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

2016 Bosnia And Herzegovina - Regional Olympiad, 3
Tags: touching circles , semicircle , collinear , geometry
Let $AB$ be a diameter of semicircle $h$. On this semicircle there is point $C$, distinct from points $A$ and $B$. Foot of perpendicular from point $C$ to side $AB$ is point $D$. Circle $k$ is outside the triangle $ADC$ and at the same time touches semicircle $h$ and sides $AB$ and $CD$. Touching point of $k$ with side $AB$ is point $E$, with semicircle $h$ is point $T$ and with side $CD$ is point $S$ $a)$ Prove that points $A$, $S$ and $T$ are collinear $b)$ Prove that $AC=AE$

1967 IMO Longlists, 49
Tags: inequality , trigonometric inequality , minimum , trigonometry
Let $n$ and $k$ be positive integers such that $1 \leq n \leq N+1$, $1 \leq k \leq N+1$. Show that: \[ \min_{n \neq k} |\sin n - \sin k| < \frac{2}{N}. \]

1997 APMO, 1
Tags: inequalities
Given: \[ S = 1 + \frac{1}{1 + \frac{1}{3}} + \frac{1}{1 + \frac{1}{3} + \frac{1} {6}} + \cdots + \frac{1}{1 + \frac{1}{3} + \frac{1}{6} + \cdots + \frac{1} {1993006}} \] where the denominators contain partial sums of the sequence of reciprocals of triangular numbers (i.e. $k=\frac{n(n+1)}{2}$ for $n = 1$, $2$, $\ldots$,$1996$). Prove that $S>1001$.

Kvant 2021, M2656
Tags: number theory
The increasing sequence of natural numbers $a_1,a_2,\ldots$ is such that for every $n>100$ the number $a_n$ is equal to the smallest natural number greater than $a_{n-1}$ and not divisible by any of the numbers $a_1,\ldots,a_{n-1}$. Prove that there is only a finite number of composite numbers in such a sequence. [i]Proposed by P. Kozhevnikov[/i]

1999 Putnam, 3
Tags: modular arithmetic , polynomial , limit , algebra
Let $A=\{(x,y): 0\le x,y < 1\}.$ For $(x,y)\in A,$ let \[S(x,y)=\sum_{\frac12\le\frac mn\le2}x^my^n,\] where the sum ranges over all pairs $(m,n)$ of positive integers satisfying the indicated inequalities. Evaluate \[\lim_{(x,y)\to(1,1),(x,y)\in A}(1-xy^2)(1-x^2y)S(x,y).\]

2017 Junior Regional Olympiad - FBH, 3
Tags: angle , triangle , compare
In acute triangle $ABC$ holds $\angle BAC=80^{\circ}$, and altitudes $h_a$ and $h_b$ intersect in point $H$. if $\angle AHB = 126^{\circ}$, which side is the smallest, and which is the biggest in $ABC$

2018 Federal Competition For Advanced Students, P1, 2
Tags: midpoint , incircle , collinear , geometry
Let $ABC$ be a triangle with incenter $I$. The incircle of the triangle is tangent to the sides $BC$ and $AC$ in points $D$ and $E$, respectively. Let $P$ denote the common point of lines $AI$ and $DE$, and let $M$ and $N$ denote the midpoints of sides $BC$ and $AB$, respectively. Prove that points $M, N$ and $P$ are collinear. [i](Proposed by Karl Czakler)[/i]

1977 IMO Shortlist, 7
Tags: trigonometric inequality , inequality , trigonometry , function
Let $a,b,A,B$ be given reals. We consider the function defined by \[ f(x) = 1 - a \cdot \cos(x) - b \cdot \sin(x) - A \cdot \cos(2x) - B \cdot \sin(2x). \] Prove that if for any real number $x$ we have $f(x) \geq 0$ then $a^2 + b^2 \leq 2$ and $A^2 + B^2 \leq 1.$

1960 AMC 12/AHSME, 3
Tags:
Applied to a bill for $\$10,000$ the difference between a discount of $40\%$ and two successive discounts of $36\%$ and $4\%$, expressed in dollars, is: $ \textbf{(A) }0\qquad\textbf{(B) }144\qquad\textbf{(C) }256\qquad\textbf{(D) }400\qquad\textbf{(E) }416 $

2016 All-Russian Olympiad, 4
Tags: coordinate geometry , tetrahedron , 3d geometry , octahedron , geometry , analytic geometry , number theory
There is three-dimensional space. For every integer $n$ we build planes $ x \pm y\pm z = n$. All space is divided on octahedrons and tetrahedrons. Point $(x_0,y_0,z_0)$ has rational coordinates but not lies on any plane. Prove, that there is such natural $k$ , that point $(kx_0,ky_0,kz_0)$ lies strictly inside the octahedron of partition.

May Olympiad L2 - geometry, 2019.3
Tags: equal angles , geometry
On the sides $AB, BC$ and $CA$ of a triangle $ABC$ are located the points $P, Q$ and $R$ respectively, such that $BQ = 2QC, CR = 2RA$ and $\angle PRQ = 90^o$. Show that $\angle APR =\angle RPQ$.

