prompt stringlengths 19 3.19k | ground_truth stringlengths 1 23 |
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For each $t\geqslant0$, let $S_{t}$ be the set of all nonnegative, increasing, convex, continuous, real-valued functions $f(x)$ defined on the closed interval $[0,1]$ for which $$ f(1)-2 f(2 / 3)+f(1 / 3)\geqslant t[f(2 / 3)-2 f(1 / 3)+f(0)] . $$ Develop necessary and sufficient conditions on $t$ for $S_{t}$ to be clos... | 1 |
for all polygons $P$, where the double integral is taken over $H(P)$. The original answer is in $\frac{m}{n}$ format, please give the value of $m + n$. | 11 |
A convex pentagon $P = ABCDE$, with vertices labeled consecutively, is inscribed in a circle of radius 1. Find the maximum area of P subject to the condition that the chords AC and BD be perpendicular. The original answer is in the format $1 + \frac{m}{n} \sqrt{3}$, please give the value of $m + n$. | 7 |
Assume $F$ is a finite field with an odd number of elements $m$. Assume $p(x)$ is an irreducible (i.e., non-factorizable) polynomial in $F$ of the form $$ x^{2}+b x+c, \quad b, c \in F $$ For how many elements $k$ in $F$, $p(x)+k$ is irreducible in $F$? How many elements are there when $m=7$. | 3 |
Let $u, f$, and g be functions defined for all real numbers x, such that $$ \frac{u(x+1)+u(x-1)}{2}=f(x) \text { and } \frac{u(x+4)+u(x-4)}{2}=g(x). $$ If $u(x)=g(x+a)-f(x+b)+f(x+c)-f(x+d)+f(x+e)$, what is the value of $a+b+c+d+e$. | 20 |
Evaluate the infinite product $$ \prod_{n=2}^{\infty} \frac{n^{3}-1}{n^{3}+1} $$ The original answer is in \frac{m}{n} format, please give the value of m + n. | 5 |
Determine all polynomials $P(x)$ such that $P\left(x^{2}+1\right)=(P(x))^{2}+1$ and $P(0)=0$. Please provide the minimum positive integer value $P (x) $that meets the criteria. | 1 |
Establish necessary and sufficient conditions on the constant $k$ for the existence of a continuous real valued function $f(x)$ satisfying $f(f(x))=k x^{9}$ for all real x. Provide the smallest integer value of k that satisfies the condition. | 0 |
For each $t\geqslant0$, let $S_t$ be the set of all nonnegative, increasing, convex, continuous, real-valued functions $f(x)$ defined on the closed interval $[0,1]$ for which $$f(1)-2f(2/3)+f(1/3)\geqslant[f(2/3)-2f(1/3)+f(0)]$$. Develop necessary and sufficient conditions on $t$ for $S_t$ to be closed under multiplica... | 1 |
Let r and s be positive integers. Derive a formula for the number of ordered quadruples $(a, b, c, d)$ of positive integers such that $$3^{r}·7^{s}= \mathrm{lcm}[a,b,c]=\mathrm{lcm}[a,b,d]=\mathrm{lcm}[a,c,d]=\mathrm{lcm}[b,c,d]$$. Please provide the value of the formula when r = 2 and s = 1. The answer should be the s... | 1089 |
Find positive integers $n$ and $a_{1}, a_{2}, \ldots, a_{n}$ such that \n$$\na_{1}+a_{2}+\cdots+a_{n}=1979\n$$\nand the product $a_{1} a_{2} \cdots a_{n}$ is as large as possible. Please provide the value of $a + n$. | 662 |
Let $C$ be the class of all real valued continuously differentiable functions $f$ on the interval $0<x<1$ with $f(0)=0$ and $f(1)=1$. Determine the largest real number $u$ such that $$ u \leqslant \int_{0}^{1}\left|f^{\prime}(x)-f(x)\right| d x$$ for all $f$ in $C$. The original answer is in \(\frac{m}{ne}\) format, pl... | 2 |
A point P moves inside a unit square in a straight line at unit speed. When it meets a corner it escapes. When it meets an edge its line of motion is reflected so that the angle of incidence equals the angle of reflection. Let $N(T)$ be the number of starting directions from a fixed interior point $P_{0}$ for which P e... | 1 |
Find the area of a convex octagon that is inscribed in a circle and has four consecutive sides of length 3 units and the remaining four sides of length 2 units. The answer is in the form $r+s \sqrt{t}$ with $r, s$, and $t$ being positive integers. Please provide the value of $r + s + t$. | 27 |
