Found problems: 85335
2014 Hanoi Open Mathematics Competitions, 11
Determine all real numbers $a, b, c, d$ such that the polynomial $f(x) = ax^3 +bx^2 + cx + d$ satis fies simultaneously the folloving conditions $\begin {cases} |f(x)| \le 1 \,for \, |x| \le 1 \\ f(2) = 26 \end {cases}$
2021 Saint Petersburg Mathematical Olympiad, 1
Let $p$ be a prime number. All natural numbers from $1$ to $p$ are written in a row in ascending order. Find all $p$ such that this sequence can be split into several blocks of consecutive numbers, such that every block has the same sum.
[i]A. Khrabov[/i]
2020 Sharygin Geometry Olympiad, 6
Circles $\omega_1$ and $\omega_2$ meet at point $P,Q$. Let $O$ be the common point of external tangents of $\omega_1$ and $\omega_2$. A line passing through $O$ meets $\omega_1$ and $\omega_2$ at points $A,B$ located on the same side with respect to line segment $PQ$.The line $PA$ meets $\omega_2$ for the second time at $C$ and the line $QB$ meets $\omega_1$ for the second time at $D$. Prove that $O-C-D$ are collinear.
2014 ASDAN Math Tournament, 2
Compute the number of positive integers less than or equal to $10000$ which are relatively prime to $2014$.
2012 Today's Calculation Of Integral, 807
Define a sequence $a_n$ satisfying :
\[a_1=1,\ \ a_{n+1}=\frac{na_n}{2+n(a_n+1)}\ (n=1,\ 2,\ 3,\ \cdots).\]
Find $\lim_{m\to\infty} m\sum_{n=m+1}^{2m} a_n.$
2021 CMIMC, 2.4
What is the $101$st smallest integer which can represented in the form $3^a+3^b+3^c$, where $a,b,$ and $c$ are integers?
[i]Proposed by Dilhan Salgado[/i]
2008 Federal Competition For Advanced Students, P1, 2
Given $a \in R^{+}$ and an integer $n > 4$ determine all n-tuples ($x_1, ...,x_n$) of positive real numbers that satisfy the following system of equations: $\begin {cases} x_1x_2(3a-2x_3) = a^3\\
x_2x_3(3a-2x_4) = a^3\\
...\\
x_{n-2}x_{n-1}(3a-2x_n) = a^3\\
x_{n-1}x_n(3a-2x_1) = a^3 \\
x_nx_1(3a-2x_2) = a^3 \end {cases}$
.
1997 Tournament Of Towns, (534) 6
Let $P$ be a point inside the triangle $ABC$ such that $AB = BC$, $\angle ABC = 80^o$, $\angle PAC = 40^o$ and $\angle ACP = 30^o$. Find $\angle BPC$.
(G Galperin)
2016 Hanoi Open Mathematics Competitions, 3
Given two positive numbers $a,b$ such that $a^3 +b^3 = a^5 +b^5$, then the greatest value of $M = a^2 + b^2 - ab$ is
(A): $\frac14$ (B): $\frac12$ (C): $2$ (D): $1$ (E): None of the above.
2023 Mexican Girls' Contest, 8
There are $3$ sticks of each color between blue, red and green, such that we can make a triangle $T$ with sides sticks with all different colors. Dana makes $2$ two arrangements, she starts with $T$ and uses the other six sticks to extend the sides of $T$, as shown in the figure. This leads to two hexagons with vertex the ends of these six sticks. Prove that the area of the both hexagons it´s the same.
[asy]size(300);
pair A, B, C, D, M, N, P, Q, R, S, T, U, V, W, X, Y, Z, K;
A = (0, 0);
B = (1, 0);
C=(-0.5,2);
D=(-1.1063,4.4254);
M=(-1.7369,3.6492);
N=(3.5,0);
P=(-2.0616,0);
Q=(0.2425,-0.9701);
R=(1.6,-0.8);
S=(7.5164,0.8552);
T=(8.5064,0.8552);
U=(7.0214,2.8352);
V=(8.1167,-1.546);
W=(9.731,-0.7776);
X=(10.5474,0.8552);
Y=(6.7813,3.7956);
Z=(6.4274,3.6272);
K=(5.0414,0.8552);
draw(A--B, blue);
label("$b$", (A + B) / 2, dir(270), fontsize(10));
label("$g$", (B+C) / 2, dir(10), fontsize(10));
label("$r$", (A+C) / 2, dir(230), fontsize(10));
draw(B--C,green);
draw(D--C,green);
label("$g$", (C + D) / 2, dir(10), fontsize(10));
draw(C--A,red);
label("$r$", (C + M) / 2, dir(200), fontsize(10));
draw(B--N,green);
label("$g$", (B + N) / 2, dir(70), fontsize(10));
draw(A--P,red);
label("$r$", (A+P) / 2, dir(70), fontsize(10));
draw(A--Q,blue);
label("$b$", (A+Q) / 2, dir(540), fontsize(10));
draw(B--R,blue);
draw(C--M,red);
label("$b$", (B+R) / 2, dir(600), fontsize(10));
draw(Q--R--N--D--M--P--Q, dashed);
draw(Y--Z--K--V--W--X--Y, dashed);
draw(S--T,blue);
draw(U--T,green);
draw(U--S,red);
draw(T--W,red);
draw(T--X,red);
draw(S--K,green);
draw(S--V,green);
draw(Y--U,blue);
draw(U--Z,blue);
label("$b$", (Y+U) / 2, dir(0), fontsize(10));
label("$b$", (U+Z) / 2, dir(200), fontsize(10));
label("$b$", (S+T) / 2, dir(100), fontsize(10));
label("$r$", (S+U) / 2, dir(200), fontsize(10));
label("$r$", (T+W) / 2, dir(70), fontsize(10));
label("$r$", (T+X) / 2, dir(70), fontsize(10));
label("$g$", (U+T) / 2, dir(70), fontsize(10));
label("$g$", (S+K) / 2, dir(70), fontsize(10));
label("$g$", (V+S) / 2, dir(30), fontsize(10));
[/asy]
2003 Regional Competition For Advanced Students, 2
Find all prime numbers $ p$ with $ 5^p\plus{}4p^4$ is the square of an integer.
