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

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Found problems: 15925

MBMT Team Rounds, 2023

[hide=B stands for Bernoulli, G stands for Germain]they had two problem sets under those two names[/hide] [b]B1[/b] What is the sum of the first $5$ positive integers? [b]B2[/b] Bread picks a number $n$. He finds out that if he multiplies $n$ by $23$ and then subtracts $20$, he gets $46279$. What is $n$? [b]B3[/b] A [i]Harshad [/i] Number is a number that is divisible by the sum of its digits. For example, $27$ is divisible by $2 + 7 = 9$. Only one two-digit multiple of $9$ is not a [i]Harshad [/i] Number. What is this number? [b]B4 / G1[/b] There are $5$ red balls and 3 blue balls in a bag. Alice randomly picks a ball out of the bag and then puts it back in the bag. Bob then randomly picks a ball out of the bag. What is the probability that Alice gets a red ball and Bob gets a blue ball, assuming each ball is equally likely to be chosen? [b]B5[/b] Let $a$ be a $1$-digit positive integer and $b$ be a $3$-digit positive integer. If the product of $a$ and $b$ is a$ 4$-digit integer, what is the minimum possible value of the sum of $a$ and $b$? [b]B6 / G2[/b] A circle has radius $6$. A smaller circle with the same center has radius $5$. What is the probability that a dart randomly placed inside the outer circle is outside the inner circle? [b]B7[/b] Call a two-digit integer “sus” if its digits sum to $10$. How many two-digit primes are sus? [b]B8 / G3[/b] Alex and Jeff are playing against Max and Alan in a game of tractor with $2$ standard decks of $52$ cards. They take turns taking (and keeping) cards from the combined decks. At the end of the game, the $5$s are worth $5$ points, the $10$s are worth $10$ points, and the kings are worth 10 points. Given that a team needs $50$ percent more points than the other to win, what is the minimal score Alan and Max need to win? [b]B9 / G4[/b] Bob has a sandwich in the shape of a rectangular prism. It has side lengths $10$, $5$, and $5$. He cuts the sandwich along the two diagonals of a face, resulting in four pieces. What is the volume of the largest piece? [b]B10 / G5[/b] Aven makes a rectangular fence of area $96$ with side lengths $x$ and $y$. John makesva larger rectangular fence of area 186 with side lengths $x + 3$ and $y + 3$. What is the value of $x + y$? [b]B11 / G6[/b] A number is prime if it is only divisible by itself and $1$. What is the largest prime number $n$ smaller than $1000$ such that $n + 2$ and $n - 2$ are also prime? Note: $1$ is not prime. [b]B12 / G7[/b] Sally has $3$ red socks, $1$ green sock, $2$ blue socks, and $4$ purple socks. What is the probability she will choose a pair of matching socks when only choosing $2$ socks without replacement? [b]B13 / G8[/b] A triangle with vertices at $(0, 0)$,$ (3, 0)$, $(0, 6)$ is filled with as many $1 \times 1$ lattice squares as possible. How much of the triangle’s area is not filled in by the squares? [b]B14 / G10[/b] A series of concentric circles $w_1, w_2, w_3, ...$ satisfy that the radius of $w_1 = 1$ and the radius of $w_n =\frac34$ times the radius of $w_{n-1}$. The regions enclosed in $w_{2n-1}$ but not in $w_{2n}$ are shaded for all integers $n > 0$. What is the total area of the shaded regions? [b]B15 / G12[/b] $10$ cards labeled 1 through $10$ lie on a table. Kevin randomly takes $3$ cards and Patrick randomly takes 2 of the remaining $7$ cards. What is the probability that Kevin’s largest card is smaller than Patrick’s largest card, and that Kevin’s second-largest card is smaller than Patrick’s smallest card? [b]G9[/b] Let $A$ and $B$ be digits. If $125A^2 + B161^2 = 11566946$. What is $A + B$? [b]G11[/b] How many ordered pairs of integers $(x, y)$ satisfy $y^2 - xy + x = 0$? [b]G13[/b] $N$ consecutive integers add to $27$. How many possible values are there for $N$? [b]G14[/b] A circle with center O and radius $7$ is tangent to a pair of parallel lines $\ell_1$ and $\ell_2$. Let a third line tangent to circle $O$ intersect $\ell_1$ and $\ell_2$ at points $A$ and $B$. If $AB = 18$, find $OA + OB$. [b]G15[/b] Let $$ M =\prod ^{42}_{i=0}(i^2 - 5).$$ Given that $43$ doesn’t divide $M$, what is the remainder when M is divided by $43$? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

