id int64 1.61k 3.1k | subfield stringclasses 4
values | context null | solution listlengths 1 5 | is_multiple_answer bool 2
classes | unit stringclasses 8
values | answer_type stringclasses 4
values | error stringclasses 1
value | problem stringlengths 49 4.42k | answer listlengths 1 1 | a_difficulty float64 1 10 | domains listlengths 1 3 |
|---|---|---|---|---|---|---|---|---|---|---|---|
1,606 | Combinatorics | null | [
"Sergey can determine Xenia's number in 2 but not fewer moves.\n\n\n\nWe first show that 2 moves are sufficient. Let Sergey provide the set $\\{17,18\\}$ on his first move, and the set $\\{18,19\\}$ on the second move. In Xenia's two responses, exactly one number occurs twice, namely, $a_{18}$. Thus, Sergey is able... | false | null | Numerical | null | Xenia and Sergey play the following game. Xenia thinks of a positive integer $N$ not exceeding 5000. Then she fixes 20 distinct positive integers $a_{1}, a_{2}, \ldots, a_{20}$ such that, for each $k=1,2, \ldots, 20$, the numbers $N$ and $a_{k}$ are congruent modulo $k$. By a move, Sergey tells Xenia a set $S$ of posit... | [
"2"
] | 7 | [
"Number Theory -> Congruences",
"Number Theory -> Greatest Common Divisors (GCD)"
] |
1,610 | Algebra | null | [
"The required maximum is $\\frac{1}{2 n+2}$. To show that the condition in the statement is not met if $\\mu>\\frac{1}{2 n+2}$, let $U=(0,1) \\times(0,1)$, choose a small enough positive $\\epsilon$, and consider the configuration $C$ consisting of the $n$ four-element clusters of points $\\left(\\frac{i}{n+1} \\pm... | false | null | Expression | null | Given a positive integer $n$, determine the largest real number $\mu$ satisfying the following condition: for every $4 n$-point configuration $C$ in an open unit square $U$, there exists an open rectangle in $U$, whose sides are parallel to those of $U$, which contains exactly one point of $C$, and has an area greater ... | [
"$\\frac{1}{2 n+2}$"
] | 8 | [
"Geometry -> Plane Geometry -> Area",
"Discrete Mathematics -> Combinatorics -> Other"
] |
1,612 | Number Theory | null | [
"For every integer $M \\geq 0$, let $A_{M}=\\sum_{n=-2^{M}+1}^{0}(-1)^{w(n)}$ and let $B_{M}=$ $\\sum_{n=1}^{2^{M}}(-1)^{w(n)}$; thus, $B_{M}$ evaluates the difference of the number of even weight integers in the range 1 through $2^{M}$ and the number of odd weight integers in that range.\n\n\n\nNotice that\n\n\n\n... | false | null | Numerical | null |
Find (in closed form) the difference between the number of positive integers at most $2^{2017}$ with even weight and the number of positive integers at most $2^{2017}$ with odd weight. | [
"$2^{1009}$"
] | 5 | [
"Discrete Mathematics -> Combinatorics -> Other",
"Algebra -> Prealgebra -> Integers"
] |
1,613 | Algebra | null | [
"There is only one such integer, namely, $n=2$. In this case, if $P$ is a constant polynomial, the required condition is clearly satisfied; if $P=X+c$, then $P(c-1)+P(c+1)=$ $P(3 c)$; and if $P=X^{2}+q X+r$, then $P(X)=P(-X-q)$.\n\n\n\nTo rule out all other values of $n$, it is sufficient to exhibit a monic polynom... | false | null | Numerical | null | Determine all positive integers $n$ satisfying the following condition: for every monic polynomial $P$ of degree at most $n$ with integer coefficients, there exists a positive integer $k \leq n$, and $k+1$ distinct integers $x_{1}, x_{2}, \ldots, x_{k+1}$ such that
$$
P\left(x_{1}\right)+P\left(x_{2}\right)+\cdots+... | [
"2"
] | 8 | [
"Algebra -> Algebra -> Algebraic Expressions",
"Number Theory -> Congruences",
"Algebra -> Algebra -> Polynomial Operations"
] |
1,614 | Combinatorics | null | [
"The required maximum is $2 n-2$. To describe a $(2 n-2)$-element collection satisfying the required conditions, write $X=\\{1,2, \\ldots, n\\}$ and set $B_{k}=\\{1,2, \\ldots, k\\}$, $k=1,2, \\ldots, n-1$, and $B_{k}=\\{k-n+2, k-n+3, \\ldots, n\\}, k=n, n+1, \\ldots, 2 n-2$. To show that no subcollection of the $B... | false | null | Expression | null | Let $n$ be an integer greater than 1 and let $X$ be an $n$-element set. A non-empty collection of subsets $A_{1}, \ldots, A_{k}$ of $X$ is tight if the union $A_{1} \cup \cdots \cup A_{k}$ is a proper subset of $X$ and no element of $X$ lies in exactly one of the $A_{i}$ s. Find the largest cardinality of a collection ... | [
"$2n-2$"
] | 8 | [
"Discrete Mathematics -> Combinatorics"
] |
1,618 | Number Theory | null | [
"Up to a swap of the first two entries, the only solutions are $(x, y, p)=(1,8,19)$, $(x, y, p)=(2,7,13)$ and $(x, y, p)=(4,5,7)$. The verification is routine.\n\n\n\nSet $s=x+y$. Rewrite the equation in the form $s\\left(s^{2}-3 x y\\right)=p(p+x y)$, and express $x y$ :\n\n\n\n$$\n\nx y=\\frac{s^{3}-p^{2}}{3 s+p}... | true | null | Tuple | null | Determine all prime numbers $p$ and all positive integers $x$ and $y$ satisfying $x^{3}+y^{3}=$ $p(x y+p)$. | [
"$(1,8,19), (2,7,13), (4,5,7)$"
] | 6 | [
"Algebra -> Algebra -> Equations and Inequalities",
"Number Theory -> Prime Numbers"
] |
1,620 | Algebra | null | [
"The smallest possible degree is $2 n$. In what follows, we will frequently write $A_{i}=$ $\\left(x_{i}, y_{i}\\right)$, and abbreviate $P\\left(x_{1}, y_{1}, \\ldots, x_{2 n}, y_{2 n}\\right)$ to $P\\left(A_{1}, \\ldots, A_{2 n}\\right)$ or as a function of any $2 n$ points.\n\n\n\nSuppose that $f$ is valid. Firs... | false | null | Expression | null | Let $n \geqslant 2$ be an integer, and let $f$ be a $4 n$-variable polynomial with real coefficients. Assume that, for any $2 n$ points $\left(x_{1}, y_{1}\right), \ldots,\left(x_{2 n}, y_{2 n}\right)$ in the plane, $f\left(x_{1}, y_{1}, \ldots, x_{2 n}, y_{2 n}\right)=0$ if and only if the points form the vertices of ... | [
"$2n$"
] | 8 | [
"Algebra -> Algebra -> Polynomial Operations",
"Algebra -> Abstract Algebra -> Group Theory",
"Geometry -> Plane Geometry -> Polygons"
] |
1,631 | Algebra | null | [
"The largest such is $k=2$. Notice first that if $y_{i}$ is prime, then $x_{i}$ is prime as well. Actually, if $x_{i}=1$ then $y_{i}=1$ which is not prime, and if $x_{i}=m n$ for integer $m, n>1$ then $2^{m}-1 \\mid 2^{x_{i}}-1=y_{i}$, so $y_{i}$ is composite. In particular, if $y_{1}, y_{2}, \\ldots, y_{k}$ are pr... | false | null | Numerical | null | For a positive integer $a$, define a sequence of integers $x_{1}, x_{2}, \ldots$ by letting $x_{1}=a$ and $x_{n+1}=2 x_{n}+1$ for $n \geq 1$. Let $y_{n}=2^{x_{n}}-1$. Determine the largest possible $k$ such that, for some positive integer $a$, the numbers $y_{1}, \ldots, y_{k}$ are all prime. | [
"2"
] | 5 | [
"Number Theory -> Prime Numbers",
"Algebra -> Algebraic Expressions"
] |
1,641 | Combinatorics | null | [
"The required number is $\\left(\\begin{array}{c}2 n \\\\ n\\end{array}\\right)$. To prove this, trace the circumference counterclockwise to label the points $a_{1}, a_{2}, \\ldots, a_{2 n}$.\n\nLet $\\mathcal{C}$ be any good configuration and let $O(\\mathcal{C})$ be the set of all points from which arrows emerge.... | false | null | Expression | null | Let $n$ be a positive integer and fix $2 n$ distinct points on a circumference. Split these points into $n$ pairs and join the points in each pair by an arrow (i.e., an oriented line segment). The resulting configuration is good if no two arrows cross, and there are no arrows $\overrightarrow{A B}$ and $\overrightarrow... | [
"$\\binom{2n}{n}$"
] | 8 | [
"Discrete Mathematics -> Combinatorics",
"Algebra -> Algebraic Expressions -> Generating Functions"
] |
1,644 | Combinatorics | null | [
"The required maximum is $m n-\\lfloor m / 2\\rfloor$ and is achieved by the brick-like vertically symmetric arrangement of blocks of $n$ and $n-1$ horizontal dominoes placed on alternate rows, so that the bottom row of the board is completely covered by $n$ dominoes.\n\n\n\nTo show that the number of dominoes in a... | false | null | Expression | null | Given positive integers $m$ and $n \geq m$, determine the largest number of dominoes $(1 \times 2$ or $2 \times 1$ rectangles) that can be placed on a rectangular board with $m$ rows and $2 n$ columns consisting of cells $(1 \times 1$ squares $)$ so that:
(i) each domino covers exactly two adjacent cells of the boar... | [
"$m n-\\lfloor m / 2\\rfloor$"
] | 7 | [
"Discrete Mathematics -> Combinatorics",
"Geometry -> Plane Geometry -> Area"
] |
1,645 | Algebra | null | [
"The only possible value of $a_{2015} \\cdot a_{2016}$ is 0 . For simplicity, by performing a translation of the sequence (which may change the defining constants $b, c$ and $d$ ), we may instead concern ourselves with the values $a_{0}$ and $a_{1}$, rather than $a_{2015}$ and $a_{2016}$.\n\n\n\nSuppose now that we... | false | null | Numerical | null | A cubic sequence is a sequence of integers given by $a_{n}=n^{3}+b n^{2}+c n+d$, where $b, c$ and $d$ are integer constants and $n$ ranges over all integers, including negative integers.
