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: 85335

1987 IMO Longlists, 65
Tags: algebra
The [i]runs[/i] of a decimal number are its increasing or decreasing blocks of digits. Thus $024379$ has three [i]runs[/i] : $024, 43$, and $379$. Determine the average number of runs for a decimal number in the set $\{d_1d_2 \cdots d_n | d_k \neq d_{k+1}, k = 1, 2,\cdots, n - 1\}$, where $n \geq 2.$

2021 Azerbaijan EGMO TST, 2
Tags: geometry , isogonal lines
Let $\omega$ be a circle with center $O,$ and let $A$ be a point with tangents $AP$ and $AQ$ to the circle. Denote by $K$ the intersection point of $AO$ and $PQ.$ $l_1$ and $l_2$ are two lines passing through $A$ that intersect $\omega.$ Call $B$ the intersection point of $l_1$ with $\omega,$ which is located nearer to $A$ on $l_1.$ Call $C$ the intersection point of $l_2$ with $\omega,$ which is located further to $A$ on $l_2.$ Prove that $\angle PAB = \angle QAC$ if and only if the points $B, K, C$ are on line.

2017 HMNT, 9
Tags: algebra
Find the minimum value of $\sqrt{58-42x}+\sqrt{149-140\sqrt{1-x^2}}$ where $-1 \le x \le 1$.

2012 Junior Balkan Team Selection Tests - Romania, 2
Tags: geometry , isosceles , semicircle , tangent circle
Consider a semicircle of center $O$ and diameter $[AB]$, and let $C$ be an arbitrary point on the segment $(OB)$. The perpendicular to the line $AB$ through $C$ intersects the semicircle in $D$. A circle centered in $P$ is tangent to the arc $BD$ in $F$ and to the segments $[AB]$ and $[CD]$ in $G$ and $E$, respectively. Prove that the triangle $ADG$ is isosceles.

2023 China Second Round, 2
Tags: algebra , logarithm
if a,b∈R+,$a^{\log b}=2$,$a^{\log a}b^{\log b}=5$,find out $(ab)^{\log ab}$

2010 AMC 12/AHSME, 15
Tags:
For how many ordered triples $ (x,y,z)$ of nonnegative integers less than $ 20$ are there exactly two distinct elements in the set $ \{i^x,(1 \plus{} i)^y,z\}$, where $ i \equal{} \sqrt { \minus{} 1}$? $ \textbf{(A)}\ 149 \qquad \textbf{(B)}\ 205 \qquad \textbf{(C)}\ 215 \qquad \textbf{(D)}\ 225 \qquad \textbf{(E)}\ 235$

2020 AMC 12/AHSME, 3
Tags:
A driver travels for $2$ hours at $60$ miles per hour, during which her car gets $30$ miles per gallon of gasoline. She is paid $\$0.50$ per mile, and her only expense is gasoline at $\$2.00$ per gallon. What is her net rate of pay, in dollars per hour, after this expense? $\textbf{(A) }20 \qquad\textbf{(B) }22 \qquad\textbf{(C) }24 \qquad\textbf{(D) } 25\qquad\textbf{(E) } 26$

2001 China Team Selection Test, 1
Tags: geometry
In an acute-angled triangle $\triangle ABC$, construct $\triangle ACD$ and $\triangle BCE$ externally on sides $CA$ and $CB$ respectively, such that $AD=CD$. Let $M$ be the midpoint of $AB$, and connect $DM$ and $EM$. Given that $DM$ is perpendicular to $EM$, set $\frac{AC}{BC} =u$ and $\frac{DM}{EM}=v$. Express $\frac{DC}{EC}$ in terms of $u$ and $v$.

2015 Vietnam National Olympiad, 1
Tags: polynomial , induction , algebra
Let ${\left\{ {f(x)} \right\}}$ be a sequence of polynomial, where ${f_0}(x) = 2$, ${f_1}(x) = 3x$, and ${f_n}(x) = 3x{f_{n - 1}}(x) + (1 - x - 2{x^2}){f_{n - 2}}(x)$ $(n \ge 2)$ Determine the value of $n$ such that ${f_n}(x)$ is divisible by $x^3-x^2+x$.

1996 Bosnia and Herzegovina Team Selection Test, 2
Tags: euler , number theory , eulers function , greatest common divisor , function
$a)$ Let $m$ and $n$ be positive integers. If $m>1$ prove that $ n \mid \phi(m^n-1)$ where $\phi$ is Euler function $b)$ Prove that number of elements in sequence $1,2,...,n$ $(n \in \mathbb{N})$, which greatest common divisor with $n$ is $d$, is $\phi\left(\frac{n}{d}\right)$

1988 Bulgaria National Olympiad, Problem 4
Tags: geometry
Let $A,B,C$ be non-collinear points. For each point $D$ of the ray $AC$, we denote by $E$ and $F$ the points of tangency of the incircle of $\triangle ABD$ with $AB$ and $AD$, respectively. Prove that, as point $D$ moves along the ray $AC$, the line $EF$ passes through a fixed point.