Let $p$ be a prime number. Let $J$ be the set of all $2 \times 2$ matrices $\binom{a b}{c d}$ whose entries are chosen from $\{0,1,2, \cdots, p-1\}$ and satisfy the conditions $a+d \equiv 1(\bmod p), a d-b c \equiv 0(\bmod p)$. Determine how many members $J$ has. The original answer is in the format of $p^2 + p$. Let p... | 30 |
Evaluate $$\sum_{k=0}^{n} (-1)^k\binom{n}{k} (x-k)^n.$$ Provide the integer value of n=3. | 6 |
Three numbers are chosen independently at random, one from each of the three intervals $[0, L_i](i=1, 2,3)$. If the distribution of each random number is uniform with respect to length in the interval it is chosen from, determine the expected value of the smallest of the three numbers chosen. The original answer is in ... | 23 |
Let $F(x)$ be a real valued function defined for all real $x$ except for $x=0$ and $x=1$ and satisfying the functional equation $F(x)+F\{(x-1) / x\}=1+x$. Find all functions $F(x)$ satisfying these conditions. The original answer is in the format \frac{m}{n}, please provide the value of m + n when x = 2. | 7 |
Let u, f, and g be functions, defined for all real numbers x, such that $$\frac{u(x+1)+u(x-1)}{2}=f(x) \text { and } \frac{u(x+4)+u(x-4)}{2}=g(x).$$ Determine u(x) in terms of f and g. Please provide the value of u(x) when x = 1. The answer should be in the format $u(x)=g(a)-f(b)+f(c)-f(d)+f(e)$, directly give the valu... | 25 |
Determine the minimum value of $$(r-1)^{2}+\left(\frac{s}{r}-1\right)^{2}+\left(\frac{t}{s}-1\right)^{2}+\left(\frac{4}{t}-1\right)^{2}$$ for all real numbers r, s, t with $1 \leq r \leq s \leq t \leq 4$.The original answer form is $a-b \ sqrt {c} $, find the value of a+b+c | 22 |
(a) Find a solution that is not identically zero, of the homogeneous linear differential equation $$( 3x^2+x-1 ){y}''- ( 9x^2+9x-2 ){y}'+( 18x+3 )y=0$$ Intelligent guessing of the form of a solution may be helpful. (b) Let $y=f(x)$ be the solution of the nonhomogeneous differential equation $$( 3x^2+x-1){y}''-( 9x^2+9x... | 21 |
Assume that the differential equation $$y^{\prime\prime\prime}+p(x)y^{\prime\prime}+q(x)y^{\prime}+r(x)y = 0$$ has solutions $y_1(x)$, $y_2(x)$, and $y_3(x)$ on the whole real line such that $$y_{1}^{2}(x)+y_{2}^{2}(x)+y_{3}^{2}(x)=1$$ for all real x. Let $$f(x)=\left(y_{1}^{\prime}(x)\right)^{2}+\left(y_{2}^{\prime}(x... | 5 |
A sequence of convex polygons $\{P_n\}$, $n\geqslant0$, is defined inductively as follows. $P_0$ is an equilateral triangle with sides of length 1. Once $P_n$ has been determined, its sides are trisected; the vertices of $P_{n+1}$ are the interior trisection points of the sides of $P_n$. Thus, $P_{n+1}$ is obtained by ... | 10 |
Let $V$ be the region in the cartesian plane consisting of all points $(x, y)$ satisfying the simultaneous conditions $$|x|\leqslant y\leqslant|x|+3\text{ and }y\leqslant4.$$ Find the centroid $(\bar{x},\bar{y})$ of $V$. The original answer for $\bar{y}$ is in the form \(\frac{m}{n}\). Please provide the value of m + n... | 18 |
(a) Let $z$ be a solution of the quadratic equation $$ a z^{2}+b z+c=0 $$ and let $n$ be a positive integer. Show that $z$ can be expressed as a rational function of $z^{n}, a, b, c$. (b) Using (a) or by any other means, express $x$ as a rational function of $x^{3}$ and $x+(1 / x)$. The original answer is in \frac{m}{n... | 13 |
Express $$\sum_{n=1}^{\infty} \sum_{m=1}^{\infty} \frac{1}{m^{2} n+m n^{2}+2 m n}$$ as a rational number. The original answer is in \(\frac{m}{n}\) format, please give the value of m + n. | 11 |
Evaluate $$\lim\limits_{x\to1^-}\prod\limits_{n=0}^\infty\left(\frac{1+x^{n+1}}{1+x^n}\right)^{x^n}.$$ Please provide the value of a-b, where the original answer is in the form of \ln a-b. | 1 |
