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

2003 Denmark MO - Mohr Contest, 3
Tags: number theory , prime
Determine the integers $n$ where $$|2n^2+9n+4|$$ is a prime number.

2023 ISI Entrance UGB, 6
Tags: inequalities , recurrence relation
Let $\{u_n\}_{n \ge 1}$ be a sequence of real numbers defined as $u_1 = 1$ and \[ u_{n+1} = u_n + \frac{1}{u_n} \text{ for all $n \ge 1$.}\] Prove that $u_n \le \frac{3\sqrt{n}}{2}$ for all $n$.

2018 District Olympiad, 4
Tags: complex number
Let $n\ge 2$ be a natural number. Find all complex numbers $z$ which simultaneously satisfy the relations $\text{a)}\ z^n + z^{n - 1} + \ldots + z^2 + |z| = n;$ $\text{b)}\ |z|^{n- 1} + |z|^{n - 2} + \ldots + |z|^2 + z = n z^n.$

2006 Victor Vâlcovici, 1
Tags: inequalities
Let be two nonnegative real numbers $ a,b, $ not both $ 0, $ satisfying $ \frac{1}{2a+b} +\frac{1}{a+2b} =1. $ Prove the following inequalities and explain the equality cases for [b]a),b).[/b] [b]a)[/b] $ 4/3\le a+b\le 3/2 $ [b]b)[/b] $ 8/9\le a^2+b^2\le 9/4 $ [b]c)[/b] $ ab<1/2 $ [i]Laurențiu Panaitopol[/i]

Ukrainian TYM Qualifying - geometry, II.16
Tags: symmetry , symmetric , geometry
Inside the circle are given three points that do not belong to one line. In one step it is allowed to replace one of the points with a symmetric one wrt the line containing the other two points. Is it always possible for a finite number of these steps to ensure that all three points are outside the circle?

2003 Tournament Of Towns, 7
Tags: combinatorics
A square is triangulated in such way that no three vertices are collinear. For every vertex (including vertices of the square) the number of sides issuing from it is counted. Can it happen that all these numbers are even?

2011 Math Prize For Girls Problems, 17
Tags: algebra , polynomial
There is a polynomial $P$ such that for every real number $x$, \[ x^{512} + x^{256} + 1 = (x^2 + x + 1) P(x). \] When $P$ is written in standard polynomial form, how many of its coefficients are nonzero?

2015 HMNT, 10
Tags:
Let $N$ be the number of functions $f$ from $\{1, 2, \dots, 101 \} \rightarrow \{1, 2, \dots, 101 \}$ such that $f^{101}(1) = 2.$ Find the remainder when $N$ is divided by $103.$

2024 Ukraine National Mathematical Olympiad, Problem 1
Tags: greatest common divisor , least common multiple , number theory
Find all pairs $a, b$ of positive integers, for which $$(a, b) + 3[a, b] = a^3 - b^3$$ Here $(a, b)$ denotes the greatest common divisor of $a, b$, and $[a, b]$ denotes the least common multiple of $a, b$. [i]Proposed by Oleksiy Masalitin[/i]

1999 Singapore Team Selection Test, 3
Tags: divide , number theory , prime number
Let $f(x) = x^{1998} - x^{199}+x^{19}+ 1$. Prove that there is an infinite set of prime numbers, each dividing at least one of the integers $f(1), f(2), f(3), f(4), ...$

2008 May Olympiad, 5
Tags: combinatorics
On a $16 x 16$ board, $25$ coins are placed, as in the figure. It is allowed to select $8$ rows and $8$ columns and remove from the board all the coins that are in those $16$ lines. Determine if it is possible to remove all coins from the board. [img]https://cdn.artofproblemsolving.com/attachments/1/5/e2c7379a6f47e2e8b8c9b989b85b96454a38e1.gif[/img] If the answer is yes, indicate the $8$ rows and $8$ columns selected, and if no, explain why.

ICMC 6, 3
Tags: probability , combinatorics , random process
The numbers $1, 2, \dots , n$ are written on a blackboard and then erased via the following process:[list] [*] Before any numbers are erased, a pair of numbers is chosen uniformly at random and circled. [*] Each minute for the next $n -1$ minutes, a pair of numbers still on the blackboard is chosen uniformly at random and the smaller one is erased. [*] In minute $n$, the last number is erased. [/list] What is the probability that the smaller circled number is erased before the larger? [i]Proposed by Ethan Tan[/i]

2012 Princeton University Math Competition, B1
Tags: combinatorics
Your friend sitting to your left (or right?) is unable to solve any of the eight problems on his or her Combinatorics $B$ test, and decides to guess random answers to each of them. To your astonishment, your friend manages to get two of the answers correct. Assuming your friend has equal probability of guessing each of the questions correctly, what is the average possible value of your friend’s score? Recall that each question is worth the point value shown at the beginning of each question.

