Found problems: 85335
1986 Canada National Olympiad, 2
A Mathlon is a competition in which there are $M$ athletic events. Such a competition was held in which only $A$, $B$, and $C$ participated. In each event $p_1$ points were awarded for first place, $p_2$ for second and $p_3$ for third, where $p_1 > p_2 > p_3 > 0$ and $p_1$, $p_2$, $p_3$ are integers. The final scores for $A$ was 22, for $B$ was 9 and for $C$ was also 9. $B$ won the 100 metres. What is the value of $M$ and who was second in the high jump?
2000 Saint Petersburg Mathematical Olympiad, 9.4
On a Cartesian plane 101 planes are drawn and all points of intersection are labeled. Is it possible, that for every line, 50 of the points have positive coordinates and 50 of the points have negative coordinates
[I]Proposed by S. Ivanov[/i]
KoMaL A Problems 2023/2024, A. 877
A convex quadrilateral $ABCD$ is circumscribed about circle $\omega$. A tangent to $\omega$ parallel to $AC$ intersects $BD$ at a point $P$ outside of $\omega$. The second tangent from $P$ to $\omega$ touches $\omega$ at a point $T$. Prove that $\omega$ and circumcircle of $ATC$ are tangent.
[i]Proposed by Nikolai Beluhov, Bulgaria[/i]
2011 All-Russian Olympiad, 4
Let $N$ be the midpoint of arc $ABC$ of the circumcircle of triangle $ABC$, let $M$ be the midpoint of $AC$ and let $I_1, I_2$ be the incentres of triangles $ABM$ and $CBM$. Prove that points $I_1, I_2, B, N$ lie on a circle.
[i]M. Kungojin[/i]
1979 AMC 12/AHSME, 18
To the nearest thousandth, $\log_{10}2$ is $.301$ and $\log_{10}3$ is $.477$. Which of the following is the best approximation of $\log_5 10$?
$\textbf{(A) }\frac{8}{7}\qquad\textbf{(B) }\frac{9}{7}\qquad\textbf{(C) }\frac{10}{7}\qquad\textbf{(D) }\frac{11}{7}\qquad\textbf{(E) }\frac{12}{7}$
2006 Korea Junior Math Olympiad, 4
In the coordinate plane, de fine $M = \{(a, b),a,b \in Z\}$. A transformation $S$, which is de fined on $M$, sends $(a,b)$
to $(a + b, b)$. Transformation $T$, also de fined on $M$, sends $(a, b)$ to $(-b, a)$. Prove that for all $(a, b) \in M$, we
can use $S,T$ denitely to map it to $(g,0)$.
2023 AMC 12/AHSME, 2
The weight of $\frac 13$ of a large pizza together with $3 \frac 12$ cups of orange slices is the same as the weight of $\frac 34$ of a large pizza together with $\frac 12$ cup of orange slices. A cup of orange slices weighs $\frac 14$ of a pound. What is the weight, in pounds, of a large pizza?
$\textbf{(A)}~1\frac45\qquad\textbf{(B)}~2\qquad\textbf{(C)}~2\frac25\qquad\textbf{(D)}~3\qquad\textbf{(E)}~3\frac35$
DMM Team Rounds, 2010
[b]p1.[/b] Find the smallest positive integer $N$ such that $N!$ is a multiple of $10^{2010}$.
[b]p2.[/b] An equilateral triangle $T$ is externally tangent to three mutually tangent unit circles, as shown in the diagram. Find the area of $T$.
[b]p3. [/b]The polynomial $p(x) = x^3 + ax^2 + bx + c$ has the property that the average of its roots, the product of its roots, and the sum of its coefficients are all equal. If $p(0) = 2$, find $b$.
[b]p4.[/b] A regular pentagon $P = A_1A_2A_3A_4A_5$ and a square $S = B_1B_2B_3B_4$ are both inscribed in the unit circle. For a given pentagon $P$ and square $S$, let $f(P, S)$ be the minimum length of the minor arcs AiBj , for $1 \le i \le 5$ and $1 \le j \le 4$. Find the maximum of $f(P, S)$ over all pairs of shapes.
[b]p5.[/b] Let $ a, b, c$ be three three-digit perfect squares that together contain each nonzero digit exactly once. Find the value of $a + b + c$.
