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

1966 IMO Longlists, 54
Tags: number theory , decimal representation , digit , modular arithmetic , last digit
We take $100$ consecutive natural numbers $a_{1},$ $a_{2},$ $...,$ $a_{100}.$ Determine the last two digits of the number $a_{1}^{8}+a_{2}^{8}+...+a_{100}^{8}.$

2017 Taiwan TST Round 2, 6
Tags: geometry
Let $I$ be the incentre of a non-equilateral triangle $ABC$, $I_A$ be the $A$-excentre, $I'_A$ be the reflection of $I_A$ in $BC$, and $l_A$ be the reflection of line $AI'_A$ in $AI$. Define points $I_B$, $I'_B$ and line $l_B$ analogously. Let $P$ be the intersection point of $l_A$ and $l_B$. [list=a] [*] Prove that $P$ lies on line $OI$ where $O$ is the circumcentre of triangle $ABC$. [*] Let one of the tangents from $P$ to the incircle of triangle $ABC$ meet the circumcircle at points $X$ and $Y$. Show that $\angle XIY = 120^{\circ}$. [/list]

2014 Stanford Mathematics Tournament, 1
Tags: geometry
The coordinates of three vertices of a parallelogram are $A(1, 1)$, $B(2, 4)$, and $C(-5, 1)$. Compute the area of the parallelogram.

1995 AMC 12/AHSME, 23
Tags: inequalities , calculus , integration , trigonometry , triangle inequality , pythagorean theorem , geometry
The sides of a triangle have lengths $11$,$15$, and $k$, where $k$ is an integer. For how many values of $k$ is the triangle obtuse? $\textbf{(A)}\ 5 \qquad \textbf{(B)}\ 7 \qquad \textbf{(C)}\ 12 \qquad \textbf{(D)}\ 13 \qquad \textbf{(E)}\ 14$

2005 China Girls Math Olympiad, 2
Tags: trigonometry , geometry , trig identities , law of sines , algebra
Find all ordered triples $ (x, y, z)$ of real numbers such that \[ 5 \left(x \plus{} \frac{1}{x} \right) \equal{} 12 \left(y \plus{} \frac{1}{y} \right) \equal{} 13 \left(z \plus{} \frac{1}{z} \right),\] and \[ xy \plus{} yz \plus{} zy \equal{} 1.\]

2023 Assam Mathematics Olympiad, 16
Tags:
$n$ is a positive integer such that the product of all its positive divisors is $n^3$. Find all such $n$ less than $100$.

2001 Bosnia and Herzegovina Team Selection Test, 1
Tags: ratio , geometry , arc , angle , circles
On circle there are points $A$, $B$ and $C$ such that they divide circle in ratio $3:5:7$. Find angles of triangle $ABC$

2003 IMAR Test, 2
Tags: geometry , angle bisector , geometric inequality , semiperimeter
Prove that in a triangle the following inequality holds: $$s\sqrt3 \ge \ell_a + \ell_b + \ell_c$$ where $\ell_a$ is the length of the angle bisector from angle $A$, and $s$ is the semiperimeter of the triangle

2017 SG Originals, N6
Tags: number theory , vieta jumping
Find the smallest positive integer $n$ or show no such $n$ exists, with the following property: there are infinitely many distinct $n$-tuples of positive rational numbers $(a_1, a_2, \ldots, a_n)$ such that both $$a_1+a_2+\dots +a_n \quad \text{and} \quad \frac{1}{a_1} + \frac{1}{a_2} + \dots + \frac{1}{a_n}$$ are integers.

2019 Polish Junior MO Second Round, 1.
Tags: algebra , inequalities
Let $x$, $y$ be real numbers, such that $x^2 + x \leq y$. Prove that $y^2 + y \geq x$.

2019 Math Prize for Girls Problems, 4
Tags:
A paper equilateral triangle with area 2019 is folded over a line parallel to one of its sides. What is the greatest possible area of the overlap of folded and unfolded parts of the triangle?

1997 Bulgaria National Olympiad, 1
Tags: function , inequalities
Let $ a$, $ b$, $ c$ be positive real numbers such that $ abc=1$. Prove that $ \frac{1}{1+b+c}+\frac{1}{1+c+a}+\frac{1}{1+a+b}\leq\frac{1}{2+a}+\frac{1}{2+b}+\frac{1}{2+c}$.

2012 Danube Mathematical Competition, 3
Tags: number theory
Let $p$ and $q, p < q,$ be two primes such that $1 + p + p^2+...+p^m$ is a power of $q$ for some positive integer $m$, and $1 + q + q^2+...+q^n$ is a power of $p$ for some positive integer $n$. Show that $p = 2$ and $q = 2^t-1$ where $t$ is prime.

