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

2020 Azerbaijan IMO TST, 2
Tags: combinatorics , induction , local argument
You are given a set of $n$ blocks, each weighing at least $1$; their total weight is $2n$. Prove that for every real number $r$ with $0 \leq r \leq 2n-2$ you can choose a subset of the blocks whose total weight is at least $r$ but at most $r + 2$.

2011 USA TSTST, 8
Tags: floor function , number theory
Let $x_0, x_1, \dots , x_{n_0-1}$ be integers, and let $d_1, d_2, \dots, d_k$ be positive integers with $n_0 = d_1 > d_2 > \cdots > d_k$ and $\gcd (d_1, d_2, \dots , d_k) = 1$. For every integer $n \ge n_0$, define \[ x_n = \left\lfloor{\frac{x_{n-d_1} + x_{n-d_2} + \cdots + x_{n-d_k}}{k}}\right\rfloor. \] Show that the sequence $\{x_n\}$ is eventually constant.

2015 Baltic Way, 8
Tags: combinatorics
With inspiration drawn from the rectilinear network of streets in [i]New York[/i] , the [i]Manhattan distance[/i] between two points $(a,b)$ and $(c,d)$ in the plane is defined to be \[|a-c|+|b-d|\] Suppose only two distinct [i]Manhattan distance[/i] occur between all pairs of distinct points of some point set. What is the maximal number of points in such a set?

2021 LMT Fall, Tie
Tags: algebra
Estimate the value of $e^f$ , where $f = e^e$ .

PEN K Problems, 32
Tags: functional equation , function
Find all functions $f: \mathbb{Z}^{2}\to \mathbb{R}^{+}$ such that for all $i, j \in \mathbb{Z}$: \[f(i,j)=\frac{f(i+1, j)+f(i,j+1)+f(i-1,j)+f(i,j-1)}{4}.\]

2005 AMC 12/AHSME, 9
Tags:
On a certain math exam, $ 10 \%$ of the students got 70 points, $ 25 \%$ got 80 points, $ 20 \%$ got 85 points, $ 15 \%$ got 90 points, and the rest got 95 points. What is the difference between the mean and the median score on this exam? $ \textbf{(A)}\ 0\qquad \textbf{(B)}\ 1\qquad \textbf{(C)}\ 2\qquad \textbf{(D)}\ 4\qquad \textbf{(E)}\ 5$

1953 Putnam, A7
Tags: trigonometry , polynomial , root
Assuming that the roots of $x^3 +px^2 +qx +r=0$ are all real and positive, find the relation between $p,q,r$ which is a necessary and sufficient condition that the roots are the cosines of the angles of a triangle.

1994 Romania TST for IMO, 3:
Tags: number theory
Prove that the sequence $a_n = 3^n- 2^n$ contains no three numbers in geometric progression.

1991 AMC 8, 12
Tags:
If $\frac{2+3+4}{3}=\frac{1990+1991+1992}{N}$, then $N=$ $\text{(A)}\ 3 \qquad \text{(B)}\ 6 \qquad \text{(C)}\ 1990 \qquad \text{(D)}\ 1991 \qquad \text{(E)}\ 1992$

I Soros Olympiad 1994-95 (Rus + Ukr), 10.2
Tags: geometry , geometric inequality
Given a triangle $ABC$ and a point $O$ inside it, it is known that $AB\le BC\le CA$. Prove that $$OA+OB+OC<BC+CA.$$

2018 Yasinsky Geometry Olympiad, 3
Tags: geometry , altitude , midpoint
In the triangle $ABC$, $\angle B = 2 \angle C$, $AD$ is altitude, $M$ is the midpoint of the side $BC$. Prove that $AB = 2DM$.

2010 Princeton University Math Competition, 4
Tags:
Let $S$ be the sum of all real $x$ such that $4^x = x^4$. Find the nearest integer to $S$.

PEN E Problems, 23
Tags: induction , number theory , prime number
Let $p_{1}=2, p_{2}={3}, p_{3}=5, \cdots, p_{n}$ be the first $n$ prime numbers, where $n \ge 3$. Prove that \[\frac{1}{{p_{1}}^{2}}+\frac{1}{{p_{2}}^{2}}+\cdots+\frac{1}{{p_{n}}^{2}}+\frac{1}{p_{1}p_{2}\cdots p_{n}}< \frac{1}{2}.\]

2015 Iran Team Selection Test, 3
Tags: combinatorics
Find the maximum number of rectangles with sides equal to 1 and 2 and parallel to the coordinate axes such that each two have an area equal to 1 in common.