Determine all real numbers $a>0$ for which there exists a nonnegative continuous function $f(x)$ defined on $[0,a]$ with the property that the region $$R=\{(x,y);0\leq x\leq a,0\leq y\leq f(x)\}$$ has perimeter $k$ units and area $k$ square units for some real number $k.$ Please provide the smallest integer value of $a... | 2 |
Evaluate $\int_{0}^{1}\frac{ln(x+1)}{x^2+1}dx$. The original answer is in the format $\frac{m\pi}{n}$, please provide the value of $m + n$. | 10 |
Let $S={(a, b) \mid a=1,2,..., n, b=1,2,3}$. A rook tour of $S$ is a polygonal path made up of line segments connecting points $p_1, p_2,\dots,p_{3n}$ in sequence such that (i) $p_i \in S$, (ii) $p_i$ and $p_{i+1}$ are a unit distance apart, for $1 \le i< 3n$, (iii) for each $p\in S$ there is a unique i such that $p_i ... | 8 |
Let $a_1, a_2, \ldots, a_n$ be real numbers, and let $b_1, b_2, \ldots, b_n$ be distinct positive integers. Suppose there is a polynomial $f(x)$ satisfying the identity $$(1-x)^n f(x)=1+\sum_{i=1}^n a_i x^{b_i}.$$ Find a simple expression (not involving any sums) for $f(1)$ in terms of $b_1, b_2, \ldots, b_n$ and $n$ (... | 7 |
Evaluate $\sum_{n=0}^{\infty} \operatorname{Arccot}\left(n^2+n+1\right)$, where $\operatorname{Arccot} t$ for $t \geqslant 0$ denotes the number $\theta$ in the interval $0<\theta \leqslant \pi / 2$ with $\cot \theta=t$. The original answer is in the format \frac{m\pi}{n}, please give the value of m + n. | 3 |
Four points are chosen uniformly and independently at random in the interior of a given circle. Find the probability that they are the vertices of a convex quadrilateral. The original answer is in the format $1-\frac{m}{n\pi^2}$, please provide the value of m + n. | 47 |
Find the volume of the region of points $(x,y,z)$ such that $$(x^2+y^2+z^2+8)^2\leq36(x^2+y^2).$$ The original answer is in the format $m\pi^{n}$, please give the value of m + n. | 8 |
For each continuous function $f:[0,1]\to\mathbb{R}$, let $I(f)=$ $\int _{0}^{1}x^{2}f( x)$ $dx$ and $J( x) = \int _0^{1}x\left ( f( x) \right ) ^{2}$ $dx.$ Find the maximum value of $I(f)-J(f)$ over all such functions $f.$ The original answer is in the format \frac{m}{n}, please give the value of m + n. | 17 |
Let $k$ be an integer greater than 1. Suppose $a_0>0$, and define $$a_{n+1}=a_n+\frac{1}{\sqrt[k]{a_n}}$$ for $n>0.$ Evaluate $$\lim_{n\to\infty}\frac{a_n^{k+1}}{n^k}.$$ The original answer is in \left(\frac{k+1}{k}\right)^{k} format, please give the value of m + n where m/n is the simplified fraction form of the origi... | 2 |
Let $n$ be a positive integer. Find the number of pairs $P,Q$ of polynomials with real coefficients such that $$(P(X))^2+(Q(X))^2=X^{2n}+1$$ and $\deg P>\deg Q.$ Provide the value when $n=3$. The answer should be an integer. | 16 |
What is the largest possible radius of a circle contained in a 4-dimensional hypercube of side length 1? The original answer is in the format $\frac{\sqrt{m}}{n}$, please provide the value of m + n. | 4 |
Consider the equation $\cos \frac{x}{4}=\cos x$. Determine the number of distinct solutions in the interval $(0,24 \pi)$. Please provide the number of distinct solutions as an integer. | 16 |
In the Cartesian coordinate plane $xOy$, circle $C_{1}: x^{2}+y^{2}-a=0$ is symmetric about the line $l$, resulting in circle $C_{2}: x^{2}+y^{2}+2x-2ay+3=0$. Find the equation of line $l$. Please provide the sum of the coefficients of the line equation. | 3 |
Determine all rational numbers $a$ for which the matrix $$ \left(\begin{array}{cccc} a & -a & -1 & 0 \\ a & -a & 0 & -1 \\ 1 & 0 & a & -a \\ 0 & 1 & a & -a \end{array}\right) $$ is the square of a matrix with all rational entries. Provide the integer value of $a$. | 0 |
