Found problems: 85335
1998 Gauss, 4
Jean writes five tests and achieves the marks shown on the
graph. What is her average mark on these five tests?
[asy]
draw(origin -- (0, 10.1));
for(int i = 0; i < 11; ++i) {
draw((0, i) -- (10.5, i));
label(string(10*i), (0, i), W);
}
filldraw((1, 0) -- (1, 8) -- (2, 8) -- (2, 0) -- cycle, black);
filldraw((3, 0) -- (3, 7) -- (4, 7) -- (4, 0) -- cycle, black);
filldraw((5, 0) -- (5, 6) -- (6, 6) -- (6, 0) -- cycle, black);
filldraw((7, 0) -- (7, 9) -- (8, 9) -- (8, 0) -- cycle, black);
filldraw((9, 0) -- (9, 8) -- (10, 8) -- (10, 0) -- cycle, black);
label("Test Marks", (5, 0), S);
label(rotate(90)*"Marks out of 100", (-2, 5), W);
[/asy]
$\textbf{(A)}\ 74 \qquad \textbf{(B)}\ 76 \qquad \textbf{(C)}\ 70 \qquad \textbf{(D)}\ 64 \qquad \textbf{(E)}\ 79$
2012 Czech-Polish-Slovak Junior Match, 2
On the circle $k$, the points $A,B$ are given, while $AB$ is not the diameter of the circle $k$. Point $C$ moves along the long arc $AB$ of circle $k$ so that the triangle $ABC$ is acute. Let $D,E$ be the feet of the altitudes from $A, B$ respectively. Let $F$ be the projection of point $D$ on line $AC$ and $G$ be the projection of point $E$ on line $BC$.
(a) Prove that the lines $AB$ and $FG$ are parallel.
(b) Determine the set of midpoints $S$ of segment $FG$ while along all allowable positions of point $C$.
2004 Unirea, 2
Let be two matrices $ A,N\in\mathcal{M}_2(\mathbb{R}) $ that commute and such that $ N $ is nilpotent. Show that:
[b]a)[/b] $ \det (A+N)=\det (A) $
[b]b)[/b] if $ A $ is general linear, then the matrix $ A+N $ is invertible and $ (A+N)^{-1}=(A-N)A^{-2} . $
2024 Korea Winter Program Practice Test, Q4
Show that there are infinitely many positive odd integers $n$ such that $n^5+2n+1$ is expressible as a sum of squares of two coprime integers.
2012 ELMO Shortlist, 4
Do there exist positive integers $b,n>1$ such that when $n$ is expressed in base $b$, there are more than $n$ distinct permutations of its digits? For example, when $b=4$ and $n=18$, $18 = 102_4$, but $102$ only has $6$ digit arrangements. (Leading zeros are allowed in the permutations.)
[i]Lewis Chen.[/i]
LMT Team Rounds 2010-20, 2020.S6
Let $\triangle ABC$ be a triangle such that $AB=6, BC=8,$ and $AC=10$. Let $M$ be the midpoint of $BC$. Circle $\omega$ passes through $A$ and is tangent to $BC$ at $M$. Suppose $\omega$ intersects segments $AB$ and $AC$ again at points $X$ and $Y$, respectively. If the area of $AXY$ can be expressed as $\frac{p}{q}$ where $p, q$ are relatively prime integers, compute $p+q$.
2022 JHMT HS, 3
Let $x$ be a variable that can take any positive real value. For certain positive real constants $s$ and $t$, the value of $x^2 + \frac{s}{x}$ is minimized at $x = t$, and the value of $t^2\ln(2 + tx) + \frac{1}{x^2}$ is minimized at $x = s$. Compute the ordered pair $(s, t)$.
2025 Spain Mathematical Olympiad, 2
The cyclic quadrilateral $ABCD$, inscribed in the circle $\Gamma$, satisfies $AB=BC$ and $CD=DA$, and $E$ is the intersection point of the diagonals $AC$ and $BD$. The circle with center $A$ and radius $AE$ intersects $\Gamma$ in two points $F$ and $G$. Prove that the line $FG$ is tangent to the circles with diameters $BE$ and $DE$.
2011 ISI B.Math Entrance Exam, 6
Let $f(x)=e^{-x}\ \forall\ x\geq 0$ and let $g$ be a function defined as for every integer $k \ge 0$, a straight line joining $(k,f(k))$ and $(k+1,f(k+1))$ . Find the area between the graphs of $f$ and $g$.
2023 SAFEST Olympiad, 4
Let $ABC$ be a triangle with incenter $I$ and let $AI$ meet $BC$ at $D$. Let $E$ be a point on the segment $AC$, such that $CD=CE$ and let $F$ be on the segment $AB$ such that $BF=BD$. Let $(CEI) \cap (DFI)=P \neq I$ and $(BFI) \cap (DEI)=Q \neq I$. Prove that $PQ \perp BC$.
[i]Proposed by Leonardo Franchi, Italy[/i]
Kvant 2023, M2774
In a $50\times 50$ checkered square, each cell is colored in one of the 100 given colors so that all colors are used and there does not exist a monochromatic domino. Galia wants to repaint all the cells of one of the colors in a different color (from the given 100 colors) so that a monochromatic domino still won't exist. Is it true that Galia will surely be able to do this
[i]Proposed by G. Sharafutdinova[/i]
2017 Assam Mathematics Olympiad, 1
1)$k, l, m\in\mathbb{N}$
$2^{k+l} +2^{l+m}+2^{m+k}\le 2^{k+l+m+1} +1$
[color=#00f]Moved to HSO. ~ oVlad[/color]
2004 Germany Team Selection Test, 1
Let $D_1$, $D_2$, ..., $D_n$ be closed discs in the plane. (A closed disc is the region limited by a circle, taken jointly with this circle.) Suppose that every point in the plane is contained in at most $2003$ discs $D_i$. Prove that there exists a disc $D_k$ which intersects at most $7\cdot 2003 - 1 = 14020$ other discs $D_i$.