2022 USEMO, 2
A function $\psi \colon {\mathbb Z} \to {\mathbb Z}$ is said to be [i]zero-requiem[/i] if for any positive integer $n$ and any integers $a_1$, $\ldots$, $a_n$ (not necessarily distinct), the sums $a_1 + a_2 + \dots + a_n$ and $\psi(a_1) + \psi(a_2) + \dots + \psi(a_n)$ are not both zero.
Let $f$ and $g$ be two zero-requiem functions for which $f \circ g$ and $g \circ f$ are both the identity function (that is, $f$ and $g$ are mutually inverse bijections). Given that $f+g$ is [i]not[/i] a zero-requiem function, prove that $f \circ f$ and $g \circ g$ are both zero-requiem.
[i]Sutanay Bhattacharya[/i]
1999 Harvard-MIT Mathematics Tournament, 3
Find \[\int_{-4\pi\sqrt{2}}^{4\pi\sqrt{2}}\left(\dfrac{\sin x}{1+x^4}+1\right)dx.\]
2013 Turkey Junior National Olympiad, 2
Find all prime numbers $p, q, r$ satisfying the equation
\[ p^4+2p+q^4+q^2=r^2+4q^3+1 \]
2011 Bosnia Herzegovina Team Selection Test, 3
Numbers $1,2, ..., 2n$ are partitioned into two sequences $a_1<a_2<...<a_n$ and $b_1>b_2>...>b_n$. Prove that number
\[W= |a_1-b_1|+|a_2-b_2|+...+|a_n-b_n|\]
is a perfect square.
1976 IMO, 1
In a convex quadrilateral (in the plane) with the area of $32 \text{ cm}^{2}$ the sum of two opposite sides and a diagonal is $16 \text{ cm}$. Determine all the possible values that the other diagonal can have.
2012 Brazil National Olympiad, 6
Find all surjective functions $f\colon (0,+\infty) \to (0,+\infty)$ such that $2x f(f(x)) = f(x)(x+f(f(x)))$ for all $x>0$.
2021 Thailand TSTST, 3
Let $1 \leq n \leq 2021$ be a positive integer. Jack has $2021$ coins arranged in a line where each coin has an $H$ on one side and a $T$ on the other. At the beginning, all coins show $H$ except the nth coin. Jack can repeatedly perform the following operation: he chooses a coin showing $T$, and turns over the coins next to it to the left and to the right (if any). Determine all $n$ such that Jack can make all coins show $T$ after a finite number of operations.
2014 Singapore Senior Math Olympiad, 29
Find the number of ordered triples of real numbers $(x,y,z)$ that satisfy the following systems of equations:
$x^2=4y-4,y^2=4z-4,z^2=4x-4$
2016 BMT Spring, 3
Let $S$ be the set of all non-degenerate triangles with integer sidelengths, such that two of the sides are $20$ and $16$. Suppose we pick a triangle, at random, from this set. What is the probability that it is acute?
1954 AMC 12/AHSME, 49
The difference of the squares of two odd numbers is always divisible by $ 8$. If $ a>b$, and $ 2a\plus{}1$ and $ 2b\plus{}1$ are the odd numbers, to prove the given statement we put the difference of the squares in the form:
$ \textbf{(A)}\ (2a\plus{}1)^2\minus{}(2b\plus{}1)^2 \\
\textbf{(B)}\ 4a^2\minus{}4b^2\plus{}4a\minus{}4b \\
\textbf{(C)}\ 4[a(a\plus{}1)\minus{}b(b\plus{}1)] \\
\textbf{(D)}\ 4(a\minus{}b)(a\plus{}b\plus{}1) \\
\textbf{(E)}\ 4(a^2\plus{}a\minus{}b^2\minus{}b)$
2023 Lusophon Mathematical Olympiad, 2
Let $D$ be a point on the inside of triangle $ABC$ such that $AD=CD$, $\angle DAB=70^{\circ}$, $\angle DBA=30^{\circ}$ and $\angle DBC=20^{\circ}$. Find the measure of angle $\angle DCB$.
2012 ELMO Shortlist, 4
A tournament on $2k$ vertices contains no $7$-cycles. Show that its vertices can be partitioned into two sets, each with size $k$, such that the edges between vertices of the same set do not determine any $3$-cycles.
[i]Calvin Deng.[/i]
2003 Gheorghe Vranceanu, 4
Prove that among any $ 16 $ numbers smaller than $ 101 $ there are four of them that have the property that the sum of two of them is equal to the sum of the other two.
2023 Estonia Team Selection Test, 3
In the acute-angled triangle $ABC$, the point $F$ is the foot of the altitude from $A$, and $P$ is a point on the segment $AF$. The lines through $P$ parallel to $AC$ and $AB$ meet $BC$ at $D$ and $E$, respectively. Points $X \ne A$ and $Y \ne A$ lie on the circles $ABD$ and $ACE$, respectively, such that $DA = DX$ and $EA = EY$.
Prove that $B, C, X,$ and $Y$ are concyclic.