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

Kvant 2023, M2763
Tags: number theory , sum of digits , modular arithmetic
Let $k\geqslant 2$ be a natural number. Prove that the natural numbers with an even sum of digits give all the possible residues when divided by $k{}$. [i]Proposed by P. Kozlov and I. Bogdanov[/i]

the 9th XMO, 4
Tags: combinatorics , graph theory
One hundred million cities lie on Planet MO. Initially, there are no air routes between any two cities. Now an airline company comes. It plans to establish $5050$ two-way routes, each route connects two different cities, and no two routes connect the same two cities. The "degree" of a city is defined to be the number of routes departing from that city. The "benefit" of a route is the product of the "degrees" of the two cities it connects. Find the maximum possible value of the sum of the benefits of these $5050$ routes.

2002 AMC 12/AHSME, 14
Tags:
Find $i+2i^2+3i^3+\ldots+2002i^{2002}$. $\textbf{(A) }-999+1002i\qquad\textbf{(B) }-1002+999i\sqrt2\qquad\textbf{(C) }-1001+1000i$ $\textbf{(D) }-1002+1001i\qquad\textbf{(E) }i$

2015 IMC, 1
Tags: matrices
For any integer $n\ge 2$ and two $n\times n$ matrices with real entries $A,\; B$ that satisfy the equation $$A^{-1}+B^{-1}=(A+B)^{-1}\;$$ prove that $\det (A)=\det(B)$. Does the same conclusion follow for matrices with complex entries? (Proposed by Zbigniew Skoczylas, Wroclaw University of Technology)

2022 Kosovo National Mathematical Olympiad, 3
Tags: parallelogram , geometry
Let $ABCD$ be a parallelogram and $l$ the line parallel to $AC$ which passes through $D$. Let $E$ and $F$ points on $l$ such that $DE=DF=DB$. Show that $EA,FC$ and $BD$ are concurrent.

2007 Moldova National Olympiad, 11.7
Tags: geometry , 3d geometry , tetrahedron , inequalities
Given a tetrahedron $VABC$ with edges $VA$, $VB$ and $VC$ perpendicular any two of them. The sum of the lengths of the tetrahedron's edges is $3p$. Find the maximal volume of $VABC$.

2010 National Olympiad First Round, 35
Tags:
Which one below is not less than $x^3+y^5$ for all reals $x,y$ such that $0<x<1$ and $0<y<1$? $ \textbf{(A)}\ x^2y \qquad\textbf{(B)}\ x^2y^2 \qquad\textbf{(C)}\ x^2y^3 \qquad\textbf{(D)}\ x^3y \qquad\textbf{(E)}\ xy^4 $

1970 Putnam, A6
Tags: probability , expected value , interval
Three numbers are chosen independently at random, one from each of the three intervals $[0, L_i ]$ ($i=1,2,3$). If the distribution of each random number is uniform with respect to the length of the interval it is chosen from, determine the expected value of the smallest number chosen.

2000 AMC 8, 16
Tags: geometry , perimeter
In order for Mateen to walk a kilometer ($1000$m) in his rectangular backyard, he must walk the length $25$ times or walk its perimeter $10$ times. What is the area of Mateen's backyard in square meters? $\text{(A)}\ 40 \qquad \text{(B)}\ 200 \qquad \text{(C)}\ 400 \qquad \text{(D)}\ 500 \qquad \text{(E)}\ 1000$

1999 Romania National Olympiad, 2
Tags: sequence
Let $k$ be a positive integer, let $z_1,z_2, \ldots, z_k \in \mathbb{C}$ be distinct and let $u_1,u_2,\ldots,u_k \in \mathbb{C}$ be such that the set $\big\{a_n=u_1z_1^n+u_2z_2^n+\ldots+u_kz_k^n : n \in \mathbb{Z}_{>0} \big\}$ is finite. Prove that there exists a positive integer $p$ such that $a_n=a_{n+p},$ for any positive integer $n.$

2009 China Western Mathematical Olympiad, 2
Tags: circumcircle , geometry
Given an acute triangle $ABC$, $D$ is a point on $BC$. A circle with diameter $BD$ intersects line $AB,AD$ at $X,P$ respectively (different from $B,D$).The circle with diameter $CD$ intersects $AC,AD$ at $Y,Q$ respectively (different from $C,D$). Draw two lines through $A$ perpendicular to $PX,QY$, the feet are $M,N$ respectively.Prove that $\triangle AMN$ is similar to $\triangle ABC$ if and only if $AD$ passes through the circumcenter of $\triangle ABC$.