1998 Tournament Of Towns, 3
Tags: tangent , geometry , tangential , circles
Segment $AB$ intersects two equal circles, is parallel to the line joining their centres, and all the points of intersection of the segment and the circles lie between $A$ and $B$. From the point $A$ tangents to the circle nearest to $A$ are drawn, and from the point $B$ tangents to the circle nearest to $B$ are also drawn. It turns out that the quadrilateral formed by the four tangents extended contains both circles. Prove that a circle can be drawn so that it touches all four sides of the quadrilateral. (P Kozhevnikov)

1973 Chisinau City MO, 68
Tags: geometry , equilateral , similar
Inside the triangle $ABC$, point $O$ was chosen so that the triangles $AOB, BOC, COA$ turned out to be similar. Prove that triangle $ABC$ is equilateral.

2005 All-Russian Olympiad Regional Round, 8.6
Tags: geometry , angle bisector
In quadrilateral $ABCD$, angles $A$ and $C$ are equal. Angle bisector of $B$ intersects line $AD$ at point $P$. Perpendicular on $BP$ passing through point $A$ intersects line $BC$ at point $Q$. Prove that the lines $PQ$ and $CD$ are parallel.

2007 Grigore Moisil Intercounty, 4
Tags: function , group theory , complex number
Consider the group $ \{f:\mathbb{C}\setminus\mathbb{Q}\longrightarrow\mathbb{C}\setminus\mathbb{Q} | f\text{ is bijective}\} $ under the composition of functions. Find the order of the smallest subgroup of it that: $ \text{(1)} $ contains the function $ z\mapsto \frac{z-1}{z+1} . $ $ \text{(2)} $ contains the function $ z\mapsto \frac{z-3}{z+1} . $ $ \text{(3)} $ contain both of the above functions.

2010 AMC 12/AHSME, 10
Tags:
The average of the numbers $ 1,2,3,...,98,99$, and $ x$ is $ 100x$. What is $ x$? $ \textbf{(A)}\ \frac{49}{101} \qquad\textbf{(B)}\ \frac{50}{101} \qquad\textbf{(C)}\ \frac12 \qquad\textbf{(D)}\ \frac{51}{101} \qquad\textbf{(E)}\ \frac{50}{99}$

2002 Germany Team Selection Test, 1
Tags: quadratic , number theory
Determine the number of all numbers which are represented as $x^2+y^2$ with $x, y \in \{1, 2, 3, \ldots, 1000\}$ and which are divisible by 121.

2013 Federal Competition For Advanced Students, Part 1, 4
Tags: ratio , perpendicular bisector , geometry
Let $A$, $B$ and $C$ be three points on a line (in this order). For each circle $k$ through the points $B$ and $C$, let $D$ be one point of intersection of the perpendicular bisector of $BC$ with the circle $k$. Further, let $E$ be the second point of intersection of the line $AD$ with $k$. Show that for each circle $k$, the ratio of lengths $\overline{BE}:\overline{CE}$ is the same.

2009 China Team Selection Test, 3
Tags: induction , number theory , prime number , relatively prime , prime factorization , greatest common divisor , mobius function
Let $ (a_{n})_{n\ge 1}$ be a sequence of positive integers satisfying $ (a_{m},a_{n}) = a_{(m,n)}$ (for all $ m,n\in N^ +$). Prove that for any $ n\in N^ + ,\prod_{d|n}{a_{d}^{\mu (\frac {n}{d})}}$ is an integer. where $ d|n$ denotes $ d$ take all positive divisors of $ n.$ Function $ \mu (n)$ is defined as follows: if $ n$ can be divided by square of certain prime number, then $ \mu (1) = 1;\mu (n) = 0$; if $ n$ can be expressed as product of $ k$ different prime numbers, then $ \mu (n) = ( - 1)^k.$

2021 New Zealand MO, 2
Tags: algebra , inequalities
Prove that $$x^2 +\frac{8}{xy}+ y^2 \ge 8$$ for all positive real numbers $x$ and $y$.

2020 Purple Comet Problems, 3
Tags: number theory
Find the number of perfect squares that divide $20^{20}$.

2000 ITAMO, 3
Tags: ratio , geometry , 3d geometry , pyramid , volume , sphere
A pyramid with the base $ABCD$ and the top $V$ is inscribed in a sphere. Let $AD = 2BC$ and let the rays $AB$ and $DC$ intersect in point $E$. Compute the ratio of the volume of the pyramid $VAED$ to the volume of the pyramid $VABCD$.