2021 Dutch IMO TST, 3

Find all functions $f : R \to R$ with $f (x + yf(x + y))= y^2 + f(x)f(y)$ for all $x, y \in R$.

2016 District Olympiad, 2

For any natural number $ n, $ denote $ x_n $ as being the number of natural numbers of $ n $ digits that are divisible by $ 4 $ and formed only with the digits $ 0,1,2 $ or $ 6. $ [b]a)[/b] Calculate $ x_1,x_2,x_3,x_4. $ [b]b)[/b] Find the natural number $ m $ such that $$ 1+\left\lfloor \frac{x_2}{x_1}\right\rfloor +\left\lfloor \frac{x_3}{x_2}\right\rfloor +\left\lfloor \frac{x_4}{x_3}\right\rfloor +\cdots +\left\lfloor \frac{x_{m+1}}{x_m}\right\rfloor =2016 , $$ where $ \lfloor\rfloor $ is the usual integer part.

1994 Cono Sur Olympiad, 1

Tags: algebra
The positive integrer number $n$ has $1994$ digits. $14$ of its digits are $0$'s and the number of times that the other digits: $1, 2, 3, 4, 5, 6, 7, 8, 9$ appear are in proportion $1: 2: 3: 4: 5: 6: 7: 8: 9$, respectively. Prove that $n$ is not a perfect square.

2014 JBMO Shortlist, 8

Let $\displaystyle {x, y, z}$ be positive real numbers such that $\displaystyle {xyz = 1}$. Prove the inequality:$$\displaystyle{\dfrac{1}{x\left(ay+b\right)}+\dfrac{1}{y\left(az+b\right)}+\dfrac{1}{z\left(ax+b\right)}\geq 3}$$ if: (A) $\displaystyle {a = 0, b = 1}$ (B) $\displaystyle {a = 1, b = 0}$ (C) $\displaystyle {a + b = 1, \; a, b> 0}$ When the equality holds?

2013 Costa Rica - Final Round, F1

Find all functions $f: R\to R$ such that for all real numbers $x, y$ is satisfied that $$f (x + y) = (f (x))^{ 2013} + f (y).$$

1901 Eotvos Mathematical Competition, 1

Tags: algebra
Prove that, for any positive integer $n$, $$1^n+2^n+3^n+4^n$$ is divisible by $5$ if and only if $n$ is not divisible by $4$.

2008 Princeton University Math Competition, A6

Tags: algebra
Let $x$ be the largest root of $x^4 - 2009x + 1$. Find the nearest integer to $\frac{1}{x^3-2009}$ .

2015 Postal Coaching, 3

Let $n\ge2$ and let $p(x)=x^n+a_{n-1}x^{n-1} \cdots a_1x+a_0$ be a polynomial with real coefficients. Prove that if for some positive integer $k(<n)$ the polynomial $(x-1)^{k+1}$ divides $p(x)$ then $$\sum_{i=0}^{n-1}|a_i| \ge 1 +\frac{2k^2}{n}$$

2023 USA IMOTST, 3

Tags: function , algebra
Let $\mathbb{N}$ denote the set of positive integers. Fix a function $f: \mathbb{N} \rightarrow \mathbb{N}$ and for any $m,n \in \mathbb{N}$ define $$\Delta(m,n)=\underbrace{f(f(\ldots f}_{f(n)\text{ times}}(m)\ldots))-\underbrace{f(f(\ldots f}_{f(m)\text{ times}}(n)\ldots)).$$ Suppose $\Delta(m,n) \neq 0$ for any distinct $m,n \in \mathbb{N}$. Show that $\Delta$ is unbounded, meaning that for any constant $C$ there exists $m,n \in \mathbb{N}$ with $\left|\Delta(m,n)\right| > C$.

