Found problems: 85335
Ukrainian TYM Qualifying - geometry, VII.12
Let $a, b$, and $c$ be the lengths of the sides of an arbitrary triangle, and let $\alpha,\beta$, and $\gamma$ be the radian measures of its corresponding angles. Prove that $$ \frac{\pi}{3}\le \frac{\alpha a +\beta b + \gamma c}{a+b+c} < \frac{\pi}{2}.$$ Suggest spatial analogues of this inequality.
2013 Today's Calculation Of Integral, 899
Find the limit as below.
\[\lim_{n\to\infty} \frac{(1^2+2^2+\cdots +n^2)(1^3+2^3+\cdots +n^3)(1^4+2^4+\cdots +n^4)}{(1^5+2^5+\cdots +n^5)^2}\]
2016 HMNT, 7
Let ABC be a triangle with $AB = 13, BC = 14, CA = 15$. The altitude from $A$ intersects $BC$ at $D$.
Let $\omega_1$ and $\omega_2$ be the incircles of $ABD$ and $ACD$, and let the common external tangent of $\omega_1$ and $\omega_2$ (other than $BC$) intersect $AD$ at $E$. Compute the length of $AE$.
2019 BMT Spring, 8
Let $(k_i)$ be a sequence of unique nonzero integers such that $x^2- 5x + k_i$ has rational solutions. Find the minimum possible value of $$\frac15 \sum_{i=1}^{\infty} \frac{1}{k_i}$$
2010 Contests, 3
Let $(x_n)_{n \in \mathbb{N}}$ be the sequence defined as $x_n = \sin(2 \pi n! e)$ for all $n \in \mathbb{N}$. Compute $\lim_{n \to \infty} x_n$.
1987 Swedish Mathematical Competition, 1
Sixteen real numbers are arranged in a magic square of side $4$ so that the sum of numbers in each row, column or main diagonal equals $k$. Prove that the sum of the numbers in the four corners of the square is also $k$.
2022 Greece National Olympiad, 3
The positive real numbers $a,b,c,d$ satisfy the equality
$$a+bc+cd+db+\frac{1}{ab^2c^2d^2}=18.$$
Find the maximum possible value of $a$.
2017 Junior Balkan MO, 4
Consider a regular 2n-gon $ P$,$A_1,A_2,\cdots ,A_{2n}$ in the plane ,where $n$ is a positive integer . We say that a point $S$ on one of the sides of $P$ can be seen from a point $E$ that is external to $P$ , if the line segment $SE$ contains no other points that lie on the sides of $P$ except $S$ .We color the sides of $P$ in 3 different colors (ignore the vertices of $P$,we consider them colorless), such that every side is colored in exactly one color, and each color is used at least once . Moreover ,from every point in the plane external to $P$ , points of most 2 different colors on $P$ can be seen .Find the number of distinct such colorings of $P$ (two colorings are considered distinct if at least one of sides is colored differently).
[i]Proposed by Viktor Simjanoski, Macedonia[/i]
JBMO 2017, Q4
2005 Indonesia MO, 2
For an arbitrary positive integer $ n$, define $ p(n)$ as the product of the digits of $ n$ (in decimal). Find all positive integers $ n$ such that $ 11p(n)\equal{}n^2\minus{}2005$.
2007 iTest Tournament of Champions, 2
In the game of [i]Winners Make Zeros[/i], a pair of positive integers $(m,n)$ is written on a sheet of paper. Then the game begins, as the players make the following legal moves:
[list]
[*] If $m\geq n$, the player choose a positive integer $c$ such that $m-cn\geq 0$, and replaces $(m,n)$ with $(m-cn,n)$.
[*] If $m<n$, the player choose a positive integer $c$ such that $n-cm\geq 0$, and replaces $(m,n)$ with $(m,n-cm)$.
[/list]
When $m$ or $n$ becomes $0$, the game ends, and the last player to have moved is declared the winner. If $m$ and $n$ are originally $2007777$ and $2007$, find the largest choice the first player can make for $c$ (on his first move) such that the first player has a winning strategy after that first move.
