Found problems: 85335
2022 Chile TST IMO, 1
The sets of rational numbers $A = \{a_1, \dots, a_5\}$ and $B = \{b_1, \dots, b_5\}$ both contain $0$ and satisfy the condition that
$$ \{a_i + b_j\}_{i,j} = \{0, 1, 2, \dots, 23, 24\}. $$
Determine these sets. (The set $\{a_i + b_j\}_{i,j}$ consists of all possible sums between an element of $A$ and an element of $B$)
2017 MIG, 4
Percy buys $3$ apples for $6$ dollars, $4$ pears for $16$ dollars, and $1$ watermelon for $5$ dollars. Assuming the rates stay the same, how much would it cost to buy $10$ apples, $3$ pears, and $2$ watermelons?
$\textbf{(A) } 38\qquad\textbf{(B) } 39\qquad\textbf{(C) } 40\qquad\textbf{(D) } 41\qquad\textbf{(E) } 42$
2001 Cuba MO, 7
Prove that the equation $x^{19} + x^{17} = x^{16 }+ x^7 + a$ for any $a \in R$ has at least two imaginary roots
2024 Princeton University Math Competition, 7
Consider a regular $24$-gon $\mathcal{P}.$ A quadrilateral is said to be inscribed in $\mathcal{P}$ if its vertices are among those of $\mathcal{P}.$ We consider two inscribed quadrilaterals equivalent if one can be obtained from the other via a rotation about the center of $\mathcal{P}.$ How many distinct (i.e. not equivalent) quadrilaterals can be inscribed in $\mathcal{P}$?
2022 VN Math Olympiad For High School Students, Problem 1
Let $ABC$ be a triangle with $\angle A,\angle B,\angle C <120^{\circ}$.
Prove that: there is exactly one point $T$ inside $\triangle ABC$ such that $\angle BTC=\angle CTA=\angle ATB=120^{\circ}$.
($T$ is called [i]Fermat-Torricelli[/i] point of $\triangle ABC$)
Kyiv City MO Juniors 2003+ geometry, 2021.7.3
There are $n$ sticks which have distinct integer length. Suppose that it's possible to form a non-degenerate triangle from any $3$ distinct sticks among them. It's also known that there are sticks of lengths $5$ and $12$ among them. What's the largest possible value of $n$ under such conditions?
[i](Proposed by Bogdan Rublov)[/i]
2006 Korea National Olympiad, 7
Points $A,B,C,D,E,F$ is on the circle $O.$ A line $\ell$ is tangent to $O$ at $E$ is parallel to $AC$ and $DE>EF.$ Let $P,Q$ be the intersection of $\ell$ and $BC,CD$ ,respectively and let $R,S$ be the intersection of $\ell$ and $CF,DF$ ,respectively. Show that $PQ=RS$ if and only if $QE=ER.$
2014 BMT Spring, 18
Monty wants to play a game with you. He shows you five boxes, one of which contains a prize and four of which contain nothing. He allows you to choose one box but not to open it. He then opens one of the other four boxes that he knows to contain nothing. Then, he makes you switch and choose a different, unopened box. However, Monty sketchily reveals the contents of another (empty) box, selected uniformly at random from the two or three closed boxes (that you do not currently have chosen) that he knows to contain no prize. He then offers you the chance to switch again. Assuming you seek to maximize your return, determine the probability you get a prize.
2003 AMC 12-AHSME, 22
Objects $A$ and $B$ move simultaneously in the coordinate plane via a sequence of steps, each of length one. Object $A$ starts at $(0,0)$ and each of its steps is either right or up, both equally likely. Object $B$ starts at $(5,7)$ and each of its steps is either left or down, both equally likely. Which of the following is closest to the probability that the objects meet?
$ \textbf{(A)}\ 0.10 \qquad
\textbf{(B)}\ 0.15 \qquad
\textbf{(C)}\ 0.20 \qquad
\textbf{(D)}\ 0.25 \qquad
\textbf{(E)}\ 0.30$
2022 CHMMC Winter (2022-23), 10
Suppose that $\xi \ne 1$ is a root of the polynomial $f(x) = x^{167} -1$. Compute
$$\left| \sum_{0<a<b<167} \xi^{a^2+b^2} \right|.$$ In the above summation $a,b$ are integers
2010 Korea National Olympiad, 1
$ x, y, z $ are positive real numbers such that $ x+y+z=1 $. Prove that
\[ \sqrt{ \frac{x}{1-x} } + \sqrt{ \frac{y}{1-y} } + \sqrt{ \frac{z}{1-z} } > 2 \]
1999 Harvard-MIT Mathematics Tournament, 7
Evaluate $\sum_{n=1}^\infty \dfrac{n^5}{n!}.$
2002 China Team Selection Test, 1
$ A$ is a set of points on the plane, $ L$ is a line on the same plane. If $ L$ passes through one of the points in $ A$, then we call that $ L$ passes through $ A$.
(1) Prove that we can divide all the rational points into $ 100$ pairwisely non-intersecting point sets with infinity elements. If for any line on the plane, there are two rational points on it, then it passes through all the $ 100$ sets.