2018 Saint Petersburg Mathematical Olympiad, 4

$f(x)$ is polynomial with integer coefficients, with module not exceeded $5*10^6$. $f(x)=nx$ has integer root for $n=1,2,...,20$. Prove that $f(0)=0$

PEN K Problems, 11

Find all functions $f: \mathbb{N}_{0}\to \mathbb{N}_{0}$ such that for all $m,n\in \mathbb{N}_{0}$: \[mf(n)+nf(m)=(m+n)f(m^{2}+n^{2}).\]

2004 Uzbekistan National Olympiad, 1

Solve the equation: $[\sqrt x+\sqrt{x+1}]+[\sqrt {4x+2}]=18$

2014 Contests, 2

Find all functions $f:R\rightarrow R$ such that \[ f(x^3)+f(y^3)=(x+y)(f(x^2)+f(y^2)-f(xy)) \] for all $x,y\in R$.

2008 Switzerland - Final Round, 2

Determine all functions $f : R^+ \to R^+$, so that for all $x, y > 0$: $$f(xy) \le \frac{xf(y) + yf(x)}{2}$$

2012 Indonesia TST, 1

Suppose a function $f : \mathbb{Z}^+ \rightarrow \mathbb{Z}^+$ satisfies $f(f(n)) + f(n+1) = n+2$ for all positive integer $n$. Prove that $f(f(n)+n) = n+1$ for all positive integer $n$.

1985 Vietnam Team Selection Test, 3

Suppose a function $ f: \mathbb R\to \mathbb R$ satisfies $ f(f(x)) \equal{} \minus{} x$ for all $ x\in \mathbb R$. Prove that $ f$ has infinitely many points of discontinuity.

2020 HMNT (HMMO), 9

Tags: algebra
While waiting for their next class on Killian Court, Alesha and Belinda both write the same sequence $S$ on a piece of paper, where $S$ is a $2020$-term strictly increasing geometric sequence with an integer common ratio . Every second, Alesha erases the two smallest terms on her paper and replaces them with their geometric mean, while Belinda erases the two largest terms in her paper and replaces them with their geometric mean. They continue this process until Alesha is left with a single value $A$ and Belinda is left with a single value $B$. Let $r_0$ be the minimal value of $r$ such that $\frac{A}{B}$ is an integer. If $d$ is the number of positive factors of $r_0$, what is the closest integer to $\log_{2} d$?

KoMaL A Problems 2024/2025, A. 898

Let $n$ be a given positive integer. Ana and Bob play the following game: Ana chooses a polynomial $p$ of degree $n$ with integer coefficients. In each round, Bob can choose a finite set $S$ of positive integers, and Ana responds with a list containing the values of the polynomial $p$ evaluated at the elements of $S$ with multiplicity (sorted in increasing order). Determine, in terms of $n$, the smallest positive integer $k$ such that Bob can always determine the polynomial $p$ in at most $k$ rounds. [i]Proposed by: Andrei Chirita, Cambridge[/i]

2001 Romania Team Selection Test, 1

Show that if $a,b,c$ are complex numbers that such that \[ (a+b)(a+c)=b \qquad (b+c)(b+a)=c \qquad (c+a)(c+b)=a\] then $a,b,c$ are real numbers.

DMM Individual Rounds, 2013(-14)Tie

[b]p1.[/b] A light beam shines from the origin into the unit square at an angle of $\theta$ to one of the sides such that $\tan \theta = \frac{13}{17}$ . The light beam is reflected by the sides of the square. How many times does the light beam hit a side of the square before hitting a vertex of the square? [img]https://cdn.artofproblemsolving.com/attachments/5/7/1db0aad33ed9bf82bee3303c7fbbe0b7c2574f.png[/img] [b]p2.[/b] Alex is given points $A_1,A_2,...,A_{150}$ in the plane such that no three are collinear and $A_1$, $A_2$, $...$, $A_{100}$ are the vertices of a convex polygon $P$ containing $A_{101}$, $A_{102}$, $ ...$, $A_{150}$ in its interior. He proceeds to draw edges $A_iA_j$ such that no two edges intersect (except possibly at their endpoints), eventually dividing $P$ up into triangles. How many triangles are there? [img]https://cdn.artofproblemsolving.com/attachments/d/5/12c757077e87809837d16128b018895a8bcc94.png[/img] [b]p3. [/b]The polynomial P(x) has the property that $P(1)$, $P(2)$, $P(3)$, $P(4)$, and $P(5)$ are equal to $1$, $2$, $3$, $4$,$5$ in some order. How many possibilities are there for the polynomial $P$, given that the degree of $P$ is strictly less than $4$? PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

