Found problems: 85335
2011 Canadian Mathematical Olympiad Qualification Repechage, 6
In the diagram, $ABDF$ is a trapezoid with $AF$ parallel to $BD$ and $AB$ perpendicular to $BD.$ The circle with center $B$ and radius $AB$ meets $BD$ at $C$ and is tangent to $DF$ at $E.$ Suppose that $x$ is equal to the area of the region inside quadrilateral $ABEF$ but outside the circle, that y is equal to the area of the region inside $\triangle EBD$ but outside the circle, and that $\alpha = \angle EBC.$ Prove that there is exactly one measure $\alpha,$ with $0^\circ \leq \alpha \leq 90^\circ,$ for which $x = y$ and that this value of $\frac 12 < \sin \alpha < \frac{1}{\sqrt 2}.$
[asy]
import graph; size(150); real lsf = 0.5; pen dp = linewidth(0.7) + fontsize(10); defaultpen(dp); pen ds = black; pen qqttff = rgb(0,0.2,1); pen fftttt = rgb(1,0.2,0.2);
draw(circle((6.04,2.8),1.78),qqttff); draw((6.02,4.58)--(6.04,2.8),fftttt); draw((6.02,4.58)--(6.98,4.56),fftttt); draw((6.04,2.8)--(8.13,2.88),fftttt); draw((6.98,4.56)--(8.13,2.88),fftttt);
dot((6.04,2.8),ds); label("$B$", (5.74,2.46), NE*lsf); dot((6.02,4.58),ds); label("$A$", (5.88,4.7), NE*lsf); dot((6.98,4.56),ds); label("$F$", (7.06,4.6), NE*lsf); dot((7.39,3.96),ds); label("$E$", (7.6,3.88), NE*lsf); dot((8.13,2.88),ds); label("$D$", (8.34,2.56), NE*lsf); dot((7.82,2.86),ds); label("$C$", (7.5,2.46), NE*lsf); clip((-4.3,-10.94)--(-4.3,6.3)--(16.18,6.3)--(16.18,-10.94)--cycle);
[/asy]
1961 Kurschak Competition, 2
$x, y, z$ are positive reals less than $1$. Show that at least one of $(1 - x)y$, $(1 - y)z$ and $(1 - z)x$ does not exceed $\frac14$ .
2000 Iran MO (3rd Round), 1
Let us denote $\prod = \{(x, y) | y > 0\}$. We call a [i]semicircle[/i] in $\prod$ with
center on the $x-\text{axis}$ a [i]semi-line[/i]. Two intersecting [i]semi-lines [/i]determine
four [i]semi-angles[/i]. A bisector of a [i]semi-angle [/i]is a [i]semi-line [/i]that bisects
the [i]semi-angle[/i]. Prove that in every [i]semi-triangle [/i](determined by three
[i]semi-lines[/i]) the bisectors are concurrent.
2004 Romania National Olympiad, 4
(a) Build a function $f : \mathbb R \to \mathbb R_+$ with the property $\left( \mathcal P \right)$, i.e. all $x \in \mathbb Q$ are local, strict minimum points.
(b) Build a function $f : \mathbb Q \to \mathbb R_+$ such that every point is a local, strict minimum point and such that $f$ is unbounded on $I \cap \mathbb Q$, where $I$ is a non-degenerate interval.
(c) Let $f: \mathbb R \to \mathbb R_+$ be a function unbounded on every $I \cap \mathbb Q$, where $I$ is a non-degenerate interval. Prove that $f$ doesn't have the property $\left( \mathcal P \right)$.
2002 Indonesia MO, 3
Find all solutions (real and complex) for $x,y,z$, given that:
\[ x+y+z = 6 \\
x^2+y^2+z^2 = 12 \\
x^3+y^3+z^3 = 24 \]
1966 IMO Shortlist, 13
Let $a_1, a_2, \ldots, a_n$ be positive real numbers. Prove the inequality
\[\binom n2 \sum_{i<j} \frac{1}{a_ia_j} \geq 4 \left( \sum_{i<j} \frac{1}{a_i+a_j} \right)^2\]
2017 Harvard-MIT Mathematics Tournament, 8
Consider all ordered pairs of integers $(a,b)$ such that $1\le a\le b\le 100$ and $$\frac{(a+b)(a+b+1)}{ab}$$ is an integer.