The Imomi archipelago consists of $n \geqslant 2$ islands. Between each pair of distinct islands is a unique ferry line that runs in both directions, and each ferry line is operated by one of $k$ companies. It is known that if any one of the $k$ companies closes all its ferry lines, then it becomes impossible for a tra... | 3 |
Determine all positive, composite integers $n$ that satisfy the following property: if the positive divisors of $n$ are $1=d_{1}<d_{2}<\cdots<d_{k}=n$, then $d_{i}$ divides $d_{i+1}+d_{i+2}$ for every $1 \leqslant i \leqslant k-2$. Please provide the minimum value of $n $that meets the criteria. | 4 |
Determine all pairs ( $a, p$ ) of positive integers with $p$ prime such that $p^{a}+a^{4}$ is a perfect square. Please provide the sum of all values of $a$ and $p$ for the pairs that satisfy the condition. | 30 |
Let $a_{1}, a_{2}, \\ldots, a_{n}, b_{1}, b_{2}, \\ldots, b_{n}$ be $2 n$ positive integers such that the $n+1$ products $$\begin{gathered} a_{1} a_{2} a_{3} \\cdots a_{n} \\ b_{1} a_{2} a_{3} \\cdots a_{n} \\ b_{1} b_{2} a_{3} \\cdots a_{n} \\ \vdots \\ b_{1} b_{2} b_{3} \\cdots b_{n} \end{gathered}$$ form a strictly ... | 1 |
A $2\times3$ rectangle has vertices at $(0,0)$, $(2,0)$, $(0,3)$, and $(2,3)$. It rotates $90^{\circ}$ clockwise about the point $(2,0)$. It then rotates $90^{\circ}$ clockwise about the point $(5,0)$, then $90^{\circ}$ clockwise about the point $(7,0)$, and finally, $90^{\circ}$ clockwise about the point $(10,0)$. (Th... | 15 |
Let P be a polynomial, with real coefficients, in three variables and F be a function of two variables such that $P(ux,uy, uz)=u^{2}F(y-x,z-x,z-x)$ for all real $x$, $y$, $z$, $u$, and such that $P(1,0,0)=4$, $P(0,1,0)=5$, and $P(0,0,1)=6$. Also let A, B, C be complex numbers with $P(A,B,C)=0$ and $|B- A|=10$. Find $|C... | 38 |
Find the maximum value of \(\int_0^y \sqrt{x^4+(y-y^2)^2} \, dx\) for \(0 \leq y \leq 1\). The original answer is in \(\frac{m}{n}\) format, please give the value of \(m + n\). | 4 |
For each positive integer $n$, let $a(n)$ be the number of zeros in the base 3 representation of $n$. For which positive real numbers $x$ does the series $$ \sum_{n=1}^{\infty} \frac{x^{a(n)}}{n^3} $$ converge? Please provide the sum of the endpoints of the interval for which the series converges. | 25 |
Four points are chosen at random on the surface of a sphere. What is the probability that the center of the sphere lies inside the tetrahedron whose vertices are at the four points? (It is understood that each point is independently chosen relative to a uniform distribution on the sphere.) The original answer is in \fr... | 9 |
Let $S$ be the smallest set of positive integers such that\n(a) 2 is in $S$,\n(b) $n$ is in $S$ whenever $n^2$ is in $S$, and\n(c) $(n+5)^2$ is in $S$ whenever $n$ is in $S$.\nWhich positive integers are not in $S$? The original answer is in the form of a set, please provide the sum of all positive integers not in $S$.... | 6 |
Find all ordered pairs $(a, b)$ of positive integers for which $$ \frac{1}{a}+\frac{1}{b}=\frac{3}{2018} . $$ Please provide the sum of all integers in the ordered pairs. | 1438383 |
Given a positive integer $n$, let $M(n)$ be the largest integer $m$ such that $$\begin{pmatrix}m\\n-1\end{pmatrix}>\begin{pmatrix}m-1\\n\end{pmatrix}.$$ Evaluate the limit $$\lim_{n\to\infty}\frac{M(n)}{n}.$$ The original answer is in the format $\frac{a + \sqrt{b}}{c}$. Please provide the integer sum of $a$, $b$, and ... | 10 |
Let $k$ be a nonnegative integer. Evaluate $$\sum_{j=0}^k2^{k-j}\binom{k+j}j.$$ Please provide the value of the expression when $k = 1$. The original answer is in $4^{k}$ format, please give the value of $m + n$ | 5 |
For every positive real number $x$, let $$g(x)=\lim\limits_{r\to0}((x+1)^{r+1}-x^{r+1})^{\frac{1}{r}}.$$\mathrm{Find}\lim_{x\to\infty}\frac{g(x)}{x}. The original answer is in the form of $e$, please provide the integer part of the result. | 1 |