2021 HMIC, 2
Tags:
Let $n$ be a positive integer. Alice writes $n$ real numbers $a_1, a_2,\dots, a_n$ in a line (in that order). Every move, she picks one number and replaces it with the average of itself and its neighbors ($a_n$ is not a neighbor of $a_1$, nor vice versa). A number [i]changes sign[/i] if it changes from being nonnegative to negative or vice versa. In terms of $n$, determine the maximum number of times that $a_1$ can change sign, across all possible values of $a_1,a_2,\dots, a_n$ and all possible sequences of moves Alice may make.

2000 Harvard-MIT Mathematics Tournament, 47
Tags:
Find an $n<100$ such that $n\cdot 2^n-1$ is prime. Score will be $n-5$ for correct $n$, $5-n$ for incorrect $n$ ($0$ points for answer $<5$)

2012 ELMO Shortlist, 4
Tags: factorial , number theory
Do there exist positive integers $b,n>1$ such that when $n$ is expressed in base $b$, there are more than $n$ distinct permutations of its digits? For example, when $b=4$ and $n=18$, $18 = 102_4$, but $102$ only has $6$ digit arrangements. (Leading zeros are allowed in the permutations.) [i]Lewis Chen.[/i]

2017 Pan-African Shortlist, A?
Tags: algebra
We consider the real sequence $(x_n)$ defined by $x_0=0, x_1=1$ and $x_{n+2}=3x_{n+1}-2x_n$ for $n=0,1,...$ We define the sequence $(y_n)$ by $y_n=x_n^2+2^{n+2}$ for every non negative integer $n$. Prove that for every $n>0$, $y_n$ is the square of an odd integer

2023 Auckland Mathematical Olympiad, 6
Tags: combinatorics , number theory
Suppose there is an infi nite sequence of lights numbered $1, 2, 3,...,$ and you know the following two rules about how the lights work: $\bullet$ If the light numbered $k$ is on, the lights numbered $2k$ and $2k + 1$ are also guaranteed to be on. $\bullet$ If the light numbered $k$ is off, then the lights numbered $4k + 1$ and $4k + 3$ are also guaranteed to be off. Suppose you notice that light number $2023$ is on. Identify all the lights that are guaranteed to be on?

1994 Miklós Schweitzer, 1
Tags: ordered set
An ordered set of numbers is mean-free if for all $x < y < z$ , $y \neq \frac{x + z}{2}$. Is it possible to order the real numbers so it becomes mean-free? related: [url]https://www.youtube.com/watch?v=ppaXUxsEjMQ[/url]

2011-2012 SDML (High School), 12
Tags: factorial
Kate multiplied all the integers from $1$ to her age and got $1,307,674,368,000$. How old is Kate? $\text{(A) }14\qquad\text{(B) }15\qquad\text{(C) }16\qquad\text{(D) }17\qquad\text{(E) }18$

2014 Kyiv Mathematical Festival, 4a
Tags: least common multiple , modular arithmetic , number theory
a) Prove that for every positive integer $y$ the equality ${\rm lcm}(x,y+1)\cdot {\rm lcm}(x+1,y)=x(x+1)$ holds for infinitely many positive integers $x.$ b) Prove that there exists positive integer $y$ such that the equality ${\rm lcm}(x,y+1)\cdot {\rm lcm}(x+1,y)=y(y+1)$ holds for at least 2014 positive integers $x.$

2019 CCA Math Bonanza, L1.4
Tags:
What is the smallest prime number $p$ such that $1+p+p^2+\ldots+p^{p-1}$ is [i]not[/i] prime? [i]2019 CCA Math Bonanza Lightning Round #1.4[/i]

1953 AMC 12/AHSME, 37
Tags: geometry , circumcircle , trigonometry , perpendicular bisector , trig identities , law of sines
The base of an isosceles triangle is $ 6$ inches and one of the equal sides is $ 12$ inches. The radius of the circle through the vertices of the triangle is: $ \textbf{(A)}\ \frac{7\sqrt{15}}{5} \qquad\textbf{(B)}\ 4\sqrt{3} \qquad\textbf{(C)}\ 3\sqrt{5} \qquad\textbf{(D)}\ 6\sqrt{3} \qquad\textbf{(E)}\ \text{none of these}$

1979 IMO Longlists, 41
Tags: 3d geometry , pyramid , geometry
Prove the following statement: There does not exist a pyramid with square base and congruent lateral faces for which the measures of all edges, total area, and volume are integers.

2016 Romania Team Selection Tests, 1
Tags: geometry
Two circles, $\omega_1$ and $\omega_2$, centered at $O_1$ and $O_2$, respectively, meet at points $A$ and $B$. A line through $B$ meet $\omega_1$ again at $C$, and $\omega_2$ again at $D$. The tangents to $\omega_1$ and $\omega_2$ at $C$ and $D$, respectively, meet at $E$, and the line $AE$ meets the circle $\omega$ through $A, O_1,O_2$ again at $F$. Prove that the length of the segment $EF$ is equal to the diameter of $\omega$.