2023 Malaysian IMO Training Camp, 3
Tags: algebra
A sequence of reals $a_1, a_2, \cdots$ satisfies for all $m>1$, $$a_{m+1}a_{m-1}=a_m^2-a_1^2$$ Prove that for all $m>n>1$, the sequence satisfies the equation $$a_{m+n}a_{m-n}=a_m^2-a_n^2$$ [i]Proposed by Ivan Chan Kai Chin[/i]

2008 Indonesia MO, 3
Tags: number theory
Find all natural number which can be expressed in $ \frac{a\plus{}b}{c}\plus{}\frac{b\plus{}c}{a}\plus{}\frac{c\plus{}a}{b}$ where $ a,b,c\in \mathbb{N}$ satisfy $ \gcd(a,b)\equal{}\gcd(b,c)\equal{}\gcd(c,a)\equal{}1$

2023 Kyiv City MO Round 1, Problem 4
Tags: number theory , algebra , digit
Let's call a pair of positive integers $\overline{a_1a_2\ldots a_k}$ and $\overline{b_1b_2\ldots b_k}$ $k$-similar if all digits $a_1, a_2, \ldots, a_k , b_1 , b_2, \ldots, b_k$ are distinct, and there exist distinct positive integers $m, n$, for which the following equality holds: $$a_1^m + a_2^m + \ldots + a_k^m = b_1^n + b_2^n + \ldots + b_k^n$$ For which largest $k$ do there exist $k$-similar numbers? [i]Proposed by Oleksiy Masalitin[/i]

2021 China Second Round A1, 1
Tags: geometry
In triangle ABC,X,Y are on the angle bisector of ∠BAC and ∠ABX=∠ACY.BX intersects CY at P and circles (BYP) and (CXP) intersect at Q different from P. Prove that A,P,Q are on a line.

2002 AMC 12/AHSME, 24
Tags: perimeter , inequalities , trigonometry , geometry
A convex quadrilateral $ ABCD$ with area $ 2002$ contains a point $ P$ in its interior such that $ PA \equal{} 24$, $ PB \equal{} 32$, $ PC \equal{} 28$, and $ PD \equal{} 45$. FInd the perimeter of $ ABCD$. $ \textbf{(A)}\ 4\sqrt {2002}\qquad \textbf{(B)}\ 2\sqrt {8465}\qquad \textbf{(C)}\ 2\left(48 \plus{} \sqrt {2002}\right)$ $ \textbf{(D)}\ 2\sqrt {8633}\qquad \textbf{(E)}\ 4\left(36 \plus{} \sqrt {113}\right)$

1998 Austrian-Polish Competition, 1
Tags: inequalities , algebra
Let $x_1, x_2,y _1,y_2$ be real numbers such that $x_1^2 + x_2^2 \le 1$. Prove the inequality $$(x_1y_1 + x_2y_2 - 1)^2 \ge (x_1^2 + x_2^2 - 1)(y_1^2 + y_2^2 -1)$$

2009 Italy TST, 3
Tags: number theory , geometry
Find all pairs of integers $(x,y)$ such that \[ y^3=8x^6+2x^3y-y^2.\]

2002 Korea - Final Round, 2
Tags: function , algebra
Find all functions $f:\mathbb{R}\to \mathbb{R}$ satisfying $f(x-y)=f(x)+xy+f(y)$ for every $x \in \mathbb{R}$ and every $y \in \{f(x) \mid x\in \mathbb{R}\}$, where $\mathbb{R}$ is the set of real numbers.

2016 Dutch BxMO TST, 1
Tags: divisor , odd , equation , number theory
For a positive integer $n$ that is not a power of two, we de fine $t(n)$ as the greatest odd divisor of $n$ and $r(n)$ as the smallest positive odd divisor of $n$ unequal to $1$. Determine all positive integers $n$ that are not a power of two and for which we have $n = 3t(n) + 5r(n)$.

1976 Dutch Mathematical Olympiad, 3
Tags: chessboard , combinatorics
In how many ways can the king in the chessboard reach the eighth rank in $7$ moves from its original square on the first row?

2012 Dutch BxMO/EGMO TST, 5
Tags: number theory , combinatorics , set
Let $A$ be a set of positive integers having the following property: for each positive integer $n$ exactly one of the three numbers $n, 2n$ and $3n$ is an element of $A$. Furthermore, it is given that $2 \in A$. Prove that $13824 \notin A$.

2017 ASDAN Math Tournament, 5
Tags: algebra test
Compute $$\sum_{i=0}^\infty(-1)^i\sum_{j=i}^\infty(-1)^j\frac{2}{j^2+4j+3}.$$

2010 Postal Coaching, 3
Tags: function , algebra
Find all functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ such that $\boxed{1} \ f(1) = 1$ $\boxed{2} \ f(m+n)(f(m)-f(n)) = f(m-n)(f(m)+f(n)) \ \forall \ m,n \in \mathbb{Z}$

1979 Romania Team Selection Tests, 5.
Tags: combinatorics , counting
In how many ways can we fill the cells of a $m\times n$ board with $+1$ and $-1$ such that the product of numbers on each line and on each column are all equal to $-1$?