Found problems: 85335
2018 China Northern MO, 1
In triangle $ABC$, let the circumcenter, incenter, and orthocenter be $O$, $I$, and $H$ respectively. Segments $AO$, $AI$, and $AH$ intersect the circumcircle of triangle $ABC$ at $D$, $E$, and $F$. $CD$ intersects $AE$ at $M$ and $CE$ intersects $AF$ at $N$. Prove that $MN$ is parallel to $BC$.
2004 USAMTS Problems, 1
The numbers 1 through 10 can be arranged along the vertices and sides of a pentagon so that the sum of the three numbers along each side is the same. The diagram below shows an arrangement with sum 16. Find, with proof, the smallest possible value for a sum and give an example of an arrangement with that sum.
[asy]
int i;
pair[] A={dir(18+72*0), dir(18+72*1), dir(18+72*2),dir(18+72*3), dir(18+72*4), dir(18+72*0)};
pair O=origin;
int[] v = {7,1,10,4,3};
int[] s = {8, 5, 2, 9, 6};
for(i=0; i<5; i=i+1) {
label(string(v[i]), A[i], dir(O--A[i]));
label(string(s[i]), A[i]--A[i+1], dir(-90)*dir(A[i]--A[i+1]));
}
draw(rotate(0)*polygon(5));[/asy]
2005 QEDMO 1st, 10 (C3)
Let $n\geq 3$ be an integer. Let also $P_1,P_2,...,P_n$ be different two-element-subsets of $M=\{1,2,...,n\}$, such that when for $i,j \in M , i\neq j$ the sets $P_i,P_j$ are not totally disjoint, then there is a $k \in M$ with $P_k = \{ i,j\}$.
Prove that every element of $M$ occurse in exactly $2$ of these subsets.
2024 Indonesia TST, C
Given a sequence of integers $A_1,A_2,\cdots A_{99}$ such that for every sub-sequence that contains $m$ consecutive elements, there exist not more than $max\{ \frac{m}{3} ,1\}$ odd integers. Let $S=\{ (i,j) \ | i<j \}$ such that $A_i$ is even and $A_j$ is odd. Find $max\{ |S|\}$.
1990 Irish Math Olympiad, 6
Let $n$ be a natural number, and suppose that the equation $$x_1x_2+x_2x_3+x_3x_4+x_4x_5+\dots +x_{n-1}x_n+x_nx_1=0$$ has a solution with all the $x_i$s equal to $\pm 1$. Prove that $n$ is divisible by $4$.
2023 Princeton University Math Competition, A7
Define $f(n)$ to be the smallest integer such that for every positive divisor $d \mid n,$ either $n \mid d^d$ or $d^d \mid n^{f(n)}.$ How many positive integers $b < 1000$ which are not squarefree satisfy the equation $f(2023) \cdot f(b) = f(2023b)$?
2022 Assam Mathematical Olympiad, 6
Prove that $n! \geq n^{\frac{n}{2}}$ for all natural numbers $n$. Also, show that the inequality is strict for $n > 2$.
1988 IMO Longlists, 85
Around a circular table an even number of persons have a discussion. After a break they sit again around the circular table in a different order. Prove that there are at least two people such that the number of participants sitting between them before and after a break is the same.
2020 New Zealand MO, 2
Find the smallest positive integer $N$ satisfying the following three properties.
$\bullet$ N leaves a remainder of $5$ when divided by $7$.
$\bullet$ N leaves a remainder of $6$ when divided by $ 8$.
$\bullet$ N leaves a remainder of $7$ when divided by $9$.
2023 BMT, 17
Let $N$ be the smallest positive integer divisble by $10^{2023} - 1$ that only has the digits $4$ and $8$ in decimal form (these digits may be repeated). Compute the sum of the digits of $\frac{N}{10^{2023}-1}$ .
2009 Turkey MO (2nd round), 3
[i]Alice[/i], who works for the [i]Graph County Electric Works[/i], is commissioned to wire the newly erected utility poles in $k$ days. Each day she either chooses a pole and runs wires from it to as many poles as she wishes, or chooses at most $17$ pairs of poles and runs wires between each pair. [i]Bob[/i], who works for the [i]Graph County Paint Works[/i], claims that, no matter how many poles there are and how [i]Alice[/i] connects them, all the poles can be painted using not more than $2009$ colors in such a way that no pair of poles connected by a wire is the same color. Determine the greatest value of $k$ for which [i]Bob[/i]'s claim is valid.
2015 AMC 10, 8
Two years ago Pete was three times as old as his cousin Claire. Two years before that, Pete was four times as old as Claire. In how many years will the ratio of their ages be $2:1$?
