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

2009 Tournament Of Towns, 2
Tags: combinatorial geometry , combinatorics
Several points on the plane are given, no three of them lie on the same line. Some of these points are connected by line segments. Assume that any line that does not pass through any of these points intersects an even number of these segments. Prove that from each point exits an even number of the segments.

1953 AMC 12/AHSME, 22
Tags: logarithm
The logarithm of $ 27\sqrt[4]{9}\sqrt[3]{9}$ to the base $ 3$ is: $ \textbf{(A)}\ 8\frac{1}{2} \qquad\textbf{(B)}\ 4\frac{1}{6} \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ \text{none of these}$

1996 Romania Team Selection Test, 4
Tags: incenter , cyclic quadrilateral , geometry
Let $ ABCD $ be a cyclic quadrilateral and let $ M $ be the set of incenters and excenters of the triangles $ BCD $, $ CDA $, $ DAB $, $ ABC $ (so 16 points in total). Prove that there exist two sets $ \mathcal{K} $ and $ \mathcal{L} $ of four parallel lines each, such that every line in $ \mathcal{K} \cup \mathcal{L} $ contains exactly four points of $ M $.

2016 India IMO Training Camp, 2
Tags: algebra
Suppose that a sequence $a_1,a_2,\ldots$ of positive real numbers satisfies \[a_{k+1}\geq\frac{ka_k}{a_k^2+(k-1)}\] for every positive integer $k$. Prove that $a_1+a_2+\ldots+a_n\geq n$ for every $n\geq2$.

2021 LMT Spring, A21 B22
Tags: number theory
A Haiku is a Japanese poem of seventeen syllables, in three lines of five, seven, and five. In how many ways Can you add three integers Summing seventeen? Order matters here. For example, eight, three, six Is not eight, six, three. All nonnegative, Do not need to be distinct. What is your answer? [i]Proposed by Derek Gao[/i]

2011 Middle European Mathematical Olympiad, 6
Tags: circumcircle , trigonometry , pythagorean theorem , trig identities , law of cosines , geometry
Let $ABC$ be an acute triangle. Denote by $B_0$ and $C_0$ the feet of the altitudes from vertices $B$ and $C$, respectively. Let $X$ be a point inside the triangle $ABC$ such that the line $BX$ is tangent to the circumcircle of the triangle $AXC_0$ and the line $CX$ is tangent to the circumcircle of the triangle $AXB_0$. Show that the line $AX$ is perpendicular to $BC$.

2019 Purple Comet Problems, 19
Tags: number theory
Find the remainder when $\prod_{n=3}^{33}2n^4 - 25n^3 + 33n^2$ is divided by $2019$.

1951 AMC 12/AHSME, 38
Tags:
A rise of $ 600$ feet is required to get a railroad line over a mountain. The grade can be kept down by lengthening the track and curving it around the mountain peak. The additional length of track required to reduce the grade from $ 3\%$ to $ 2\%$ is approximately: $ \textbf{(A)}\ 10000 \text{ ft.} \qquad\textbf{(B)}\ 20000 \text{ ft.} \qquad\textbf{(C)}\ 30000 \text{ ft.} \qquad\textbf{(D)}\ 12000 \text{ ft.} \qquad\textbf{(E)}\ \text{none of these}$

2015 Princeton University Math Competition, A8
Tags: algebra
Let $P(x)$ be a polynomial with positive integer coefficients and degree 2015. Given that there exists some $\omega \in \mathbb{C}$ satisfying $$\omega^{73} = 1\quad \text{and}$$ $$P(\omega^{2015}) + P(\omega^{2015^2}) + P(\omega^{2015^3}) + \ldots + P(\omega^{2015^{72}}) = 0,$$ what is the minimum possible value of $P(1)$?

2022 Kazakhstan National Olympiad, 3
Tags: functional equation , algebra
Given $m\in\mathbb{N}$. Find all functions $f:\mathbb{R^{+}}\rightarrow\mathbb{R^{+}}$ such that $$f(f(x)+y)-f(x)=\left( \frac{f(y)}{y}-1\right)x+f^m(y)$$ holds for all $x,y\in\mathbb{R^{+}}.$ ($f^m(x) =$ $f$ applies $m$ times.)

2008 Switzerland - Final Round, 10
Tags: algebra , inequalities
Find all pairs$ (a, b)$ of positive real numbers with the following properties: (i) For all positive real numbers $x, y, z,w$ holds $x + y^2 + z^3 + w^6 \ge a (xyzw)^{b}$ . (ii) There is a quadruple $(x, y, z,w)$ of positive real numbers such that in equality (i) applies.

