Found problems: 85335
2021 Junior Balkan Team Selection Tests - Moldova, 8
In a box there are $n$ balls, each colored in one of the following colors: green, red, blue or yellow. It is known that among any $28$ balls in the box at least one is green. Among any $26$ balls at least one is red. Among any $24$ balls at least one is blue. Among any $23$ balls at least one is yellow. Find the largest possible value of the number $n$.
1994 Austrian-Polish Competition, 9
On the plane are given four distinct points $A,B,C,D$ on a line $g$ in this order, at the mutual distances $AB = a, BC = b, CD = c$.
(a) Construct (if possible) a point $P$ outside line $g$ such that $\angle APB =\angle BPC =\angle CPD$.
(b) Prove that such a point $P$ exists if and only if $ (a+b)(b+c) < 4ac$
2023 Macedonian Mathematical Olympiad, Problem 2
Let $p$ and $q$ be odd prime numbers and $a$ a positive integer so that $p|a^q+1$ and $q|a^p+1$. Show that $p|a+1$ or $q|a+1$.
[i]Authored by Nikola Velov[/i]
2017 CMI B.Sc. Entrance Exam, 6
You are given a regular hexagon. We say that a square is inscribed in the hexagon if it can be drawn in the interior such that all the four vertices lie on the perimeter of the hexagon.
[b](a)[/b] A line segment has its endpoints on opposite edges of the hexagon. Show that, it passes through the centre of the hexagon if and only if it divides the two edges in the same ratio.
[b](b)[/b] Suppose, a square $ABCD$ is inscribed in the hexagon such that $A$ and $C$ are on the opposite sides of the hexagon. Prove that, centre of the square is same as that of the hexagon.
[b](c)[/b] Suppose, the side of the hexagon is of length $1$. Then find the length of the side of the inscribed square whose one pair of opposite sides is parallel to a pair of opposite sides of the hexagon.
[b](d)[/b] Show that, up to rotation, there is a unique way of inscribing a square in a regular hexagon.
1993 Miklós Schweitzer, 3
Let K be the field formed by the addition of a root of the polynomial $x^4 - 2x^2 - 1$ to the rational field. Prove that there are no exceptional units in the ring of integers of K. (A unit $\varepsilon$ is called exceptional if $1-\varepsilon$ is also a unit.)
2017 Saudi Arabia BMO TST, 4
Let $p$ be a prime number and a table of size $(p^2+ p+1)\times (p^2+p + 1)$ which is divided into unit cells. The way to color some cells of this table is called nice if there are no four colored cells that form a rectangle (the sides of rectangle are parallel to the sides of given table).
1. Let $k$ be the number of colored cells in some nice coloring way. Prove that $k \le (p + 1)(p^2 + p + 1)$. Denote this number as $k_{max}$.
2. Prove that all ordered tuples $(a, b, c)$ with $0 \le a, b, c < p$ and $a + b + c > 0$ can be partitioned into $p^2 + p + 1$ sets $S_1, S_2, .. . S_{p^2+p+1}$ such that two tuples $(a_1, b_1, c_1)$ and $(a_2, b_2, c_2)$ belong to the same set if and only if $a_1 \equiv ka_2, b_1 \equiv kb_2, c_1 \equiv kc_2$ (mod $p$) for some $k \in \{1,2, 3, ... , p - 1\}$.
3. For $1 \le i, j \le p^2+p+1$, if there exist $(a_1, b_1, c_1) \in S_i$ and $(a_2, b_2, c_2) \in S_j$ such that $a_1a_2 + b_1b_2 + c_1c_2 \equiv 0$ (mod $p$), we color the cell $(i, j)$ of the given table. Prove that this coloring way is nice with $k_{max}$ colored cells
2004 National High School Mathematics League, 7
In rectangular coordinate system, the area which is surrounded by the figure of $f(x)=a\sin ax+\cos ax(a>0)$ on a complete period and the figure of $g(x)=\sqrt{a^2+1}$ is________.
2019 Purple Comet Problems, 12
Find the number of ordered triples of positive integers $(a, b, c)$, where $a, b,c$ is a strictly increasing arithmetic progression, $a + b + c = 2019$, and there is a triangle with side lengths $a, b$, and $c$.
