Found problems: 85335
Find all real solutions of the equation $\sin^{5}x+\cos^{3}x=1$.
Determine all triples $(a,b,c)$ of real numbers with
\[a+\frac{4}{b}=b+\frac{4}{c}=c+\frac{4}{a}.\]
Is it possible that the heights of a tetrahedron (that is, a triangular pyramid) would be equal to the numbers $1$, $2$, $3$ and $6$?
Let $a,b,c$ be positive reals such that $a+b+c=ab+bc+ca$. Prove that \[ (a+b)^{ab-bc}(b+c)^{bc-ca}(c+a)^{ca-ab} \ge a^{ca}b^{ab}c^{bc}. \][i]Proposed by Sammy Luo[/i]
Let $n \ge 2$ be an integer. Let $x_1 \ge x_2 \ge ... \ge x_n$ and $y_1 \ge y_2 \ge ... \ge y_n$ be $2n$ real numbers such that $$0 = x_1 + x_2 + ... + x_n = y_1 + y_2 + ... + y_n $$ $$\text{and} \hspace{2mm} 1 =x_1^2 + x_2^2 + ... + x_n^2 = y_1^2 + y_2^2 + ... + y_n^2.$$ Prove that $$\sum_{i = 1}^n (x_iy_i - x_iy_{n + 1 - i}) \ge \frac{2}{\sqrt{n-1}}.$$
[i]Proposed by David Speyer and Kiran Kedlaya[/i]
Let $m,n \geqslant 2$ be integers, let $X$ be a set with $n$ elements, and let $X_1,X_2,\ldots,X_m$ be pairwise distinct non-empty, not necessary disjoint subset of $X$. A function $f \colon X \to \{1,2,\ldots,n+1\}$ is called [i]nice[/i] if there exists an index $k$ such that \[\sum_{x \in X_k} f(x)>\sum_{x \in X_i} f(x) \quad \text{for all } i \ne k.\] Prove that the number of nice functions is at least $n^n$.
The diagram below shows a rectangle with side lengths $36$ and $48$. Each of the sides is trisected and edges are added between the trisection points as shown. Then the shaded corner regions are removed, leaving the octagon which is not shaded in the diagram. Find the perimeter of this octagon.
[asy]
size(4cm);
dotfactor=3.5;
pair A,B,C,D,E,F,G,H,W,X,Y,Z;
A=(0,12);
B=(0,24);
C=(16,36);
D=(32,36);
E=(48,24);
F=(48,12);
G=(32,0);
H=(16,0);
W=origin;
X=(0,36);
Y=(48,36);
Z=(48,0);
filldraw(W--A--H--cycle^^B--X--C--cycle^^D--Y--E--cycle^^F--Z--G--cycle,rgb(.76,.76,.76));
draw(W--X--Y--Z--cycle,linewidth(1.2));
dot(A);
dot(B);
dot(C);
dot(D);
dot(E);
dot(F);
dot(G);
dot(H);
[/asy]
Let $B, M$, and $T$ be the three roots of the equation $x^3 + 20x^2 -18x-19 = 0$. What is the value of $|(B + 1)(M + 1)(T + 1)|$?
In acute triangle $ABC$, $AB > AC$.
$D$ and $E$ are the midpoints of $AB$, $AC$ respectively.
The circumcircle of $ADE$ intersects the circumcircle of $BCE$ again at $P$.
The circumcircle of $ADE$ intersects the circumcircle $BCD$ again at $Q$.
Prove that $AP = AQ$.
Let $AA_1$ and $CC_1$ be the altitudes of the acute-angled triangle $ABC$. Let $K,L$ and $M$ be the midpoints of the sides $AB,BC$ and $CA$ respectively. Prove that if $\angle C_1MA_1 =\angle ABC$, then $C_1 K = A_1L$.
The angles at the vertices $A$ and $C$ in the convex quadrilateral $ABCD$ are not acute. Points $K, L, M$ and $N$ are marked on the sides $AB, BC, CD$ and $DA$ respectively. Prove that the perimeter of $KLMN$ is not less than the double length of the diagonal $AC$.
