Found problems: 85335
2014 Math Prize For Girls Problems, 13
Deepali has a bag containing 10 red marbles and 10 blue marbles (and nothing else). She removes a random marble from the bag. She keeps doing so until all of the marbles remaining in the bag have the same color. Compute the probability that Deepali ends with exactly 3 marbles remaining in the bag.
2019 Pan-African, 4
The tangents to the circumcircle of $\triangle ABC$ at $B$ and $C$ meet at $D$. The circumcircle of $\triangle BCD$ meets sides $AC$ and $AB$ again at $E$ and $F$ respectively. Let $O$ be the circumcentre of $\triangle ABC$. Show that $AO$ is perpendicular to $EF$.
1990 All Soviet Union Mathematical Olympiad, 533
A game is played in three moves. The first player picks any real number, then the second player makes it the coefficient of a cubic, except that the coefficient of $x^3$ is already fixed at $1$. Can the first player make his choices so that the final cubic has three distinct integer roots?
2005 Today's Calculation Of Integral, 28
Evaluate
\[\int_0^{\frac{\pi}{4}} \frac{x\cos 5x}{\cos x}dx\]
1975 All Soviet Union Mathematical Olympiad, 213
Three flies are crawling along the perimeter of the triangle $ABC$ in such a way, that the centre of their masses is a constant point. One of the flies has already passed along all the perimeter. Prove that the centre of the flies' masses coincides with the centre of masses of the triangle $ABC$ . (The centre of masses for the triangle is the point of medians intersection.
1999 AMC 12/AHSME, 1
$ 1 \minus{} 2 \plus{} 3 \minus{} 4 \plus{} \cdots \minus{} 98 \plus{} 99 \equal{}$
$ \textbf{(A)}\minus{}\! 50 \qquad \textbf{(B)}\minus{}\! 49 \qquad \textbf{(C)}\ 0 \qquad \textbf{(D)}\ 49 \qquad \textbf{(E)}\ 50$
2019 USA IMO Team Selection Test, 1
Let $ABC$ be a triangle and let $M$ and $N$ denote the midpoints of $\overline{AB}$ and $\overline{AC}$, respectively. Let $X$ be a point such that $\overline{AX}$ is tangent to the circumcircle of triangle $ABC$. Denote by $\omega_B$ the circle through $M$ and $B$ tangent to $\overline{MX}$, and by $\omega_C$ the circle through $N$ and $C$ tangent to $\overline{NX}$. Show that $\omega_B$ and $\omega_C$ intersect on line $BC$.
[i]Merlijn Staps[/i]
2005 China Team Selection Test, 3
Let $n$ be a positive integer, set $S_n = \{ (a_1,a_2,\cdots,a_{2^n}) \mid a_i=0 \ \text{or} \ 1, 1 \leq i \leq 2^n\}$. For any two elements $a=(a_1,a_2,\cdots,a_{2^n})$ and $b=(b_1,b_2,\cdots,b_{2^n})$ of $S_n$, define
\[ d(a,b)= \sum_{i=1}^{2^n} |a_i - b_i| \]
We call $A \subseteq S_n$ a $\textsl{Good Subset}$ if $d(a,b) \geq 2^{n-1}$ holds for any two distinct elements $a$ and $b$ of $A$. How many elements can the $\textsl{Good Subset}$ of $S_n$ at most have?
2016 Belarus Team Selection Test, 2
A point $A_1$ is marked inside an acute non-isosceles triangle $ABC$ such that $\angle A_1AB = \angle A_1BC$ and $\angle A_1AC=\angle A_1CB$. Points $B_1$ and $C_1$ are defined same way. Let $G$ be the gravity center if the triangle $ABC$.
Prove that the points $A_1,B_1,C_1,G$ are concyclic.
2025 All-Russian Olympiad, 9.3
Find all natural numbers \(n\) for which there exists an even natural number \(a\) such that the number
\[
(a - 1)(a^2 - 1)\cdots(a^n - 1)
\]
is a perfect square.