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 Saint Petersburg Mathematical Olympiad, 2.
Tags: number theory
Find all positive integers $n$ such that the sum of the squares of the five smallest divisors of $n$ is a square.

2016 Balkan MO Shortlist, N2
Tags: divisor , maximum , number theory , odd
Find all odd natural numbers $n$ such that $d(n)$ is the largest divisor of the number $n$ different from $n$. ($d(n)$ is the number of divisors of the number n including $1$ and $n$ ).

1979 Bundeswettbewerb Mathematik, 4
Tags: algebra , polynomial
Prove that the polynomial $P(x) = x^5-x+a$ is irreducible over $\mathbb{Z}$ if $5 \nmid a$.

2001 Greece JBMO TST, 3
Tags: minimum , algebra , combinatorics
$4$ men stand at the entrance of a dark tunnel. Man $A$ needs $10$ minutes to pass through the tunnel, man $B$ needs $5$ minutes, man $C$ needs $2$ minutes and man $D$ needs $1$ minute. There is only one torch, that may be used from anyone that passes through the tunnel. Additionaly, at most $2$ men can pass through at the same time using the existing torch. Determine the smallest possible time the four men need to reach the exit of the tunnel.

1996 ITAMO, 3
Tags: geometry , 3d geometry , sphere , analytic geometry , symmetry
Given a cube of unit side. Let $A$ and $B$ be two opposite vertex. Determine the radius of the sphere, with center inside the cube, tangent to the three faces of the cube with common point $A$ and tangent to the three sides with common point $B$.

2017 European Mathematical Cup, 1
Tags: functional equation , natural number
Find all functions $f:\mathbb{N}\rightarrow \mathbb{N}$ such that the inequality $$f(x)+yf(f(x))\le x(1+f(y))$$ holds for all positive integers $x, y$. Proposed by Adrian Beker.

2017 HMNT, 4
Tags: algebra
[b]M[/b]ary has a sequence $m_2,m_3,m_4,...$ , such that for each $b \ge 2$, $m_b$ is the least positive integer m for which none of the base-$b$ logarithms $log_b(m),log_b(m+1),...,log_b(m+2017)$ are integers. Find the largest number in her sequence.

2014 Portugal MO, 5
Tags: geometry
Let $[ABCD]$ be a convex quadrilateral with area $2014$, and let $P$ be a point on $[AB]$ and $Q$ a point on $[AD]$ such that triangles $[ABQ]$ and $[ADP]$ have area $1$. Let $R$ be the intersection of $[AC]$ and $[PQ]$. Determine $\frac{\overline{RC}}{\overline{RA}}$.

2006 India IMO Training Camp, 1
Tags: linear algebra , matrix , combinatorics
Let $n$ be a positive integer divisible by $4$. Find the number of permutations $\sigma$ of $(1,2,3,\cdots,n)$ which satisfy the condition $\sigma(j)+\sigma^{-1}(j)=n+1$ for all $j \in \{1,2,3,\cdots,n\}$.

2015 Caucasus Mathematical Olympiad, 1
Tags: combinatorics
At the round table, $10$ people are sitting, some of them are knights, and the rest are liars (knights always say pride, and liars always lie) . It is clear thath I have at least one knight and at least one liar. What is the largest number of those sitting at the table can say: ''Both of my neighbors are knights '' ? (A statement that is at least partially false is considered false.)

2024 BMT, 1
Tags: geometry
Andrew has three identical semicircular mooncake halves, each with radius $3,$ and uses them to construct the following shape, which contains an equilateral triangle in the center. Compute the perimeter around this shape, in bold below. [center] [img] https://cdn.artofproblemsolving.com/attachments/7/2/2314ac2d34cd0706f47bace3eedbb87a91582a.png [/img] [/center]

2004 AIME Problems, 7
Tags:
Let $C$ be the coefficient of $x^2$ in the expansion of the product \[(1 - x)(1 + 2x)(1 - 3x)\cdots(1 + 14x)(1 - 15x).\] Find $|C|$.

2009 Indonesia TST, 1
Tags: pigeonhole principle , search , number theory
a. Does there exist 4 distinct positive integers such that the sum of any 3 of them is prime? b. Does there exist 5 distinct positive integers such that the sum of any 3 of them is prime?

2023 HMNT, 10
Tags:
A real number $x$ is chosen uniformly at random from the interval $(0,10).$ Compute the probability that $\sqrt{x}, \sqrt{x+7},$ and $\sqrt{10-x}$ are the side lengths of a non-degenerate triangle.