Find the number of all real functions $f$ which map the sum of $n$ elements into the sum of their images, such that $f^{n-1}$ is a constant function and $f^{n-2}$ is not. Here $f^0(x) = x$ and $f^k = f \circ f^{k-1}$ for $k \ge 1$.
Given is a square board with dimensions $2023 \times 2023$, in which each unit cell is colored blue or red. There are exactly $1012$ rows in which the majority of cells are blue, and exactly $1012$ columns in which the majority of cells are red.
What is the maximal possible side length of the largest monochromatic square?
Suppose circles $\mathit{W}_1$ and $\mathit{W}2$, with centres $\mathit{O}_1$ and $\mathit{O}_2$ respectively, intersect at points $\mathit{M}$ and $\mathit{N}$. Let the tangent on $\mathit{W}_2$ at point $\mathit{N}$ intersect $\mathit{W}_1$ for the second time at $\mathit{B}_1$. Similarly, let the tangent on $\mathit{W}_1$ at point $\mathit{N}$ intersect $\mathit{W}_2$ for the second time at $\mathit{B}_2$. Let $\mathit{A}_1$ be a point on $\mathit{W}_1$ which is on arc $\mathit{B}_1\mathit{N}$ not containing $\mathit{M}$ and suppose line $\mathit{A}_1\mathit{N}$ intersects $\mathit{W}_2$ at point $\mathit{A}_2$. Denote the incentres of triangles $\mathit{B}_1\mathit{A}_1\mathit{N}$ and $\mathit{B}_2\mathit{A}_2\mathit{N}$ by $\mathit{I}_1$ and $\mathit{I}_2$, respectively.*
[asy]
/* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki, go to User:Azjps/geogebra */
import graph; size(10.1cm);
real labelscalefactor = 0.5; /* changes label-to-point distance */
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */
pen dotstyle = black; /* point style */
real xmin = -0.9748626324969808, xmax = 13.38440254515721, ymin = 0.5680051903627492, ymax = 10.99430986899034; /* image dimensions */
pair O_2 = (7.682929606970993,6.084708172218866), O_1 = (2.180000000000002,6.760000000000007), M = (4.560858774883258,8.585242858926296), B_2 = (10.07334553576748,9.291873850408265), A_2 = (11.49301008867042,4.866805580476367), B_1 = (2.113311869970955,9.759258690628950), A_1 = (0.2203184186713625,4.488514120712773);
/* draw figures */
draw(circle(O_2, 4.000000000000000));
draw(circle(O_1, 3.000000000000000));
draw((4.048892687647541,4.413249028538064)--B_2);
draw(B_2--A_2);
draw(A_2--(4.048892687647541,4.413249028538064));
draw((4.048892687647541,4.413249028538064)--B_1);
draw(B_1--A_1);
draw(A_1--(4.048892687647541,4.413249028538064));
/* dots and labels */
dot(O_2,dotstyle);
label("$O_2$", (7.788512439159622,6.243082420501817), NE * labelscalefactor);
dot(O_1,dotstyle);
label("$O_1$", (2.298205165350667,6.929370829727937), NE * labelscalefactor);
dot(M,dotstyle);
label("$M$", (4.383466101076183,8.935444641311980), NE * labelscalefactor);
dot((4.048892687647541,4.413249028538064),dotstyle);
label("$N$", (3.855551940133015,3.761885864068922), NE * labelscalefactor);
dot(B_2,dotstyle);
label("$B_2$", (10.19052187145104,9.463358802255147), NE * labelscalefactor);
dot(A_2,dotstyle);
label("$A_2$", (11.80066006232771,4.659339937672310), NE * labelscalefactor);
dot(B_1,dotstyle);
label("$B_1$", (1.981456668784765,10.09685579538695), NE * labelscalefactor);
dot(A_1,dotstyle);
label("$A_1$", (0.08096568938935705,3.973051528446190), NE * labelscalefactor);
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
/* end of picture */[/asy]
Show that \[\angle\mathit{I}_1\mathit{MI}_2=\angle\mathit{O}_1\mathit{MO}_2.\]
*[size=80]Given a triangle ABC, the incentre of the triangle is defined to be the intersection of the angle bisectors of A, B, and C. To avoid cluttering, the incentre is omitted in the provided diagram. Note also that the diagram serves only as an aid and is not necessarily drawn to scale.[/size]
The plane is divided by three infinite sets of parallel lines into equilateral triangles of equal area. Let $M$ be the set of their vertices, and $A$ and $B$ be two vertices of such an equilateral triangle. One may rotate the plane through $120^o$ around any vertex of the set $M$. Is it possible to move the point $A$ to the point $B$ by a number of such rotations
(N Vasiliev, Moscow)
If there is a natural number $n$ such that the number $n!$ has exactly $11$ zeros at the end?