2023 India Regional Mathematical Olympiad, 6
Tags: circumcircle , geometry , cyclic quadrilateral
The diagonals $AC$ and $BD$ of a cyclic quadrilateral $ABCD$ meet at $P$. The point $Q$ is chosen on the segment $BC$ so that $PQ$ is perpendicular to $AC$. Prove that the line joining the centres of the circumcircles of triangles $APD$ and $BQD$ is parallel to $AD$.

2016 CCA Math Bonanza, L3.1
Tags:
How many 3-digit positive integers have the property that the sum of their digits is greater than the product of their digits? [i]2016 CCA Math Bonanza Lightning #3.1[/i]

Durer Math Competition CD 1st Round - geometry, 2022.C4
Tags: geometry , rectangle
We inscribed in triangle $ABC$ the rectangle $DEFG$ such that $D$ and $E$ fall on side $AB$, $F$ on side $BC$, and $G$ on side $AC$. We know that $AF$ bisects angle $\angle BAC$, and that $\frac{AD}{DE} = \frac12$. What is the measure of angle $\angle CAB$?

2010 Vietnam Team Selection Test, 3
Tags: combinatorics , rectangle , geometry
We call a rectangle of the size $1 \times 2$ a domino. Rectangle of the $2 \times 3$ removing two opposite (under center of rectangle) corners we call tetramino. These figures can be rotated. It requires to tile rectangle of size $2008 \times 2010$ by using dominoes and tetraminoes. What is the minimal number of dominoes should be used?

2015 BMT Spring, 9
Tags: number theory
The number $2^{29}$ has a $9$-digit decimal representation that contains all but one of the $10$ (decimal) digits. Determine which digit is missing

2016 Putnam, A3
Tags: mapping
Suppose that $f$ is a function from $\mathbb{R}$ to $\mathbb{R}$ such that \[f(x)+f\left(1-\frac1x\right)=\arctan x\] for all real $x\ne 0.$ (As usual, $y=\arctan x$ means $-\pi/2<y<\pi/2$ and $\tan y=x.$) Find \[\int_0^1f(x)\,dx.\]

2016 AMC 8, 10
Tags:
Suppose that $a * b$ means $3a-b.$ What is the value of $x$ if $$2 * (5 * x)=1?$$ $\textbf{(A) }\frac{1}{10} \qquad\textbf{(B) }2\qquad\textbf{(C) }\frac{10}{3} \qquad\textbf{(D) }10\qquad \textbf{(E) }14$

2017 Harvard-MIT Mathematics Tournament, 25
Tags: algebra
Find all real numbers $x$ satisfying the equation $x^3 - 8 = 16 \sqrt[3]{x + 1}$.

2004 AMC 12/AHSME, 11
Tags:
All the students in an algebra class took a $ 100$-point test. Five students scored $ 100$, each student scored at least $ 60$, and the mean score was $ 76$. What is the smallest possible number of students in the class? $ \textbf{(A)}\ 10 \qquad \textbf{(B)}\ 11 \qquad \textbf{(C)}\ 12 \qquad \textbf{(D)}\ 13 \qquad \textbf{(E)}\ 14$

2014 Finnish National High School Mathematics, 3
Tags: coordinates , broken line , lattice points , combinatorics
The points $P = (a, b)$ and $Q = (c, d)$ are in the first quadrant of the $xy$ plane, and $a, b, c$ and $d$ are integers satisfying $a < b, a < c, b < d$ and $c < d$. A route from point $P$ to point $Q$ is a broken line consisting of unit steps in the directions of the positive coordinate axes. An allowed route is a route not touching the line $x = y$. Tetermine the number of allowed routes.

2005 Baltic Way, 3
Tags: algebra
Consider the sequence $\{a_k\}_{k \geq 1}$ defined by $a_1 = 1$, $a_2 = \frac{1}{2}$ and \[ a_{k + 2} = a_k + \frac{1}{2}a_{k + 1} + \frac{1}{4a_ka_{k + 1}}\ \textrm{for}\ k \geq 1. \] Prove that \[ \frac{1}{a_1a_3} + \frac{1}{a_2a_4} + \frac{1}{a_3a_5} + \cdots + \frac{1}{a_{98}a_{100}} < 4. \]

2023 Oral Moscow Geometry Olympiad, 3
Tags: geometry
In an acute triangle $ABC$ the line $OI$ is parallel to side $BC$. Prove that the center of the nine-point circle of triangle $ABC$ lies on the line $MI$, where $M$ is the midpoint of $BC$.

2000 Slovenia National Olympiad, Problem 1
Tags: digit
In the expression $4\cdot\text{RAKEC}=\text{CEKAR}$, each letter represents a (decimal) digit. Replace the letters so that the equality is true.

2017 Mediterranean Mathematics Olympiad, Problem 4
Tags: inequalities
Let $x,y,z$ and $a,b,c$ be positive real numbers with $a+b+c=1$. Prove that $$\left(x^2+y^2+z^2\right) \left( \frac{a^3}{x^2+2y^2} + \frac{b^3}{y^2+2z^2} + \frac{c^3}{z^2+2x^2} \right) \ge\frac19.$$