1987 Greece National Olympiad, 4
In rectangular coodinate system $Oxy$, consider the line $y=3x$ and point $A(4,3)$. Find on the line $y=3x$, point $B\ne O$ such that the area of triangle $OBC$ is the minimum possible, where $C= AB\cap Ox$.
2016 ASDAN Math Tournament, 10
Using the fact that
$$\sum_{n=1}^\infty\frac{1}{n^2}=\frac{\pi^2}{6},$$
compute
$$\int_0^1(\ln x)\ln(1-x)dx.$$
KoMaL A Problems 2020/2021, A. 795
The following game is played with a group of $n$ people and $n+1$ hats are numbered from $1$ to $n+1.$ The people are blindfolded and each of them puts one of the $n+1$ hats on his head (the remaining hat is hidden). Now, a line is formed with the $n$ people, and their eyes are uncovered: each of them can see the numbers on the hats of the people standing in front of him. Now, starting from the last person (who can see all the other players) the players take turns to guess the number of the hat on their head, but no two players can guess the same number (each player hears all the guesses from the other players).
What is the highest number of guaranteed correct guesses, if the $n$ people can discuss a common strategy?
[i]Proposed by Viktor Kiss, Budapest[/i]
2016 AMC 10, 25
How many ordered triples $(x,y,z)$ of positive integers satisfy $\text{lcm}(x,y) = 72, \text{lcm}(x,z) = 600$ and $\text{lcm}(y,z)=900$?
$\textbf{(A)}\ 15\qquad\textbf{(B)}\ 16\qquad\textbf{(C)}\ 24\qquad\textbf{(D)}\ 27\qquad\textbf{(E)}\ 64$
2014 BAMO, 3
Suppose that for two real numbers $x$ and $y$ the following equality is true:
$$(x+ \sqrt{1+ x^2})(y+\sqrt{1+y^2})=1$$
Find (with proof) the value of $x+y$.
1977 Vietnam National Olympiad, 3
Into how many regions do $n$ circles divide the plane, if each pair of circles intersects in two points and no point lies on three circles?
2010 Indonesia TST, 4
$300$ parliament members are divided into $3$ chambers, each chamber consists of $100$ members. For every $2$ members, they either know each other or are strangers to each other.Show that no matter how they are divided into these $3$ chambers, it is always possible to choose $2$ members, each from different chamber such that there exist $17$ members from the third chamber so that all of them knows these two members, or all of them are strangers to these two members.
2017 Dutch Mathematical Olympiad, 2
A parallelogram $ABCD$ with $|AD| =|BD|$ has been given. A point $E$ lies on line segment $|BD|$ in such a way that $|AE| = |DE|$. The (extended) line $AE$ intersects line segment $BC$ in $F$. Line $DF$ is the angle bisector of angle $CDE$. Determine the size of angle $ABD$.
[asy]
unitsize (3 cm);
pair A, B, C, D, E, F;
D = (0,0);
A = dir(250);
B = dir(290);
C = B + D - A;
E = extension((A + D)/2, (A + D)/2 + rotate(90)*(A - D), B, D);
F = extension(A, E, B, C);
draw(A--B--C--D--cycle);
draw(A--F--D--B);
dot("$A$", A, SW);
dot("$B$", B, SE);
dot("$C$", C, NE);
dot("$D$", D, NW);
dot("$E$", E, S);
dot("$F$", F, SE);
[/asy]
KoMaL A Problems 2024/2025, A. 886
Let $k$ and $n$ be two given distinct positive integers greater than $1$. There are finitely many (not necessarily distinct) integers written on the blackboard. Kázmér is allowed to erase $k$ consecutive elements of an arithmetic sequence with a difference not divisible by $k$. Similarly, Nándor is allowed to erase $n$ consecutive elements of an arithmetic sequence with a difference that is not divisible by $n$. The initial numbers on the blackboard have the property that both Kázmér and Nándor can erase all of them (independently from each other) in a finite number of steps. Prove that the difference of biggest and the smallest number on the blackboard is at least $\varphi(n)+\varphi(k)$.