Among these pairs, find the one with largest value of $b$. If multiple pairs have this maximal value of $b$, choose the one with largest $a$. For example choose $(3,85)$ over $(2,85)$ over $(4,84)$. Note that your answer should be an ordered pair.
2010 Math Prize For Girls Problems, 9
Lynnelle took 10 tests in her math class at Stanford. Her score on each test was an integer from 0 through 100. She noticed that, for every four consecutive tests, her average score on those four tests was at most 47.5. What is the largest possible average score she could have on all 10 tests?
2012 Online Math Open Problems, 29
How many positive integers $a$ with $a\le 154$ are there such that the coefficient of $x^a$ in the expansion of \[(1+x^{7}+x^{14}+ \cdots +x^{77})(1+x^{11}+x^{22}+\cdots +x^{77})\] is zero?
[i]Author: Ray Li[/i]
2023 HMIC, P1
Let $\mathbb{Q}^{+}$ denote the set of positive rational numbers. Find, with proof, all functions $f:\mathbb{Q}^+ \to \mathbb{Q}^+$ such that, for all positive rational numbers $x$ and $y,$ we have \[f(x)=f(x+y)+f(x+x^2f(y)).\]
2010 AMC 8, 24
What is the correct ordering of the three numbers, $10^8$, $5{}^1{}^2$, and $2{}^2{}^4$?
$ \textbf{(A)}\ 2{}^2{}^4<10^8<5{}^1{}^2 $
$ \textbf{(B)}\ 2{}^2{}^4<5{}^1{}^2<10^8 $
$ \textbf{(C)}\ 5{}^1{}^2<2{}^2{}^4<10^8 $
$ \textbf{(D)}\ 10^8<5{}^1{}^2<2{}^2{}^4$
$ \textbf{(E)}\ 10^8<2{}^2{}^4<5{}^1{}^2 $
1996 South africa National Olympiad, 6
The function $f$ is increasing and convex (i.e. every straight line between two points on the graph of $f$ lies above the graph) and satisfies $f(f(x))=3^x$ for all $x\in\mathbb{R}$. If $f(0)=0.5$ determine $f(0.75)$ with an error of at most $0.025$. The following are corrent to the number of digits given:
\[3^{0.25}=1.31607,\quad 3^{0.50}=1.73205,\quad 3^{0.75}=2.27951.\]
1987 Canada National Olympiad, 4
On a large, flat field $n$ people are positioned so that for each person the distances to all the other people are different. Each person holds a water pistol and at a given signal fires and hits the person who is closest. When $n$ is odd show that there is at least one person left dry. Is this always true when $n$ is even?
2016 Greece Team Selection Test, 1
Given is the sequence $(a_n)_{n\geq 0}$ which is defined as follows:$a_0=3$ and $a_{n+1}-a_n=n(a_n-1) \ , \ \forall n\geq 0$.
Determine all positive integers $m$ such that $\gcd (m,a_n)=1 \ , \ \forall n\geq 0$.
2018 Kazakhstan National Olympiad, 2
The natural number $m\geq 2$ is given.Sequence of natural numbers $(b_0,b_1,\ldots,b_m)$ is called concave if $b_k+b_{k-2}\le2b_{k-1}$ for all $2\le k\le m.$ Prove that there exist not greater than $2^m$ concave sequences starting with $b_0 =1$ or $b_0 =2$
1997 Moscow Mathematical Olympiad, 2
$9$ different pieces of cheese are placed on a plate. Is it always possible to cut one of them into two parts so that the $10$ pieces obtained were divisible into two portions of equal mass of $5$ pieces each?
1999 Mongolian Mathematical Olympiad, Problem 3
At each vertex of a $4\times5$ rectangle there is a house. Find the path of the minimum length connecting all these houses.
