm b.~2\qquad \mathrm c. ~4 \qquad \mathrm d. ~6 \qquad \mathrm e. ~8 $$ | 4 | 0.5 |
Which number is greater: $$ A=\frac{2.00\ldots04}{1.00\ldots04^2+2.00\ldots04},\text{ or }B=\frac{2.00\ldots02}{1.00\ldots02^2+2.00\ldots02}, $$ where each of the numbers above contains $1998$ zeros? | B | 0.875 |
The target below is made up of concentric circles with diameters $4$ , $8$ , $12$ , $16$ , and $20$ . The area of the dark region is $n\pi$ . Find $n$ .
[asy]
size(150);
defaultpen(linewidth(0.8));
int i;
for(i=5;i>=1;i=i-1)
{
if (floor(i/2)==i/2)
{
filldraw(circle(origin,4*i),white);
}
else
{
filldraw(circle(origin,4*i),red);
}
}
[/asy] | 60 | 0.875 |
In circle $\Omega$ , let $\overline{AB}=65$ be the diameter and let points $C$ and $D$ lie on the same side of arc $\overarc{AB}$ such that $CD=16$ , with $C$ closer to $B$ and $D$ closer to $A$ . Moreover, let $AD, BC, AC,$ and $BD$ all have integer lengths. Two other circles, circles $\omega_1$ and $\omega_2$ , have $\overline{AC}$ and $\overline{BD}$ as their diameters, respectively. Let circle $\omega_1$ intersect $AB$ at a point $E \neq A$ and let circle $\omega_2$ intersect $AB$ at a point $F \neq B$ . Then $EF=\frac{m}{n}$ , for relatively prime integers $m$ and $n$ . Find $m+n$ .
[asy]
size(7cm);
pair A=(0,0), B=(65,0), C=(117/5,156/5), D=(125/13,300/13), E=(23.4,0), F=(9.615,0);
draw(A--B--C--D--cycle);
draw(A--C);
draw(B--D);
dot(" $A$ ", A, SW);
dot(" $B$ ", B, SE);
dot(" $C$ ", C, NE);
dot(" $D$ ", D, NW);
dot(" $E$ ", E, S);
dot(" $F$ ", F, S);
draw(circle((A + C)/2, abs(A - C)/2));
draw(circle((B + D)/2, abs(B - D)/2));
draw(circle((A + B)/2, abs(A - B)/2));
label(" $\mathcal P$ ", (A + B)/2 + abs(A - B)/2 * dir(-45), dir(-45));
label(" $\mathcal Q$ ", (A + C)/2 + abs(A - C)/2 * dir(-210), dir(-210));
label(" $\mathcal R$ ", (B + D)/2 + abs(B - D)/2 * dir(70), dir(70));
[/asy]
*Proposed by **AOPS12142015*** | 961 | 0.25 |
Let $ABCD$ be a square, and let $M$ be the midpoint of side $BC$ . Points $P$ and $Q$ lie on segment $AM$ such that $\angle BPD=\angle BQD=135^\circ$ . Given that $AP<AQ$ , compute $\tfrac{AQ}{AP}$ . | \sqrt{5} | 0.25 |
Starting from 37, adding 5 before each previous term, forms the following sequence:
\[37,537,5537,55537,555537,...\]
How many prime numbers are there in this sequence? | 1 | 0.375 |
Let $f(t)$ be the cubic polynomial for $t$ such that $\cos 3x=f(\cos x)$ holds for all real number $x$ .
Evaluate
\[\int_0^1 \{f(t)\}^2 \sqrt{1-t^2}dt\] | \frac{\pi}{8} | 0.625 |
Ana has $22$ coins. She can take from her friends either $6$ coins or $18$ coins, or she can give $12$ coins to her friends. She can do these operations many times she wants. Find the least number of coins Ana can have. | 4 | 0.875 |
Find the least $k$ for which the number $2010$ can be expressed as the sum of the squares of $k$ integers. | k=3 | 0.75 |
Let $x_0,x_1,x_2,\dots$ be the sequence such that $x_0=1$ and for $n\ge 0,$ \[x_{n+1}=\ln(e^{x_n}-x_n)\]
(as usual, the function $\ln$ is the natural logarithm). Show that the infinite series
\[x_0+x_1+x_2+\cdots\]
converges and find its sum. | e - 1 | 0.875 |
$n$ balls are placed independently uniformly at random into $n$ boxes. One box is selected at random, and is found to contain $b$ balls. Let $e_n$ be the expected value of $b^4$ . Find $$ \lim_{n \to
\infty}e_n. $$ | 15 | 0.375 |
To any triangle with side lengths $a,b,c$ and the corresponding angles $\alpha, \beta, \gamma$ (measured in radians), the 6-tuple $(a,b,c,\alpha, \beta, \gamma)$ is assigned. Find the minimum possible number $n$ of distinct terms in the 6-tuple assigned to a scalene triangle.
