This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 15925

2000 Manhattan Mathematical Olympiad, 1

Prove there exists no polynomial $f(x)$, with integer coefficients, such that $f(7) = 11$ and $f(11) = 13$.

2023 LMT Fall, 9

Tags: algebra
Find the least positive integer $k$ such that when $\frac{k}{2023}$ is written in simplest form, the sum of the numerator and denominator is divisible by $7$. [i]Proposed byMuztaba Syed[/i]

2019 Purple Comet Problems, 6

Tags: algebra
Find the value of $n$ such that $\frac{2019 + n}{2019 - n}= 5$

1978 Miklós Schweitzer, 6

Suppose that the function $ g : (0,1) \rightarrow \mathbb{R}$ can be uniformly approximated by polynomials with nonnegative coefficients. Prove that $ g$ must be analytic. Is the statement also true for the interval $ (\minus{}1,0)$ instead of $ (0,1)$? [i]J. Kalina, L. Lempert[/i]

1994 All-Russian Olympiad Regional Round, 11.6

Find all functions satisfying the equality $$(x-1)f \left(\dfrac{x+1}{x-1}\right)- f(x) = x$$ for all $x \ne 1$.

1984 IMO Shortlist, 20

Determine all pairs $(a, b)$ of positive real numbers with $a \neq 1$ such that \[\log_a b < \log_{a+1} (b + 1).\]

2009 Saint Petersburg Mathematical Olympiad, 3

Tags: algebra
$f(x),g(x),h(x)$ are square trinomials with discriminant, that equals $2$. And $f(x)+g(x),f(x)+h(x),g(x)+h(x)$ are square trinomials with discriminant, that equals $1$. Prove,that $f(x)+g(x)+h(x)$ has not roots.

2010 Puerto Rico Team Selection Test, 4

Find the largest possible value in the real numbers of the term $$\frac{3x^2 + 16xy + 15y^2}{x^2 + y^2}$$ with $x^2 + y^2 \ne 0$.

2016 Costa Rica - Final Round, F3

Let $f: Z^+ \to Z^+ \cup \{0\}$ a function that meets the following conditions: a) $f (a b) = f (a) + f (b)$, b) $f (a) = 0$ provided that the digits of the unit of $a$ are $7$, c) $f (10) = 0$. Find $f (2016).$

2012 IMO Shortlist, N2

Find all triples $(x,y,z)$ of positive integers such that $x \leq y \leq z$ and \[x^3(y^3+z^3)=2012(xyz+2).\]

2002 HKIMO Preliminary Selection Contest, 17

Tags: algebra
Let $a_0=2$ and for $n\geq 1$, $a_n=\frac{\sqrt3 a_{n-1}+1}{\sqrt3-a_{n-1}}$. Find the value of $a_{2002}$ in the form $p+q\sqrt3$ where $p$ and $q$ are rational numbers

2016 Belarus Team Selection Test, 2

Tags: algebra
Suppose that a sequence $a_1,a_2,\ldots$ of positive real numbers satisfies \[a_{k+1}\geq\frac{ka_k}{a_k^2+(k-1)}\] for every positive integer $k$. Prove that $a_1+a_2+\ldots+a_n\geq n$ for every $n\geq2$.

2008 Postal Coaching, 4

Tags: root , polynomial , algebra
Find all real numbers$p, q$ for which the polynomial equation $P(x) = x^4 - \frac{8p^2}{q}x^3 + 4qx^2 - 3px + p^2 = 0$ has four positive roots.

2018 Bosnia And Herzegovina - Regional Olympiad, 1

if $a$, $b$ and $c$ are real numbers such that $(a-b)(b-c)(c-a) \neq 0$, prove the equality: $\frac{b^2c^2}{(a-b)(a-c)}+\frac{c^2a^2}{(b-c)(b-a)}+\frac{a^2b^2}{(c-a)(c-b)}=ab+bc+ca$

