Found problems: 85335
2006 Junior Balkan Team Selection Tests - Moldova, 3
The convex polygon $A_{1}A_{2}\ldots A_{2006}$ has opposite sides parallel $(A_{1}A_{2}||A_{1004}A_{1005}, \ldots)$.
Prove that the diagonals $A_{1}A_{1004}, A_{2}A_{1005}, \ldots A_{1003}A_{2006}$ are concurrent if and only if opposite sides are equal.
2018 Indonesia MO, 8
Let $I, O$ be the incenter and circumcenter of the triangle $ABC$ respectively. Let the excircle $\omega_A$ of $ABC$ be tangent to the side $BC$ on $N$, and tangent to the extensions of the sides $AB, AC$ on $K, M$ respectively. If the midpoint of $KM$ lies on the circumcircle of $ABC$, prove that $O, I, N$ are collinear.
1995 Poland - First Round, 5
Given triangle $ABC$ in the plane such that $\angle CAB = a > \pi/2$. Let $PQ$ be a segment whose midpoint is the point $A$. Prove that
$(BP+CQ) \tan a/2 \geq BC$.
1996 USAMO, 6
Determine (with proof) whether there is a subset $X$ of the integers with the following property: for any integer $n$ there is exactly one solution of $a + 2b = n$ with $a,b \in X$.
2019 Purple Comet Problems, 5
Evaluate
$$\frac{(2 + 2)^2}{2^2} \cdot \frac{(3 + 3 + 3 + 3)^3}{(3 + 3 + 3)^3} \cdot \frac{(6 + 6 + 6 + 6 + 6 + 6)^6}{(6 + 6 + 6 + 6)^6}$$
2000 District Olympiad (Hunedoara), 2
[b]a)[/b] Let $ a,b $ two non-negative integers such that $ a^2>b. $ Show that the equation
$$ \left\lfloor\sqrt{x^2+2ax+b}\right\rfloor =x+a-1 $$
has an infinite number of solutions in the non-negative integers. Here, $ \lfloor\alpha\rfloor $ denotes the floor of $ \alpha. $
[b]b)[/b] Find the floor of $ m=\sqrt{2+\sqrt{2+\underbrace{\cdots}_{\text{n times}}+\sqrt{2}}} , $ where $ n $ is a natural number. Justify.
2008 Baltic Way, 18
Let $ AB$ be a diameter of a circle $ S$, and let $ L$ be the tangent at $ A$. Furthermore, let $ c$ be a fixed, positive real, and consider all pairs of points $ X$ and $ Y$ lying on $ L$, on opposite sides of $ A$, such that $ |AX|\cdot |AY| \equal{} c$. The lines $ BX$ and $ BY$ intersect $ S$ at points $ P$ and $ Q$, respectively. Show that all the lines $ PQ$ pass through a common point.
1984 National High School Mathematics League, 2
Which figure's shaded part satisfies the inequality $\log_x(\log_x y^2)>0$?
[img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvNi83LzY2ZWE5OGJmZjlhNzI1NDM5ZjdiNjZmYTcyZTFkMzEzZjUzMzk5LnBuZw==&rn=NGRkLnBuZw==[/img]
2002 Federal Math Competition of S&M, Problem 3
Find all pairs $(n,k)$ of positive integers such that $\binom nk=2002$.
2013 Harvard-MIT Mathematics Tournament, 7
Find the number of positive divisors $d$ of $15!=15\cdot 14\cdot\cdots\cdot 2\cdot 1$ such that $\gcd(d,60)=5$.
2018 BMT Spring, 8
What is the largest possible area of a triangle with largest side length $39$ and inradius $10$?
2011 IFYM, Sozopol, 6
Define a sequence {$a_n$}$^{\infty}_{n=1}$ by $a_1 = 4, a_2 = a_3 = (a^2 - 2)^2$ and
$a_n = a_{n-1}.a_{n-2} - 2(a_{n-1} + a_{n-2}) - a_{n-3} + 8, n \ge 4$, where $a > 2$ is a natural number.