(2) Find the biggest integer $ r$, so that if we divide all the rational points on the plane into $ 100$ pairwisely non-intersecting point sets with infinity elements with any method, then there is at least one line that passes through $ r$ sets of the $ 100$ point sets.
2014 AMC 12/AHSME, 17
Let $P$ be the parabola with equation $y = x^2$ and let $Q = (20, 14)$ There are real numbers $r$ and $s$ such that the line through $Q$ with slope $m$ does not intersect $P$ if and only if $r < m < s$. What is $r + s?$
$ \textbf{(A)} 1 \qquad \textbf{(B)} 26 \qquad \textbf{(C)} 40 \qquad \textbf{(D)} 52 \qquad \textbf{(E)} 80 \qquad $
2010 Tournament Of Towns, 4
A rectangle is divided into $2\times 1$ and $1\times 2$ dominoes. In each domino, a diagonal is drawn, and no two diagonals have common endpoints. Prove that exactly two corners of the rectangle are endpoints of these diagonals.
2022 CCA Math Bonanza, I15
Let $P$, $A$, $B$, $C$, $D$ be points on a plane such that $PA = 9$, $PB = 19$, $PC = 9$, $PD = 5$, $\angle APB = 120^\circ$, $\angle BPC = 45^\circ$, $\angle CPD = 60^\circ$, and $\angle DPA = 135^\circ$. Let $G_1$, $G_2$, $G_3$, and $G_4$ be the centroids of triangles $PAB$, $PBC$, $PCD$, $PDA$. $[G_1G_2G_3G_4]$ can be expressed as $a\sqrt{b} + c\sqrt{d}$. Find $a+b+c+d$.
[i]2022 CCA Math Bonanza Individual Round #15[/i]
JOM 2023, 2
Ruby has a non-negative integer $n$. In each second, Ruby replaces the number she has with the product of all its digits. Prove that Ruby will eventually have a single-digit number or $0$. (e.g. $86\rightarrow 8\times 6=48 \rightarrow 4 \times 8 =32 \rightarrow 3 \times 2=6$)
[i]Proposed by Wong Jer Ren[/i]
2021 Indonesia MO, 4
Let $x,y$ and $z$ be positive reals such that $x + y + z = 3$. Prove that
\[ 2 \sqrt{x + \sqrt{y}} + 2 \sqrt{y + \sqrt{z}} + 2 \sqrt{z + \sqrt{x}} \le \sqrt{8 + x - y} + \sqrt{8 + y - z} + \sqrt{8 + z - x} \]
2018 Yasinsky Geometry Olympiad, 6
Given a triangle $ABC$, in which $AB = BC$. Point $O$ is the center of the circumcircle, point $I$ is the center of the incircle. Point $D$ lies on the side $BC$, such that the lines $DI$ and $AB$ parallel. Prove that the lines $DO$ and $CI$ are perpendicular.
(Vyacheslav Yasinsky)
2017 BMT Spring, 3
Let $ABCDEF$ be a regular hexagon with side length $ 1$. Now, construct square $AGDQ$. What is the area of the region inside the hexagon and not the square?
2022 Yasinsky Geometry Olympiad, 6
Let $AD$, $BE$ and $CF$ be the diameters of the circle circumscribed around the acute angle triangle $ABC$. Point $N$ is the midpoint of the arc $CAD$, and point $M$ is the midpoint of arc $BAD$. Prove that the lines $EN$ and $MF$ intersect at the angle bisector of $\angle BAC$.
(Matvii Kurskyi)
2020-21 KVS IOQM India, 28
For a natural number $n$, let $n'$ denote the number obtained by deleting zero digits, if any. (For example, if $n = 260$, $n' = 26$, if $n = 2020$, $n' = 22$.),Find the number of $3$-digit numbers $n$ for which $n'$ is a divisor of $n$, different from $n$.
2023 Argentina National Olympiad Level 2, 4
Initially, Igna distributes $1000$ balls into $30$ boxes. Then, Igna and Mica alternate turns, starting with Igna. Each player, on their turn, chooses a box and removes one ball. When a player removes the last ball from a box, they earn a coin. Find the maximum integer $k$ such that, regardless of how Mica plays, Igna can earn at least $k$ coins.
PEN L Problems, 11
Let the sequence $\{K_{n}\}_{n \ge 1}$ be defined by \[K_{1}=2, K_{2}=8, K_{n+2}=3K_{n+1}-K_{n}+5(-1)^{n}.\] Prove that if $K_{n}$ is prime, then $n$ must be a power of $3$.
2021 Sharygin Geometry Olympiad, 14
Let $\gamma_A, \gamma_B, \gamma_C$ be excircles of triangle $ABC$, touching the sides $BC$, $CA$, $AB$ respectively. Let $l_A$ denote the common external tangent to $\gamma_B$ and $\gamma_C$ distinct from $BC$. Define $l_B, l_C$ similarly. The tangent from a point $P$ of $l_A$ to $\gamma_B$ distinct from $l_A$ meets $l_C$ at point $X$. Similarly the tangent from $P$ to $\gamma_C$ meets $l_B$ at $Y$. Prove that $XY$ touches $\gamma_A$.