2013 Cono Sur Olympiad, 4
Let $M$ be the set of all integers from $1$ to $2013$. Each subset of $M$ is given one of $k$ available colors, with the only condition that if the union of two different subsets $A$ and $B$ is $M$, then $A$ and $B$ are given different colors. What is the least possible value of $k$?
1990 IMO Longlists, 6
Let function $f : \mathbb Z_{\geq 0}^0 \to \mathbb N$ satisfy the following conditions:
(i) $ f(0, 0, 0) = 1;$
(ii) $f(x, y, z) = f(x - 1, y, z) + f(x, y - 1, z) + f(x, y, z - 1);$
(iii) when applying above relation iteratively, if any of $x', y', z$' is negative, then $f(x', y', z') = 0.$
Prove that if $x, y, z$ are the side lengths of a triangle, then $\frac{\left(f(x,y,z) \right) ^k}{ f(mx ,my, mz)}$ is not an integer for any integers $k, m > 1.$
2014 CIIM, Problem 2
Let $n$ be an integer and $p$ a prime greater than 2. Show that: $$(p-1)^nn!|(p^n-1)(p^n-p)(p^n-p^2)\cdots(p^n-p^{n-1}).$$
2024 CCA Math Bonanza, L5.2
You have a 1 by 2024 grid of squares in a column, vertices labelled with coordinates $(0,0)$ to $(1,2024)$. Place a weed at $(0,0)$. When a weed is attempting to be placed at coordinates $(x,y)$, it will be placed with a $50\%$ probability if and only if exactly one of the vertices $(x-1, y)$ or $(x, y-1)$ has a weed on it, otherwise the attempt will fail with probability $1$. The placement attempts are made in the following order: For each vertex with $x$ coordinate $0$, attempt a placement for each vertex starting from $y$ coordinate $0$, incrementing by $1$ until $2024$. Then, attempts will be made on the vertices with $x$ coordinate $1$ in the same fashion. Each placement attempt is made exactly once.
The probability that a weed appears on $(1,2024)$ after placing the weed at $(0,0)$ and attempting to place weeds on every vertex is $p$. Estimate $9p\cdot2^{2025}$ to the nearest integer.
\\\\ Your score will be calculated by the function $\max(0, \lfloor\frac{2000\log_{10}A}{(A - S)^2+100\log_{10}A}\rfloor)$, where $S$ is your submission and $A$ is the true answer.
[i]Lightning 5.2[/i]
2017 239 Open Mathematical Olympiad, 1
Denote every permutation of $1,2,\dots, n$ as $\sigma =(a_1,a_2,\dots,n)$. Prove that the sum $$\sum \frac{1}{(a_1)(a_1+a_2)(a_1+a_2+a_3)\dots(a_1+a_2+\dots+a_n)}$$ taken over all possible permutations $\sigma$ equals $\frac{1}{n!}$.
2019 India PRMO, 23
Let $ABCD$ be a convex cyclic quadilateral. Suppose $P$ is a point in the plane of the quadilateral such that the sum of its distances from the vertices of $ABCD$ is the least. If $$\{PC, PB, PC, PD\} = \{3, 4, 6, 8\}$$, what is the maxumum possible area of $ABCD$?
1978 AMC 12/AHSME, 16
In a room containing $N$ people, $N > 3$, at least one person has not shaken hands with everyone else in the room. What is the maximum number of people in the room that could have shaken hands with everyone else?
$\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }N-1\qquad\textbf{(D) }N\qquad \textbf{(E) }\text{none of these}$
2014 National Olympiad First Round, 12
If one can find a student with at least $k$ friends in any class which has $21$ students such that at least two of any three of these students are friends, what is the largest possible value of $k$?
$
\textbf{(A)}\ 8
\qquad\textbf{(B)}\ 9
\qquad\textbf{(C)}\ 10
\qquad\textbf{(D)}\ 11
\qquad\textbf{(E)}\ 12
$