Let $n$ be an integer with $n\geq2.$ Over all real polynomials $p(x)$ of degree $n$, what is the largest possible number of negative coefficients of $p(x)^2?$ Provide the value when $n=2$. directly give the value of the answer when n=4 | 6 |
Let $n$ be an even positive integer. Let $p$ be a monic, real polynomial of degree $2n;$ that is to say, $p(x)=x^{2n}+a_{2n-1}x^{2n-1}+\cdots+a_{1}x+a_{0}$ for some real coefficients $a_{0},\ldots,a_{2n-1}.$ Suppose that $p(1/k)=k^{2}$ for all integers $k$ such that $1\leq|k|\leq n.$ Find all other real numbers $x$ for... | 2 |
Let $a_0=5/2$ and $a_k=a^2_{k-1}-2$ for $k\geq1.$ Compute $$\prod\limits_{k=0}^{\infty}\left(1-\frac{1}{a_k}\right)$$ in closed form. The original answer is in \frac{m}{n} format, please give the value of m + n. | 10 |
Suppose that $f$ is a function from $\mathbb{R}$ to $\mathbb{R}$ such that $$f(x)+f\left(1-\dfrac{1}{x}\right)=\arctan x$$ for all real $x\neq0.$ (As usual, y=arctan $x$ means $-\pi/2<$ $y<\pi/2$ and tan$y=x.)$ Find $$\int_0^1f(x)\:dx.$$ The original answer is in the format \(\frac{m \pi}{n}\). Please give the value of... | 11 |
Let $n$ be a positive integer. Two players $A$ and $B$ play a game in which they take turns choosing positive integers $k\leqslant n.$ The rules of the game are: $(i)$ $A$ player cannot choose a number that has been chosen by either player on any previous turn. $(ii)$ $A$ player cannot choose a number consecutive to an... | 13 |
Determine all triples $(a,b,c)$ of positive integers for which $ab-c,bc-a$, and $ca-b$ are powers of $2$. Explanation:$A$ power of $2$ is an integer of the form $2^n$, where $n$ denotes some nonnegative integer. How many triples $(a, b, c)$ satisfy the conditions? | 16 |
Let $\mathbb{Z}_{>0}$ denote the set of positive integers. For any positive integer $k$, a function $f:\mathbb{Z}_{>0}\to\mathbb{Z}_{>0}$ is called $k$-good if $\gcd(f(m)+n,f(n)+m)\leqslant k$ for all $m\neq n.$ Find the smallest positive integer $k$ such that there exists a $k$-good function. | 2 |
Determine the largest real number $a$ such that for all $n\geqslant1$ 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 $$\begin{aligned}\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+\frac{n+1}{x_n}\right).\... | 13 |
The leader of an IMO team chooses positive integers $n$ and $k$ with $n>k$, and announces them to the deputy leader and a contestant. The leader then secretly tells the deputy leader an $n$-digit binary string, and the deputy leader writes down all $n$-digit binary strings which differ from the leader's in exactly $k$ ... | 2 |
Denote by $\mathbb{N}$ the set of all positive integers. Find all functions $f: \mathbb{N} \to \mathbb{N}$ such that for all positive integers $m$ and $n$, the integer $f(m)+f(n)-mn$ is nonzero and divides $mf(m)+nf(n).$ Please provide the value of $f(n)$ when $n=2$. Directly give the value of $f(n)$. | 4 |
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: $\bullet$ 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. $... | 0 |
For each permutation $y_1,y_2,\cdots,y_n$ of these numbers we have $\sum\limits_{i=1}^{n-1}y_iy_{i+1}=y_1y_2+y_2y_3+y_3y_4+\cdots+y_{n-1}y_n\geqslant-1$. Find the largest constant $K=K(n)$ such that $\sum\limits_{1\leqslant i<j\leqslant n}x_ix_j\geqslant K$ holds for every Shiny$n$-tuple $(x_1,x_2,\cdots,x_n)$. Please ... | 1 |
Sir Alex plays the following game on a row of $9$ cells. Initially, all cells are empty. In each move, Sir Alex is allowed to perform exactly one of the following two operations: (1) Choose any number of the form $2^j$, where $j$ is a non-negative integer, and put it into an empty cell. (2) Choose two (not necessarily ... | 2025 |