Determine the possible values of $a_{2015} \cdot a_{2016}$ for a cubic sequence satisfying the condition in part (a). | [
"0"
] | 6 | [
"Algebra -> Algebra -> Polynomial Operations",
"Number Theory -> Other"
] |
1,659 | Algebra | null | [
"First we show that $f(y)>y$ for all $y \\in \\mathbb{R}^{+}$. Functional equation (1) yields $f(x+f(y))>f(x+y)$ and hence $f(y) \\neq y$ immediately. If $f(y)<y$ for some $y$, then setting $x=y-f(y)$ we get\n\n$$\nf(y)=f((y-f(y))+f(y))=f((y-f(y))+y)+f(y)>f(y),\n$$\n\ncontradiction. Therefore $f(y)>y$ for all $y \\... | false | null | Expression | null | Find all functions $f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ such that
$$
f(x+f(y))=f(x+y)+f(y)\tag{1}
$$
for all $x, y \in \mathbb{R}^{+}$. (Symbol $\mathbb{R}^{+}$denotes the set of all positive real numbers.) | [
"$f(x)=2 x$"
] | 6 | [
"Algebra -> Algebra -> Equations and Inequalities",
"Algebra -> Algebra -> Algebraic Expressions",
"Algebra -> Abstract Algebra -> Other"
] |
1,662 | Combinatorics | null | [
"It is easy to find $3 n$ such planes. For example, planes $x=i, y=i$ or $z=i$ $(i=1,2, \\ldots, n)$ cover the set $S$ but none of them contains the origin. Another such collection consists of all planes $x+y+z=k$ for $k=1,2, \\ldots, 3 n$.\n\nWe show that $3 n$ is the smallest possible number.\n\nLemma 1. Consider... | false | null | Expression | null | Let $n>1$ be an integer. In the space, consider the set
$$
S=\{(x, y, z) \mid x, y, z \in\{0,1, \ldots, n\}, x+y+z>0\}
$$
Find the smallest number of planes that jointly contain all $(n+1)^{3}-1$ points of $S$ but none of them passes through the origin. | [
"$3 n$"
] | 8 | [
"Geometry -> Solid Geometry -> 3D Shapes",
"Algebra -> Algebra -> Algebraic Expressions",
"Discrete Mathematics -> Combinatorics -> Other"
] |
1,664 | Combinatorics | null | [
"Suppose that the numbers $1,2, \\ldots, n$ are colored red and blue. Denote by $R$ and $B$ the sets of red and blue numbers, respectively; let $|R|=r$ and $|B|=b=n-r$. Call a triple $(x, y, z) \\in S \\times S \\times S$ monochromatic if $x, y, z$ have the same color, and bichromatic otherwise. Call a triple $(x, ... | true | null | Numerical | null | Find all positive integers $n$, for which the numbers in the set $S=\{1,2, \ldots, n\}$ can be colored red and blue, with the following condition being satisfied: the set $S \times S \times S$ contains exactly 2007 ordered triples $(x, y, z)$ such that (i) $x, y, z$ are of the same color and (ii) $x+y+z$ is divisible b... | [
"$69$,$84$"
] | 6 | [
"Discrete Mathematics -> Combinatorics",
"Number Theory -> Congruences",
"Algebra -> Algebraic Expressions"
] |
1,675 | Geometry | null | [
"Throughout the solution, triangles $A A_{1} D_{1}, B B_{1} A_{1}, C C_{1} B_{1}$, and $D D_{1} C_{1}$ will be referred to as border triangles. We will denote by $[\\mathcal{R}]$ the area of a region $\\mathcal{R}$.\n\nFirst, we show that $k \\geq 1$. Consider a triangle $A B C$ with unit area; let $A_{1}, B_{1}, K... | false | null | Numerical | null | Determine the smallest positive real number $k$ with the following property.
Let $A B C D$ be a convex quadrilateral, and let points $A_{1}, B_{1}, C_{1}$ and $D_{1}$ lie on sides $A B, B C$, $C D$ and $D A$, respectively. Consider the areas of triangles $A A_{1} D_{1}, B B_{1} A_{1}, C C_{1} B_{1}$, and $D D_{1} C_{1... | [
"$k=1$"
] | 7 | [
"Geometry -> Plane Geometry -> Area"
] |
1,678 | Number Theory | null | [
"Suppose that a pair $(k, n)$ satisfies the condition of the problem. Since $7^{k}-3^{n}$ is even, $k^{4}+n^{2}$ is also even, hence $k$ and $n$ have the same parity. If $k$ and $n$ are odd, then $k^{4}+n^{2} \\equiv 1+1=2(\\bmod 4)$, while $7^{k}-3^{n} \\equiv 7-3 \\equiv 0(\\bmod 4)$, so $k^{4}+n^{2}$ cannot be d... | false | null | Tuple | null | Find all pairs $(k, n)$ of positive integers for which $7^{k}-3^{n}$ divides $k^{4}+n^{2}$. | [
"$(2,4)$"
] | 6 | [
"Number Theory -> Factorization",
"Algebra -> Algebraic Expressions"
] |
1,681 | Number Theory | null | [
"Suppose that function $f: \\mathbb{N} \\rightarrow \\mathbb{N}$ satisfies the problem conditions.\n\nLemma. For any prime $p$ and any $x, y \\in \\mathbb{N}$, we have $x \\equiv y(\\bmod p)$ if and only if $f(x) \\equiv f(y)$ $(\\bmod p)$. Moreover, $p \\mid f(x)$ if and only if $p \\mid x$.\n\nProof. Consider an ... | false | null | Expression | null | Find all surjective functions $f: \mathbb{N} \rightarrow \mathbb{N}$ such that for every $m, n \in \mathbb{N}$ and every prime $p$, the number $f(m+n)$ is divisible by $p$ if and only if $f(m)+f(n)$ is divisible by $p$.
( $\mathbb{N}$ is the set of all positive integers.) | [
"$f(n)=n$"
] | 7 | [
"Algebra -> Algebra -> Equations and Inequalities",
"Number Theory -> Congruences",
"Number Theory -> Prime Numbers"
] |
1,687 | Algebra | null | [
"The given relation implies\n\n$$\nf\\left(f^{g(n)}(n)\\right)<f(n+1) \\quad \\text { for all } n\n\\tag{1}\n$$\n\nwhich will turn out to be sufficient to determine $f$.\n\nLet $y_{1}<y_{2}<\\ldots$ be all the values attained by $f$ (this sequence might be either finite or infinite). We will prove that for every po... | false | null | Expression | null | Determine all pairs $(f, g)$ of functions from the set of positive integers to itself that satisfy
$$
f^{g(n)+1}(n)+g^{f(n)}(n)=f(n+1)-g(n+1)+1
$$
for every positive integer $n$. Here, $f^{k}(n)$ means $\underbrace{f(f(\ldots f}_{k}(n) \ldots))$. | [
"$f(n)=n$, $g(n)=1$"
] | 8 | [
"Algebra -> Algebra -> Equations and Inequalities",
"Discrete Mathematics -> Combinatorics -> Other"
] |
1,694 | Combinatorics | null | [
"There are various examples showing that $k=3$ does indeed have the property under consideration. E.g. one can take\n\n$$\n\\begin{gathered}\nA_{1}=\\{1,2,3\\} \\cup\\{3 m \\mid m \\geq 4\\} \\\\\nA_{2}=\\{4,5,6\\} \\cup\\{3 m-1 \\mid m \\geq 4\\} \\\\\nA_{3}=\\{7,8,9\\} \\cup\\{3 m-2 \\mid m \\geq 4\\}\n\\end{gath... | false | null | Numerical | null | Determine the greatest positive integer $k$ that satisfies the following property: The set of positive integers can be partitioned into $k$ subsets $A_{1}, A_{2}, \ldots, A_{k}$ such that for all integers $n \geq 15$ and all $i \in\{1,2, \ldots, k\}$ there exist two distinct elements of $A_{i}$ whose sum is $n$. | [
"3"
] | 8 | [
"Discrete Mathematics -> Combinatorics",
"Discrete Mathematics -> Graph Theory"
] |
1,695 | Combinatorics | null | [
"For $m=1$ the answer is clearly correct, so assume $m>1$. In the sequel, the word collision will be used to denote meeting of exactly two ants, moving in opposite directions.\n\nIf at the beginning we place an ant on the southwest corner square facing east and an ant on the southeast corner square facing west, the... | false | null | Expression | null | Let $m$ be a positive integer and consider a checkerboard consisting of $m$ by $m$ unit squares. At the midpoints of some of these unit squares there is an ant. At time 0, each ant starts moving with speed 1 parallel to some edge of the checkerboard. When two ants moving in opposite directions meet, they both turn $90^... | [
"$\\frac{3 m}{2}-1$"
] | 7 | [
"Discrete Mathematics -> Combinatorics",
"Geometry -> Plane Geometry -> Area"
] |
1,697 | Combinatorics | null | [
"Let $m=39$, then $2011=52 m-17$. We begin with an example showing that there can exist 3986729 cells carrying the same positive number.\n\n<img_3188>\n\nTo describe it, we number the columns from the left to the right and the rows from the bottom to the top by $1,2, \\ldots, 2011$. We will denote each napkin by th... | false | null | Numerical | null | On a square table of 2011 by 2011 cells we place a finite number of napkins that each cover a square of 52 by 52 cells. In each cell we write the number of napkins covering it, and we record the maximal number $k$ of cells that all contain the same nonzero number. Considering all possible napkin configurations, what is... | [
"3986729"
] | 8 | [
"Discrete Mathematics -> Combinatorics",
"Algebra -> Algebraic Expressions"
] |
1,709 | Number Theory | null | [
"A pair $(a, n)$ satisfying the condition of the problem will be called a winning pair. It is straightforward to check that the pairs $(1,1),(3,1)$, and $(5,4)$ are winning pairs.\n\nNow suppose that $a$ is a positive integer not equal to 1,3 , and 5 . We will show that there are no winning pairs $(a, n)$ by distin... | true | null | Numerical | null | For each positive integer $k$, let $t(k)$ be the largest odd divisor of $k$. Determine all positive integers $a$ for which there exists a positive integer $n$ such that all the differences
$$
t(n+a)-t(n), \quad t(n+a+1)-t(n+1), \quad \ldots, \quad t(n+2 a-1)-t(n+a-1)
$$
are divisible by 4 . | [
"1,3,5"
] | 6 | [
"Number Theory -> Factorization",
"Number Theory -> Congruences"
] |
1,716 | Algebra | null | [
"Let $x_{2 i}=0, x_{2 i-1}=\\frac{1}{2}$ for all $i=1, \\ldots, 50$. Then we have $S=50 \\cdot\\left(\\frac{1}{2}\\right)^{2}=\\frac{25}{2}$. So, we are left to show that $S \\leq \\frac{25}{2}$ for all values of $x_{i}$ 's satisfying the problem conditions.\n\nConsider any $1 \\leq i \\leq 50$. By the problem cond... | false | null | Numerical | null | Let $x_{1}, \ldots, x_{100}$ be nonnegative real numbers such that $x_{i}+x_{i+1}+x_{i+2} \leq 1$ for all $i=1, \ldots, 100$ (we put $x_{101}=x_{1}, x_{102}=x_{2}$ ). Find the maximal possible value of the sum
$$
S=\sum_{i=1}^{100} x_{i} x_{i+2}
$$ | [
"$\\frac{25}{2}$"
] | 5 | [
"Algebra -> Algebra -> Inequalities",
"Algebra -> Algebraic Expressions -> Other",
"Discrete Mathematics -> Combinatorics -> Other"
] |
1,718 | Algebra | null | [
"By substituting $y=1$, we get\n\n$$\nf\\left(f(x)^{2}\\right)=x^{3} f(x)\\tag{2}\n$$\n\nThen, whenever $f(x)=f(y)$, we have\n\n$$\nx^{3}=\\frac{f\\left(f(x)^{2}\\right)}{f(x)}=\\frac{f\\left(f(y)^{2}\\right)}{f(y)}=y^{3}\n$$\n\nwhich implies $x=y$, so the function $f$ is injective.\n\nNow replace $x$ by $x y$ in (... | false | null | Expression | null | Denote by $\mathbb{Q}^{+}$the set of all positive rational numbers. Determine all functions $f: \mathbb{Q}^{+} \rightarrow \mathbb{Q}^{+}$ which satisfy the following equation for all $x, y \in \mathbb{Q}^{+}$:
$$
f\left(f(x)^{2} y\right)=x^{3} f(x y)
\tag{1}
$$ | [
"$f(x)=\\frac{1}{x}$"
] | 7 | [
"Algebra -> Algebra -> Equations and Inequalities",
"Algebra -> Abstract Algebra -> Group Theory",
"Number Theory -> Factorization"
] |
1,723 | Combinatorics | null | [
"When speaking about the diagonal of a square, we will always mean the main diagonal.\n\nLet $M_{N}$ be the smallest positive integer satisfying the problem condition. First, we show that $M_{N}>2^{N-2}$. Consider the collection of all $2^{N-2}$ flags having yellow first squares and blue second ones. Obviously, bot... | false | null | Expression | null | On some planet, there are $2^{N}$ countries $(N \geq 4)$. Each country has a flag $N$ units wide and one unit high composed of $N$ fields of size $1 \times 1$, each field being either yellow or blue. No two countries have the same flag.