Prove that for all $n$ the number $2 + \sqrt{a_n}$ is a perfect square.
2016 Japan Mathematical Olympiad Preliminary, 6
Integers $1 \le n \le 200$ are written on a blackboard just one by one. We surrounded just $100$ integers with circle. We call a square of the sum of surrounded integers minus the sum of not surrounded integers $score$ of this situation. Calculate the average score in all ways.
1998 Switzerland Team Selection Test, 1
A function $f : R -\{0\} \to R$ has the following properties:
(i) $f(x)- f(y) = f(x)f\left(\frac{1}{y}\right)- f(y)f\left(\frac{1}{x}\right)$ for all $x,y \ne 0$,
(ii) $f$ takes the value $\frac12$ at least once.
Determine $f(-1)$.
Prove that $f$ is a periodic function
2015 JHMT, 5
Points $ABCDEF$ are evenly spaced on a unit circle and line segments $AD$, $DF$, $FB$, $BE$, $EC$, $CA$ are drawn. The line segments intersect each other at seven points inside the circle. Denote these intersections $p_1$, $p_2$, $...$,$p_7$, where $p_7$ is the center of the circle. What is the area of the $12$-sided shape $A_{p_1}B_{p_2}C_{p_3}D_{p_4}E_{p_5}F_{p_6}$?
[img]https://cdn.artofproblemsolving.com/attachments/9/2/645b3c31b0bb7f81d0c0b86e13b27ec5b32864.png[/img]
2023 China National Olympiad, 3
Given positive integer $m,n$, color the points of the regular $(2m+2n)$-gon in black and white, $2m$ in black and $2n$ in white.
The [i]coloring distance[/i] $d(B,C) $ of two black points $B,C$ is defined as the smaller number of white points in the two paths linking the two black points.
The [i]coloring distance[/i] $d(W,X) $ of two white points $W,X$ is defined as the smaller number of black points in the two paths linking the two white points.
We define the matching of black points $\mathcal{B}$ : label the $2m$ black points with $A_1,\cdots,A_m,B_1,\cdots,B_m$ satisfying no $A_iB_i$ intersects inside the gon.
We define the matching of white points $\mathcal{W}$ : label the $2n$ white points with $C_1,\cdots,C_n,D_1,\cdots,D_n$ satisfying no $C_iD_i$ intersects inside the gon.
We define $P(\mathcal{B})=\sum^m_{i=1}d(A_i,B_i), P(\mathcal{W} )=\sum^n_{j=1}d(C_j,D_j) $.
Prove that: $\max_{\mathcal{B}}P(\mathcal{B})=\max_{\mathcal{W}}P(\mathcal{W})$
2000 Bulgaria National Olympiad, 2
Let $D$ be the midpoint of the base $AB$ of the isosceles acute triangle $ABC$. Choose point $E$ on segment $AB$, and let $O$ be the circumcenter of triangle $ACE$. Prove that the line through $D$ perpendicular to $DO$, the line through $E$ perpendicular to $BC$, and the line through $B$ parallel to $AC$ are concurrent.
2019 CCA Math Bonanza, T6
Compute $\displaystyle\sum_{n=3}^{\infty}\frac{n^2-2}{\left(n^2-1\right)\left(n^2-4\right)}$.
[i]2019 CCA Math Bonanza Team Round #6[/i]
2020 Costa Rica - Final Round, 4
Consider the function $ h$, defined for all positive real numbers, such that:
$$10x -6h(x) = 4h \left(\frac{2020}{x}\right) $$
for all $x > 0$. Find $h(x)$ and the value of $h(4)$.
