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

1967 IMO Longlists, 49
Tags: trigonometry , inequality , trigonometric inequality , minimum
Let $n$ and $k$ be positive integers such that $1 \leq n \leq N+1$, $1 \leq k \leq N+1$. Show that: \[ \min_{n \neq k} |\sin n - \sin k| < \frac{2}{N}. \]

2007-2008 SDML (Middle School), 3
Tags:
If $n$ is a positive integer such that $1+2+3+\cdots+n=190$, then what is $n$?

1972 Swedish Mathematical Competition, 5
Tags: integration , algebra , analysis
Show that \[ \int\limits_0^1 \frac{1}{(1+x)^n} dx > 1-\frac{1}{n} \] for all positive integers $n$.

1973 Putnam, A6
Tags: geometry , line
Prove that it is impossible for seven distinct straight lines to be situated in the euclidean plane so as to have at least six points where exactly three of these lines intersect and at least four points where exactly two of these lines intersect.

2024 LMT Fall, 1
Tags: team
A positive integer $n$ is called "foursic'' if there exists a placement of $0$ in the digits of $n$ such that the resulting number a multiple of $4.$ For example, $14$ is foursic because $104$ is a multiple of $4.$ Find the number of two-digit foursic numbers.

2017 Rioplatense Mathematical Olympiad, Level 3, 2
Tags: geometry , combinatorial geometry , combinatorics
One have $n$ distinct circles(with the same radius) such that for any $k+1$ circles there are (at least) two circles that intersects in two points. Show that for each line $l$ one can make $k$ lines, each one parallel with $l$, such that each circle has (at least) one point of intersection with some of these lines.

2021 239 Open Mathematical Olympiad, 3
Tags: combinatorics
Given is a simple graph with $239$ vertices, such that it is not bipartite and each vertex has degree at least $3$. Find the smallest $k$, such that each odd cycle has length at most $k$.

2013 Saudi Arabia IMO TST, 4
Tags: algebra , polynomial , integer , divisible
Find all polynomials $p(x)$ with integer coefficients such that for each positive integer $n$, the number $2^n - 1$ is divisible by $p(n)$.

1994 Moldova Team Selection Test, 2
Tags: number theory , power series , algebra
Prove that every positive rational number can be expressed uniquely as a finite sum of the form $$a_1+\frac{a_2}{2!}+\frac{a_3}{3!}+\dots+\frac{a_n}{n!},$$ where $a_n$ are integers such that $0 \leq a_n \leq n-1$ for all $n > 1$.

2009 Today's Calculation Of Integral, 493
Tags: calculus , integration , ellipse , geometry , trigonometry , calculus computations , conic
In the $ x \minus{} y$ plane, let $ l$ be the tangent line at the point $ A\left(\frac {a}{2},\ \frac {\sqrt {3}}{2}b\right)$ on the ellipse $ \frac {x^2}{a^2} \plus{} \frac {y^2}{b^2}\equal{}1\ (0 < b < 1 < a)$. Let denote $ S$ be the area of the figure bounded by $ l,$ the $ x$ axis and the ellipse. (1) Find the equation of $ l$. (2) Express $ S$ in terms of $ a,\ b$. (3) Find the maximum value of $ S$ with the constraint $ a^2 \plus{} 3b^2 \equal{} 4$.

2012 May Olympiad, 4
Tags: combinatorics , game
Pedro has $111$ blue chips and $88$ white chips. There is a machine that for every $14$ blue chips , it gives $11$ white pieces and for every $7$ white chips, it gives $13$ blue pieces. Decide if Pedro can achieve, through successive operations with the machine, increase the total number of chips by $33$, so that the number of blue chips equals $\frac53$ of the amount of white chips. If possible, indicate how to do it. If not, indicate why.

2008 Thailand Mathematical Olympiad, 6
Tags: inequalities , algebra , functional inequality , functional
Let $f : R \to R$ be a function satisfying the inequality $|f(x + y) -f(x) - f(y)| < 1$ for all reals $x, y$. Show that $\left| f\left( \frac{x}{2008 }\right) - \frac{f(x)}{2008} \right| < 1$ for all real numbers $x$.