We say that a set of $N$ flags is diverse if these flags can be arranged into an $... | [
"$M=2^{N-2}+1$"
] | 8 | [
"Discrete Mathematics -> Combinatorics",
"Discrete Mathematics -> Graph Theory"
] |
1,724 | Combinatorics | null | [
"Suppose that we have an arrangement satisfying the problem conditions. Divide the board into $2 \\times 2$ pieces; we call these pieces blocks. Each block can contain not more than one king (otherwise these two kings would attack each other); hence, by the pigeonhole principle each block must contain exactly one k... | false | null | Numerical | null | 2500 chess kings have to be placed on a $100 \times 100$ chessboard so that
(i) no king can capture any other one (i.e. no two kings are placed in two squares sharing a common vertex);
(ii) each row and each column contains exactly 25 kings.
Find the number of such arrangements. (Two arrangements differing by rotati... | [
"2"
] | 8 | [
"Discrete Mathematics -> Combinatorics",
"Applied Mathematics -> Math Word Problems"
] |
1,736 | Number Theory | null | [
"Suppose that for some $n$ there exist the desired numbers; we may assume that $s_{1}<s_{2}<\\cdots<s_{n}$. Surely $s_{1}>1$ since otherwise $1-\\frac{1}{s_{1}}=0$. So we have $2 \\leq s_{1} \\leq s_{2}-1 \\leq \\cdots \\leq s_{n}-(n-1)$, hence $s_{i} \\geq i+1$ for each $i=1, \\ldots, n$. Therefore\n\n$$\n\\begin{... | false | null | Numerical | null | Find the least positive integer $n$ for which there exists a set $\left\{s_{1}, s_{2}, \ldots, s_{n}\right\}$ consisting of $n$ distinct positive integers such that
$$
\left(1-\frac{1}{s_{1}}\right)\left(1-\frac{1}{s_{2}}\right) \ldots\left(1-\frac{1}{s_{n}}\right)=\frac{51}{2010}
$$ | [
"39"
] | 5 | [
"Number Theory -> Other",
"Algebra -> Algebraic Expressions -> Other"
] |
1,737 | Number Theory | null | [
"For fixed values of $n$, the equation (1) is a simple quadratic equation in $m$. For $n \\leq 5$ the solutions are listed in the following table.\n\n| case | equation | discriminant | integer roots |\n| :--- | :--- | :--- | :--- |\n| $n=0$ | $m^{2}-m+2=0$ | -7 | none |\n| $n=1$ | $m^{2}-3 m+6=0$ | -15 | none |\n| ... | true | null | Tuple | null | Find all pairs $(m, n)$ of nonnegative integers for which
$$
m^{2}+2 \cdot 3^{n}=m\left(2^{n+1}-1\right)
\tag{1}
$$ | [
"$(6,3),(9,3),(9,5),(54,5)$"
] | 5 | [
"Algebra -> Algebra -> Equations and Inequalities",
"Number Theory -> Factorization"
] |
1,738 | Number Theory | null | [
"The equality $x^{2}+7=x^{2}+2^{2}+1^{2}+1^{2}+1^{2}$ shows that $n \\leq 5$. It remains to show that $x^{2}+7$ is not a sum of four (or less) squares of polynomials with rational coefficients.\n\nSuppose by way of contradiction that $x^{2}+7=f_{1}(x)^{2}+f_{2}(x)^{2}+f_{3}(x)^{2}+f_{4}(x)^{2}$, where the coefficie... | false | null | Numerical | null | Find the smallest number $n$ such that there exist polynomials $f_{1}, f_{2}, \ldots, f_{n}$ with rational coefficients satisfying
$$
x^{2}+7=f_{1}(x)^{2}+f_{2}(x)^{2}+\cdots+f_{n}(x)^{2} .
$$ | [
"5"
] | 6 | [
"Algebra -> Algebra -> Equations and Inequalities",
"Algebra -> Abstract Algebra -> Field Theory",
"Number Theory -> Factorization"
] |
1,747 | Algebra | null | [
"We first consider the cubic polynomial\n\n$$\nP(t)=t b\\left(t^{2}-b^{2}\\right)+b c\\left(b^{2}-c^{2}\\right)+c t\\left(c^{2}-t^{2}\\right) .\n$$\n\nIt is easy to check that $P(b)=P(c)=P(-b-c)=0$, and therefore\n\n$$\nP(t)=(b-c)(t-b)(t-c)(t+b+c)\n$$\n\nsince the cubic coefficient is $b-c$. The left-hand side of t... | false | null | Numerical | null | Determine the smallest number $M$ such that the inequality
$$
\left|a b\left(a^{2}-b^{2}\right)+b c\left(b^{2}-c^{2}\right)+c a\left(c^{2}-a^{2}\right)\right| \leq M\left(a^{2}+b^{2}+c^{2}\right)^{2}
$$
holds for all real numbers $a, b, c$. | [
"$M=\\frac{9}{32} \\sqrt{2}$"
] | 7 | [
"Algebra -> Algebra -> Equations and Inequalities"
] |
1,750 | Combinatorics | null | [
"Call an isosceles triangle odd if it has two odd sides. Suppose we are given a dissection as in the problem statement. A triangle in the dissection which is odd and isosceles will be called iso-odd for brevity.\n\nLemma. Let $A B$ be one of dissecting diagonals and let $\\mathcal{L}$ be the shorter part of the bou... | false | null | Numerical | null | A diagonal of a regular 2006-gon is called odd if its endpoints divide the boundary into two parts, each composed of an odd number of sides. Sides are also regarded as odd diagonals.
Suppose the 2006-gon has been dissected into triangles by 2003 nonintersecting diagonals. Find the maximum possible number of isosceles ... | [
"$1003$"
] | 8 | [
"Geometry -> Plane Geometry -> Polygons",
"Discrete Mathematics -> Combinatorics -> Other"
] |
1,760 | Geometry | null | [
"Let $K$ be the intersection point of lines $J C$ and $A_{1} B_{1}$. Obviously $J C \\perp A_{1} B_{1}$ and since $A_{1} B_{1} \\perp A B$, the lines $J K$ and $C_{1} D$ are parallel and equal. From the right triangle $B_{1} C J$ we obtain $J C_{1}^{2}=J B_{1}^{2}=J C \\cdot J K=J C \\cdot C_{1} D$ from which we in... | true | ^{\circ} | Numerical | null | In triangle $A B C$, let $J$ be the centre of the excircle tangent to side $B C$ at $A_{1}$ and to the extensions of sides $A C$ and $A B$ at $B_{1}$ and $C_{1}$, respectively. Suppose that the lines $A_{1} B_{1}$ and $A B$ are perpendicular and intersect at $D$. Let $E$ be the foot of the perpendicular from $C_{1}$ to... | [
"$\\angle B E A_{1}=90$,$\\angle A E B_{1}=90$"
] | 7 | [
"Geometry -> Plane Geometry -> Angles",
"Geometry -> Plane Geometry -> Polygons",
"Geometry -> Plane Geometry -> Triangulations"
] |
1,766 | Number Theory | null | [
"If $(x, y)$ is a solution then obviously $x \\geq 0$ and $(x,-y)$ is a solution too. For $x=0$ we get the two solutions $(0,2)$ and $(0,-2)$.\n\nNow let $(x, y)$ be a solution with $x>0$; without loss of generality confine attention to $y>0$. The equation rewritten as\n\n$$\n2^{x}\\left(1+2^{x+1}\\right)=(y-1)(y+1... | true | null | Tuple | null | Determine all pairs $(x, y)$ of integers satisfying the equation
$$
1+2^{x}+2^{2 x+1}=y^{2}
$$ | [
"$(0,2),(0,-2),(4,23),(4,-23)$"
] | 5 | [
"Algebra -> Algebra -> Equations and Inequalities",
"Number Theory -> Factorization"
] |
1,775 | Algebra | null | [
"Suppose that $2^{k} \\leqslant n<2^{k+1}$ with some nonnegative integer $k$. First we show a permutation $\\left(a_{1}, a_{2}, \\ldots, a_{n}\\right)$ such that $\\left\\lfloor\\frac{a_{1}}{1}\\right\\rfloor+\\left\\lfloor\\frac{a_{2}}{2}\\right\\rfloor+\\cdots+\\left\\lfloor\\frac{a_{n}}{n}\\right\\rfloor=k+1$; t... | false | null | Expression | null | Given a positive integer $n$, find the smallest value of $\left\lfloor\frac{a_{1}}{1}\right\rfloor+\left\lfloor\frac{a_{2}}{2}\right\rfloor+\cdots+\left\lfloor\frac{a_{n}}{n}\right\rfloor$ over all permutations $\left(a_{1}, a_{2}, \ldots, a_{n}\right)$ of $(1,2, \ldots, n)$. | [
"$\\left\\lfloor\\log _{2} n\\right\\rfloor+1$"
] | 6 | [
"Algebra -> Algebra -> Algebraic Expressions",
"Algebra -> Algebra -> Equations and Inequalities",
"Discrete Mathematics -> Combinatorics -> Other"
] |
1,782 | Combinatorics | null | [
"First suppose that there are $n(n-1)-1$ marbles. Then for one of the colours, say blue, there are at most $n-2$ marbles, which partition the non-blue marbles into at most $n-2$ groups with at least $(n-1)^{2}>n(n-2)$ marbles in total. Thus one of these groups contains at least $n+1$ marbles and this group does not... | false | null | Expression | null | Let $n \geqslant 3$ be an integer. An integer $m \geqslant n+1$ is called $n$-colourful if, given infinitely many marbles in each of $n$ colours $C_{1}, C_{2}, \ldots, C_{n}$, it is possible to place $m$ of them around a circle so that in any group of $n+1$ consecutive marbles there is at least one marble of colour $C_... | [
"$m_{\\max }=n^{2}-n-1$"
] | 7 | [
"Discrete Mathematics -> Combinatorics"
] |
1,789 | Combinatorics | null | [
"Non-existence of a larger table. Let us consider some fixed row in the table, and let us replace (for $k=1,2, \\ldots, 50$ ) each of two numbers $2 k-1$ and $2 k$ respectively by the symbol $x_{k}$. The resulting pattern is an arrangement of 50 symbols $x_{1}, x_{2}, \\ldots, x_{50}$, where every symbol occurs exa... | false | null | Expression | null | Determine the largest $N$ for which there exists a table $T$ of integers with $N$ rows and 100 columns that has the following properties:
(i) Every row contains the numbers 1,2, ., 100 in some order.
(ii) For any two distinct rows $r$ and $s$, there is a column $c$ such that $|T(r, c)-T(s, c)| \geqslant 2$.
