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

1961 Polish MO Finals, 6
Tags: combinatorics
Someone wrote six letters to six people and addressed six envelopes to them. How many ways can the letters be put into the envelopes so that none of the letters end up in the correct envelope?

1969 AMC 12/AHSME, 17
Tags: logarithm
The equation $2^{2x}-8\cdot 2^x+12=0$ is satisfied by: $\textbf{(A) }\log3\qquad \textbf{(B) }\tfrac12\log6\qquad \textbf{(C) }1+\log\tfrac34\qquad$ $\textbf{(D) }1+\tfrac{\log3}{\log2}\qquad \textbf{(E) }\text{none of these}$

2021 CMIMC, 4
Tags: combinatorics
How many four-digit positive integers $\overline{a_1a_2a_3a_4}$ have only nonzero digits and have the property that $|a_i-a_j| \neq 1$ for all $1 \leq i<j \leq 4?$ [i]Proposed by Kyle Lee[/i]

2024 Myanmar IMO Training, 6
Tags: induction , number theory
Prove that for all integers $n \geq 3$, there exist odd positive integers $x$, $y$ such that $7x^2 + y^2 = 2^n$.

2014 China Second Round Olympiad, 2
Tags: geometry , circumcircle
Let $ABC$ be an acute triangle such that $\angle BAC \neq 60^\circ$. Let $D,E$ be points such that $BD,CE$ are tangent to the circumcircle of $ABC$ and $BD=CE=BC$ ($A$ is on one side of line $BC$ and $D,E$ are on the other side). Let $F,G$ be intersections of line $DE$ and lines $AB,AC$. Let $M$ be intersection of $CF$ and $BD$, and $N$ be intersection of $CE$ and $BG$. Prove that $AM=AN$.

1986 Traian Lălescu, 1.4
Tags: real analysis , calculus , integration , set theory , function
Let be a parametric set: $$ \mathcal{F}_{\lambda } =\left\{ f:[1,\infty)\longrightarrow\mathbb{R}\bigg| x\in(1,\infty )\implies \int_{x}^{x^2+\lambda^2 x} f\left( \xi\right) d\xi =1\right\} . $$ [b]a)[/b] Show that $ \mathcal{F}_0 =\emptyset . $ [b]b)[/b] Prove that $ \lambda\neq 0 $ implies $ \mathcal{F}_{\lambda }\neq\emptyset . $

1981 Spain Mathematical Olympiad, 8
Tags: number theory , sum of squares , divisible
If $a$ is an odd number, show that $$a^4 + 4a^3 + 11a^2 + 6a+ 2$$ is a sum of three squares and is divisible by $4$.

2003 China Team Selection Test, 2
Tags: function , induction , algebra
Let $S$ be a finite set. $f$ is a function defined on the subset-group $2^S$ of set $S$. $f$ is called $\textsl{monotonic decreasing}$ if when $X \subseteq Y\subseteq S$, then $f(X) \geq f(Y)$ holds. Prove that: $f(X \cup Y)+f(X \cap Y ) \leq f(X)+ f(Y)$ for $X, Y \subseteq S$ if and only if $g(X)=f(X \cup \{ a \}) - f(X)$ is a $\textsl{monotonic decreasing}$ funnction on the subset-group $2^{S \setminus \{a\}}$ of set $S \setminus \{a\}$ for any $a \in S$.

1984 AMC 12/AHSME, 5
Tags:
The largest integer $n$ for which $n^{200} < 5^{300}$ is $\textbf{(A) }8\qquad \textbf{(B) }9\qquad \textbf{(C) }10\qquad \textbf{(D) }11\qquad \textbf{(E) }12$

2014 Greece Team Selection Test, 3
Tags: circumcircle , trapezoid , geometry
Let $ABC$ be an acute,non-isosceles triangle with $AB<AC<BC$.Let $D,E,Z$ be the midpoints of $BC,AC,AB$ respectively and segments $BK,CL$ are altitudes.In the extension of $DZ$ we take a point $M$ such that the parallel from $M$ to $KL$ crosses the extensions of $CA,BA,DE$ at $S,T,N$ respectively (we extend $CA$ to $A$-side and $BA$ to $A$-side and $DE$ to $E$-side).If the circumcirle $(c_{1})$ of $\triangle{MBD}$ crosses the line $DN$ at $R$ and the circumcirle $(c_{2})$ of $\triangle{NCD}$ crosses the line $DM$ at $P$ prove that $ST\parallel PR$.

2023 Iran MO (3rd Round), 2
Tags: algebra
Does there exist bijections $f,g$ from positive integers to themselves st: $$g(n)=\frac{f(1)+f(2)+ \cdot \cdot \cdot +f(n)}{n}$$ holds for any $n$?