2008 China Team Selection Test, 2
Tags: polynomial , modular arithmetic , absolute value , algebra
Prove that for all $ n\geq 2,$ there exists $ n$-degree polynomial $ f(x) \equal{} x^n \plus{} a_{1}x^{n \minus{} 1} \plus{} \cdots \plus{} a_{n}$ such that (1) $ a_{1},a_{2},\cdots, a_{n}$ all are unequal to $ 0$; (2) $ f(x)$ can't be factorized into the product of two polynomials having integer coefficients and positive degrees; (3) for any integers $ x, |f(x)|$ isn't prime numbers.

2013 Regional Competition For Advanced Students, 3
Tags: inequalities , arithmetic mean-geometric mean , sequence
For non-negative real numbers $a,$ $b$ let $A(a, b)$ be their arithmetic mean and $G(a, b)$ their geometric mean. We consider the sequence $\langle a_n \rangle$ with $a_0 = 0,$ $a_1 = 1$ and $a_{n+1} = A(A(a_{n-1}, a_n), G(a_{n-1}, a_n))$ for $n > 0.$ (a) Show that each $a_n = b^2_n$ is the square of a rational number (with $b_n \geq 0$). (b) Show that the inequality $\left|b_n - \frac{2}{3}\right| < \frac{1}{2^n}$ holds for all $n > 0.$

1990 IMO Longlists, 31
Tags: symmetry , combinatorics
Let $S = \{1, 2, \ldots, 1990\}$. A $31$-element subset of $S$ is called "good" if the sum of its elements is divisible by $5$. Find the number of good subsets of $S.$

2012 Czech-Polish-Slovak Match, 3
Tags: inequalities
Let $a,b,c,d$ be positive real numbers such that $abcd=4$ and \[a^2+b^2+c^2+d^2=10.\] Find the maximum possible value of $ab+bc+cd+da$.

1985 Traian Lălescu, 1.4
Tags: equation , algebra , floor , logarithm
Let $ a $ be a non-negative real number distinct from $ 1. $ [b]a)[/b] For which positive values $ x $ the equation $$ \left\lfloor\log_a x\right\rfloor +\left\lfloor \frac{1}{3} +\log_a x\right\rfloor =\left\lfloor 2\cdot\log_a x\right\rfloor $$ is true? [b]b)[/b] Solve $ \left\lfloor\log_3 x\right\rfloor +\left\lfloor \frac{1}{3} +\log_3 x\right\rfloor =3. $

1999 Portugal MO, 3
Tags: chord , geometry
If two parallel chords of a circumference, $10$ mm and $14$ mm long, with distance $6$ mm from each other, how long is the chord equidistant from these two?

1994 Vietnam Team Selection Test, 3
Tags: calculus , derivative , function , combinatorics
Calculate \[T = \sum \frac{1}{n_1! \cdot n_2! \cdot \cdots n_{1994}! \cdot (n_2 + 2 \cdot n_3 + 3 \cdot n_4 + \ldots + 1993 \cdot n_{1994})!}\] where the sum is taken over all 1994-tuples of the numbers $n_1, n_2, \ldots, n_{1994} \in \mathbb{N} \cup \{0\}$ satisfying $n_1 + 2 \cdot n_2 + 3 \cdot n_3 + \ldots + 1994 \cdot n_{1994} = 1994.$

2007 Paraguay Mathematical Olympiad, 3
Tags: ratio , analytic geometry , graphing lines , slope
Let $ABCD$ be a square, $E$ and $F$ midpoints of $AB$ and $AD$ respectively, and $P$ the intersection of $CF$ and $DE$. a) Show that $DE \perp CF$. b) Determine the ratio $CF : PC : EP$

2004 Iran MO (3rd Round), 5
Tags: function , combinatorics
assume that k,n are two positive integer $k\leq n$count the number of permutation $\{\ 1,\dots ,n\}\ $ st for any $1\leq i,j\leq k$and any positive integer m we have $f^m(i)\neq j$ ($f^m$ meas iterarte function,)

2000 China National Olympiad, 2
Tags: algebra
A sequence $(a_n)$ is defined recursively by $a_1=0, a_2=1$ and for $n\ge 3$, \[a_n=\frac12na_{n-1}+\frac12n(n-1)a_{n-2}+(-1)^n\left(1-\frac{n}{2}\right).\] Find a closed-form expression for $f_n=a_n+2\binom{n}{1}a_{n-1}+3\binom{n}{2}a_{n-2}+\ldots +(n-1)\binom{n}{n-2}a_2+n\binom{n}{n-1}a_1$.

2024 Francophone Mathematical Olympiad, 1
Tags: polynomial , game , algebra
Let $d$ and $m$ be two fixed positive integers. Pinocchio and Geppetto know the values of $d$ and $m$ and play the following game: In the beginning, Pinocchio chooses a polynomial $P$ of degree at most $d$ with integer coefficients. Then Geppetto asks him questions of the following form "What is the value of $P(n)$?'' for $n \in \mathbb{Z}$. Pinocchio usually says the truth, but he can lie up to $m$ times. What is, as a function of $d$ and $m$, the minimal number of questions that Geppetto needs to ask to be sure to determine $P$, no matter how Pinocchio chooses to reply?

2013 CHMMC (Fall), 2
Tags: geometry
Two circles of radii $7$ and $17$ have a distance of $25$ between their centers. What is the difference between the lengths of their common internal and external tangents (positive difference)?