Here $T(... | [
"$\\frac{100!}{2^{50}}$"
] | 8 | [
"Discrete Mathematics -> Combinatorics",
"Algebra -> Algebraic Expressions -> Other"
] |
1,798 | Number Theory | null | [
"As $b \\equiv-a^{2}-3\\left(\\bmod a^{2}+b+3\\right)$, the numerator of the given fraction satisfies\n\n$$\na b+3 b+8 \\equiv a\\left(-a^{2}-3\\right)+3\\left(-a^{2}-3\\right)+8 \\equiv-(a+1)^{3} \\quad\\left(\\bmod a^{2}+b+3\\right)\n$$\n\nAs $a^{2}+b+3$ is not divisible by $p^{3}$ for any prime $p$, if $a^{2}+b+... | false | null | Numerical | null | Determine all integers $n \geqslant 1$ for which there exists a pair of positive integers $(a, b)$ such that no cube of a prime divides $a^{2}+b+3$ and
$$
\frac{a b+3 b+8}{a^{2}+b+3}=n
$$ | [
"2"
] | 6 | [
"Algebra -> Algebra -> Equations and Inequalities",
"Number Theory -> Factorization"
] |
1,800 | Number Theory | null | [
"For $i=1,2, \\ldots, k$ let $d_{1}+\\ldots+d_{i}=s_{i}^{2}$, and define $s_{0}=0$ as well. Obviously $0=s_{0}<s_{1}<s_{2}<\\ldots<s_{k}$, so\n\n$$\ns_{i} \\geqslant i \\quad \\text { and } \\quad d_{i}=s_{i}^{2}-s_{i-1}^{2}=\\left(s_{i}+s_{i-1}\\right)\\left(s_{i}-s_{i-1}\\right) \\geqslant s_{i}+s_{i-1} \\geqslan... | true | null | Numerical | null | Find all positive integers $n$ with the following property: the $k$ positive divisors of $n$ have a permutation $\left(d_{1}, d_{2}, \ldots, d_{k}\right)$ such that for every $i=1,2, \ldots, k$, the number $d_{1}+\cdots+d_{i}$ is a perfect square. | [
"1,3"
] | 6 | [
"Number Theory -> Factorization",
"Algebra -> Prealgebra -> Integers"
] |
1,807 | Algebra | null | [
"Call a number $q$ good if every number in the second line appears in the third line unconditionally. We first show that the numbers 0 and \\pm 2 are good. The third line necessarily contains 0 , so 0 is good. For any two numbers $a, b$ in the first line, write $a=x-y$ and $b=u-v$, where $x, y, u, v$ are (not neces... | true | null | Numerical | null | Let $q$ be a real number. Gugu has a napkin with ten distinct real numbers written on it, and he writes the following three lines of real numbers on the blackboard:
- In the first line, Gugu writes down every number of the form $a-b$, where $a$ and $b$ are two (not necessarily distinct) numbers on his napkin.
- In the... | [
"$-2,0,2$"
] | 8 | [
"Algebra -> Algebraic Expressions -> Other",
"Number Theory -> Other"
] |
1,810 | Algebra | null | [
"First of all, we show that we may not take a larger constant $K$. Let $t$ be a positive number, and take $x_{2}=x_{3}=\\cdots=t$ and $x_{1}=-1 /(2 t)$. Then, every product $x_{i} x_{j}(i \\neq j)$ is equal to either $t^{2}$ or $-1 / 2$. Hence, for every permutation $y_{i}$ of the $x_{i}$, we have\n\n$$\ny_{1} y_{2... | false | null | Expression | null | An integer $n \geqslant 3$ is given. We call an $n$-tuple of real numbers $\left(x_{1}, x_{2}, \ldots, x_{n}\right)$ Shiny if for each permutation $y_{1}, y_{2}, \ldots, y_{n}$ of these numbers we have
$$
\sum_{i=1}^{n-1} y_{i} y_{i+1}=y_{1} y_{2}+y_{2} y_{3}+y_{3} y_{4}+\cdots+y_{n-1} y_{n} \geqslant-1
$$
Find the l... | [
"$-(n-1) / 2$"
] | 8 | [
"Algebra -> Algebra -> Equations and Inequalities",
"Discrete Mathematics -> Combinatorics",
"Number Theory -> Other"
] |
1,818 | Combinatorics | null | [
"Call a $n \\times n \\times 1$ box an $x$-box, a $y$-box, or a $z$-box, according to the direction of its short side. Let $C$ be the number of colors in a valid configuration. We start with the upper bound for $C$.\n\nLet $\\mathcal{C}_{1}, \\mathcal{C}_{2}$, and $\\mathcal{C}_{3}$ be the sets of colors which appe... | false | null | Expression | null | Let $n>1$ be an integer. An $n \times n \times n$ cube is composed of $n^{3}$ unit cubes. Each unit cube is painted with one color. For each $n \times n \times 1$ box consisting of $n^{2}$ unit cubes (of any of the three possible orientations), we consider the set of the colors present in that box (each color is listed... | [
"$\\frac{n(n+1)(2 n+1)}{6}$"
] | 8 | [
"Discrete Mathematics -> Combinatorics",
"Geometry -> Solid Geometry -> 3D Shapes"
] |
1,820 | Combinatorics | null | [
"We always identify a butterfly with the lattice point it is situated at. For two points $p$ and $q$, we write $p \\geqslant q$ if each coordinate of $p$ is at least the corresponding coordinate of $q$. Let $O$ be the origin, and let $\\mathcal{Q}$ be the set of initially occupied points, i.e., of all lattice point... | false | null | Expression | null | Let $n$ be a given positive integer. In the Cartesian plane, each lattice point with nonnegative coordinates initially contains a butterfly, and there are no other butterflies. The neighborhood of a lattice point $c$ consists of all lattice points within the axis-aligned $(2 n+1) \times$ $(2 n+1)$ square centered at $c... | [
"$n^{2}+1$"
] | 9 | [
"Discrete Mathematics -> Combinatorics -> Other",
"Geometry -> Plane Geometry -> Polygons",
"Algebra -> Algebraic Expressions -> Other"
] |
1,828 | Geometry | null | [
"First, consider a particular arrangement of circles $C_{1}, C_{2}, \\ldots, C_{n}$ where all the centers are aligned and each $C_{i}$ is eclipsed from the other circles by its neighbors - for example, taking $C_{i}$ with center $\\left(i^{2}, 0\\right)$ and radius $i / 2$ works. Then the only tangent segments that... | false | null | Numerical | null | There are 2017 mutually external circles drawn on a blackboard, such that no two are tangent and no three share a common tangent. A tangent segment is a line segment that is a common tangent to two circles, starting at one tangent point and ending at the other one. Luciano is drawing tangent segments on the blackboard,... | [
"6048"
] | 8 | [
"Geometry -> Plane Geometry -> Other",
"Discrete Mathematics -> Graph Theory",
"Discrete Mathematics -> Combinatorics"
] |
1,832 | Number Theory | null | [
"First notice that $x \\in \\mathbb{Q}$ is short if and only if there are exponents $a, b \\geqslant 0$ such that $2^{a} \\cdot 5^{b} \\cdot x \\in \\mathbb{Z}$. In fact, if $x$ is short, then $x=\\frac{n}{10^{k}}$ for some $k$ and we can take $a=b=k$; on the other hand, if $2^{a} \\cdot 5^{b} \\cdot x=q \\in \\mat... | false | null | Numerical | null | Call a rational number short if it has finitely many digits in its decimal expansion. For a positive integer $m$, we say that a positive integer $t$ is $m$-tastic if there exists a number $c \in\{1,2,3, \ldots, 2017\}$ such that $\frac{10^{t}-1}{c \cdot m}$ is short, and such that $\frac{10^{k}-1}{c \cdot m}$ is not sh... | [
"807"
] | 8 | [
"Number Theory -> Congruences",
"Number Theory -> Factorization"
] |
1,833 | Number Theory | null | [
"Let $M=(p+q)^{p-q}(p-q)^{p+q}-1$, which is relatively prime with both $p+q$ and $p-q$. Denote by $(p-q)^{-1}$ the multiplicative inverse of $(p-q)$ modulo $M$.\n\nBy eliminating the term -1 in the numerator,\n\n$$\n(p+q)^{p+q}(p-q)^{p-q}-1 \\equiv(p+q)^{p-q}(p-q)^{p+q}-1 \\quad(\\bmod M)\\\\\n(p+q)^{2 q} \\equiv... | false | null | Tuple | null | Find all pairs $(p, q)$ of prime numbers with $p>q$ for which the number
$$
\frac{(p+q)^{p+q}(p-q)^{p-q}-1}{(p+q)^{p-q}(p-q)^{p+q}-1}
$$
is an integer. | [
"$(3,2)$"
] | 6 | [
"Number Theory -> Prime Numbers",
"Algebra -> Algebraic Expressions",
"Algebra -> Equations and Inequalities"
] |
1,834 | Number Theory | null | [
"For $n=1, a_{1} \\in \\mathbb{Z}_{>0}$ and $\\frac{1}{a_{1}} \\in \\mathbb{Z}_{>0}$ if and only if $a_{1}=1$. Next we show that\n\n(i) There are finitely many $(x, y) \\in \\mathbb{Q}_{>0}^{2}$ satisfying $x+y \\in \\mathbb{Z}$ and $\\frac{1}{x}+\\frac{1}{y} \\in \\mathbb{Z}$\n\nWrite $x=\\frac{a}{b}$ and $y=\\fra... | false | null | Numerical | null | Find the smallest positive integer $n$, or show that no such $n$ exists, with the following property: there are infinitely many distinct $n$-tuples of positive rational numbers $\left(a_{1}, a_{2}, \ldots, a_{n}\right)$ such that both
$$
a_{1}+a_{2}+\cdots+a_{n} \quad \text { and } \quad \frac{1}{a_{1}}+\frac{1}{a_{2}... | [
"3"
] | 8 | [
"Algebra -> Algebra -> Equations and Inequalities",
"Number Theory -> Other"
] |
1,838 | Algebra | null | [
"We first show that $C \\leqslant \\frac{1}{2}$. For any positive real numbers $a_{1} \\leqslant a_{2} \\leqslant a_{3} \\leqslant a_{4} \\leqslant a_{5}$, consider the five fractions\n\n$$\n\\frac{a_{1}}{a_{2}}, \\frac{a_{3}}{a_{4}}, \\frac{a_{1}}{a_{5}}, \\frac{a_{2}}{a_{3}}, \\frac{a_{4}}{a_{5}}\\tag{2}\n$$\n\nE... | false | null | Numerical | null | Find the smallest real constant $C$ such that for any positive real numbers $a_{1}, a_{2}, a_{3}, a_{4}$ and $a_{5}$ (not necessarily distinct), one can always choose distinct subscripts $i, j, k$ and $l$ such that
$$
\left|\frac{a_{i}}{a_{j}}-\frac{a_{k}}{a_{l}}\right| \leqslant C .\tag{1}
$$ | [
"$\\frac{1}{2}$"
] | 5 | [
"Algebra -> Algebra -> Equations and Inequalities",
"Discrete Mathematics -> Combinatorics"
] |
1,843 | Algebra | null | [
"Since there are 2016 common linear factors on both sides, we need to erase at least 2016 factors. We claim that the equation has no real roots if we erase all factors $(x-k)$ on the left-hand side with $k \\equiv 2,3(\\bmod 4)$, and all factors $(x-m)$ on the right-hand side with $m \\equiv 0,1(\\bmod 4)$. Therefo... | false | null | Numerical | null | The equation
$$
(x-1)(x-2) \cdots(x-2016)=(x-1)(x-2) \cdots(x-2016)
$$
is written on the board. One tries to erase some linear factors from both sides so that each side still has at least one factor, and the resulting equation has no real roots. Find the least number of linear factors one needs to erase to achieve th... | [
"2016"
] | 5 | [
"Algebra -> Algebra -> Equations and Inequalities",
"Algebra -> Algebra -> Polynomial Operations"
] |
1,845 | Algebra | null | [
"We first show that $a=\\frac{4}{9}$ is admissible. For each $2 \\leqslant k \\leqslant n$, by the CauchySchwarz Inequality, we have\n\n$$\n\\left(x_{k-1}+\\left(x_{k}-x_{k-1}\\right)\\right)\\left(\\frac{(k-1)^{2}}{x_{k-1}}+\\frac{3^{2}}{x_{k}-x_{k-1}}\\right) \\geqslant(k-1+3)^{2},\n$$\n\nwhich can be rewritten a... | false | null | Numerical | null | Determine the largest real number $a$ such that for all $n \geqslant 1$ and for all real numbers $x_{0}, x_{1}, \ldots, x_{n}$ satisfying $0=x_{0}<x_{1}<x_{2}<\cdots<x_{n}$, we have
$$