2006 Italy TST, 2
Tags: modular arithmetic , euler , number theory
Let $n$ be a positive integer, and let $A_{n}$ be the the set of all positive integers $a\le n$ such that $n|a^{n}+1$. a) Find all $n$ such that $A_{n}\neq \emptyset$ b) Find all $n$ such that $|{A_{n}}|$ is even and non-zero. c) Is there $n$ such that $|{A_{n}}| = 130$?

2025 Kosovo National Mathematical Olympiad`, P1
Tags: combinatorics
Anna wants to form a four-digit number with four different digits from the digits $1, 2, 3, 4, 5, 6, 7, 8, 9$. She wants the first digit of that number to be bigger than the sum of the other three digits. How many such numbers can she form?

2012 Princeton University Math Competition, A7
Tags: number theory
Let $a, b$, and $c$ be positive integers satisfying $a^4 + a^2b^2 + b^4 = 9633$ $2a^2 + a^2b^2 + 2b^2 + c^5 = 3605$. What is the sum of all distinct values of $a + b + c$?

2010 Purple Comet Problems, 24
Tags: sfft , special factorizations
Find the number of ordered pairs of integers $(m, n)$ that satisfy $20m-10n = mn$.

1973 Canada National Olympiad, 1
Tags: inequalities , calculus , integration , logarithm
(i) Solve the simultaneous inequalities, $x<\frac{1}{4x}$ and $x<0$; i.e. find a single inequality equivalent to the two simultaneous inequalities. (ii) What is the greatest integer that satisfies both inequalities $4x+13 < 0$ and $x^{2}+3x > 16$. (iii) Give a rational number between $11/24$ and $6/13$. (iv) Express 100000 as a product of two integers neither of which is an integral multiple of 10. (v) Without the use of logarithm tables evaluate \[\frac{1}{\log_{2}36}+\frac{1}{\log_{3}36}.\]

1989 AMC 8, 3
Tags:
Which of the following numbers is the largest? $\text{(A)}\ .99 \qquad \text{(B)}\ .9099 \qquad \text{(C)}\ .9 \qquad \text{(D)}\ .909 \qquad \text{(E)}\ .9009$

2023 HMNT, 6
Tags:
There are five people in a room. They each simultaneously pick two of the other people in the room independently and uniformly at random and point at them. Compute the probability that there exists a group of three people such that each of them is pointing at the other two in the group.

2019 IFYM, Sozopol, 2
Tags: combinatorics , game strategy , game theory , board , coloring
Let $n$ be a natural number. At first the cells of a table $2n$ x $2n$ are colored in white. Two players $A$ and $B$ play the following game. First is $A$ who has to color $m$ arbitrary cells in red and after that $B$ chooses $n$ rows and $n$ columns and color their cells in black. Player $A$ wins, if there is at least one red cell on the board. Find the least value of $m$ for which $A$ wins no matter how $B$ plays.

2008 ITest, 19
Tags:
Let $A$ be the set of positive integers that are the product of two consecutive integers. Let $B$ the set of positive integers that are the product of three consecutive integers. Find the sum of the two smallest elements of $A\cap B$.

2021 JHMT HS, 9
Tags: general , geometry
Squares of side lengths $1,$ $2,$ $3,$ and $4,$ are placed on a line segment $\ell$ from left to right, respectively, and these squares lie on the same side of $\ell,$ forming a polygon $P.$ An equilateral triangle whose base is $\ell$ is drawn around the squares such that its other two sides intersect $P$ at its leftmost and rightmost vertices (that are not on $\ell$). The area of the triangle can be written in the form $\tfrac{a + b\sqrt{3}}{c},$ where $a,$ $b,$ and $c$ are positive integers, and $b$ and $c$ are relatively prime. Find $a + b + c.$

2021 Malaysia IMONST 1, 18
Tags: algebra , equation
How many real numbers $x$ are solutions to the equation $|x - 2| - 4 =\frac{1}{|x - 3|}$ ?

2012 Princeton University Math Competition, B3
Tags: number theory
Find, with proof, all pairs $(x, y)$ of integers satisfying the equation $3x^2+ 4 = 2y^3$.

2020 SJMO, 6
Tags: number theory , combinatorics
We say a positive integer $n$ is [i]$k$-tasty[/i] for some positive integer $k$ if there exists a permutation $(a_0, a_1, a_2, \ldots , a_n)$ of $(0,1,2, \ldots, n)$ such that $|a_{i+1} - a_i| \in \{k, k+1\}$ for all $0 \le i \le n-1$. Prove that for all positive integers $k$, there exists a constant $N$ such that all integers $n \geq N$ are $k$-tasty. [i]Proposed by Anthony Wang[/i]

2003 Korea Junior Math Olympiad, 5
Tags: integer , number theory
Four odd positive intgers $a, b, c, d (a\leq b \leq c\leq d)$ are given. Choose any three numbers among them and divide their sum by the un-chosen number, and you will always get the remainder as $1$. Find all $(a, b, c, d)$ that satisfies this.