2004 Thailand Mathematical Olympiad, 15
Tags: divide , divisible , number theory
Find the largest positive integer $n \le 2004$ such that $3^{3n+3} - 27$ is divisible by $169$.

1965 Kurschak Competition, 1
Tags: number theory , diophantine , inequalities
What integers $a, b, c$ satisfy $a^2 + b^2 + c^2 + 3 < ab + 3b + 2c$ ?

2009 Romania National Olympiad, 1
Tags: limit , real analysis
Let $(t_n)_n$ a convergent sequence of real numbers, $t_n\in (0,1),\ (\forall)n\in \mathbb{N}$ and $\lim_{n\to \infty} t_n\in (0,1)$. Define the sequences $(x_n)_n$ and $(y_n)_n$ by \[x_{n+1}=t_nx_n+(1-t_n)y_n,\ y_{n+1}=(1-t_n)x_n+t_n y_n,\ (\forall)n\in \mathbb{N}\] and $x_0,y_0$ are given real numbers. a) Prove that the sequences $(x_n)_n$ and $(y_n)_n$ are convergent and have the same limit. b) Prove that if $\lim_{n\to \infty} t_n\in \{0,1\}$, then the question is false.

2007 Postal Coaching, 5
Tags: combinatorics , combinatorial geometry , distance
There are $N$ points in the plane such that the [b]total number[/b] of pairwise distances of these $N$ points is at most $n$. Prove that $N \le (n + 1)^2$.

2016 PUMaC Algebra Individual B, B1
Tags:
If $x$ is a positive real number such that $(x^2 - 1)^2 - 1 = 9800$, compute $x$.

1959 Poland - Second Round, 2
Tags: median , geometry , similar
What relationship between the sides of a triangle makes it similar to the triangle formed by its medians?

2023 HMNT, 32
Tags:
Compute $$\sum_{\underset{a \ge 6, b,c \ge 0}{a+b+c=12}} \frac{a!}{b!c!(a-b-c)!},$$ where the sum runs over all triples of nonnegative integers $(a,b,c)$ such that $a+b+c=12$ and $a \ge 6.$

2016 Tournament Of Towns, 4
Tags: combinatorics , geometry , 3d geometry
A designer took a wooden cube $5 \times 5 \times 5$, divided each face into unit squares and painted each square black, white or red so that any two squares with a common side have different colours. What is the least possible number of black squares? (Squares with a common side may belong to the same face of the cube or to two different faces.) [i](8 points)[/i] [i]Mikhail Evdokimov[/i]

1989 Chile National Olympiad, 1
Tags: number theory , sum of digits , digit
Writing $1989$ in base $b$, we obtain a three-digit number: $xyz$. It is known that the sum of the digits is the same in base $10$ and in base $b$, that is, $1 + 9 + 8 + 9 = x + y + z$. Determine $x,y,z,b.$

1969 AMC 12/AHSME, 8
Tags:
Triangle $ABC$ is inscribed in a circle. The measure of the non-overlapping minor arcs $AB$, $BC$, and $CA$ are, respectively, $x+75^\circ$, $2x+25^\circ$, $3x-22^\circ$. Then one interior angle of the triangle, in degrees, is: $\textbf{(A) }57\tfrac12\qquad \textbf{(B) }59\qquad \textbf{(C) }60\qquad \textbf{(D) }61\qquad \textbf{(E) }122$

2004 USA Team Selection Test, 3
Tags: floor function , perimeter , inequalities , vector , calculus , integration , geometry
Draw a $2004 \times 2004$ array of points. What is the largest integer $n$ for which it is possible to draw a convex $n$-gon whose vertices are chosen from the points in the array?

2017 AIME Problems, 4
Tags:
A pyramid has a triangular base with side lengths $20$, $20$, and $24$. The three edges of the pyramid from the three corners of the base to the fourth vertex of the pyramid all have length $25$. The volume of the pyramid is $m\sqrt{n}$, where $m$ and $n$ are positive integers, and $n$ is not divisible by the square of any prime. Find $m+n$.

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 $ST\parallel PR$.