\frac{1}{x_{1}-x_{0}}+\frac{1}{x_{2}-x_{1}}+\cdots+\frac{1}{x_{n}-x_{n-1}} \geqslant a\left(\frac{2}{x_{1}}+\frac{3}{x_{2}}+\cdots+\fr... | [
"$\\frac{4}{9}$"
] | 7 | [
"Algebra -> Algebra -> Inequalities",
"Algebra -> Algebraic Expressions -> Other",
"Discrete Mathematics -> Combinatorics -> Other"
] |
1,847 | Combinatorics | null | [
"Suppose all positive divisors of $n$ can be arranged into a rectangular table of size $k \\times l$ where the number of rows $k$ does not exceed the number of columns $l$. Let the sum of numbers in each column be $s$. Since $n$ belongs to one of the columns, we have $s \\geqslant n$, where equality holds only when... | false | null | Numerical | null | Find all positive integers $n$ for which all positive divisors of $n$ can be put into the cells of a rectangular table under the following constraints:
- each cell contains a distinct divisor;
- the sums of all rows are equal; and
- the sums of all columns are equal. | [
"1"
] | 7 | [
"Number Theory -> Factorization",
"Number Theory -> Other"
] |
1,854 | Combinatorics | null | [
"We first construct an example of marking $2 n$ cells satisfying the requirement. Label the rows and columns $1,2, \\ldots, 2 n$ and label the cell in the $i$-th row and the $j$-th column $(i, j)$.\n\nFor $i=1,2, \\ldots, n$, we mark the cells $(i, i)$ and $(i, i+1)$. We claim that the required partition exists and... | false | null | Expression | null | Let $n$ be a positive integer. Determine the smallest positive integer $k$ with the following property: it is possible to mark $k$ cells on a $2 n \times 2 n$ board so that there exists a unique partition of the board into $1 \times 2$ and $2 \times 1$ dominoes, none of which contains two marked cells. | [
"$2 n$"
] | 8 | [
"Discrete Mathematics -> Combinatorics",
"Geometry -> Plane Geometry -> Polygons"
] |
1,866 | Number Theory | null | [
"We have the following observations.\n\n(i) $(P(n), P(n+1))=1$ for any $n$.\n\nWe have $(P(n), P(n+1))=\\left(n^{2}+n+1, n^{2}+3 n+3\\right)=\\left(n^{2}+n+1,2 n+2\\right)$. Noting that $n^{2}+n+1$ is odd and $\\left(n^{2}+n+1, n+1\\right)=(1, n+1)=1$, the claim follows.\n\n(ii) $(P(n), P(n+2))=1$ for $n \\not \\eq... | false | null | Numerical | null | Define $P(n)=n^{2}+n+1$. For any positive integers $a$ and $b$, the set
$$
\{P(a), P(a+1), P(a+2), \ldots, P(a+b)\}
$$
is said to be fragrant if none of its elements is relatively prime to the product of the other elements. Determine the smallest size of a fragrant set. | [
"6"
] | 6 | [
"Number Theory -> Greatest Common Divisors (GCD)",
"Number Theory -> Congruences",
"Algebra -> Algebraic Expressions"
] |
1,869 | Number Theory | null | [
"It is given that\n\n$$\nf(m)+f(n)-m n \\mid m f(m)+n f(n) .\n\\tag{1}\n$$\n\nTaking $m=n=1$ in (1), we have $2 f(1)-1 \\mid 2 f(1)$. Then $2 f(1)-1 \\mid 2 f(1)-(2 f(1)-1)=1$ and hence $f(1)=1$.\n\nLet $p \\geqslant 7$ be a prime. Taking $m=p$ and $n=1$ in (1), we have $f(p)-p+1 \\mid p f(p)+1$ and hence\n\n$$\nf(... | false | null | Expression | null | Denote by $\mathbb{N}$ the set of all positive integers. Find all functions $f: \mathbb{N} \rightarrow \mathbb{N}$ such that for all positive integers $m$ and $n$, the integer $f(m)+f(n)-m n$ is nonzero and divides $m f(m)+n f(n)$. | [
"$f(n)=n^{2}$"
] | 8 | [
"Algebra -> Algebra -> Equations and Inequalities",
"Number Theory -> Other"
] |
1,872 | Algebra | null | [
"First of all, assume that $a_{n}<N / 2$ satisfies the condition. Take $x=1+t$ for $t>0$, we should have\n\n$$\n\\frac{(1+t)^{2 N}+1}{2} \\leqslant\\left(1+t+a_{n} t^{2}\\right)^{N}\n$$\n\nExpanding the brackets we get\n\n$$\n\\left(1+t+a_{n} t^{2}\\right)^{N}-\\frac{(1+t)^{2 N}+1}{2}=\\left(N a_{n}-\\frac{N^{2}}{2... | false | null | Expression | null | Let $n$ be a positive integer, and set $N=2^{n}$. Determine the smallest real number $a_{n}$ such that, for all real $x$,
$$
\sqrt[N]{\frac{x^{2 N}+1}{2}} \leqslant a_{n}(x-1)^{2}+x
$$ | [
"$\\frac{N}{2}$"
] | 8 | [
"Algebra -> Algebra -> Equations and Inequalities",
"Algebra -> Polynomial Operations",
"Mathematics -> Other"
] |
1,873 | Algebra | null | [
"We start by showing that $n \\leqslant 4$, i.e., any monomial $f=x^{i} y^{j} z^{k}$ with $i+j+k \\geqslant 4$ belongs to $\\mathcal{B}$. Assume that $i \\geqslant j \\geqslant k$, the other cases are analogous.\n\nLet $x+y+z=p, x y+y z+z x=q$ and $x y z=r$. Then\n\n$$\n0=(x-x)(x-y)(x-z)=x^{3}-p x^{2}+q x-r\n$$\n\n... | false | null | Numerical | null | Let $\mathcal{A}$ denote the set of all polynomials in three variables $x, y, z$ with integer coefficients. Let $\mathcal{B}$ denote the subset of $\mathcal{A}$ formed by all polynomials which can be expressed as
$$
(x+y+z) P(x, y, z)+(x y+y z+z x) Q(x, y, z)+x y z R(x, y, z)
$$
with $P, Q, R \in \mathcal{A}$. Find t... | [
"4"
] | 6 | [
"Algebra -> Algebra -> Algebraic Expressions"
] |
1,874 | Algebra | null | [
"To show that $S \\geqslant 8$, apply the AM-GM inequality twice as follows:\n\n$$\n\\left(\\frac{a}{b}+\\frac{c}{d}\\right)+\\left(\\frac{b}{c}+\\frac{d}{a}\\right) \\geqslant 2 \\sqrt{\\frac{a c}{b d}}+2 \\sqrt{\\frac{b d}{a c}}=\\frac{2(a c+b d)}{\\sqrt{a b c d}}=\\frac{2(a+c)(b+d)}{\\sqrt{a b c d}} \\geqslant 2... | false | null | Numerical | null | Suppose that $a, b, c, d$ are positive real numbers satisfying $(a+c)(b+d)=a c+b d$. Find the smallest possible value of
$$
S=\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}
$$ | [
"8"
] | 5 | [
"Algebra -> Algebra -> Equations and Inequalities",
"Algebra -> Algebraic Expressions",
"Algebra -> Intermediate Algebra -> Other"
] |
1,879 | Algebra | null | [
"A straightforward check shows that $f(x)=x+1$ satisfies (*). We divide the proof of the converse statement into a sequence of steps.\n\nStep 1: $f$ is injective.\n\nPut $x=1$ in (*) and rearrange the terms to get\n\n$$\ny=f(1) f(y)+1-f(1+f(y))\n$$\n\nTherefore, if $f\\left(y_{1}\\right)=f\\left(y_{2}\\right)$, the... | false | null | Expression | null | Let $\mathbb{R}^{+}$be the set of positive real numbers. Determine all functions $f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ such that, for all positive real numbers $x$ and $y$,
$$
f(x+f(x y))+y=f(x) f(y)+1
\tag{*}
$$ | [
"$f(x)=x+1$"
] | 8 | [
"Algebra -> Algebra -> Equations and Inequalities",
"Algebra -> Algebra -> Algebraic Expressions"
] |
1,882 | Combinatorics | null | [
"We start with showing that for any $k \\leqslant n^{2}-n$ there may be no pair of checkpoints linked by both companies. Clearly, it suffices to provide such an example for $k=n^{2}-n$.\n\nLet company $A$ connect the pairs of checkpoints of the form $(i, i+1)$, where $n \\nmid i$. Then all pairs of checkpoints $(i,... | false | null | Expression | null | Let $n$ be an integer with $n \geqslant 2$. On a slope of a mountain, $n^{2}$ checkpoints are marked, numbered from 1 to $n^{2}$ from the bottom to the top. Each of two cable car companies, $A$ and $B$, operates $k$ cable cars numbered from 1 to $k$; each cable car provides a transfer from some checkpoint to a higher o... | [
"$n^{2}-n+1$"
] | 7 | [
"Discrete Mathematics -> Combinatorics",
"Algebra -> Prealgebra -> Integers"
] |
1,883 | Combinatorics | null | [
"First we show that if a set $S \\subset \\mathbb{Z}$ satisfies the conditions then $|S| \\geqslant \\frac{n}{2}+1$.\n\nLet $d=\\lceil n / 2\\rceil$, so $n \\leqslant 2 d \\leqslant n+1$. In order to prove that $|S| \\geqslant d+1$, construct a graph as follows. Let the vertices of the graph be the elements of $S$.... | false | null | Expression | null | The Fibonacci numbers $F_{0}, F_{1}, F_{2}, \ldots$ are defined inductively by $F_{0}=0, F_{1}=1$, and $F_{n+1}=F_{n}+F_{n-1}$ for $n \geqslant 1$. Given an integer $n \geqslant 2$, determine the smallest size of a set $S$ of integers such that for every $k=2,3, \ldots, n$ there exist some $x, y \in S$ such that $x-y=F... | [
"$\\lceil n / 2\\rceil+1$"
] | 7 | [
"Discrete Mathematics -> Combinatorics",
"Number Theory -> Other"
] |
1,887 | Combinatorics | null | [
"For a positive integer $n$, we denote by $S_{2}(n)$ the sum of digits in its binary representation. We prove that, in fact, if a board initially contains an even number $n>1$ of ones, then $A$ can guarantee to obtain $S_{2}(n)$, but not more, cookies. The binary representation of 2020 is $2020=\\overline{111111001... | false | null | Numerical | null | Players $A$ and $B$ play a game on a blackboard that initially contains 2020 copies of the number 1. In every round, player $A$ erases two numbers $x$ and $y$ from the blackboard, and then player $B$ writes one of the numbers $x+y$ and $|x-y|$ on the blackboard. The game terminates as soon as, at the end of some round,... | [
"7"
] | 7 | [
"Discrete Mathematics -> Combinatorics",
"Number Theory -> Other"
] |
1,914 | Combinatorics | null | [
"We represent the problem using a directed graph $G_{n}$ whose vertices are the length- $n$ strings of $H$ 's and $T$ 's. The graph features an edge from each string to its successor (except for $T T \\cdots T T$, which has no successor). We will also write $\\bar{H}=T$ and $\\bar{T}=H$.\n\nThe graph $G_{0}$ consis... | false | null | Expression | null | Let $n$ be a positive integer. Harry has $n$ coins lined up on his desk, each showing heads or tails. He repeatedly does the following operation: if there are $k$ coins showing heads and $k>0$, then he flips the $k^{\text {th }}$ coin over; otherwise he stops the process. (For example, the process starting with THT wou... | [
"$\\frac{1}{4} n(n+1)$"
] | 7 | [
"Applied Mathematics -> Probability -> Combinations",
"Discrete Mathematics -> Combinatorics"
] |
1,915 | Combinatorics | null | [
"First we show by induction that the $n$ walls divide the plane into $\\left(\\begin{array}{c}n+1 \\\\ 2\\end{array}\\right)+1$ regions. The claim is true for $n=0$ as, when there are no walls, the plane forms a single region. When placing the $n^{\\text {th }}$ wall, it intersects each of the $n-1$ other walls exa... | false | null | Expression | null | On a flat plane in Camelot, King Arthur builds a labyrinth $\mathfrak{L}$ consisting of $n$ walls, each of which is an infinite straight line. No two walls are parallel, and no three walls have a common point. Merlin then paints one side of each wall entirely red and the other side entirely blue.
