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

2019 IFYM, Sozopol, 4
Tags: combinatorics , inequalities
On a competition called [i]"Mathematical duels"[/i] students were given $n$ problems and each student solved exactly 3 of them. For each two students there is at most one problem that is solved from both of them. Prove that, if $s\in \mathbb{N}$ is a number for which $s^2-s+1<2n$, then there are $s$ problems among the $n$, no three of which solved by one student.

MOAA Team Rounds, 2023.15
Tags:
Triangle $ABC$ has circumcircle $\omega$. Let $D$ be the foot of the altitude from $A$ to $BC$ and let $AD$ intersect $\omega$ at $E \neq A$. Let $M$ be the midpoint of $AD$. If $\angle{BMC} = 90^\circ$, $AB = 9$ and $AE = 10$, the area of $\triangle{ABC}$ can be expressed in the form $\frac{a\sqrt{b}}{c}$ where $a,b,c$ are positive integers and $b$ is square-free. Find $a+b+c$. [i]Proposed by Andy Xu[/i]

2022 Junior Balkan Team Selection Tests - Romania, P2
Tags: geometry
Let $ABC$ be a triangle such that $\angle A=30^\circ$ and $\angle B=80^\circ$. Let $D$ and $E$ be points on sides $AC$ and $BC$ respectively so that $\angle ABD=\angle DBC$ and $DE\parallel AB$. Determine the measure of $\angle EAC$.

1958 AMC 12/AHSME, 20
Tags:
If $ 4^x \minus{} 4^{x \minus{} 1} \equal{} 24$, then $ (2x)^x$ equals: $ \textbf{(A)}\ 5\sqrt{5}\qquad \textbf{(B)}\ \sqrt{5}\qquad \textbf{(C)}\ 25\sqrt{5}\qquad \textbf{(D)}\ 125\qquad \textbf{(E)}\ 25$

2016 APMC, 8
Tags: number theory
Let be $n\geq 3$ fixed positive integer.Let be real numbers $a_1,a_2,...,a_n,b_1,b_2,...,b_n$ such that satisfied this conditions: [b]$i)$[/b] $ $ $a_n\geq a_{n-1}$ and $b_n\geq b_{n-1}$ [b]$ii)$[/b] $ $ $0<a_1\leq b_1\leq a_2\leq b_2\leq ... \leq a_{n-1}\leq b_{n-1}$ [b]$iii)$[/b] $ $ $a_1+a_2+...+a_n=b_1+b_2+...+b_n$ [b]$iv)$[/b] $ $ $a_{1}\cdot a_2\cdot ...\cdot a_n=b_1\cdot b_2\cdot ...\cdot b_n$ Show that $a_i=b_i$ for all $i=1,2,...,n$

2015 Saint Petersburg Mathematical Olympiad, 6
Tags: number theory , relatively prime
A sequence of integers is defined as follows: $a_1=1,a_2=2,a_3=3$ and for $n>3$, $$a_n=\textsf{The smallest integer not occurring earlier, which is relatively prime to }a_{n-1}\textsf{ but not relatively prime to }a_{n-2}.$$Prove that every natural number occurs exactly once in this sequence. [i]M. Ivanov[/i]

2010 China Western Mathematical Olympiad, 5
Tags: inequalities , algebra
Let $k$ be an integer and $k > 1$. Define a sequence $\{a_n\}$ as follows: $a_0 = 0$, $a_1 = 1$, and $a_{n+1} = ka_n + a_{n-1}$ for $n = 1,2,...$. Determine, with proof, all possible $k$ for which there exist non-negative integers $l,m (l \not= m)$ and positive integers $p,q$ such that $a_l + ka_p = a_m + ka_q$.

1986 National High School Mathematics League, 3
Tags: inequalities
For real numbers $a,b,c$, if $$a^2-bc-8a+7=b^2+c^2+bc-6a-6=0,$$ then the range value of $a$ is $\text{(A)}(-\infty,+\infty)\qquad\text{(B)}(-\infty,1]\cup[9,+\infty)\qquad\text{(C)}(0,7)\qquad\text{(D)}[1,9]$

2015 Cono Sur Olympiad, 1
Tags: divisibility , number theory
Show that, for any integer $n$, the number $n^3 - 9n + 27$ is not divisible by $81$.

1998 Irish Math Olympiad, 1
Tags: integration , inequalities
Prove that if $ x \not\equal{} 0$ is a real number, then: $ x^8\minus{}x^5\minus{}\frac{1}{x}\plus{}\frac{1}{x^4} \ge 0$.

2003 India IMO Training Camp, 5
Tags: geometry , geometric transformation , reflection , combinatorics
On the real number line, paint red all points that correspond to integers of the form $81x+100y$, where $x$ and $y$ are positive integers. Paint the remaining integer point blue. Find a point $P$ on the line such that, for every integer point $T$, the reflection of $T$ with respect to $P$ is an integer point of a different colour than $T$.

2009 Princeton University Math Competition, 4
Tags: geometry , 3d geometry , tetrahedron , inequalities
Tetrahedron $ABCD$ has sides of lengths, in increasing order, $7, 13, 18, 27, 36, 41$. If $AB=41$, then what is the length of $CD$?