[i]Proposed by Boldizsár Varga, Budapest[/i]
2012 Princeton University Math Competition, A6
Two white pentagonal pyramids, with side lengths all the same, are glued to each other at their regular pentagon bases. Some of the resulting $10$ faces are colored black. How many rotationally distinguishable colorings may result?
2011 BAMO, 4
Three circles $k_1, k_2$, and $k_3$ intersect in point $O$. Let $A, B$, and $C$ be the second intersection points (other than $O$) of $k_2$ and $k_3, k_1$ and $k_3$, and $k_1$ and $k_2$, respectively. Assume that $O$ lies inside of the triangle $ABC$. Let lines $AO,BO$, and $CO$ intersect circles $k_1, k_2$, and $k_3$ for a second time at points $A', B'$, and $C'$, respectively. If $|XY|$ denotes the length of segment $XY$, prove that $\frac{|AO|}{|AA'|}+\frac{|BO|}{|BB'|}+\frac{|CO|}{|CC'|}= 1$
2005 IberoAmerican Olympiad For University Students, 3
Consider the sequence defined recursively by $(x_1,y_1)=(0,0)$,
$(x_{n+1},y_{n+1})=\left(\left(1-\frac{2}{n}\right)x_n-\frac{1}{n}y_n+\frac{4}{n},\left(1-\frac{1}{n}\right)y_n-\frac{1}{n}x_n+\frac{3}{n}\right)$.
Find $\lim_{n\to \infty}(x_n,y_n)$.
2002 All-Russian Olympiad Regional Round, 10.5
Various points $x_1,..., x_n$ ($n \ge 3$) are randomly located on the $Ox$ axis. Construct all parabolas defined by the monic square trinomials and intersecting the Ox axis at these points (and not intersecting axis at other points). Let$ y = f_1$, $...$ , $y = f_m$ are functions that define these parabolas. Prove that the parabola $y = f_1 +...+ f_m$ intersects the $Ox$ axis at two points.
2008 F = Ma, 5
Which of the following acceleration [i]vs.[/i] time graphs most closely represents the acceleration of the toy car?
[asy]
size(300);
Label f;
f.p=fontsize(8);
xaxis(0,3);
yaxis(-2,2);
label("Time (s)",(2.8,0),0.5*N);
label(rotate(90)*"Acceleration",(-0.2,0),W);
label("$0$",(-0,0),SW,fontsize(9));
label("1",(1,0),2*S,fontsize(9));
label("2",(2,0),2*S,fontsize(9));
label("3",(3,0),2*S,fontsize(9));
draw((0.5,0)--(0.5,-0.1));
draw((1,0)--(1,-0.1));
draw((1.5,0)--(1.5,-0.1));
draw((2,0)--(2,-0.1));
draw((2.5,0)--(2.5,-0.1));
draw((3,0)--(3,-0.1));
label("(a)",(1.5,-2),N);
pair A, B, C, D, E, F;
A = (0,1);
B = (1,1);
C = (1,0);
D = (1.5,0);
E = (1.5, 0.5);
F = (3, 0.5);
draw(A--B--C--D--E--F);
real x=6;
Label f;
f.p=fontsize(8);
draw((x+3,0)--(x+0,0));
draw((x,-2)--(x,2));
label("Time (s)",(x+2.8,0.03),0.5*N);
label(rotate(90)*"Acceleration",(x-0.2,0),W);
label("$0$",(x+0,0),SW,fontsize(9));
label("1",(x+1,0),2*S,fontsize(9));
label("2",(x+2,0),2*S,fontsize(9));
label("3",(x+3,0),2*S,fontsize(9));
draw((x+0.5,0)--(x+0.5,-0.1));
draw((x+1,0)--(x+1,-0.1));
draw((x+1.5,0)--(x+1.5,-0.1));
draw((x+2,0)--(x+2,-0.1));