At the intersection o... | [
"$k=n+1$"
] | 9 | [
"Geometry -> Plane Geometry -> Polygons",
"Discrete Mathematics -> Graph Theory",
"Discrete Mathematics -> Combinatorics"
] |
1,918 | Combinatorics | null | [
"We present solutions for the general case of $N>1$ boxes, and write $M=\\left\\lfloor\\frac{N}{2}+1\\right\\rfloor\\left\\lceil\\frac{N}{2}+1\\right\\rceil-1$ for the claimed answer. For $1 \\leqslant k<N$, say that Bob makes a $k$-move if he splits the boxes into a left group $\\left\\{B_{1}, \\ldots, B_{k}\\righ... | false | null | Numerical | null | There are 60 empty boxes $B_{1}, \ldots, B_{60}$ in a row on a table and an unlimited supply of pebbles. Given a positive integer $n$, Alice and Bob play the following game.
In the first round, Alice takes $n$ pebbles and distributes them into the 60 boxes as she wishes. Each subsequent round consists of two steps:
(... | [
"960"
] | 7 | [
"Discrete Mathematics -> Combinatorics -> Other",
"Discrete Mathematics -> Logic -> Other"
] |
1,920 | Combinatorics | null | [
"We first construct a set $\\mathcal{F}$ with $2^{k}$ members, each member having at most $k$ different scales in $\\mathcal{F}$. Take $\\mathcal{F}=\\left\\{0,1,2, \\ldots, 2^{k}-1\\right\\}$. The scale between any two members of $\\mathcal{F}$ is in the set $\\{0,1, \\ldots, k-1\\}$.\n\nWe now show that $2^{k}$ i... | false | null | Expression | null | For any two different real numbers $x$ and $y$, we define $D(x, y)$ to be the unique integer $d$ satisfying $2^{d} \leqslant|x-y|<2^{d+1}$. Given a set of reals $\mathcal{F}$, and an element $x \in \mathcal{F}$, we say that the scales of $x$ in $\mathcal{F}$ are the values of $D(x, y)$ for $y \in \mathcal{F}$ with $x \... | [
"$2^{k}$"
] | 8 | [
"Discrete Mathematics -> Combinatorics",
"Algebra -> Algebraic Expressions -> Other"
] |
1,929 | Number Theory | null | [
"For any prime $p$ and positive integer $N$, we will denote by $v_{p}(N)$ the exponent of the largest power of $p$ that divides $N$. The left-hand side of (1) will be denoted by $L_{n}$; that is, $L_{n}=\\left(2^{n}-1\\right)\\left(2^{n}-2\\right)\\left(2^{n}-4\\right) \\cdots\\left(2^{n}-2^{n-1}\\right)$.\n\nWe wi... | true | null | Tuple | null | Find all pairs $(m, n)$ of positive integers satisfying the equation
$$
\left(2^{n}-1\right)\left(2^{n}-2\right)\left(2^{n}-4\right) \cdots\left(2^{n}-2^{n-1}\right)=m !
\tag{1}
$$ | [
"$(1,1), (3,2)$"
] | 7 | [
"Number Theory -> Factorization",
"Algebra -> Algebraic Expressions",
"Discrete Mathematics -> Combinatorics"
] |
1,930 | Number Theory | null | [
"Note that the equation is symmetric. We will assume without loss of generality that $a \\geqslant b \\geqslant c$, and prove that the only solution is $(a, b, c)=(3,2,1)$.\n\n\nWe will start by proving that $c=1$. Note that\n\n$$\n3 a^{3} \\geqslant a^{3}+b^{3}+c^{3}>a^{3} .\n$$\n\nSo $3 a^{3} \\geqslant(a b c)^{2... | true | null | Tuple | null | Find all triples $(a, b, c)$ of positive integers such that $a^{3}+b^{3}+c^{3}=(a b c)^{2}$. | [
"$(1,2,3),(1,3,2),(2,1,3),(2,3,1),(3,1,2),(3,2,1)$"
] | 5 | [
"Algebra -> Algebra -> Equations and Inequalities",
"Number Theory -> Other"
] |
1,938 | Algebra | null | [
"It is immediately checked that both functions mentioned in the answer are as desired.\n\nNow let $f$ denote any function satisfying (1) for all $x, y \\in \\mathbb{Z}$. Substituting $x=0$ and $y=f(0)$ into (1) we learn that the number $z=-f(f(0))$ satisfies $f(z)=-1$. So by plugging $y=z$ into (1) we deduce that\n... | true | null | Expression | null | Determine all functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ with the property that
$$
f(x-f(y))=f(f(x))-f(y)-1
\tag{1}
$$
holds for all $x, y \in \mathbb{Z}$. | [
"$f(x)=-1$,$f(x)=x+1$"
] | 6 | [
"Algebra -> Algebra -> Equations and Inequalities",
"Algebra -> Algebra -> Algebraic Expressions",
"Discrete Mathematics -> Other"
] |
1,939 | Algebra | null | [
"Let $Z$ be the expression to be maximized. Since this expression is linear in every variable $x_{i}$ and $-1 \\leqslant x_{i} \\leqslant 1$, the maximum of $Z$ will be achieved when $x_{i}=-1$ or 1 . Therefore, it suffices to consider only the case when $x_{i} \\in\\{-1,1\\}$ for all $i=1,2, \\ldots, 2 n$.\n\nFor ... | false | null | Expression | null | Let $n$ be a fixed positive integer. Find the maximum possible value of
$$
\sum_{1 \leqslant r<s \leqslant 2 n}(s-r-n) x_{r} x_{s}
$$
where $-1 \leqslant x_{i} \leqslant 1$ for all $i=1,2, \ldots, 2 n$. | [
"$n(n-1)$"
] | 6 | [
"Algebra -> Algebra -> Equations and Inequalities",
"Algebra -> Algebraic Expressions",
"Discrete Mathematics -> Combinatorics"
] |
1,940 | Algebra | null | [
"Clearly, each of the functions $x \\mapsto x$ and $x \\mapsto 2-x$ satisfies (1). It suffices now to show that they are the only solutions to the problem.\n\nSuppose that $f$ is any function satisfying (1). Then setting $y=1$ in (1), we obtain\n\n$$\nf(x+f(x+1))=x+f(x+1)\\tag{2}\n$$\n\nin other words, $x+f(x+1)$ i... | true | null | Expression | null | Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying the equation
$$
f(x+f(x+y))+f(x y)=x+f(x+y)+y f(x)\tag{1}
$$
for all real numbers $x$ and $y$. | [
"$f(x)=x$,$f(x)=2-x$"
] | 7 | [
"Algebra -> Algebra -> Equations and Inequalities",
"Algebra -> Algebra -> Algebraic Expressions"
] |
1,945 | Combinatorics | null | [
"Let $A=\\left\\{a_{1}, a_{2}, \\ldots, a_{n}\\right\\}$, where $a_{1}<a_{2}<\\cdots<a_{n}$. For a finite nonempty set $B$ of positive integers, denote by $\\operatorname{lcm} B$ and $\\operatorname{gcd} B$ the least common multiple and the greatest common divisor of the elements in $B$, respectively.\n\nConsider a... | false | null | Numerical | null | For a finite set $A$ of positive integers, we call a partition of $A$ into two disjoint nonempty subsets $A_{1}$ and $A_{2}$ good if the least common multiple of the elements in $A_{1}$ is equal to the greatest common divisor of the elements in $A_{2}$. Determine the minimum value of $n$ such that there exists a set of... | [
"3024"
] | 7 | [
"Number Theory -> Greatest Common Divisors (GCD)",
"Number Theory -> Least Common Multiples (LCM)",
"Discrete Mathematics -> Combinatorics"
] |
1,953 | Geometry | null | [
"Let $S$ be the center of the parallelogram $B P T Q$, and let $B^{\\prime} \\neq B$ be the point on the ray $B M$ such that $B M=M B^{\\prime}$ (see Figure 1). It follows that $A B C B^{\\prime}$ is a parallelogram. Then, $\\angle A B B^{\\prime}=\\angle P Q M$ and $\\angle B B^{\\prime} A=\\angle B^{\\prime} B C=... | false | null | Numerical | null | Let $A B C$ be an acute triangle, and let $M$ be the midpoint of $A C$. A circle $\omega$ passing through $B$ and $M$ meets the sides $A B$ and $B C$ again at $P$ and $Q$, respectively. Let $T$ be the point such that the quadrilateral $B P T Q$ is a parallelogram. Suppose that $T$ lies on the circumcircle of the triang... | [
"$\\sqrt{2}$"
] | 7 | [
"Geometry -> Plane Geometry -> Polygons",
"Geometry -> Plane Geometry -> Angles"
] |
1,962 | Number Theory | null | [
"It can easily be verified that these sixteen triples are as required. Now let $(a, b, c)$ be any triple with the desired property. If we would have $a=1$, then both $b-c$ and $c-b$ were powers of 2 , which is impossible since their sum is zero; because of symmetry, this argument shows $a, b, c \\geqslant 2$.\n\nCa... | true | null | Tuple | null | Determine all triples $(a, b, c)$ of positive integers for which $a b-c, b c-a$, and $c a-b$ are powers of 2 .
Explanation: A power of 2 is an integer of the form $2^{n}$, where $n$ denotes some nonnegative integer. | [
"$(2,2,2),(2,2,3),(2,3,2),(3,2,2),(2,6,11),(2,11,6),(6,2,11),(6,11,2),(11,2,6),(11,6,2),(3,5,7),(3,7,5),(5,3,7),(5,7,3),(7,3,5),(7,5,3)$"
] | 6 | [
"Algebra -> Algebra -> Equations and Inequalities",
"Number Theory -> Other"
] |
1,964 | Number Theory | null | [
"For any function $f: \\mathbb{Z}_{>0} \\rightarrow \\mathbb{Z}_{>0}$, let $G_{f}(m, n)=\\operatorname{gcd}(f(m)+n, f(n)+m)$. Note that a $k$-good function is also $(k+1)$-good for any positive integer $k$. Hence, it suffices to show that there does not exist a 1-good function and that there exists a 2-good functio... | false | null | Expression | null | Let $\mathbb{Z}_{>0}$ denote the set of positive integers. For any positive integer $k$, a function $f: \mathbb{Z}_{>0} \rightarrow \mathbb{Z}_{>0}$ is called $k$-good if $\operatorname{gcd}(f(m)+n, f(n)+m) \leqslant k$ for all $m \neq n$. Find all $k$ such that there exists a $k$-good function. | [
"$k \\geqslant 2$"
] | 8 | [
"Number Theory -> Greatest Common Divisors (GCD)",
"Algebra -> Algebraic Expressions -> Other"
] |
1,965 | Number Theory | null | [
"A straightforward check shows that all the functions listed in the answer satisfy the problem condition. It remains to show the converse.\n\nAssume that $f$ is a function satisfying the problem condition. Notice that the function $g(x)=f(x)-f(0)$ also satisfies this condition. Replacing $f$ by $g$, we assume from ... | true | null | Expression | null | For every positive integer $n$ with prime factorization $n=\prod_{i=1}^{k} p_{i}^{\alpha_{i}}$, define
$$
\mho(n)=\sum_{i: p_{i}>10^{100}} \alpha_{i}\tag{1}
$$
That is, $\mho(n)$ is the number of prime factors of $n$ greater than $10^{100}$, counted with multiplicity.