2023 Indonesia TST, 2
Tags: combinatorics , coin , invariant , monovariant
Let $n > 3$ be a positive integer. Suppose that $n$ children are arranged in a circle, and $n$ coins are distributed between them (some children may have no coins). At every step, a child with at least 2 coins may give 1 coin to each of their immediate neighbors on the right and left. Determine all initial distributions of the coins from which it is possible that, after a finite number of steps, each child has exactly one coin.

2024 PErA, P4
Tags: geometry
Let $ABC$ be a triangle, and let $E$, $F$ be the feet of the altitudes from $B$ and $C$ to sides $AC$ and $AB$, respectively. Let $P$ and $Q$ be the intersections of $EF$ with the tangents from $B$ and $C$ to $(ABC)$, respectively. If $M$ is the midpoint of $BC$, prove that $(PQM)$ is tangent to $BC$ at $M$.

2016 NIMO Summer Contest, 10
Tags: geometry
In rectangle $ABCD$, point $M$ is the midpoint of $AB$ and $P$ is a point on side $BC$. The perpendicular bisector of $MP$ intersects side $DA$ at point $X$. Given that $AB = 33$ and $BC = 56$, find the least possible value of $MX$. [i]Proposed by Michael Tang[/i]

2008 Romania National Olympiad, 2
Tags:
a) We call [i]admissible sequence[/i] a sequence of 4 even digits in which no digits appears more than two times. Find the number of admissible sequences. b) For each integer $ n\geq 2$ we denote $ d_n$ the number of possibilities of completing with even digits an array with $ n$ rows and 4 columns, such that (1) any row is an admissible sequence; (2) the sequence 2, 0, 0, 8 appears exactly ones in the array. Find the values of $ n$ for which the number $ \frac {d_{n\plus{}1}}{d_n}$ is an integer.

1991 AIME Problems, 2
Tags: geometry , rectangle , geometric transformation
Rectangle $ABCD$ has sides $\overline {AB}$ of length 4 and $\overline {CB}$ of length 3. Divide $\overline {AB}$ into 168 congruent segments with points $A=P_0, P_1, \ldots, P_{168}=B$, and divide $\overline {CB}$ into 168 congruent segments with points $C=Q_0, Q_1, \ldots, Q_{168}=B$. For $1 \le k \le 167$, draw the segments $\overline {P_kQ_k}$. Repeat this construction on the sides $\overline {AD}$ and $\overline {CD}$, and then draw the diagonal $\overline {AC}$. Find the sum of the lengths of the 335 parallel segments drawn.

1992 Tournament Of Towns, (338) 6
Tags: number theory , combinatorics
For natural numbers $n$ and $b$, let $V(n, b)$ denote the number of decompositions of $n$ into the product of integers each of which is greater than $b$: for example $$36 = 6\times 6 = 4\times 9 = 3\times 3\times 4 = 3\times 12,$$ i.e. $V(36,2) = 5$. Prove that $V(n, b) < n/b$ for all $n$ and $b$. (N.B. Vasiliev, Moscow)

1993 Romania Team Selection Test, 1
Tags: inequalities
Find max. numbers $A$ wich is true ineq.: $\frac{x}{\sqrt{y^{2}+z^{2}}}+\frac{y}{\sqrt{x^{2}+z^{2}}}+\frac{z}{\sqrt{x^{2}+y^{2}}}\geq A$ $x,y,z$ are positve reals numberes! :wink:

1999 Gauss, 4
Tags: gauss
$1+\dfrac{1}{2}+\dfrac{1}{4}+\dfrac{1}{8}$ is equal to $\textbf{(A)}\ \dfrac{15}{8} \qquad \textbf{(B)}\ 1\dfrac{3}{14} \qquad \textbf{(C)}\ \dfrac{11}{8} \qquad \textbf{(D)}\ 1\dfrac{3}{4} \qquad \textbf{(E)}\ \dfrac{7}{8}$

1972 Canada National Olympiad, 10
Tags: ratio , calculus , integration , geometric sequence , algebra
What is the maximum number of terms in a geometric progression with common ratio greater than 1 whose entries all come from the set of integers between 100 and 1000 inclusive?

2004 Regional Olympiad - Republic of Srpska, 2
Tags: inequalities
The positive real numbers $x,y,z$ satisfy $x+y+z=1$. Show that \[\sqrt{3xyz}\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}+\frac{1}{1-x}+\frac{1}{1-y}+\frac{1}{1-z}\right)\geq4+ \frac{4xyz}{(1-x)(1-y)(1-z)}.\]

1964 Poland - Second Round, 2
Tags: perpendicular , arc midpoint , geometry
The circle is divided into four non-overlapping gaps $ AB $, $ BC $, $ CD $ and $ DA $. Prove that the segment joining the midpoints of the arcs $AB$ and $CD$ is perpendicular to the segment joining the midpoints of the arcs $BC$ and $DA$.

2019 CCA Math Bonanza, L1.1
Tags:
How many integers divide either $2018$ or $2019$? Note: $673$ and $1009$ are both prime. [i]2019 CCA Math Bonanza Lightning Round #1.1[/i]

1989 USAMO, 2
Tags: combinatorics
The 20 members of a local tennis club have scheduled exactly 14 two-person games among themselves, with each member playing in at least one game. Prove that within this schedule there must be a set of 6 games with 12 distinct players.