| 4 | 0.125 |
Rectangle $ABCD$ has $AB = 8$ and $BC = 13$ . Points $P_1$ and $P_2$ lie on $AB$ and $CD$ with $P_1P_2 \parallel BC$ . Points $Q_1$ and $Q_2$ lie on $BC$ and $DA$ with $Q_1Q_2 \parallel AB$ . Find the area of quadrilateral $P_1Q_1P_2Q_2$ . | 52 | 0.875 |
Points $C\neq D$ lie on the same side of line $AB$ so that $\triangle ABC$ and $\triangle BAD$ are congruent with $AB = 9$ , $BC=AD=10$ , and $CA=DB=17$ . The intersection of these two triangular regions has area $\tfrac mn$ , where $m$ and $n$ are relatively prime positive integers. Find $m+n$ . | 59 | 0.375 |
A thousand integer divisions are made: $2018$ is divided by each of the integers from $ 1$ to $1000$ . Thus, a thousand integer quotients are obtained with their respective remainders. Which of these thousand remainders is the bigger? | 672 | 0.25 |
How many different ways are there to write 2004 as a sum of one or more positive integers which are all "aproximately equal" to each other? Two numbers are called aproximately equal if their difference is at most 1. The order of terms does not matter: two ways which only differ in the order of terms are not considered different. | 2004 | 0.75 |
Define an ordered triple $ (A, B, C)$ of sets to be minimally intersecting if $ |A \cap B| \equal{} |B \cap C| \equal{} |C \cap A| \equal{} 1$ and $ A \cap B \cap C \equal{} \emptyset$ . For example, $ (\{1,2\},\{2,3\},\{1,3,4\})$ is a minimally intersecting triple. Let $ N$ be the number of minimally intersecting ordered triples of sets for which each set is a subset of $ \{1,2,3,4,5,6,7\}$ . Find the remainder when $ N$ is divided by $ 1000$ .
**Note**: $ |S|$ represents the number of elements in the set $ S$ . | 760 | 0.25 |
Black and white checkers are placed on an $8 \times 8$ chessboard, with at most one checker on each cell. What is the maximum number of checkers that can be placed such that each row and each column contains twice as many white checkers as black ones? | 48 | 0.375 |
A triangle is composed of circular cells arranged in $5784$ rows: the first row has one cell, the second has two cells, and so on (see the picture). The cells are divided into pairs of adjacent cells (circles touching each other), so that each cell belongs to exactly one pair. A pair of adjacent cells is called **diagonal** if the two cells in it *aren't* in the same row. What is the minimum possible amount of diagonal pairs in the division?
An example division into pairs is depicted in the image. | 2892 | 0.375 |
A gambler plays the following coin-tossing game. He can bet an arbitrary positive amount of money. Then a fair coin is tossed, and the gambler wins or loses the amount he bet depending on the outcome. Our gambler, who starts playing with $ x$ forints, where $ 0<x<2C$ , uses the following strategy: if at a given time his capital is $ y<C$ , he risks all of it; and if he has $ y>C$ , he only bets $ 2C\minus{}y$ . If he has exactly $ 2C$ forints, he stops playing. Let $ f(x)$ be the probability that he reaches $ 2C$ (before going bankrupt). Determine the value of $ f(x)$ . | \frac{x}{2C} | 0.75 |
In the real axis, there is bug standing at coordinate $x=1$ . Each step, from the position $x=a$ , the bug can jump to either $x=a+2$ or $x=\frac{a}{2}$ . Show that there are precisely $F_{n+4}-(n+4)$ positions (including the initial position) that the bug can jump to by at most $n$ steps.
Recall that $F_n$ is the $n^{th}$ element of the Fibonacci sequence, defined by $F_0=F_1=1$ , $F_{n+1}=F_n+F_{n-1}$ for all $n\geq 1$ .
| F_{n+4} - (n+4) | 0.875 |
Let $ABCDEF$ be a convex hexagon of area $1$ , whose opposite sides are parallel. The lines $AB$ , $CD$ and $EF$ meet in pairs to determine the vertices of a triangle. Similarly, the lines $BC$ , $DE$ and $FA$ meet in pairs to determine the vertices of another triangle. Show that the area of at least one of these two triangles is at least $3/2$ . | \frac{3}{2} | 0.875 |
Given that
\begin{eqnarray*}&(1)& \text{x and y are both integers between 100 and 999, inclusive;}\qquad \qquad \qquad \qquad \qquad &(2)& \text{y is the number formed by reversing the digits of x; and} &(3)& z=|x-y|. \end{eqnarray*}How many distinct values of $z$ are possible? | 9 | 0.75 |
Let $n,k$ be positive integers such that $n>k$ . There is a square-shaped plot of land, which is divided into $n\times n$ grid so that each cell has the same size. The land needs to be plowed by $k$ tractors; each tractor will begin on the lower-left corner cell and keep moving to the cell sharing a common side until it reaches the upper-right corner cell. In addition, each tractor can only move in two directions: up and right. Determine the minimum possible number of unplowed cells. | (n-k)^2 | 0.125 |
Daniel writes over a board, from top to down, a list of positive integer numbers less or equal to 10. Next to each number of Daniel's list, Martin writes the number of times exists this number into the Daniel's list making a list with the same length. If we read the Martin's list from down to top, we get the same
list of numbers that Daniel wrote from top to down. Find the greatest length of the Daniel's list can have. | 10 | 0.375 |
Define a list of number with the following properties:
- The first number of the list is a one-digit natural number.
- Each number (since the second) is obtained by adding $9$ to the number before in the list.
- The number $2012$ is in that list.
Find the first number of the list. | 5 | 0.875 |
A solid in the shape of a right circular cone is 4 inches tall and its base has a 3-inch radius. The entire surface of the cone, including its base, is painted. A plane parallel to the base of the cone divides the cone into two solids, a smaller cone-shaped solid $C$ and a frustum-shaped solid $F$ , in such a way that the ratio between the areas of the painted surfaces of $C$ and $F$ and the ratio between the volumes of $C$ and $F$ are both equal to $k$ . Given that $k=m/n$ , where $m$ and $n$ are relatively prime positive integers, find $m+n$ . | 512 | 0.125 |
Bob chooses a $4$ -digit binary string uniformly at random, and examines an infinite sequence of uniformly and independently random binary bits. If $N$ is the least number of bits Bob has to examine in order to find his chosen string, then find the expected value of $N$ . For example, if Bob’s string is $0000$ and the stream of bits begins $101000001 \dots$ , then $N = 7$ . | 30 | 0.75 |
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