Find all strictly increasing functions $f: \math... | [
"$f(x)=a x+b$, where $b$ is an arbitrary integer, and $a$ is an arbitrary positive integer with $\\mho(a)=0$"
] | 8 | [
"Number Theory -> Prime Numbers",
"Number Theory -> Factorization",
"Algebra -> Algebra -> Equations and Inequalities"
] |
1,968 | Algebra | null | [
"If the initial numbers are $1,-1,2$, and -2 , then Dave may arrange them as $1,-2,2,-1$, while George may get the sequence $1,-1,2,-2$, resulting in $D=1$ and $G=2$. So we obtain $c \\geqslant 2$.\n\nTherefore, it remains to prove that $G \\leqslant 2 D$. Let $x_{1}, x_{2}, \\ldots, x_{n}$ be the numbers Dave and ... | false | null | Numerical | null | For a sequence $x_{1}, x_{2}, \ldots, x_{n}$ of real numbers, we define its price as
$$
\max _{1 \leqslant i \leqslant n}\left|x_{1}+\cdots+x_{i}\right|
$$
Given $n$ real numbers, Dave and George want to arrange them into a sequence with a low price. Diligent Dave checks all possible ways and finds the minimum possib... | [
"2"
] | 8 | [
"Algebra -> Algebra -> Equations and Inequalities",
"Discrete Mathematics -> Combinatorics -> Other"
] |
1,969 | Algebra | null | [
"Let $f$ be a function satisfying (1). Set $C=1007$ and define the function $g: \\mathbb{Z} \\rightarrow \\mathbb{Z}$ by $g(m)=f(3 m)-f(m)+2 C$ for all $m \\in \\mathbb{Z}$; in particular, $g(0)=2 C$. Now (1) rewrites as\n\n$$\nf(f(m)+n)=g(m)+f(n)\n$$\n\nfor all $m, n \\in \\mathbb{Z}$. By induction in both directi... | false | null | Expression | null | Determine all functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ satisfying
$$
f(f(m)+n)+f(m)=f(n)+f(3 m)+2014
\tag{1}
$$
for all integers $m$ and $n$. | [
"$f(n) = 2 n+1007$"
] | 7 | [
"Algebra -> Algebra -> Equations and Inequalities",
"Algebra -> Algebra -> Algebraic Expressions"
] |
1,970 | Algebra | null | [
"Part I. We begin by verifying that these numbers are indeed possible values of $P(0)$. To see that each negative real number $-C$ can be $P(0)$, it suffices to check that for every $C>0$ the polynomial $P(x)=-\\left(\\frac{2 x^{2}}{C}+C\\right)$ has the property described in the statement of the problem. Due to sy... | false | null | Interval | null | Consider all polynomials $P(x)$ with real coefficients that have the following property: for any two real numbers $x$ and $y$ one has
$$
\left|y^{2}-P(x)\right| \leqslant 2|x| \text { if and only if }\left|x^{2}-P(y)\right| \leqslant 2|y|
\tag{1}
$$
Determine all possible values of $P(0)$. | [
"$(-\\infty, 0) \\cup\\{1\\}$."
] | 8 | [
"Algebra -> Algebra -> Equations and Inequalities",
"Algebra -> Algebra -> Polynomial Operations",
"Algebra -> Algebra -> Algebraic Expressions"
] |
1,974 | Combinatorics | null | [
"Let $\\ell$ be a positive integer. We will show that (i) if $n>\\ell^{2}$ then each happy configuration contains an empty $\\ell \\times \\ell$ square, but (ii) if $n \\leqslant \\ell^{2}$ then there exists a happy configuration not containing such a square. These two statements together yield the answer.\n\n(i). ... | false | null | Expression | null | Let $n \geqslant 2$ be an integer. Consider an $n \times n$ chessboard divided into $n^{2}$ unit squares. We call a configuration of $n$ rooks on this board happy if every row and every column contains exactly one rook. Find the greatest positive integer $k$ such that for every happy configuration of rooks, we can find... | [
"$\\lfloor\\sqrt{n-1}\\rfloor$"
] | 6 | [
"Discrete Mathematics -> Combinatorics",
"Algebra -> Linear Algebra -> Matrices"
] |
1,977 | Combinatorics | null | [
"We prove a more general statement for sets of cardinality $n$ (the problem being the special case $n=100$, then the answer is $n$ ). In the following, we write $A>B$ or $B<A$ for \" $A$ beats $B$ \".\n\nPart I. Let us first define $n$ different rules that satisfy the conditions. To this end, fix an index $k \\in\\... | false | null | Numerical | null | We are given an infinite deck of cards, each with a real number on it. For every real number $x$, there is exactly one card in the deck that has $x$ written on it. Now two players draw disjoint sets $A$ and $B$ of 100 cards each from this deck. We would like to define a rule that declares one of them a winner. This rul... | [
"100"
] | 8 | [
"Discrete Mathematics -> Combinatorics",
"Algebra -> Abstract Algebra -> Other"
] |
1,988 | Number Theory | null | [
"Part I. First we show that every integer greater than $(n-2) 2^{n}+1$ can be represented as such a sum. This is achieved by induction on $n$.\n\nFor $n=2$, the set $A_{n}$ consists of the two elements 2 and 3 . Every positive integer $m$ except for 1 can be represented as the sum of elements of $A_{n}$ in this cas... | false | null | Expression | null | Let $n \geqslant 2$ be an integer, and let $A_{n}$ be the set
$$
A_{n}=\left\{2^{n}-2^{k} \mid k \in \mathbb{Z}, 0 \leqslant k<n\right\} .
$$
Determine the largest positive integer that cannot be written as the sum of one or more (not necessarily distinct) elements of $A_{n}$. | [
"$(n-2) 2^{n}+1$"
] | 7 | [
"Number Theory -> Other",
"Algebra -> Prealgebra -> Integers",
"Discrete Mathematics -> Combinatorics"
] |
1,997 | Algebra | null | [
"First we show that $n \\geqslant k+4$. Suppose that there exists such a set with $n$ numbers and denote them by $a_{1}<a_{2}<\\cdots<a_{n}$.\n\nNote that in order to express $a_{1}$ as a sum of $k$ distinct elements of the set, we must have $a_{1} \\geqslant a_{2}+\\cdots+a_{k+1}$ and, similarly for $a_{n}$, we mu... | false | null | Expression | null | Let $k \geqslant 2$ be an integer. Find the smallest integer $n \geqslant k+1$ with the property that there exists a set of $n$ distinct real numbers such that each of its elements can be written as a sum of $k$ other distinct elements of the set. | [
"$n=k+4$"
] | 6 | [
"Algebra -> Algebra -> Algebraic Expressions",
"Discrete Mathematics -> Combinatorics"
] |
1,998 | Algebra | null | [
"First we prove that the function $f(x)=1 / x$ satisfies the condition of the problem statement. The AM-GM inequality gives\n\n$$\n\\frac{x}{y}+\\frac{y}{x} \\geqslant 2\n$$\n\nfor every $x, y>0$, with equality if and only if $x=y$. This means that, for every $x>0$, there exists a unique $y>0$ such that\n\n$$\n\\fr... | false | null | Expression | null | Let $\mathbb{R}_{>0}$ be the set of positive real numbers. Find all functions $f: \mathbb{R}_{>0} \rightarrow \mathbb{R}_{>0}$ such that, for every $x \in \mathbb{R}_{>0}$, there exists a unique $y \in \mathbb{R}_{>0}$ satisfying
$$
x f(y)+y f(x) \leqslant 2 .
$$ | [
"$f(x)=\\frac{1}{x}$"
] | 7 | [
"Algebra -> Algebra -> Equations and Inequalities",
"Algebra -> Algebra -> Algebraic Expressions",
"Other"
] |
2,000 | Algebra | null | [
"We first show a solution for each $n \\in\\{2,3,4\\}$. We will later show the impossibility of finding such a solution for $n \\geqslant 5$.\n\nFor $n=2$, take for example $\\left(a_{1}, a_{2}\\right)=(1,3)$ and $r=2$.\n\nFor $n=3$, take the root $r>1$ of $x^{2}-x-1=0$ (the golden ratio) and set $\\left(a_{1}, a_{... | true | null | Numerical | null | Find all positive integers $n \geqslant 2$ for which there exist $n$ real numbers $a_{1}<\cdots<a_{n}$ and a real number $r>0$ such that the $\frac{1}{2} n(n-1)$ differences $a_{j}-a_{i}$ for $1 \leqslant i<j \leqslant n$ are equal, in some order, to the numbers $r^{1}, r^{2}, \ldots, r^{\frac{1}{2} n(n-1)}$. | [
"2,3,4"
] | 8 | [
"Algebra -> Algebraic Expressions -> Other",
"Number Theory -> Other"
] |
2,004 | Combinatorics | null | [
"First, we prove that this can always be achieved. Without loss of generality, suppose at least $\\frac{2022}{2}=1011$ terms of the \\pm 1 -sequence are +1 . Define a subsequence as follows: starting at $t=0$, if $a_{t}=+1$ we always include $a_{t}$ in the subsequence. Otherwise, we skip $a_{t}$ if we can (i.e. if ... | false | null | Numerical | null | $A \pm 1 \text{-}sequence$ is a sequence of 2022 numbers $a_{1}, \ldots, a_{2022}$, each equal to either +1 or -1 . Determine the largest $C$ so that, for any $\pm 1 -sequence$, there exists an integer $k$ and indices $1 \leqslant t_{1}<\ldots<t_{k} \leqslant 2022$ so that $t_{i+1}-t_{i} \leqslant 2$ for all $i$, and
... | [
"506"
] | 6 | [
"Discrete Mathematics -> Combinatorics"
] |
2,006 | Combinatorics | null | [
"We solve the problem for a general $3 N \\times 3 N$ board. First, we prove that the lumberjack has a strategy to ensure there are never more than $5 N^{2}$ majestic trees. Giving the squares of the board coordinates in the natural manner, colour each square where at least one of its coordinates are divisible by 3... | false | null | Numerical | null | In each square of a garden shaped like a $2022 \times 2022$ board, there is initially a tree of height 0 . A gardener and a lumberjack alternate turns playing the following game, with the gardener taking the first turn:
- The gardener chooses a square in the garden. Each tree on that square and all the surrounding squ... | [
"2271380"
] | 8 | [
"Discrete Mathematics -> Combinatorics",
"Geometry -> Plane Geometry -> Polygons"
] |
2,010 | Combinatorics | null | [
"We solve the problem for $n$-tuples for any $n \\geqslant 3$ : we will show that the answer is $s=3$, regardless of the value of $n$.\n\nFirst, let us briefly introduce some notation. For an $n$-tuple $\\mathbf{v}$, we will write $\\mathbf{v}_{i}$ for its $i$-th coordinate (where $1 \\leqslant i \\leqslant n$ ). F... | false | null | Numerical | null | Lucy starts by writing $s$ integer-valued 2022-tuples on a blackboard. After doing that, she can take any two (not necessarily distinct) tuples $\mathbf{v}=\left(v_{1}, \ldots, v_{2022}\right)$ and $\mathbf{w}=\left(w_{1}, \ldots, w_{2022}\right)$ that she has already written, and apply one of the following operations ... | [
"3"
] | 8 | [
"Discrete Mathematics -> Combinatorics",
"Algebra -> Linear Algebra -> Vectors"
] |
2,011 | Combinatorics | null | [
"We will call any field that is only adjacent to fields with larger numbers a well. Other fields will be called non-wells. Let us make a second $n \\times n$ board $B$ where in each field we will write the number of good sequences which end on the corresponding field in the original board $A$. We will thus look for... | false | null | Expression | null | Alice fills the fields of an $n \times n$ board with numbers from 1 to $n^{2}$, each number being used exactly once. She then counts the total number of good paths on the board. A good path is a sequence of fields of arbitrary length (including 1) such that:
(i) The first field in the sequence is one that is only adja... | [
"$2 n^{2}-2 n+1$"
] | 10 | [
"Discrete Mathematics -> Combinatorics",
"Applied Mathematics -> Math Word Problems"
] |
2,012 | Combinatorics | null | [
"We defer the constructions to the end of the solution. Instead, we begin by characterizing all such functions $f$, prove a formula and key property for such functions, and then solve the problem, providing constructions.\n\n**Characterization** Suppose $f$ satisfies the given relation. The condition can be written... | true | null | Numerical | null | Let $\mathbb{Z}_{\geqslant 0}$ be the set of non-negative integers, and let $f: \mathbb{Z}_{\geqslant 0} \times \mathbb{Z}_{\geqslant 0} \rightarrow \mathbb{Z}_{\geqslant 0}$ be a bijection such that whenever $f\left(x_{1}, y_{1}\right)>f\left(x_{2}, y_{2}\right)$, we have $f\left(x_{1}+1, y_{1}\right)>f\left(x_{2}+1, ... | [
"2500,7500"
] | 8 | [
"Discrete Mathematics -> Combinatorics",
"Algebra -> Algebraic Expressions -> Other"
] |
2,021 | Number Theory | null | [
"Observe that 1344 is a Norwegian number as 6, 672 and 1344 are three distinct divisors of 1344 and $6+672+1344=2022$. It remains to show that this is the smallest such number.\n\nAssume for contradiction that $N<1344$ is Norwegian and let $N / a, N / b$ and $N / c$ be the three distinct divisors of $N$, with $a<b<... | false | null | Numerical | null | A number is called Norwegian if it has three distinct positive divisors whose sum is equal to 2022. Determine the smallest Norwegian number.