draw((x+2.5,0)--(x+2.5,-0.1));
draw((x+3,0)--(x+3,-0.1));
label("(b)",(x+1.5,-2),N);
/*The lines*/
pair G, H, I, J, K, L;
G = (x+0,1);
H = (x+1,1);
I = (x+1,0);
J = (x+1.5,0);
K = (x+1.5, -0.5);
L = (x+3, -0.5);
draw(G--H--I--J--K--L);[/asy][asy]
size(300);
Label f;
f.p=fontsize(8);
xaxis(0,3);
yaxis(-2,2);
label("Time (s)",(2.8,0),0.5*N);
label(rotate(90)*"Acceleration",(-0.1,0),W);
label("$0$",(-0,0),SW,fontsize(9));
label("1",(1,0),2*S,fontsize(9));
label("2",(2,0),2*S,fontsize(9));
label("3",(3,0),2*S,fontsize(9));
draw((0.5,0)--(0.5,-0.1));
draw((1,0)--(1,-0.1));
draw((1.5,0)--(1.5,-0.1));
draw((2,0)--(2,-0.1));
draw((2.5,0)--(2.5,-0.1));
draw((3,0)--(3,-0.1));
label("(c)",(1.5,-2),N);
pair A, B, C, D, E, F;
A = (0,0.5);
B = (1,0.5);
C = (1,0);
D = (1.5,0);
E = (1.5, -1);
F = (3, -1);
draw(A--B--C--D--E--F);
real x = 6;
Label f;
f.p=fontsize(8);
draw((x+3,0)--(x+0,0));
draw((x,-2)--(x,2));
label("Time (s)",(x+3.4,0),0.5*N);
label(rotate(90)*"Acceleration",(x-0.2,0),W);
label("$0$",(x+0,0),SW,fontsize(9));
label("1",(x+1,0),2*S,fontsize(9));
label("2",(x+2,0),2*S,fontsize(9));
label("3",(x+3,0),2*S,fontsize(9));
draw((x+0.5,0)--(x+0.5,-0.1));
draw((x+1,0)--(x+1,-0.1));
draw((x+1.5,0)--(x+1.5,-0.1));
draw((x+2,0)--(x+2,-0.1));
draw((x+2.5,0)--(x+2.5,-0.1));
draw((x+3,0)--(x+3,-0.1));
label("(d)",(x+1.5,-2),N);
/*The lines*/
pair K, L, M, N, O, P, Q, R;
K = (x+0,1);
L = (x+1,1);
M = (x+1,0.5);
N= (x+1.5,0.5);
O= (x+1.5, -0.5);
P = (x+2.5, -0.5);
Q = (x+2.5, 0.5);
R = (x+3, 0.5);
draw(K--L--M--N--O--P--Q--R);[/asy][asy]
size(150);
Label f;
f.p=fontsize(8);
xaxis(0,3);
yaxis(-2,2);
label("Time (s)",(3.2,0.03),N);
label(rotate(90)*"Acceleration",(-0.1,0),W);
label("$0$",(-0,0),SW,fontsize(9));
label("1",(1,0),2*S,fontsize(9));
label("2",(2,0),2*S,fontsize(9));
label("3",(3,0),2*S,fontsize(9));
draw((0.5,0)--(0.5,-0.1));
draw((1,0)--(1,-0.1));
draw((1.5,0)--(1.5,-0.1));
draw((2,0)--(2,-0.1));
draw((2.5,0)--(2.5,-0.1));
draw((3,0)--(3,-0.1));
label(rotate(90)*"Acceleration",(-0.1,0),W);
label("(e)",(1.5,-2),N);
/*The lines*/
pair A, B, C, D, E, F, G, H;
A = (0,1);
B = (1,1);
C = (1,0.5);
D = (1.5,0.5);
E = (1.5, -0.5);
F = (2.5, -0.5);
G = (2.5, 0.5);
H = (3, 0.5);
draw(A--B--C--D--E--F--G--H);
[/asy]
1984 Tournament Of Towns, (063) O4
Prove that, for any natural number $n$, the graph of any increasing function $f : [0,1] \to [0, 1]$ can be covered by $n$ rectangles each of area whose sides are parallel to the coordinate axes. Assume that a rectangle includes both its interior and boundary points.
(a) Assume that $f(x)$ is continuous on $[0,1]$.
(b) Do not assume that $f(x)$ is continuous on $[0,1]$.
(A Andjans, Riga)
PS. (a) for O Level, (b) for A Level