2019 Pan-African, 1
Let $(a_n)_{n=0}^{\infty}$ be a sequence of real numbers defined as follows:
[list]
[*] $a_0 = 3$, $a_1 = 2$, and $a_2 = 12$; and
[*] $2a_{n + 3} - a_{n + 2} - 8a_{n + 1} + 4a_n = 0$ for $n \geq 0$.
[/list]
Show that $a_n$ is always a strictly positive integer.
1993 IMO Shortlist, 1
a) Show that the set $ \mathbb{Q}^{ + }$ of all positive rationals can be partitioned into three disjoint subsets. $ A,B,C$ satisfying the following conditions:
\[ BA = B; \& B^2 = C; \& BC = A;
\]
where $ HK$ stands for the set $ \{hk: h \in H, k \in K\}$ for any two subsets $ H, K$ of $ \mathbb{Q}^{ + }$ and $ H^2$ stands for $ HH.$
b) Show that all positive rational cubes are in $ A$ for such a partition of $ \mathbb{Q}^{ + }.$
c) Find such a partition $ \mathbb{Q}^{ + } = A \cup B \cup C$ with the property that for no positive integer $ n \leq 34,$ both $ n$ and $ n + 1$ are in $ A,$ that is,
\[ \text{min} \{n \in \mathbb{N}: n \in A, n + 1 \in A \} > 34.
\]
2011 AMC 8, 9
Carmen takes a long bike ride on a hilly highway. The graph indicates the miles traveled during the time of her ride. What is Carmen's average speed for her entire ride in miles per hour?
[asy]
import graph; size(8.76cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-3.58,xmax=10.19,ymin=-4.43,ymax=9.63;
draw((0,0)--(0,8)); draw((0,0)--(8,0)); draw((0,1)--(8,1)); draw((0,2)--(8,2)); draw((0,3)--(8,3)); draw((0,4)--(8,4)); draw((0,5)--(8,5)); draw((0,6)--(8,6)); draw((0,7)--(8,7)); draw((1,0)--(1,8)); draw((2,0)--(2,8)); draw((3,0)--(3,8)); draw((4,0)--(4,8)); draw((5,0)--(5,8)); draw((6,0)--(6,8)); draw((7,0)--(7,8)); label("$1$",(0.95,-0.24),SE*lsf); label("$2$",(1.92,-0.26),SE*lsf); label("$3$",(2.92,-0.31),SE*lsf); label("$4$",(3.93,-0.26),SE*lsf); label("$5$",(4.92,-0.27),SE*lsf); label("$6$",(5.95,-0.29),SE*lsf); label("$7$",(6.94,-0.27),SE*lsf); label("$5$",(-0.49,1.22),SE*lsf); label("$10$",(-0.59,2.23),SE*lsf); label("$15$",(-0.61,3.22),SE*lsf); label("$20$",(-0.61,4.23),SE*lsf); label("$25$",(-0.59,5.22),SE*lsf); label("$30$",(-0.59,6.2),SE*lsf); label("$35$",(-0.56,7.18),SE*lsf); draw((0,0)--(1,1),linewidth(1.6)); draw((1,1)--(2,3),linewidth(1.6)); draw((2,3)--(4,4),linewidth(1.6)); draw((4,4)--(7,7),linewidth(1.6)); label("HOURS",(3.41,-0.85),SE*lsf); label("M",(-1.39,5.32),SE*lsf); label("I",(-1.34,4.93),SE*lsf); label("L",(-1.36,4.51),SE*lsf); label("E",(-1.37,4.11),SE*lsf); label("S",(-1.39,3.7),SE*lsf);
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
[/asy]
$ \textbf{(A)}2\qquad\textbf{(B)}2.5\qquad\textbf{(C)}4\qquad\textbf{(D)}4.5\qquad\textbf{(E)}5 $
1976 IMO Longlists, 37
From a square board $11$ squares long and $11$ squares wide, the central square is removed. Prove that the remaining $120$ squares cannot be covered by $15$ strips each $8$ units long and one unit wide.
2007 Sharygin Geometry Olympiad, 1
In an acute triangle $ABC$, altitudes at vertices $A$ and $B$ and bisector line at angle $C$ intersect the circumcircle again at points $A_1, B_1$ and $C_0$. Using the straightedge and compass, reconstruct the triangle by points $A_1, B_1$ and $C_0$.