(Note: The total number of positive divisors of a Norwegian number is allowed to be larger than 3.) | [
"1344"
] | 5 | [
"Number Theory -> Factorization",
"Number Theory -> Other"
] |
2,022 | Number Theory | null | [
"Assume that $n$ satisfies $n ! \\mid \\prod_{p<q \\leqslant n}(p+q)$ and let $2=p_{1}<p_{2}<\\cdots<p_{m} \\leqslant n$ be the primes in $\\{1,2, \\ldots, n\\}$. Each such prime divides $n$ !. In particular, $p_{m} \\mid p_{i}+p_{j}$ for some $p_{i}<p_{j} \\leqslant n$. But\n\n$$\n0<\\frac{p_{i}+p_{j}}{p_{m}}<\\fr... | false | null | Numerical | null | Find all positive integers $n>2$ such that
$$
n ! \mid \prod_{\substack{p<q \leqslant n \\ p, q \text { primes }}}(p+q)
$$ | [
"7"
] | 8 | [
"Number Theory -> Prime Numbers",
"Number Theory -> Factorization"
] |
2,024 | Number Theory | null | [
"Clearly, $a>1$. We consider three cases.\n\nCase 1: We have $a<p$. Then we either have $a \\leqslant b$ which implies $a \\mid a^{p}-b$ ! $=p$ leading to a contradiction, or $a>b$ which is also impossible since in this case we have $b ! \\leqslant a !<a^{p}-p$, where the last inequality is true for any $p>a>1$.\n\... | true | null | Tuple | null | Find all triples of positive integers $(a, b, p)$ with $p$ prime and
$$
a^{p}=b !+p
$$ | [
"$(2,2,2),(3,4,3)$"
] | 7 | [
"Number Theory -> Prime Numbers",
"Number Theory -> Factorization",
"Algebra -> Algebraic Expressions"
] |
2,029 | Algebra | null | [
"Take any $a, b \\in \\mathbb{Q}_{>0}$. By substituting $x=f(a), y=b$ and $x=f(b), y=a$ into $(*)$ we get\n\n$$\nf(f(a))^{2} f(b)=f\\left(f(a)^{2} f(b)^{2}\\right)=f(f(b))^{2} f(a)\n$$\n\nwhich yields\n\n$$\n\\frac{f(f(a))^{2}}{f(a)}=\\frac{f(f(b))^{2}}{f(b)} \\quad \\text { for all } a, b \\in \\mathbb{Q}>0\n$$\n\... | false | null | Expression | null | Let $\mathbb{Q}_{>0}$ denote the set of all positive rational numbers. Determine all functions $f: \mathbb{Q}_{>0} \rightarrow \mathbb{Q}_{>0}$ satisfying
$$
f\left(x^{2} f(y)^{2}\right)=f(x)^{2} f(y)
\tag{*}
$$
for all $x, y \in \mathbb{Q}_{>0}$. | [
"$f(x)=1$"
] | 7 | [
"Algebra -> Algebra -> Equations and Inequalities",
"Algebra -> Algebraic Expressions -> Other"
] |
2,032 | Algebra | null | [
"The claimed maximal value is achieved at\n\n$$\n\\begin{gathered}\na_{1}=a_{2}=\\cdots=a_{2016}=1, \\quad a_{2017}=\\frac{a_{2016}+\\cdots+a_{0}}{2017}=1-\\frac{1}{2017}, \\\\\na_{2018}=\\frac{a_{2017}+\\cdots+a_{1}}{2017}=1-\\frac{1}{2017^{2}} .\n\\end{gathered}\n$$\n\nNow we need to show that this value is optim... | false | null | Numerical | null | Let $a_{0}, a_{1}, a_{2}, \ldots$ be a sequence of real numbers such that $a_{0}=0, a_{1}=1$, and for every $n \geqslant 2$ there exists $1 \leqslant k \leqslant n$ satisfying
$$
a_{n}=\frac{a_{n-1}+\cdots+a_{n-k}}{k}
$$
Find the maximal possible value of $a_{2018}-a_{2017}$. | [
"$\\frac{2016}{2017^{2}}$"
] | 8 | [
"Algebra -> Algebra -> Equations and Inequalities",
"Algebra -> Algebra -> Algebraic Expressions"
] |
2,035 | Algebra | null | [
"Since the value $8 / \\sqrt[3]{7}$ is reached, it suffices to prove that $S \\leqslant 8 / \\sqrt[3]{7}$.\n\nAssume that $x, y, z, t$ is a permutation of the variables, with $x \\leqslant y \\leqslant z \\leqslant t$. Then, by the rearrangement inequality,\n\n$$\nS \\leqslant\\left(\\sqrt[3]{\\frac{x}{t+7}}+\\sqrt... | false | null | Numerical | null | Find the maximal value of
$$
S=\sqrt[3]{\frac{a}{b+7}}+\sqrt[3]{\frac{b}{c+7}}+\sqrt[3]{\frac{c}{d+7}}+\sqrt[3]{\frac{d}{a+7}}
$$
where $a, b, c, d$ are nonnegative real numbers which satisfy $a+b+c+d=100$. | [
"$\\frac{8}{\\sqrt[3]{7}}$"
] | 6 | [
"Algebra -> Algebra -> Equations and Inequalities",
"Algebra -> Algebraic Expressions"
] |
2,037 | Combinatorics | null | [
"We show two strategies, one for Horst to place at least 100 knights, and another strategy for Queenie that prevents Horst from putting more than 100 knights on the board.\n\nA strategy for Horst: Put knights only on black squares, until all black squares get occupied.\n\nColour the squares of the board black and w... | false | null | Numerical | null | Queenie and Horst play a game on a $20 \times 20$ chessboard. In the beginning the board is empty. In every turn, Horst places a black knight on an empty square in such a way that his new knight does not attack any previous knights. Then Queenie places a white queen on an empty square. The game gets finished when someb... | [
"100"
] | 3 | [
"Discrete Mathematics -> Combinatorics",
"Discrete Mathematics -> Graph Theory"
] |
2,040 | Combinatorics | null | [
"Enumerate the days of the tournament $1,2, \\ldots,\\left(\\begin{array}{c}2 k \\\\ 2\\end{array}\\right)$. Let $b_{1} \\leqslant b_{2} \\leqslant \\cdots \\leqslant b_{2 k}$ be the days the players arrive to the tournament, arranged in nondecreasing order; similarly, let $e_{1} \\geqslant \\cdots \\geqslant e_{2 ... | false | null | Expression | null | Let $k$ be a positive integer. The organising committee of a tennis tournament is to schedule the matches for $2 k$ players so that every two players play once, each day exactly one match is played, and each player arrives to the tournament site the day of his first match, and departs the day of his last match. For eve... | [
"$k\\left(4 k^{2}+k-1\\right) / 2$"
] | 8 | [
"Applied Mathematics -> Math Word Problems",
"Discrete Mathematics -> Combinatorics",
"Discrete Mathematics -> Algorithms"
] |
2,045 | Geometry | null | [
"First, we show how to construct a good collection of $n$ triangles, each of perimeter greater than 4 . This will show that all $t \\leqslant 4$ satisfy the required conditions.\n\nConstruct inductively an $(n+2)$-gon $B A_{1} A_{2} \\ldots A_{n} C$ inscribed in $\\omega$ such that $B C$ is a diameter, and $B A_{1}... | false | null | Interval | null | A circle $\omega$ of radius 1 is given. A collection $T$ of triangles is called good, if the following conditions hold:
(i) each triangle from $T$ is inscribed in $\omega$;
(ii) no two triangles from $T$ have a common interior point.
Determine all positive real numbers $t$ such that, for each positive integer $n$, t... | [
"$t(0,4]$"
] | 7 | [
"Geometry -> Plane Geometry -> Polygons",
"Geometry -> Plane Geometry -> Area"
] |
2,063 | Algebra | null | [
"If $d=2 n-1$ and $a_{1}=\\cdots=a_{2 n-1}=n /(2 n-1)$, then each group in such a partition can contain at most one number, since $2 n /(2 n-1)>1$. Therefore $k \\geqslant 2 n-1$. It remains to show that a suitable partition into $2 n-1$ groups always exists.\n\nWe proceed by induction on $d$. For $d \\leqslant 2 n... | false | null | Expression | null | Let $n$ be a positive integer. Find the smallest integer $k$ with the following property: Given any real numbers $a_{1}, \ldots, a_{d}$ such that $a_{1}+a_{2}+\cdots+a_{d}=n$ and $0 \leqslant a_{i} \leqslant 1$ for $i=1,2, \ldots, d$, it is possible to partition these numbers into $k$ groups (some of which may be empty... | [
"$k=2 n-1$"
] | 6 | [
"Discrete Mathematics -> Combinatorics",
"Applied Mathematics -> Math Word Problems"
] |
2,064 | Combinatorics | null | [
"Firstly, let us present an example showing that $k \\geqslant 2013$. Mark 2013 red and 2013 blue points on some circle alternately, and mark one more blue point somewhere in the plane. The circle is thus split into 4026 arcs, each arc having endpoints of different colors. Thus, if the goal is reached, then each ar... | false | null | Numerical | null | In the plane, 2013 red points and 2014 blue points are marked so that no three of the marked points are collinear. One needs to draw $k$ lines not passing through the marked points and dividing the plane into several regions. The goal is to do it in such a way that no region contains points of both colors.
Find the mi... | [
"2013"
] | 7 | [
"Geometry -> Plane Geometry -> Polygons",
"Discrete Mathematics -> Combinatorics -> Other"
] |
2,075 | Number Theory | null | [
"Setting $m=n=2$ tells us that $4+f(2) \\mid 2 f(2)+2$. Since $2 f(2)+2<2(4+f(2))$, we must have $2 f(2)+2=4+f(2)$, so $f(2)=2$. Plugging in $m=2$ then tells us that $4+f(n) \\mid 4+n$, which implies that $f(n) \\leqslant n$ for all $n$.\n\nSetting $m=n$ gives $n^{2}+f(n) \\mid n f(n)+n$, so $n f(n)+n \\geqslant n^... | false | null | Expression | null | Let $\mathbb{Z}_{>0}$ be the set of positive integers. Find all functions $f: \mathbb{Z}_{>0} \rightarrow \mathbb{Z}_{>0}$ such that
$$
m^{2}+f(n) \mid m f(m)+n
$$
for all positive integers $m$ and $n$. | [
"$f(n)=n$"
] | 8 | [
"Algebra -> Algebra -> Equations and Inequalities",
"Number Theory -> Factorization"
] |
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