MMPC Part II 1958 - 95, 1965
[b]p1.[/b] For what integers $x$ is it possible to find an integer $y$ such that $$x(x + 1) (x + 2) (x + 3) + 1 = y^2 ?$$
[b]p2.[/b] Two tangents to a circle are parallel and touch the circle at points $A$ and $B$, respectively. A tangent to the circle at any point $X$, other than $A$ or $B$, meets the first tangent at $Y$ and the second tangent at $Z$. Prove $AY \cdot BZ$ is independent of the position of $X$.
[b]p3.[/b] If $a, b, c$ are positive real numbers, prove that $$8abc \le (b + c) (c + a) (a + b)$$ by first verifying the relation in the special case when $c = b$.
[b]p4.[/b] Solve the equation $$\frac{x^2}{3}+\frac{48}{x^2}=10 \left( \frac{x}{3}-\frac{4}{x}\right)$$
[b]p5.[/b] Tom and Bill live on the same street. Each boy has a package to deliver to the other boy’s house. The two boys start simultaneously from their own homes and meet $600$ yards from Bill's house. The boys continue on their errand and they meet again $700$ yards from Tom's house. How far apart do the boy's live?
[b]p6.[/b] A standard set of dominoes consists of $28$ blocks of size $1$ by $2$. Each block contains two numbers from the set $0,1,2,...,6$. We can denote the block containing $2$ and $3$ by $[2, 3]$, which is the same block as $[3, 2]$. The blocks $[0, 0]$, $[1, 1]$,..., $[6, 6]$ are in the set but there are no duplicate blocks.
a) Show that it is possible to arrange the twenty-eight dominoes in a line, end-to-end, with adjacent ends matching, e. g., $... [3, 1]$ $[1, 1]$ $[1, 0]$ $[0, 6] ...$ .
b) Consider the set of dominoes which do not contain $0$. Show that it is impossible to arrange this set in such a line.
c) Generalize the problem and prove your generalization.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2022 BMT, 9
What is the measure of the largest convex angle formed by the hour and minute hands of a clock between $1:45$ PM and $2:40$ PM, in degrees? Convex angles always have a measure of less than $180$ degrees.
1994 China National Olympiad, 1
Let $ABCD$ be a trapezoid with $AB\parallel CD$. Points $E,F$ lie on segments $AB,CD$ respectively. Segments $CE,BF$ meet at $H$, and segments $ED,AF$ meet at $G$. Show that $S_{EHFG}\le \dfrac{1}{4}S_{ABCD}$. Determine, with proof, if the conclusion still holds when $ABCD$ is just any convex quadrilateral.
2012 AIME Problems, 13
Three concentric circles have radii $3$, $4$, and $5$. An equilateral triangle with one vertex on each circle has side length $s$. The largest possible area of the triangle can be written as $a+\frac{b}{c}\sqrt{d}$, where $a,b,c$ and $d$ are positive integers, $b$ and $c$ are relatively prime, and $d$ is not divisible by the square of any prime. Find $a+b+c+d$.
2012 NIMO Problems, 4
Let $S = \{(x, y) : x, y \in \{1, 2, 3, \dots, 2012\}\}$. For all points $(a, b)$, let $N(a, b) = \{(a - 1, b), (a + 1, b), (a, b - 1), (a, b + 1)\}$. Kathy constructs a set $T$ by adding $n$ distinct points from $S$ to $T$ at random. If the expected value of $\displaystyle \sum_{(a, b) \in T} | N(a, b) \cap T |$ is 4, then compute $n$.
[i]Proposed by Lewis Chen[/i]
2018 Online Math Open Problems, 27
Let $p=2^{16}+1$ be a prime. Let $N$ be the number of ordered tuples $(A,B,C,D,E,F)$ of integers between $0$ and $p-1$, inclusive, such that there exist integers $x,y,z$ not all divisible by $p$ with $p$ dividing all three of $Ax+Ez+Fy, By+Dz+Fx, Cz+Dy+Ex$. Compute the remainder when $N$ is divided by $10^6$.
[i]Proposed by Vincent Huang[/i]