[b]p6. [/b]There is a big circle $P$ of radius $2$. Two smaller circles $Q$ and $R$ are drawn tangent to the big circle $P$ and tangent to each other at the center of the big circle $P$. A fourth circle $S$ is drawn externally tangent to the smaller circles $Q$ and $R$ and internally tangent to the big circle $P$. Finally, a tiny fifth circle $T$ is drawn externally tangent to the $3$ smaller circles $Q, R, S$. What is the radius of the tiny circle $T$?
[b]p7.[/b] Let $P(x) = (1 +x)(1 +x^2)(1 +x^4)(1 +x^8)(...)$. This infinite product converges when $|x| < 1$.
Find $P\left( \frac{1}{2010}\right)$.
[b]p8.[/b] $P(x)$ is a polynomial of degree four with integer coefficients that satisfies $P(0) = 1$ and $P(\sqrt2 + \sqrt3) = 0$. Find $P(5)$.
[b]p9.[/b] Find all positive integers $n \ge 3$ such that both roots of the equation $$(n - 2)x^2 + (2n^2 - 13n + 38)x + 12n - 12 = 0$$ are integers.
[b]p10.[/b] Let $a, b, c, d, e, f$ be positive integers (not necessarily distinct) such that $$a^4 + b^4 + c^4 + d^4 + e^4 = f^4.$$ Find the largest positive integer $n$ such that $n$ is guaranteed to divide at least one of $a, b, c, d, e, f$.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
1999 Chile National Olympiad, 1
Pedrito's lucky number is $34117$. His friend Ramanujan points out that $34117 = 166^2 + 81^2 = 159^2 + 94^2$ and $166-159 = 7$, $94- 81 = 13$. Since his lucky number is large, Pedrito decides to find a smaller one, but that satisfies the same properties, that is, write in two different ways as the sum of squares of positive integers, and the difference of the first integers that occur in that sum is $7$ and in the difference between the seconds it gives $13$. Which is the least lucky number that Pedrito can find? Find a way to generate all the positive integers with the properties mentioned above.
2015 Online Math Open Problems, 28
Let $N$ be the number of $2015$-tuples of (not necessarily distinct) subsets $(S_1, S_2, \dots, S_{2015})$ of $\{1, 2, \dots, 2015 \}$ such that the number of permutations $\sigma$ of $\{1, 2, \dots, 2015 \}$ satisfying $\sigma(i) \in S_i$ for all $1 \le i \le 2015$ is odd. Let $k_2, k_3$ be the largest integers such that $2^{k_2} | N$ and $3^{k_3} | N$ respectively. Find $k_2 + k_3.$
[i]Proposed by Yang Liu[/i]
2018 PUMaC Individual Finals B, 1
Let a positive integer $n$ have at least four positive divisors. Let the least four positive divisors be $1=d_1<d_2<d_3<d_4$. Find, with proof, all solutions to $n^2=d_1^3+d_2^3+d_4^3$.
2012 Pre - Vietnam Mathematical Olympiad, 1
Let $n \geq 2$ be a positive integer. Suppose there exist non-negative integers ${n_1},{n_2},\ldots,{n_k}$ such that $2^n - 1 \mid \sum_{i = 1}^k {{2^{{n_i}}}}$. Prove that $k \ge n$.
1950 Kurschak Competition, 3
$(x_1, y_1,z_1)$ and $(x_2, y_2, z_2)$ are triples of real numbers such that for every pair of integers $(m,n)$ at least one of $x_{1m} + y_{1n} + z_1$, $x_{2m} + y_{2n} + z_2$ is an even integer. Prove that one of the triples consists of three integers.
2017 Online Math Open Problems, 26
Define a sequence of polynomials $P_0,P_1,...$ by the recurrence $P_0(x)=1, P_1(x)=x, P_{n+1}(x) = 2xP_n(x)-P_{n-1}(x)$. Let $S=\left|P_{2017}'\left(\frac{i}{2}\right)\right|$ and $T=\left|P_{17}'\left(\frac{i}{2}\right)\right|$, where $i$ is the imaginary unit. Then $\frac{S}{T}$ is a rational number with fractional part $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $m$.
[i]Proposed by Tristan Shin[/i]
2010 Contests, 4
The sequence of Fibonnaci's numbers if defined from the two first digits $f_1=f_2=1$ and the formula $f_{n+2}=f_{n+1}+f_n$, $\forall n \in N$.