(With $n!$ is denoted the number $1\cdot 2\cdot 3 \cdot ... (n - )1 \cdot n$).
An infinite number of squares on an infinitely square grid paper are painted red. Show that you can draw a number of squares on the paper, with sides along the grid lines, such that:
(1) no square in the grid belongs to more than one square (an edge, on the other hand, may belong to more than one square)
(2) each red square is located in one of the squares and the number of red squares in such square is at least $1/5$ and at most $4/5$ of the number of squares in the square.
[hide=original wording] Ett andligt antal rutor pa ett oandligt rutat papper ar malade roda. Visa att man pa papperet kan rita in ett antal kvadrater, med sidor utefter rutnatets linjer, sadana att :
(1) ingen ruta i natet tillhor mer an en kvadrat (en kant kan daremot tillhora mer an en kvadrat),
(2) varje rod ruta ligger i nagon av kvadraterna och antalet roda rutor i en sadan kvadratar minst 1/5 och hogst 4/5 av antalet rutor i kvadraten.
[url=http://www.mattetavling.se/wp-content/uploads/2011/01/Final10.pdf]source[/url][/hide]
PS. I always post the original wording when I doubt about my (using Google) translation.
$\triangle ABC$ is an isosceles triangle with $AC = BC$ and $\angle ACB < 60^{\circ}$. $I$ and $O$ are the incenter and circumcenter of $\triangle ABC$. The circumcircle of $\triangle BIO$ intersects $BC$ at $D \neq B$.
(a) Do the lines $AC$ and $DI$ intersect? Give a proof.
(b) What is the angle of intersection between the lines $OD$ and $IB$?
Let $P(x,y)$ be a polynomial such that $\deg_x(P), \deg_y(P)\le 2020$ and \[P(i,j)=\binom{i+j}{i}\] over all $2021^2$ ordered pairs $(i,j)$ with $0\leq i,j\leq 2020$. Find the remainder when $P(4040, 4040)$ is divided by $2017$.
Note: $\deg_x (P)$ is the highest exponent of $x$ in a nonzero term of $P(x,y)$. $\deg_y (P)$ is defined similarly.
[i]Proposed by Michael Ren[/i]
Find the number of permutations of the letters $ABCDE$ where the letters $A$ and $B$ are not adjacent and the letters $C$ and $D$ are not adjacent. For example, count the permutations $ACBDE$ and $DEBCA$ but not $ABCED$ or $EDCBA$.
$ABCD$ is a trapezoid with $BC\parallel AD$ and $BC = 2AD$. Point $M$ is chosen on the side $CD$ such that $AB = AM$. Prove that $BM \perp CD$.
[i]Proposed by Bogdan Rublov[/i]
A circle which is tangent to sides $AB$ and $BC$ of triangle $ABC$ is also tangent to its circumcircle at point $T$. If $I$ in the incenter of triangle $ABC$, show that $\angle ATI=\angle CTI$.
Inside a right angle is given a point $A$. Construct an equilateral triangle, one of the vertices of which is point $A$, and two others lie on the sides of the angle (one on each side).
Find the sum of all positive integers $n$ such that \[ \frac{2n+1}{n(n-1)} \] has a terminating decimal representation.
[i]Proposed by Evan Chen[/i]
Let $ABC$ be a scalene triangle with $\angle A = 60^{\circ}$. Let $E$ and $F$ be the feet of the angle bisectors of $\angle ABC$ and $\angle ACB$, respectively, and let $I$ be the incenter of $\triangle ABC$. Let $P,Q$ be distinct points such that $\triangle PEF$ and $\triangle QEF$ are equilateral. If $O$ is the circumcenter of of $\triangle APQ$, show that $\overline{OI}\perp \overline{BC}$.
[i]Proposed by Vincent Huang