Found problems: 85335
2011 AMC 12/AHSME, 5
Let $N$ be the second smallest positive integer that is divisible by every positive integer less than 7. What is the sum of the digits of $N$?
$ \textbf{(A)}\ 3 \qquad
\textbf{(B)}\ 4 \qquad
\textbf{(C)}\ 5 \qquad
\textbf{(D)}\ 6 \qquad
\textbf{(E)}\ 9 $
2024 HMIC, 1
In an empty $100 \times 100$ grid, $300$ cells are colored blue, $3$ in each row and each column. Compute the largest positive integer $k$ such that you can always recolor $k$ of these blue cells red so that no contiguous $2 \times 2$ square has four red cells.
[i]Arul Kolla[/i]
2014 ELMO Shortlist, 6
Show that the numerator of \[ \frac{2^{p-1}}{p+1} - \left(\sum_{k = 0}^{p-1}\frac{\binom{p-1}{k}}{(1-kp)^2}\right) \] is a multiple of $p^3$ for any odd prime $p$.
[i]Proposed by Yang Liu[/i]
2013 Today's Calculation Of Integral, 881
Evaluate $\int_{-\pi}^{\pi} \left(\sum_{k=1}^{2013} \sin kx\right)^2dx$.
2005 India IMO Training Camp, 3
Consider a matrix of size $n\times n$ whose entries are real numbers of absolute value not exceeding $1$. The sum of all entries of the matrix is $0$. Let $n$ be an even positive integer. Determine the least number $C$ such that every such matrix necessarily has a row or a column with the sum of its entries not exceeding $C$ in absolute value.
[i]Proposed by Marcin Kuczma, Poland[/i]
2020 Thailand TST, 3
Let $a$ be a positive integer. We say that a positive integer $b$ is [i]$a$-good[/i] if $\tbinom{an}{b}-1$ is divisible by $an+1$ for all positive integers $n$ with $an \geq b$. Suppose $b$ is a positive integer such that $b$ is $a$-good, but $b+2$ is not $a$-good. Prove that $b+1$ is prime.
2003 Spain Mathematical Olympiad, Problem 4
Let ${x}$ be a real number such that ${x^3 + 2x^2 + 10x = 20.}$ Demonstrate that both ${x}$ and ${x^2}$ are irrational.
2024 Irish Math Olympiad, P2
A non-negative integer $p$ is a [i]3-choice[/i] if $\dfrac{k(k-1)(k-2)}{6}$ for some positive integer $k$. Let $p$ and $q$ be 3-choices with $p<q$. Show there is an integer $n$ such that $p \leq n^2 < q$.
2014 Grand Duchy of Lithuania, 1
Determine all functions $f : R \to R$ such that $f(xy + f(x)) = xf(y) + f(x)$ holds for any $x, y \in R$.
2019 Korea National Olympiad, 5
Find all functions $f$ such that $f:\mathbb{R}\rightarrow \mathbb{R}$ and $f(f(x)-x+y^2)=yf(y)$
2016 Stars of Mathematics, 1
Find the minimum number of perfect cubes such that their sum is equal to $ 346^{346} . $
2013 AMC 8, 23
Angle $ABC$ of $\triangle ABC$ is a right angle. The sides of $\triangle ABC$ are the diameters of semicircles as shown. The area of the semicircle on $\overline{AB}$ equals $8\pi$, and the arc of the semicircle on $\overline{AC}$ has length $8.5\pi$. What is the radius of the semicircle on $\overline{BC}$?
[asy]
import graph;
draw((0,8)..(-4,4)..(0,0)--(0,8));
draw((0,0)..(7.5,-7.5)..(15,0)--(0,0));
real theta = aTan(8/15);
draw(arc((15/2,4),17/2,-theta,180-theta));
draw((0,8)--(15,0));
label("$A$", (0,8), NW);
label("$B$", (0,0), SW);
label("$C$", (15,0), SE);[/asy]
$\textbf{(A)}\ 7 \qquad \textbf{(B)}\ 7.5 \qquad \textbf{(C)}\ 8 \qquad \textbf{(D)}\ 8.5 \qquad \textbf{(E)}\ 9$
2010 Dutch IMO TST, 2
Let $A$ and $B$ be positive integers. De fine the arithmetic sequence $a_0, a_1, a_2, ...$ by $a_n = A_n + B$. Suppose that there exists an $n\ge 0$ such that $a_n$ is a square. Let $M$ be a positive integer such that $M^2$ is the smallest square in the sequence. Prove that $M < A +\sqrt{B}$.