2010 National Olympiad First Round, 24
Tags:
How many $7$-digit positive integers are there such that the number remains same when its digits are reversed and is multiple of $11$? $ \textbf{(A)}\ 900 \qquad\textbf{(B)}\ 854 \qquad\textbf{(C)}\ 818 \qquad\textbf{(D)}\ 726 \qquad\textbf{(E)}\ \text{None} $

2013 Pan African, 3
Tags: geometric transformation , reflection , trigonometry , geometry
Let $ABCDEF$ be a convex hexagon with $\angle A= \angle D$ and $\angle B=\angle E$ . Let $K$ and $L$ be the midpoints of the sides $AB$ and $DE$ respectively. Prove that the sum of the areas of triangles $FAK$, $KCB$ and $CFL$ is equal to half of the area of the hexagon if and only if \[\frac{BC}{CD}=\frac{EF}{FA}.\]

2012 Putnam, 6
Tags: modular arithmetic , euler , quadratic , algebra , polynomial , linear algebra
Let $p$ be an odd prime number such that $p\equiv 2\pmod{3}.$ Define a permutation $\pi$ of the residue classes modulo $p$ by $\pi(x)\equiv x^3\pmod{p}.$ Show that $\pi$ is an even permutation if and only if $p\equiv 3\pmod{4}.$

2020 LIMIT Category 1, 18
Tags: limit , geometry
Let $\triangle ABC$ be a right triangle with $\angle C=90^{\circ}$. Two squares $S_1$ and $S_2$ are inscribed in the triangle $ABC$ such that $S_1$ and $ABC$ share a common vertex $C$ and $S_2$ has one of its sides on $AB$. Suppose that $\text{Area}(S_1)=1+\text{Area}(S_2)=441$, then calculate $AC+BC$ (A)$400$ (B)$420$ (C)$441$ (D)$462$

1999 Baltic Way, 17
Tags: number theory
Does there exist a finite sequence of integers $c_1,c_2,\ldots ,c_n$ such that all the numbers $a+c_1,a+c_2,\ldots ,a+c_n$ are primes for more than one but not infinitely many different integers $a$?

2024 Iranian Geometry Olympiad, 1
Tags: geometry
An equilateral triangle $\bigtriangleup ABC$ is split into $4$ triangles with equal area; three congruent triangles $\bigtriangleup ABX,\bigtriangleup BCY, \bigtriangleup CAZ$, and a smaller equilateral triangle $\bigtriangleup XYZ$, as shown. Prove that the points $X, Y, Z$ lie on the incircle of triangle $\bigtriangleup ABC$. [i]Proposed by Josef Tkadlec - Czech Republic[/i]

2018 NZMOC Camp Selection Problems, 4
Tags: geometry , right angle , equal angles
Let $P$ be a point inside triangle $ABC$ such that $\angle CPA = 90^o$ and $\angle CBP = \angle CAP$. Prove that $\angle P XY = 90^o$, where $X$ and $Y$ are the midpoints of $AB$ and $AC$ respectively.

2023-IMOC, A4
Tags: algebra
Find all functions $f:\mathbb{R^{+}} \rightarrow \mathbb{R^{+}}$, such that $$xf(1+xf(y))=f(f(x)+f(y))$$ for all positive reals $x, y$.

2020 Bundeswettbewerb Mathematik, 3
Tags: circumcircle , cyclic quadrilateral , moving points , geometry
Two lines $m$ and $n$ intersect in a unique point $P$. A point $M$ moves along $m$ with constant speed, while another point $N$ moves along $n$ with the same speed. They both pass through the point $P$, but not at the same time. Show that there is a fixed point $Q \ne P$ such that the points $P,Q,M$ and $N$ lie on a common circle all the time.

2012 Poland - Second Round, 2
Tags: 3d geometry , tetrahedron , pyramid , geometry
Prove that for tetrahedron $ABCD$; vertex $D$, center of insphere and centroid of $ABCD$ are collinear iff areas of triangles $ABD,BCD,CAD$ are equal.

2014 Junior Regional Olympiad - FBH, 4
Tags: number theory , prime number
Find all prime numbers $p$ and $q$ such that $3p^2q+2pq^2=483$

2017 Hong Kong TST, 1
Tags: algebra , calculus , integration
Given that $\{a_n\}$ is a sequence of integers satisfying the following condition for all positive integral values of $n$: $a_n+a_{n+1}=2a_{n+2}a_{n+3}+2016$. Find all possible values of $a_1$ and $a_2$

2017 Saudi Arabia BMO TST, 3
Tags: number theory , multiple , digit
How many ways are there to insert plus signs $+$ between the digits of number $111111 ...111$ which includes thirty of digits $1$ so that the result will be a multiple of $30$?

2003 National Olympiad First Round, 20
Tags:
How many real numbers $x$ are there such that $\sqrt{ x + 1 - 4\sqrt{x-3}} + \sqrt{ x + 6 - 6\sqrt{x-3}} = 1$? $ \textbf{(A)}\ 3 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}\ 7 \qquad\textbf{(E)}\ \text{None of the preceding} $