EMCC Accuracy Rounds, 2016

[b]p1.[/b] A right triangle has a hypotenuse of length $25$ and a leg of length $16$. Compute the length of the other leg of this triangle. [b]p2.[/b] Tanya has a circular necklace with $5$ evenly-spaced beads, each colored red or blue. Find the number of distinct necklaces in which no two red beads are adjacent. If a necklace can be transformed into another necklace through a series of rotations and reflections, then the two necklaces are considered to be the same. [b]p3.[/b] Find the sum of the digits in the decimal representation of $10^{2016} - 2016$. [b]p4.[/b] Let $x$ be a real number satisfying $$x^1 \cdot x^2 \cdot x^3 \cdot x^4 \cdot x^5 \cdot x^6 = 8^7.$$ Compute the value of $x^7$. [b]p5.[/b] What is the smallest possible perimeter of an acute, scalene triangle with integer side lengths? [b]p6.[/b] Call a sequence $a_1, a_2, a_3,..., a_n$ mountainous if there exists an index $t$ between $1$ and $n$ inclusive such that $$a_1 \le a_2\le ... \le a_t \,\,\,\, and \,\,\,\, a_t \ge a_{t+1} \ge ... \ge a_n$$ In how many ways can Bishal arrange the ten numbers $1$, $1$, $2$, $2$, $3$, $3$, $4$, $4$, $5$, and $5$ into a mountainous sequence? (Two possible mountainous sequences are $1$, $1$, $2$, $3$, $4$, $4$, $5$, $5$, $3$, $2$ and $5$, $5$, $4$, $4$, $3$, $3$, $2$, $2$, $1$, $1$.) [b]p7.[/b] Find the sum of the areas of all (non self-intersecting) quadrilaterals whose vertices are the four points $(-3,-6)$, $(7,-1)$, $(-2, 9)$, and $(0, 0)$. [b]p8.[/b] Mohammed Zhang's favorite function is $f(x) =\sqrt{x^2 - 4x + 5} +\sqrt{x^2 + 4x + 8}$. Find the minumum possible value of $f(x)$ over all real numbers $x$. [b]p9.[/b] A segment $AB$ with length $1$ lies on a plane. Find the area of the set of points $P$ in the plane for which $\angle APB$ is the second smallest angle in triangle $ABP$. [b]p10.[/b] A binary string is a dipalindrome if it can be produced by writing two non-empty palindromic strings one after the other. For example, $10100100$ is a dipalindrome because both $101$ and $00100$ are palindromes. How many binary strings of length $18$ are both palindromes and dipalindromes? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

2010 China Team Selection Test, 3

Given integer $n\geq 2$ and real numbers $x_1,x_2,\cdots, x_n$ in the interval $[0,1]$. Prove that there exist real numbers $a_0,a_1,\cdots,a_n$ satisfying the following conditions: (1) $a_0+a_n=0$; (2) $|a_i|\leq 1$, for $i=0,1,\cdots,n$; (3) $|a_i-a_{i-1}|=x_i$, for $i=1,2,\cdots,n$.

2008 Moldova MO 11-12, 5

Find the least positive integer $ n$ so that the polynomial $ P(X)\equal{}\sqrt3\cdot X^{n\plus{}1}\minus{}X^n\minus{}1$ has at least one root of modulus $ 1$.

1960 IMO Shortlist, 2

For what values of the variable $x$ does the following inequality hold: \[ \dfrac{4x^2}{(1-\sqrt{2x+1})^2}<2x+9 \ ? \]