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

2023 ISL, N2
Tags: number theory
Determine all ordered pairs $(a,p)$ of positive integers, with $p$ prime, such that $p^a+a^4$ is a perfect square. [i]Proposed by Tahjib Hossain Khan, Bangladesh[/i]

2019 AMC 12/AHSME, 8
Tags: counting
For a set of four distinct lines in a plane, there are exactly $N$ distinct points that lie on two or more of the lines. What is the sum of all possible values of $N$? $\textbf{(A) } 14 \qquad \textbf{(B) } 16 \qquad \textbf{(C) } 18 \qquad \textbf{(D) } 19 \qquad \textbf{(E) } 21$

1964 AMC 12/AHSME, 11
Tags: logarithm
Given $2^x=8^{y+1}$ and $9^y=3^{x-9}$, find the value of $x+y$. ${{ \textbf{(A)}\ 18 \qquad\textbf{(B)}\ 21 \qquad\textbf{(C)}\ 24 \qquad\textbf{(D)}\ 27 }\qquad\textbf{(E)}\ 30 } $

1958 AMC 12/AHSME, 39
Tags: quadratic , function , absolute value
We may say concerning the solution of \[ |x|^2 \plus{} |x| \minus{} 6 \equal{} 0 \] that: $ \textbf{(A)}\ \text{there is only one root}\qquad \textbf{(B)}\ \text{the sum of the roots is }{\plus{}1}\qquad \textbf{(C)}\ \text{the sum of the roots is }{0}\qquad \\ \textbf{(D)}\ \text{the product of the roots is }{\plus{}4}\qquad \textbf{(E)}\ \text{the product of the roots is }{\minus{}6}$

PEN D Problems, 6
Tags: congruence , modular arithmetic , induction , euler , search
Show that, for any fixed integer $\,n \geq 1,\,$ the sequence \[2, \; 2^{2}, \; 2^{2^{2}}, \; 2^{2^{2^{2}}}, \cdots \pmod{n}\] is eventually constant.

2009 Harvard-MIT Mathematics Tournament, 10
Tags: geometry , trigonometry , trig identities , law of cosines , triangle inequality , power of a point , inequalities
Points $A$ and $B$ lie on circle $\omega$. Point $P$ lies on the extension of segment $AB$ past $B$. Line $\ell$ passes through $P$ and is tangent to $\omega$. The tangents to $\omega$ at points $A$ and $B$ intersect $\ell$ at points $D$ and $C$ respectively. Given that $AB=7$, $BC=2$, and $AD=3$, compute $BP$.

1999 AMC 12/AHSME, 5
Tags: percent
The marked price of a book was $ 30\%$ less than the suggested retail price. Alice purchased the book for half the marked price at a Fiftieth Anniversary sale. What percent of the suggested retail price did Alice pay? $ \textbf{(A)}\ 25\% \qquad \textbf{(B)}\ 30\% \qquad \textbf{(C)}\ 35\% \qquad \textbf{(D)}\ 60\% \qquad \textbf{(E)}\ 65\%$

1984 Tournament Of Towns, (062) O3
Tags: combinatorics , combinatorial geometry , square table
From a squared sheet of paper of size $29 \times 29, 99$ pieces, each a $2\times 2$ square, are cut off (all cutting is along the lines bounding the squares). Prove that at least one more piece of size $2\times 2$ may be cut from the remaining part of the sheet. (S Fomin, Leningrad)

1997 All-Russian Olympiad Regional Round, 11.6
Tags: algebra , logarithm , inequalities
Prove that if $1 < a < b < c$, then $$\log_a(\log_a b) + \log_b(\log_b c) + \log_c(\log_c a) > 0.$$

2024 HMNT, 10
Tags: team
For each positive integer $n,$ let $f(n)$ be either the unique integer $r \in \{0,1, \ldots, n-1\}$ such that $n$ divides $15r-1,$ or $0$ if such $r$ does not exist. Compute $$f(16)+f(17)+f(18)+\cdots+f(300).$$

2016 Belarus Team Selection Test, 1
Tags: inequality , inequalities , algebra
Prove for positive $a,b,c$ that $$ (a^2+\frac{b^2}{c^2})(b^2+\frac{c^2}{a^2})(c^2+\frac{a^2}{b^2}) \geq abc (a+\frac{1}{a})(b+\frac{1}{b})(c+\frac{1}{c})$$

2010 Today's Calculation Of Integral, 530
Tags: calculus , integration , geometry , calculus computations , logarithm
Answer the following questions. (1) By setting $ x\plus{}\sqrt{x^2\minus{}1}\equal{}t$, find the indefinite integral $ \int \sqrt{x^2\minus{}1}\ dx$. (2) Given two points $ P(p,\ q)\ (p>1,\ q>0)$ and $ A(1,\ 0)$ on the curve $ x^2\minus{}y^2\equal{}1$. Find the area $ S$ of the figure bounded by two lines $ OA,\ OP$ and the curve in terms of $ p$. (3) Let $ S\equal{}\frac{\theta}{2}$. Express $ p,\ q$ in terms of $ \theta$.