2021 BmMT, Pacer Round

[b]p1.[/b] $17.5\%$ of what number is $4.5\%$ of $28000$? [b]p2.[/b] Let $x$ and $y$ be two randomly selected real numbers between $-4$ and $4$. The probability that $(x - 1)(y - 1)$ is positive can be written in the form $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m + n$. [b]p3.[/b] In the $xy$-plane, Mallen is at $(-12, 7)$ and Anthony is at $(3,-14)$. Mallen runs in a straight line towards Anthony, and stops when she has traveled $\frac23$ of the distance to Anthony. What is the sum of the $x$ and $y$ coordinates of the point that Mallen stops at? [b]p4.[/b] What are the last two digits of the sum of the first $2021$ positive integers? [b]p5.[/b] A bag has $19$ blue and $11$ red balls. Druv draws balls from the bag one at a time, without replacement. The probability that the $8$th ball he draws is red can be written in the form $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m + n$. [b]p6.[/b] How many terms are in the arithmetic sequence $3$, $11$, $...$, $779$? [b]p7.[/b] Ochama has $21$ socks and $4$ drawers. She puts all of the socks into drawers randomly, making sure there is at least $1$ sock in each drawer. If $x$ is the maximum number of socks in a single drawer, what is the difference between the maximum and minimum possible values of $x$? [b]p8.[/b] What is the least positive integer $n$ such that $\sqrt{n + 1} - \sqrt{n} < \frac{1}{20}$? [b]p9.[/b] Triangle $\vartriangle ABC$ is an obtuse triangle such that $\angle ABC > 90^o$, $AB = 10$, $BC = 9$, and the area of $\vartriangle ABC$ is $36$. Compute the length of $AC$. [img]https://cdn.artofproblemsolving.com/attachments/a/c/b648d0d60c186d01493fcb4e21b5260c46606e.png[/img] [b]p10.[/b] If $x + y - xy = 4$, and $x$ and $y$ are integers, compute the sum of all possible values of$ x + y$. [b]p11.[/b] What is the largest number of circles of radius $1$ that can be drawn inside a circle of radius $2$ such that no two circles of radius $1$ overlap? [b]p12.[/b] $22.5\%$ of a positive integer $N$ is a positive integer ending in $7$. Compute the smallest possible value of $N$. [b]p13.[/b] Alice and Bob are comparing their ages. Alice recognizes that in five years, Bob's age will be twice her age. She chuckles, recalling that five years ago, Bob's age was four times her age. How old will Alice be in five years? [b]p14.[/b] Say there is $1$ rabbit on day $1$. After each day, the rabbit population doubles, and then a rabbit dies. How many rabbits are there on day $5$? [b]15.[/b] Ajit draws a picture of a regular $63$-sided polygon, a regular $91$-sided polygon, and a regular $105$-sided polygon. What is the maximum number of lines of symmetry Ajit's picture can have? [b]p16.[/b] Grace, a problem-writer, writes $9$ out of $15$ questions on a test. A tester randomly selects $3$ of the $15$ questions, without replacement, to solve. The probability that all $3$ of the questions were written by Grace can be written in the form $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m + n$. [b]p17.[/b] Compute the number of anagrams of the letters in $BMMTBMMT$ with no two $M$'s adjacent. [b]p18.[/b] From a $15$ inch by $15$ inch square piece of paper, Ava cuts out a heart such that the heart is a square with two semicircles attached, and the arcs of the semicircles are tangent to the edges of the piece of paper, as shown in the below diagram. The area (in square inches) of the remaining pieces of paper, after the heart is cut out and removed, can be written in the form $a-b\pi$, where $a$ and $b$ are positive integers. Compute $a + b$. [b]p19.[/b] Bayus has $2021$ marbles in a bag. He wants to place them one by one into $9$ different buckets numbered $1$ through $9$. He starts by putting the first marble in bucket $1$, the second marble in bucket $2$, the third marble in bucket $3$, etc. After placing a marble in bucket $9$, he starts back from bucket $1$ again and repeats the process. In which bucket will Bayus place the last marble in the bag? [img]https://cdn.artofproblemsolving.com/attachments/9/8/4c6b1bd07367101233385b3ffebc5e0abba596.png[/img] [b]p20.[/b] What is the remainder when $1^5 + 2^5 + 3^5 +...+ 2021^5$ is divided by $5$? PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

2019 SAFEST Olympiad, 3

Let $m,n\geq 2$ be integers. Let $f(x_1,\dots, x_n)$ be a polynomial with real coefficients such that $$f(x_1,\dots, x_n)=\left\lfloor \frac{x_1+\dots + x_n}{m} \right\rfloor\text{ for every } x_1,\dots, x_n\in \{0,1,\dots, m-1\}.$$ Prove that the total degree of $f$ is at least $n$.

2014 Benelux, 1

Find the smallest possible value of the expression \[\left\lfloor\frac{a+b+c}{d}\right\rfloor+\left\lfloor\frac{b+c+d}{a}\right\rfloor+\left\lfloor\frac{c+d+a}{b}\right\rfloor+\left\lfloor\frac{d+a+b}{c}\right\rfloor\] in which $a,~ b,~ c$, and $d$ vary over the set of positive integers. (Here $\lfloor x\rfloor$ denotes the biggest integer which is smaller than or equal to $x$.)

1980 Poland - Second Round, 4

Prove that if $ a $ and $ b $ are real numbers and the polynomial $ ax^3 - ax^2 + 9bx - b $ has three positive roots, then they are equal.

1998 Tournament Of Towns, 4

For some positive numbers $A, B, C$ and $D$, the system of equations $$\begin{cases} x^2 + y^2 = A \\ |x| + |y| = B \end{cases}$$ has $m$ solutions, while the system of equations $$\begin{cases} x^2 + y^2 +z^2= X\\ |x| + |y| +|z| = D \end{cases}$$ has $n$ solutions. If $m > n > 1$, find $m$ and $n$. ( G Galperin)

2003 Estonia National Olympiad, 2

Solve the equation $\sqrt{x} = \log_2 x$.

1992 Baltic Way, 12

Tags: function , limit , algebra
Let $ N$ denote the set of natural numbers. Let $ \phi: N\rightarrow N$ be a bijective function and assume that there exists a finite limit \[ \lim_{n\rightarrow\infty}\frac{\phi(n)}{n}\equal{}L. \] What are the possible values of $ L$?

2021 Vietnam National Olympiad, 2

Find all function $f:\mathbb{R}\to \mathbb{R}$ such that \[f(x)f(y)=f(xy-1)+yf(x)+xf(y)\] for all $x,y \in \mathbb{R}$

2011 IFYM, Sozopol, 6

Define a sequence {$a_n$}$^{\infty}_{n=1}$ by $a_1 = 4, a_2 = a_3 = (a^2 - 2)^2$ and $a_n = a_{n-1}.a_{n-2} - 2(a_{n-1} + a_{n-2}) - a_{n-3} + 8, n \ge 4$, where $a > 2$ is a natural number. Prove that for all $n$ the number $2 + \sqrt{a_n}$ is a perfect square.

2002 IMO Shortlist, 4

Is there a positive integer $m$ such that the equation \[ {1\over a}+{1\over b}+{1\over c}+{1\over abc}={m\over a+b+c} \] has infinitely many solutions in positive integers $a,b,c$?

1985 IMO Longlists, 92

Find a method by which one can compute the coefficients of $P(x) = x^6 + a_1x^5 + \cdots+ a_6$ from the roots of $P(x) = 0$ by performing not more than $15$ additions and $15$ multiplications.