$\textbf{(A) }2\qquad\textbf{(B) }4\qquad\textbf{(C) }5\qquad\textbf{(D) }6\qquad\textbf{(E) }8$
2011 USA TSTST, 4
Acute triangle $ABC$ is inscribed in circle $\omega$. Let $H$ and $O$ denote its orthocenter and circumcenter, respectively. Let $M$ and $N$ be the midpoints of sides $AB$ and $AC$, respectively. Rays $MH$ and $NH$ meet $\omega$ at $P$ and $Q$, respectively. Lines $MN$ and $PQ$ meet at $R$. Prove that $OA\perp RA$.
2019 USA TSTST, 3
On an infinite square grid we place finitely many [i]cars[/i], which each occupy a single cell and face in one of the four cardinal directions. Cars may never occupy the same cell. It is given that the cell immediately in front of each car is empty, and moreover no two cars face towards each other (no right-facing car is to the left of a left-facing car within a row, etc.). In a [i]move[/i], one chooses a car and shifts it one cell forward to a vacant cell. Prove that there exists an infinite sequence of valid moves using each car infinitely many times.
[i]Nikolai Beluhov[/i]
2004 IMO, 2
Find all polynomials $f$ with real coefficients such that for all reals $a,b,c$ such that $ab+bc+ca = 0$ we have the following relations
\[ f(a-b) + f(b-c) + f(c-a) = 2f(a+b+c). \]
1962 German National Olympiad, 2
Let $u, v$ and$ w$ be any positive numbers smaller than $1$. Prove that among the numbers $u(1 -v)$, $v(1 -w)$, $w(1 - u)$ there is always at least one value not greater than $\frac14$ .
2001 Iran MO (3rd Round), 2
Does there exist a sequence $ \{b_{i}\}_{i=1}^\infty$ of positive real numbers such that for each natural $ m$: \[ b_{m}+b_{2m}+b_{3m}+\dots=\frac1m\]
2021 Baltic Way, 20
Let $n\ge 2$ be an integer. Given numbers $a_1, a_2, \ldots, a_n \in \{1,2,3,\ldots,2n\}$ such that $\operatorname{lcm}(a_i,a_j)>2n$ for all $1\le i<j\le n$, prove that
$$a_1a_2\ldots a_n \mid (n+1)(n+2)\ldots (2n-1)(2n).$$
2020 Novosibirsk Oral Olympiad in Geometry, 4
The altitudes $AN$ and $BM$ are drawn in triangle $ABC$. Prove that the perpendicular bisector to the segment $NM$ divides the segment $AB$ in half.
2022 Taiwan Mathematics Olympiad, 3
Find all functions $f,g:\mathbb{R}^2\to\mathbb{R}$ satisfying that
\[|f(a,b)-f(c,d)|+|g(a,b)-g(c,d)|=|a-c|+|b-d|\]
for all real numbers $a,b,c,d$.
[i]Proposed by usjl[/i]
2017 Moscow Mathematical Olympiad, 5
$8$ points lie on the faces of unit cube and form another cube. What can be length of edge of this cube?
1996 AMC 8, 20
Suppose there is a special key on a calculator that replaces the number $x$ currently displayed with the number given by the formula $\frac{1}{1-x}$. For example, if the calculator is displaying $2$ and the special key is pressed, then the calculator will display $-1$ since $\frac{1}{1-2}=-1$. Now suppose that the calculator is displaying $5$. After the special key is pressed 100 times in a row, the calculator will display
$\text{(A)}\ -0.25 \qquad \text{(B)}\ 0 \qquad \text{(C)}\ 0.8 \qquad \text{(D)}\ 1.25 \qquad \text{(E)}\ 5$
2014 PUMaC Geometry B, 1
Triangle $ABC$ has lengths $AB=20$, $AC=14$, $BC=22$. The median from $B$ intersects $AC$ at $M$ and the angle bisector from $C$ intersects $AB$ at $N$ and the median from $B$ at $P$. Let $\dfrac pq=\dfrac{[AMPN]}{[ABC]}$ for positive integers $p$, $q$ coprime. Note that $[ABC]$ denotes the area of triangle $ABC$. Find $p+q$.
2021 Romania Team Selection Test, 1
Consider a fixed triangle $ABC$ such that $AB=AC.$ Let $M$ be the midpoint of $BC.$ Let $P$ be a variable point inside $\triangle ABC,$ such that $\angle PBC=\angle PCA.$ Prove that the sum of the measures of $\angle BPM$ and $\angle APC$ is constant.
2014 Contests, Problem 1
Consider a square of side length $12$ centimeters. Irina draws another square that has $8$ centimeters more of perimeter than the original square. What is the area of the square drawn by Irina?