2008 Korea Junior Math Olympiad, 1
In a $\triangle XYZ$, points $A,B$ lie on segment $ZX, C,D$ lie on segment $XY , E, F$ lie on segment $YZ$. $A, B, C, D$ lie on a circle, and $\frac{AZ \cdot EY \cdot ZB \cdot Y F}{EZ \cdot CY \cdot ZF \cdot Y D}= 1$ . Let $L = ZX \cap DE$, $M = XY \cap AF$, $N = Y Z \cap BC$. Prove that $L,M,N$ are collinear.
Kyiv City MO Seniors Round2 2010+ geometry, 2016.10.2
On the horizontal line from left to right are the points $P, \, \, Q, \, \, R, \, \, S$. Construct a square $ABCD$, for which on the line $AD$ lies lies the point $P$, on the line $BC$ lies the point $Q$, on the line $AB$ lies the point $R$, on the line $CD$ lies the point $S $.
2016 Poland - Second Round, 6
$n$ ($n \ge 4$) green points are in a data space and no $4$ green points lie on one plane. Some segments which connect green points have been colored red. Number of red segments is even. Each two green points are connected with polyline which is build from red segments. Show that red segments can be split on pairs, such that segments from one pair have common end.
2007 Sharygin Geometry Olympiad, 21
There are two pipes on the plane (the pipes are circular cylinders of equal size, $4$ m around). Two of them are parallel and, being tangent one to another in the common generatrix, form a tunnel over the plane. The third pipe is perpendicular to two others and cuts out a chamber in the tunnel. Determine the area of the surface of this chamber.
1989 IMO Shortlist, 24
For points $ A_1, \ldots ,A_5$ on the sphere of radius 1, what is the maximum value that $ min_{1 \leq i,j \leq 5} A_iA_j$ can take? Determine all configurations for which this maximum is attained. (Or: determine the diameter of any set $ \{A_1, \ldots ,A_5\}$ for which this maximum is attained.)
2024 AMC 8 -, 1
What is the ones digit of \[222{,}222-22{,}222-2{,}222-222-22-2?\]
$\textbf{(A) }0\qquad\textbf{(B) }2\qquad\textbf{(C) }4\qquad\textbf{(D) }6\qquad\textbf{(E) }8$
1989 Greece National Olympiad, 3
If $a\ge 0$ prove that $a^4+ a^3-10 a^2+9 a+4>0$.
2020 Iran MO (3rd Round), 1
$1)$. Prove a graph with $2n$ vertices and $n+2$ edges has an independent set of size $n$ (there are $n$ vertices such that no two of them are adjacent ).
$2)$.Find the number of graphs with $2n$ vertices and $n+3$ edges , such that among any $n$ vertices there is an edge connecting two of them
Novosibirsk Oral Geo Oly IX, 2023.4
In a trapezoid, the length of one of the diagonals is equal to the sum of the lengths of the bases, and the angle between the diagonals is $60$ degrees. Prove that this trapezoid is isosceles.
2023 AIME, 6
Alice knows that $3$ red cards and $3$ black cards will be revealed to her one at a time in random order. Before each card is revealed, Alice must guess its color. If Alice plays optimally, the expected number of cards she will guess correctly is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
2016 ITAMO, 6
A mysterious machine contains a secret combination of $2016$ integer numbers $x_1,x_2,\ldots,x_{2016}$. It is known that all the numbers in the combination are equal but one. One may ask questions to the machine by giving to it a sequence of $2016$ integer numbers $y_1,\ldots,y_{2016}$, and the machine answers by telling the value of the sum
\[
x_1y_1+\dots+x_{2016}y_{2016}.
\]
After answering the first question, the machine accepts a second question and then a third one, and so on.
Determine how many questions are necessary to determine the combination:
(a) knowing that the number which is different from the others is equal to zero;
(b) not knowing what the number different from the others is.
2016-2017 SDML (Middle School), 1
A "domino" is made up of two small squares:
[asy]
unitsize(10);
draw((0,0) -- (2,0) -- (2,1) -- (0,1) -- cycle);
fill((0,0) -- (1,0) -- (1,1) -- (0,1) -- cycle);
[/asy]
Which of the "checkerboards" illustrated below CANNOT be covered exactly and completely by a whole number of non-overlapping dominoes?
[diagram requires in-line asy]