[b](a)[/b] Prove that $f_{2010} $ is divisible by $10$.
[b](b)[/b] Is $f_{1005}$ divisible by $4$?
Albanian National Mathematical Olympiad 2010---12 GRADE Question 4.
2011 Grand Duchy of Lithuania, 4
In the cyclic quadrilateral $ABCD$ with $AB = AD$, points $M$ and $N$ lie on the sides $CD$ and $BC$ respectively so that $MN = BN + DM$. Lines $AM$ and $AN$ meet the circumcircle of $ABCD$ again at points $P$ and $Q$ respectively. Prove that the orthocenter of the triangle $APQ$ lies on the segment $MN$.
2024 Kyiv City MO Round 2, Problem 3
For a given positive integer $n$, we consider the set $M$ of all intervals of the form $[l, r]$, where the integers $l$ and $r$ satisfy the condition $0 \leq l < r \leq n$. What largest number of elements of $M$ can be chosen so that each chosen interval completely contains at most one other selected interval?
[i]Proposed by Anton Trygub[/i]
2007 China Girls Math Olympiad, 5
Point $D$ lies inside triangle $ABC$ such that $\angle DAC = \angle DCA = 30^{\circ}$ and $\angle DBA = 60^{\circ}$. Point $E$ is the midpoint of segment $BC$. Point $F$ lies on segment $AC$ with $AF = 2FC$. Prove that $DE \perp EF$.
2001 Moldova National Olympiad, Problem 8
If $a_1,a_2,\ldots,a_n$ are positive real numbers, prove the inequality
$$\dfrac1{\dfrac1{1+a_1}+\dfrac1{1+a_2}+\ldots+\dfrac1{1+a_n}}-\dfrac1{\dfrac1{a_1}+\dfrac1{a_2}+\ldots+\dfrac1{a_n}}\ge\frac1n.$$
2011 Indonesia TST, 1
Let $Q^+$ denote the set of positive rationals. Determine all functions $f : Q^+ \to Q^+$ that satisfy both of these conditions:
(i) $f(x)$ is an integer if and only if $x$ is an integer;
(ii) $f(f(xf(y)) + x) = yf(x) + x$ for all $x, y \in Q^+$.
2023 LMT Fall, 9
Find the least positive integer $k$ such that when $\frac{k}{2023}$ is written in simplest form, the sum of the numerator and denominator is divisible by $7$.
[i]Proposed byMuztaba Syed[/i]
2003 AMC 12-AHSME, 16
A point $ P$ is chosen at random in the interior of equilateral triangle $ ABC$. What is the probability that $ \triangle ABP$ has a greater area than each of $ \triangle ACP$ and $ \triangle BCP$?
$ \textbf{(A)}\ \frac{1}{6} \qquad
\textbf{(B)}\ \frac{1}{4} \qquad
\textbf{(C)}\ \frac{1}{3} \qquad
\textbf{(D)}\ \frac{1}{2} \qquad
\textbf{(E)}\ \frac{2}{3}$
2014 Romania National Olympiad, 2
Let $ I,J $ be two intervals, $ \varphi :J\longrightarrow\mathbb{R} $ be a continuous function whose image doesn't contain $ 0, $ and $ f,g:I\longrightarrow J $ be two differentiable functions such that $ f'=\varphi\circ f,g'=\varphi\circ g $ and such that the image of $ f-g $ contains $ 0. $
Show that $ f $ and $ g $ are the same function.
2006 Cezar Ivănescu, 3
[b]a)[/b] Let be a sequence $ \left( x_n \right)_{n\ge 1} $ defined by the recursion $ x_{n+1}=\frac{1+x_n}{1-x_n} , $ with $ x_1=2006. $ Calculate $ \lim_{n\to\infty } \frac{x_1+x_2+\cdots +x_n}{n} . $
[b]b)[/b] Prove that if a convergent sequence $ \left( s_n \right)_{n\ge 1} $ verifies $ a_{2^n} =na_n , $ for any natural numbers $ n, $ then $ a_n=0, $ for any natural numbers $ n. $
[i]Cornel Stoicescu[/i]
2018 Flanders Math Olympiad, 4
Determine all three-digit numbers N such that $N^2$ has six digits and so that the sum of the number formed by the first three digits of $N^2$ and the number formed by the latter three digits of $N^2$ equals $N$.