MBMT Team Rounds, 2020.19
Tags:
In a regular hexagon $ABCDEF$ of side length $8$ and center $K$, points $W$ and $U$ are chosen on $\overline{AB}$ and $\overline{CD}$ respectively such that $\overline{KW} = 7$ and $\angle WKU = 120^{\circ}$. Find the area of pentagon $WBCUK$. [i]Proposed by Bradley Guo[/i]

Kyiv City MO 1984-93 - geometry, 1992.9.3
Tags: symmetry , geometry
Prove that a bounded figure cannot have more than one center of symmetry.

2018 Sharygin Geometry Olympiad, 8
Tags: geometry
Two triangles $ABC$ and $A'B'C'$ are given. The lines $AB$ and $A'B'$ meet at $C_1$ and the lines parallel to them and passing through $C$ and $C'$ meet at $C_2$. The points $A_1,A_2$, $B_1,B_2$ are defined similarly. Prove that $A_1A_2,B_1B_2,C_1C_1$ are either parallel or concurrent.

2022 CCA Math Bonanza, T2
Tags:
CCA's B building has 6 rooms on the second floor, labeled B201 to B206, as well as 8 rooms on the first floor, labeled B101 to B108. Annie is currently in room B205. Each minute, she chooses to stay or change floors with equal probability, and chooses a classroom on that floor to go to at random (she can stay in the classroom that she's already in). B104, B108, and B203 are the only rooms that have teachers who will scold her for randomly walking around during class time. The probability that she is first scolded in room B203 can be expressed as $\frac{p}{q}$. Compute $p+q$. [i]2022 CCA Math Bonanza Team Round #2[/i]

1992 Swedish Mathematical Competition, 2
Tags: combinatorics
The squares in a $9\times 9$ grid are numbered from $11$ to $99$, where the first digit is the row and the second the column. Each square is colored black or white. Squares $44$ and $49$ are black. Every black square shares an edge with at most one other black square, and each white square shares an edge with at most one other white square. What color is square $99$?

2021 AMC 12/AHSME Fall, 14
Tags:
Suppose that $P(z), Q(z)$, and $R(z)$ are polynomials with real coefficients, having degrees $2$, $3$, and $6$, respectively, and constant terms $1$, $2$, and $3$, respectively. Let $N$ be the number of distinct complex numbers $z$ that satisfy the equation $P(z) \cdot Q(z)=R(z)$. What is the minimum possible value of $N$? $\textbf{(A)}\: 0\qquad\textbf{(B)} \: 1\qquad\textbf{(C)} \: 2\qquad\textbf{(D)} \: 3\qquad\textbf{(E)} \: 5$

2018-IMOC, N3
Tags: number theory , divisibility
Find all pairs of positive integers $(x,y)$ so that $$\frac{(x^2-x+1)(y^2-y+1)}{xy}\in\mathbb N.$$

2023 Princeton University Math Competition, A8
Tags: number theory
Let $S_0 = 0, S_1 = 1,$ and for $n \ge 2,$ let $S_n = S_{n-1}+5S_{n-2}.$ What is the sum of the five smallest primes $p$ such that $p \mid S_{p-1}$?

2011 Paraguay Mathematical Olympiad, 4
Tags: algebra
A positive integer $N$ is divided in $n$ parts inversely proportional to the numbers $2, 6, 12, 20, \ldots$ The smallest part is equal to $\frac{1}{400} N$. Find the value of $n$.

2018 India PRMO, 28
Tags: sum of digits , combinatorics
Let $N$ be the number of ways of distributing $8$ chocolates of different brands among $3$ children such that each child gets at least one chocolate, and no two children get the same number of chocolates. Find the sum of the digits of $N$.

2013 Online Math Open Problems, 46
Tags: geometry , circumcircle , ratio , geometric transformation , homothety , rhombus
Let $ABC$ be a triangle with $\angle B - \angle C = 30^{\circ}$. Let $D$ be the point where the $A$-excircle touches line $BC$, $O$ the circumcenter of triangle $ABC$, and $X,Y$ the intersections of the altitude from $A$ with the incircle with $X$ in between $A$ and $Y$. Suppose points $A$, $O$ and $D$ are collinear. If the ratio $\frac{AO}{AX}$ can be expressed in the form $\frac{a+b\sqrt{c}}{d}$ for positive integers $a,b,c,d$ with $\gcd(a,b,d)=1$ and $c$ not divisible by the square of any prime, find $a+b+c+d$. [i]James Tao[/i]

2003 China Girls Math Olympiad, 6
Tags: inequalities
Let $ n \geq 2$ be an integer. Find the largest real number $ \lambda$ such that the inequality \[ a^2_n \geq \lambda \sum^{n\minus{}1}_{i\equal{}1} a_i \plus{} 2 \cdot a_n.\] holds for any positive integers $ a_1, a_2, \ldots a_n$ satisfying $ a_1 < a_2 < \ldots < a_n.$

2015 Oral Moscow Geometry Olympiad, 2
Tags: geometry , angle , equilateral , square
The square $ABCD$ and the equilateral triangle $MKL$ are located as shown in the figure. Find the angle $\angle PQD$. [img]https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjQKgjvzy1WhwkMJbcV_C0iveelYmm75FpaGlWgZ-Ap_uQUiegaKYafelo-J_3rMgKMgpMp5soYc1LVYLI8H4riC6R-f8eq2DiWTGGII08xQkwu7t2KVD4pKX4_IN-gC7DVRhdVZSjbaj2S/s1600/oral+moscow+geometry+2015+8.9+p2.png[/img]