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\times1$ 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 only onc... | 91 |
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 $(2n+1)\times (2n+1)$ square centered at $c$, ap... | 10 |
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}$$ is an integer. Please provide the value of $p+q$ for such pairs. | 5 |
Let $a_0, a_1, a_2, \ldots$ be a sequence of real numbers such that $a_0=0, a_1=1$, and for every $n \geqslant 2$ there exists $1 \leqslant k \leqslant n$ satisfying $$ a_n=\frac{a_{n-1}+\cdots+a_{n-k}}{k} $$ Find the maximal possible value of $a_{2018}-a_{2017}$. The original answer is in the form \frac{m}{n^2}, pleas... | 4033 |
Let $k$ be a positive integer. The organising committee of a tennis tournament is to schedule the matches for $2 k$ players so that every two players play once, each day exactly one match is played, and each player arrives to the tournament site the day of his first match, and departs the day of his last match. For eve... | 134 |
A circle $\omega$ of radius 1 is given. A collection $T$ of triangles is called good, if the following conditions hold: (i) each triangle from $T$ is inscribed in $\omega$; (ii) no two triangles from $T$ have a common interior point. Determine all positive real numbers $t$ such that, for each positive integer $n$, ther... | 4 |
Find all pairs $(m, n)$ of positive integers satisfying the equation $$ \begin{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} \end{equation*} $$ Please provide the sum of all integers in the pairs $(m, n)$. | 7 |
Find all triples $(a, b, c)$ of positive integers such that $a^{3}+b^{3}+c^{3}=(a b c)^{2}$. Please provide the sum of all integers in the solutions. | 18 |
Version 1. Let $n$ be a positive integer, and set $N=2^{n}$. Determine the smallest real number $a_{n}$ such that, for all real $x$, $$\sqrt[N]{\frac{x^{2N}+1}{2}}\leqslant a_n(x-1)^2+x.$$ Please provide the value of N when $a_n = N/2$.\n\nVersion 2. For every positive integer $N$, determine the smallest real number $b... | 5 |
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 one. ... | 7 |
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. Please provide the sum of all such $n$. | 4 |
Find all positive integers $n\geqslant2$ 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)}$. Please provide the... | 9 |
In each square of a garden shaped like a $2022\times2022$ 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 squares (o... | 2271380 |
Let $n>3$ be a positive integer. Suppose that $n$ children are arranged in a circle, and $n$ coins are distributed between them (some children may have no coins). At every step, a child with at least 2 coins may give 1 coin to each of their immediate neighbours on the right and left. Determine all initial distributions... | 15 |
Let $n$ be a positive integer. We start with $n$ piles of pebbles, each initially containing a single pebble. One can perform moves of the following form: choose two piles, take an equal number of pebbles from each pile and form a new pile out of these pebbles. For positive integer $n=8$, find the smallest number of non... | 1 |
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, y_2\right)$ an... | 10000 |
Find all triples of positive integers $(a, b, p)$ with $p$ prime such that $a^p=b!+p$. Please provide the sum of all elements in these triples. | 16 |
Determine the maximal length $L$ of a sequence $a_{1}, \ldots, a_{L}$ of positive integers satisfying both the following properties: - every term in the sequence is less than or equal to $2^{2023}$, and - there does not exist a consecutive subsequence $a_{i}, a_{i+1}, \ldots, a_{j}$ (where $1 \leqslant i \leqslant j \l... | 2025 |