2016 Costa Rica - Final Round, G2
Let $ABCD$ be a convex quadrilateral, such that $ A$, $ B$, $C$, and $D$ lie on a circle, with $\angle DAB < \angle ABC$. Let $I$ be the intersection of the bisector of $\angle ABC$ with the bisector of $\angle BAD$. Let $\ell$ be the parallel line to $CD$ passing through point $I$. Suppose $\ell$ cuts segments $DA$ and $BC$ at $ L$ and $J$, respectively. Prove that $AL + JB = LJ$.
1978 Germany Team Selection Test, 2
Let $S$ be a convex quadrilateral $ABCD$ and $O$ a point inside it. The feet of the perpendiculars from $O$ to $AB, BC, CD, DA$ are $A_1, B_1, C_1, D_1$ respectively. The feet of the perpendiculars from $O$ to the sides of $S_i$, the quadrilateral $A_iB_iC_iD_i$, are $A_{i+1}B_{i+1}C_{i+1}D_{i+1}$, where $i = 1, 2, 3.$ Prove that $S_4$ is similar to S.
1985 IMO Longlists, 12
Find the maximum value of
\[\sin^2 \theta_1+\sin^2 \theta_2+\cdots+\sin^2 \theta_n\]
subject to the restrictions $0 \leq \theta_i , \theta_1+\theta_2+\cdots+\theta_n=\pi.$
2010 HMNT, 10
You are given two diameters $AB$ and $CD$ of circle $\Omega$ with radius $1$. A circle is drawn in one of the smaller sectors formed such that it is tangent to $AB$ at $E$, tangent to $CD$ at $F$, and tangent to $\Omega$ at $P$. Lines $PE$ and $PF$ intersect $\Omega$ again at $X$ and $Y$ . What is the length of $XY$ , given that $AC = \frac23$ ?
2010 Princeton University Math Competition, 6
All the diagonals of a regular decagon are drawn. A regular decagon satisfies the property that if three diagonals concur, then one of the three diagonals is a diameter of the circumcircle of the decagon. How many distinct intersection points of diagonals are in the interior of the decagon?
2021 Pan-African, 5
Find all functions $f$ $:$ $\mathbb{R} \rightarrow \mathbb{R}$ such that $\forall x,y \in \mathbb{R}$ :
$$(f(x)+y)(f(y)+x)=f(x^2)+f(y^2)+2f(xy)$$
2019 Durer Math Competition Finals, 3
Let $P$ be an interior point of triangle $ABC$. The lines $AP$, $BP$ and $CP$ divide each of the three sides into two segments. If the so-obtained six segments all have distinct integer lengths, what is the minimum possible perimeter of $ABC$?
2011 USA Team Selection Test, 5
Let $c_n$ be a sequence which is defined recursively as follows: $c_0 = 1$, $c_{2n+1} = c_n$ for $n \geq 0$, and $c_{2n} = c_n + c_{n-2^e}$ for $n > 0$ where $e$ is the maximal nonnegative integer such that $2^e$ divides $n$. Prove that
\[\sum_{i=0}^{2^n-1} c_i = \frac{1}{n+2} {2n+2 \choose n+1}.\]
1982 Bundeswettbewerb Mathematik, 2
Decide whether every triangle $ABC$ in space can be orthogonally projected onto a plane such that the projection is an equilateral triangle $A'B'C'$.
2024 Princeton University Math Competition, B2
Alien Tanvi has a favorite number, but somehow she’s managed to forget it. She remembers that it can be written as $x^2+\tfrac{1}{x^2},$ where $x$ is a real number satisfying $x^4+4x^2+\tfrac{4}{x^2}+\tfrac{1}{x^4}=523.$ What is Alien Tanvi's favorite number?
2012 Belarus Team Selection Test, 3
For each positive integer $k,$ let $t(k)$ be the largest odd divisor of $k.$ Determine all positive integers $a$ for which there exists a positive integer $n,$ such that all the differences
\[t(n+a)-t(n); t(n+a+1)-t(n+1), \ldots, t(n+2a-1)-t(n+a-1)\] are divisible by 4.
[i]Proposed by Gerhard Wöginger, Austria[/i]
2009 District Olympiad, 4
Let $ABC$ be an equilateral $ABC$. Points $M, N, P$ are located on the sides $AC, AB, BC$, respectively, such that $\angle CBM= \frac{1}{2} \angle AMN = \frac{1}{3} \angle BNP$ and $\angle CMP = 90 ^o$.
a) Show that $\vartriangle NMB$ is isosceles.
b) Determine $\angle CBM$.