1980 Canada National Olympiad, 5
Tags: perimeter , geometry
A parallelepiped has the property that all cross sections, which are parallel to any fixed face $F$, have the same perimeter as $F$. Determine whether or not any other polyhedron has this property. Typesetter's Note: I believe that proof of existence or non-existence suffices.

2022 Princeton University Math Competition, B1
Tags: geometry
A triangle $\vartriangle ABC$ is situated on the plane and a point $E$ is given on segment $AC$. Let $D$ be a point in the plane such that lines $AD$ and $BE$ are parallel. Suppose that $\angle EBC = 25^o$, $\angle BCA = 32^o$, and $\angle CAB = 60^o$. Find the smallest possible value of $\angle DAB$ in degrees.

2018 239 Open Mathematical Olympiad, 10-11.4
Tags: combinatorics
In a $9\times 9$ table, all cells contain zeros. The following operations can be performed on the table: 1. Choose an arbitrary row, add one to all the numbers in that row, and shift all these numbers one cell to the right (and place the last number in the first position). 2. Choose an arbitrary column, subtract one from all its numbers, and shift all these numbers one cell down (and place the bottommost number in the top cell). Is it possible to obtain a table in which all cells, except two, contain zeros, with 1 in the bottom-left cell and -1 in the top-right cell after several such operations? [i]Proposed by N. Vlasova[/i]

2018 Sharygin Geometry Olympiad, 21
Tags: geometry
In the plane a line $l$ and a point $A$ outside it are given. Find the locus of the incenters of acute-angled triangles having a vertex $A$ and an opposite side lying on $l$.

2017 Sharygin Geometry Olympiad, 5
Tags: geometry
10.5 Let $BB'$, $CC'$ be the altitudes of an acute triangle $ABC$. Two circles through $A$ and $C'$ are tangent to $BC$ at points $P$ and $Q$. Prove that $A, B', P, Q$ are concyclic.

2005 Today's Calculation Of Integral, 67
Tags: calculus , integration , geometry , calculus computations
Evaluate \[\frac{2005\displaystyle \int_0^{1002}\frac{dx}{\sqrt{1002^2-x^2}+\sqrt{1003^2-x^2}}+\int_{1002}^{1003}\sqrt{1003^2-x^2}dx}{\displaystyle \int_0^1\sqrt{1-x^2}dx}\]

2019 Irish Math Olympiad, 4
Tags: algebra , equation
Find the set of all quadruplets $(x,y, z,w)$ of non-zero real numbers which satisfy $$1 +\frac{1}{x}+\frac{2(x + 1)}{xy}+\frac{3(x + 1)(y + 2)}{xyz}+\frac{4(x + 1)(y + 2)(z + 3)}{xyzw}= 0$$

2004 India IMO Training Camp, 3
Tags: trigonometry , function , functional equation , system of equations , algebra
Determine all functionf $f : \mathbb{R} \mapsto \mathbb{R}$ such that \[ f(x+y) = f(x)f(y) - c \sin{x} \sin{y} \] for all reals $x,y$ where $c> 1$ is a given constant.

2020 Taiwan TST Round 1, 1
Tags: combinatorics , induction , local argument
You are given a set of $n$ blocks, each weighing at least $1$; their total weight is $2n$. Prove that for every real number $r$ with $0 \leq r \leq 2n-2$ you can choose a subset of the blocks whose total weight is at least $r$ but at most $r + 2$.

2014-2015 SDML (High School), 4
Tags:
What is the maximum number of points that can be placed in the interior of an equilateral triangle of side length $2$ such that the distance between any two points is greater than one? $\text{(A) }3\qquad\text{(B) }4\qquad\text{(C) }5\qquad\text{(D) }6\qquad\text{(E) }7$

2020 Kazakhstan National Olympiad, 1
Tags: divide , number theory
Find all pairs $ (m, n) $ of natural numbers such that $ n ^ 4 \ | \ 2m ^ 5 - 1 $ and $ m ^ 4 \ | \ 2n ^ 5 + 1 $.

2020 AMC 12/AHSME, 8
Tags: algebra
How many ordered pairs of integers $(x, y)$ satisfy the equation$$x^{2020}+y^2=2y?$$ $\textbf{(A) } 1 \qquad\textbf{(B) } 2 \qquad\textbf{(C) } 3 \qquad\textbf{(D) } 4 \qquad\textbf{(E) } \text{infinitely many}$

1951 Putnam, A6
Tags:
Determine the position of a normal chord of a parabola such that it cuts off of the parabola a segment of minimum area.

1973 IMO Longlists, 6
Tags: geometry , analytic geometry , linear algebra , matrix , pigeonhole principle , combinatorics
Let $P_i (x_i, y_i)$ (with $i = 1, 2, 3, 4, 5$) be five points with integer coordinates, no three collinear. Show that among all triangles with vertices at these points, at least three have integer areas.