2022 Bulgaria EGMO TST, 4
Tags: shortlist , number theory , sequence , recursion , existence , admiration
Denote by $l(n)$ the largest prime divisor of $n$. Let $a_{n+1} = a_n + l(a_n)$ be a recursively defined sequence of integers with $a_1 = 2$. Determine all natural numbers $m$ such that there exists some $i \in \mathbb{N}$ with $a_i = m^2$. [i]Proposed by Nikola Velov, North Macedonia[/i]

1964 AMC 12/AHSME, 5
Tags:
If $y$ varies directly as $x$, and if $y=8$ when $x=4$, the value of $y$ when $x=-8$ is: ${{ \textbf{(A)}\ -16} \qquad\textbf{(B)}\ -4 \qquad\textbf{(C)}\ -2 \qquad\textbf{(D)}\ 4k, k= \pm1, \pm2, \dots}$ ${\qquad\textbf{(E)}\ 16k, k=\pm1,\pm2,\dots } $

I Soros Olympiad 1994-95 (Rus + Ukr), 10.6
Tags: geometry , exradius , geometric inequality
The radius of the circle inscribed in triangle $ABC$ is equal to $r$, and the radius of the circle tangent to the segment $BC$ and the extensions of sides $AB$ and $AC$ (the exscribed circle corresponding to angle $A$) is equal to $R$. A circle with radius $x < r$ is inscribed in angle $\angle BAC$. Tangents to this circles passing through points $B$ and $C$ and different from $BA$ and $AC$ intersect at point $A'$. Let $y$ be the radius of the circle inscribed in triangle $BCK$. Find the greatest value of the sum $x + y$ as x changes from $0$ to $r$. (In this case, it is necessary to prove that this largest value is the same in any triangle with given $r$ and $R$).

May Olympiad L2 - geometry, 2007.5
Tags: geometry
In the triangle $ABC$ we have $\angle A = 2\angle C$ and $2\angle B = \angle A + \angle C$. The angle bisector of $\angle C$ intersects the segment $AB$ in $E$, let $F$ be the midpoint of $AE$, let $AD$ be the altitude of the triangle $ABC$. The perpendicular bisector of $DF$ intersects $AC$ in $M$. Prove that $AM = CM$.

Russian TST 2022, P2
Tags: functional equation , algebra
Determine all functions $f:\mathbb{R}\to\mathbb{R}$ satisfying \[f(xy+f(x))+f(y)=xf(y)+f(x+y),\]for all real numbers $x,y$.

2003 Finnish National High School Mathematics Competition, 3
Tags: combinatorics
There are six empty purses on the table. How many ways are there to put 12 two-euro coins in purses in such a way that at most one purse remains empty?

1960 Kurschak Competition, 1
Tags: combinatorics
Among any four people at a party there is one who has met the three others before the party. Show that among any four people at the party there must be one who has met everyone at the party before the party

MathLinks Contest 5th, 5.1
Tags: number theory
Find all real numbers $a > 1$ such that there exists an integer $k \ge 1$ such that the sequence $\{x_n\}_{n\ge 1}$ formed with the first $k$ digits of the number $\lfloor a^n\rfloor$ is periodical.

2009 Turkey MO (2nd round), 1
Tags: geometry
Let $H$ be the orthocenter of an acute triangle $ABC,$ and let $A_1, \: B_1, \: C_1$ be the feet of the altitudes belonging to the vertices $A, \: B, \: C,$ respectively. Let $K$ be a point on the smaller $AB_1$ arc of the circle with diameter $AB$ satisfying the condition $\angle HKB = \angle C_1KB.$ Let $M$ be the point of intersection of the line segment $AA_1$ and the circle with center $C$ and radius $CL$ where $KB \cap CC_1=\{L\}.$ Let $P$ and $Q$ be the points of intersection of the line $CC_1$ and the circle with center $B$ and radius $BM.$ Show that $A, \: K, \: P, \: Q$ are concyclic.

2004 APMO, 4
Tags: function , floor function , modular arithmetic , relatively prime , number theory
For a real number $x$, let $\lfloor x\rfloor$ stand for the largest integer that is less than or equal to $x$. Prove that \[ \left\lfloor{(n-1)!\over n(n+1)}\right\rfloor \] is even for every positive integer $n$.

2014 IPhOO, 1
Tags: percent
A capacitor is made with two square plates, each with side length $L$, of negligible thickness, and capacitance $C$. The two-plate capacitor is put in a microwave which increases the side length of each square plate by $ 1 \% $. By what percent does the voltage between the two plates in the capacitor change? $ \textbf {(A) } \text {decreases by } 2\% \\ \textbf {(B) } \text {decreases by } 1\% \\ \textbf {(C) } \text {it does not change} \\ \textbf {(D) } \text {increases by } 1\% \\ \textbf {(E) } \text {increases by } 2\% $ [i]Problem proposed by Ahaan Rungta[/i]