2005 AMC 10, 1
A scout troop buys $ 1000$ candy bars at a price of five for $ \$2$. They sell all the candy bars at a price of two for $ \$1$. What was their profit, in dollars?
$ \textbf{(A)}\ 100 \qquad
\textbf{(B)}\ 200 \qquad
\textbf{(C)}\ 300 \qquad
\textbf{(D)}\ 400 \qquad
\textbf{(E)}\ 500$
2022 Saudi Arabia IMO TST, 2
Let $n$ and $k$ be two integers with $n>k\geqslant 1$. There are $2n+1$ students standing in a circle. Each student $S$ has $2k$ [i]neighbors[/i] - namely, the $k$ students closest to $S$ on the left, and the $k$ students closest to $S$ on the right.
Suppose that $n+1$ of the students are girls, and the other $n$ are boys. Prove that there is a girl with at least $k$ girls among her neighbors.
[i]Proposed by Gurgen Asatryan, Armenia[/i]
2010 Iran MO (3rd Round), 3
If $p$ is a prime number, what is the product of elements like $g$ such that $1\le g\le p^2$ and $g$ is a primitive root modulo $p$ but it's not a primitive root modulo $p^2$, modulo $p^2$?($\frac{100}{6}$ points)
1959 Miklós Schweitzer, 8
[b]8.[/b] An Oblique lattice-cubs is a lattice-cube of the three-dimensional fundamental lattice no edge of which is perpendicular to any coordinate axis. Prove that for any integer $h= 8n-1$ ($n= 1, 2, \dots $) there existis an oblique lattice-cube with edges of length $h$. Propose a method for finding such a cube. [b](N. 20)[/b]
2008 Thailand Mathematical Olympiad, 4
Let $n$ be a positive integer. Show that
$${2n+1 \choose 1} -{2n+1 \choose 3}2008 + {2n+1 \choose 5}2008^2- ...+(-1)^{n}{2n+1 \choose 2n+1}2008^n $$ is not divisible by $19$.
1991 AMC 12/AHSME, 4
Which of the following triangles cannot exist?
$\textbf{(A)}\ \text{An acute isosceles triangle}$
$\textbf{(B)}\ \text{An isosceles right triangle}$
$\textbf{(C)}\ \text{An obtuse right triangle}$
$\textbf{(D)}\ \text{A scalene right triangle}$
$\textbf{(E)}\ \text{A scalene obtuse triangle}$
2018 JHMT, 6
$\vartriangle ABC$ is inscribed in a unit circle. The three angle bisectors of $A$,$B$,$C$ are extended to intersect the circle at $A_1$,$B_1$,$C_1$, respectively. Find $$\frac{AA_1 \cos \frac{A}{2} + BB_1 \cos \frac{B}{2} + CC_1 \cos \frac{C}{2}}{\sin A + \sin B + \sin C}.$$
2000 Saint Petersburg Mathematical Olympiad, 10.5
Cells of a $2000\times2000$ board are colored according to the following rules:
1)At any moment a cell can be colored, if none of its neighbors are colored
2)At any moment a $1\times2$ rectangle can be colored, if exactly two of its neighbors are colored.
3)At any moment a $2\times2$ squared can be colored, if 8 of its neighbors are colored
(Two cells are considered to be neighboring, if they share a common side). Can the entire $2000\times2000$ board be colored?
[I]Proposed by K. Kohas[/i]
2016 Indonesia TST, 5
For a finite set $A$ of positive integers, a partition of $A$ into two disjoint nonempty subsets $A_1$ and $A_2$ is $\textit{good}$ if the least common multiple of the elements in $A_1$ is equal to the greatest common divisor of the elements in $A_2$. Determine the minimum value of $n$ such that there exists a set of $n$ positive integers with exactly $2015$ good partitions.
2021 Kosovo National Mathematical Olympiad, 3
Find all real numbers $a,b,c$ and $d$ such that:
$a^2+b^2+c^2+d^2=a+b+c+d-ab=3.$
1997 Romania National Olympiad, 4
Let two bijective and continuous functions$f,g: \mathbb{R}\to\mathbb{R}$ such that : $\left(f\circ g^{-1}\right)(x)+\left(g\circ f^{-1}\right)(x)=2x$ for any real $x$. Show that If we have a value $x_{0}\in\mathbb{R}$ such that $f(x_{0})=g(x_{0})$, then $f=g$.
2006 Moldova National Olympiad, 10.5
Let $x_{1}$, $x_{2}$, $\ldots$, $x_{n}$ be $n$ real numbers in $\left(\frac{1}{4},\frac{2}{3}\right)$. Find the minimal value of the expression: \[ \log_{\frac 32x_{1}}\left(\frac{1}{2}-\frac{1}{36x_{2}^{2}}\right)+\log_{\frac 32x_{2}}\left(\frac{1}{2}-\frac{1}{36x_{3}^{2}}\right)+\cdots+ \log_{\frac 32x_{n}}\left(\frac{1}{2}-\frac{1}{36x_{1}^{2}}\right). \]