Let $n$ be a positive integer. We arrange $1+2+\cdots+n$ circles in a triangle with $n$ rows, such that the $i^{\text {th }}$ row contains exactly $i$ circles. The following figure shows the case $n=6$. In this triangle, a ninja-path is a sequence of circles obtained by repeatedly going from a circle to one of the two ... | 3 |
Let $n \geqslant 2$ be a positive integer. Paul has a $1 \times n^{2}$ rectangular strip consisting of $n^{2}$ unit squares, where the $i^{\text {th }}$ square is labelled with $i$ for all $1 \leqslant i \leqslant n^{2}$. He wishes to cut the strip into several pieces, where each piece consists of a number of consecuti... | 5 |
The Imomi archipelago consists of $n \geqslant 2$ islands. Between each pair of distinct islands is a unique ferry line that runs in both directions, and each ferry line is operated by one of $k$ companies. It is known that if any one of the $k$ companies closes all its ferry lines, then it becomes impossible for a tra... | 8 |
Determine all pairs $(a, p)$ of positive integers with $p$ prime such that $p^{a}+a^{4}$ is a perfect square. Please provide the sum of all $a + p$ for the solutions. | 30 |
Let $\mathbb{Z}_{>0}$ be the set of positive integers. Determine all functions $f: \mathbb{Z}_{>0} \rightarrow \mathbb{Z}_{>0}$ such that $$ f^{b f(a)}(a+1)=(a+1) f(b) $$ holds for all $a, b \in \mathbb{Z}_{>0}$, where $f^{k}(n)=f(f(\cdots f(n) \cdots))$ denotes the composition of $f$ with itself $k$ times.Find the val... | 4 |
At a tennis tournament there were $2n$ boys and $n$ girls participating. Every player played every other player. The boys won $\frac{7}{5}$ times as many matches as the girls. It is known that there were no draws. Find the smallest positive integer value of $n$ that satisfies the given condition. | 3 |
Let $ABC$ be an isosceles triangle with $AC=BC$, let $M$ be the midpoint of its side $AC$, and let $Z$ be the line through $C$ perpendicular to $AB$. The circle through the points $B$, $C$, and $M$ intersects the line $Z$ at the points $C$ and $Q$. Find the radius of the circumcircle of the triangle $ABC$ in terms of $... | 5 |
Find all prime numbers $p, q, r$ such that $\frac{p}{q} - \frac{4}{r+1} = 1$. Please provide the sum of all $p+q+r$ | 37 |
Find all primes $p$ such that there exist positive integers $x,y$ that satisfy $x(y^2-p)+y(x^2-p)=5p$. Please provide the sum of all such primes $p$. | 12 |
Let $n>3$ be a positive integer. Equilateral triangle $ABC$ is divided into $n^2$ smaller congruent equilateral triangles (with sides parallel to its sides). Let $m$ be the number of rhombuses that contain two small equilateral triangles and $d$ the number of rhombuses that contain eight small equilateral triangles. Fi... | 21 |
Let the circles $k_1$ and $k_2$ intersect at two points $A$ and $B$, and let $t$ be a common tangent of $k_1$ and $k_2$ that touches $k_1$ and $k_2$ at $M$ and $N$ respectively. If $t\perp AM$ and $MN=2AM$, evaluate the angle $NMB$. The original answer is in the format of an angle in degrees. Please provide the integer... | 45 |
Find all ordered pairs $(a,b)$ of positive integers for which the numbers $\dfrac{a^3b-1}{a+1}$ and $\dfrac{b^3a+1}{b-1}$ are both positive integers. Please provide the sum of all $a + b$ values from the solutions. | 14 |
The base of a right pyramid is a quadrilateral whose sides are each of length $a$. The planar angles at the vertex of the pyramid are equal to the angles between the lateral edges and the base. When \(a = 10\), find the volume of the pyramid. Round it to the nearest whole number. | 300 |
Find four consecutive terms $a, b, c, d$ of an arithmetic progression and four consecutive terms $a_1,b_1,c_1,d_1$ of a geometric progression such that $a+a_1=27,b+b_1=27,c+c_1=39$, and $d+d_1=87.$ Please provide the sum of all terms in both progressions. | 180 |
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